Impulsive Manipulation - The Robotics Institute - Carnegie Mellon ...

Impulsive Manipulation
Wesley H. Huang
August 1997
CMU–RI–TR–97–29
Submitted in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy in Robotics
The Robotics Institute
Carnegie Mellon University
Pittsburgh, Pennsylvania 15213
Copyright c1997 Wesley H. Huang. All rights reserved.
This research was funded in part by NASA through the Graduate Student Researchers
Program and through grant NAGW 1175 and by the NSF under grants IRI–9318496and IRI-
9114208. The views and conclusions contained in this document are those of the author and
should not be interpreted as representing the official policies, either expressed or implied,
of the NSF, NASA, or the U.S. Government.

Abstract
Impact is a complicated phenomenon. A collision lasts for only a short period of time but
results in tremendous forces and accelerations during that time. The fact that so many
sports and games involve impact reflects the both the power and the difficulty in using it.
This thesis studies the use of impulsive forces to manipulate objects in order that the
methods, techniques, and strategies of this mode of manipulation can be transferred to
robotic manipulators. The particular form of impulsive manipulation that I have studied
in this thesis is tapping planar objects which then slide on a support surface, coming to rest
due to friction.
The mechanics of impact and friction are first analyzed in order to plan single taps
that achieve some desired translation and rotation. This solution is then extended to plan
multiple tap sequences and to characterize the controllability of objects under this form of
manipulation. The analysis culminates with the study of vibratory manipulation — the
use repeated impacts for manipulation, particularly a high frequency low amplitude series
of impacts.
Experiments are a major part of this work. The experimental effort started with the design
of specialized tapping devices that deliver a single controlled repeatable impact. Over
the course of this work, a number of different devices were designed, built, and tested; this
thesis describes these devices and the issues that were important in their design.
The first experiments were single tap experiments, designed to test how well the models
predict object motion for a variety of different objects and support surfaces. The next
experiments demonstrated positioning via tapping. These experiments required the adaptation
of the basic planning method developed in the analysis in order to explicitly consider
practical issues such as errors and tapping device limitations. The resulting planning
methods lie somewhere in between traditional planners and feedback control systems. The
culmination of the experimental effort was a demonstration that it is possible to position
an object via tapping more precisely than the tapping device itself is positioned.

Acknowledgements
I am indebted to many people for their help along the way.
I would like to thank Fritz Morgan, Costa Nikou, Kevin Dowling and the Medical
Robotics and Computer Assisted Surgery Lab for assistance and use of their OptoTrak system;
Arthur Quaid and the Microdynamic Systems Laboratory for assistance and use of
their laser interferometer; Ben Brown for many years of advice and assistance on mechanisms
and mechanical design; Al Rizzi for numerous suggestions, particularly on the planning
methods and sensitivity analysis; and Jason Powell for building and helping design
the tapping devices.
I have had a great group of peers in the Manipulation Lab: Kevin Lynch, Srinivas
Akella, Nina Zumel, Tammy Abel, Yan-Bin Jia, Garth Zeglin, and Mark Moll. I have also
been fortunate to be a part of the Vision and Autonomous Systems Center (VASC).
I have had the benefit of interaction with many other students in the Robotics PhD
program. In particular, I have had valuable interactions with Brad Nelson, Dan Morrow,
and Rich Voyles.
Thanks to Jeff Trinkle and Yangsheng Xu, two of my committee members, and to Mike
Erdmann and Takeo Kanade for their contributions to this work; also to Garth Zeglin for
comments on a draft of this dissertation.
Finally, I would like to thank my advisors Matt Mason and Eric Krotkov for their years
of support, advice, faith, and patience.

Contents
1 Introduction 1
1.1 Motivations ..................................... 1
1.1.1 Potentialapplications ........................... 2
1.2 Thesisoutline .................................... 3
1.3 Assumptions..................................... 4
1.3.1 Impact .................................... 4
1.3.2 Friction.................................... 5
1.4 RelatedWork .................................... 5
1.4.1 Impulsivemanipulation.......................... 5
1.4.2 Friction.................................... 6
1.4.3 Impact .................................... 6
1.4.4 Nonprehensilemanipulation ....................... 7
1.4.5 Impact-baseddynamicsimulation.................... 7
1.4.6 Minimalisminrobotics .......................... 7
1.4.7 Partsfeeding ................................ 8
1.4.8 Roboticjuggling&ballbouncing..................... 8
2 The Inverse Sliding Problem 9
2.1 Axisymmetriccase ................................. 9
2.1.1 Forceandtorqueduetofriction ..................... 10
2.1.2 Equationsofmotionanddisplacementfunctions............ 11
2.1.3 Monotonicityofthedisplacementfunctions .............. 12
2.1.4 Levelcurvesofdisplacementfunctions ................. 14
2.1.5 Solvingforinitialvelocities ........................ 17
2.1.6 Additionalaxisymmetricproperties ................... 19
2.2 Nonaxisymmetriccase ............................... 20
2.2.1 Generalsolution .............................. 21
2.2.2 Anexample ................................. 21
2.2.3 Apracticalsolution............................. 23
3 The Impact Problem 25
3.1 Impactmodels.................................... 25
3.2 Impactproblemconstructs ............................ 27
3.2.1 Fromimpulsetovelocities......................... 28
i

ii
CONTENTS
3.2.2 Directedvelocityratiosets&impactcones ............... 28
3.2.3 Fromvelocitiestoimpulse......................... 29
3.3 Examples....................................... 29
3.3.1 Circularobject................................ 29
3.3.2 Squareobject ................................ 29
4 Planning & Controllability 31
4.1 Axisymmetriccase ................................. 31
4.1.1 Planning................................... 32
4.1.2 Controllability ................................ 33
4.2 Nearlyaxisymmetriccase ............................. 33
4.2.1 Planning................................... 34
4.2.2 Controllability ................................ 38
5 Generating Impact 39
5.1 Backgroundandphilosophy............................ 39
5.1.1 Methodsofgeneratingmotionandimpact ............... 40
5.1.2 Otherimpactgeneratingdevicesformanipulation........... 40
5.1.3 Preliminarytappingexperiments..................... 41
5.1.4 Designgoals,criteria,andconstraints .................. 41
5.2 Thetappingdevices................................. 42
5.2.1 Firstspring-loadedtappingdevice.................... 42
5.2.2 Pneumatictappingdevice......................... 43
5.2.3 Electromagnetictappingdevice...................... 44
5.2.4 Wreckingballtapper............................ 45
5.2.5 Secondspring-loadedtappingdevice .................. 45
5.3 Issuesintappingdevicedesign.......................... 46
6 Single Tap Experiments 51
6.1 Experimentalmaterials,setup,andprocedures ................. 52
6.1.1 Measurementissues ............................ 53
6.1.2 Measuringmaterialparameters...................... 54
6.2 Plexiglasdiskexperiment ............................. 54
6.2.1 Evaluationoftheslidingmodel...................... 55
6.2.2 Evaluationoftheimpactmodel...................... 58
6.3 Aluminumsquareexperiment........................... 62
6.3.1 Evaluationoftheslidingmodel...................... 63
6.3.2 Evaluationoftheimpactmodel...................... 64
6.4 Aluminumdiskexperiment............................ 68
6.4.1 Evaluationoftheslidingmodel...................... 70
6.4.2 Evaluationoftheimpactmodel...................... 70
6.5 Experimentalobservations............................. 72
6.5.1 Strikingheight ............................... 72
6.5.2 Objectoverhang .............................. 73
6.5.3 Objectpath ................................. 74

CONTENTS
iii
6.5.4 Breakingstaticfriction........................... 74
6.5.5 Threedimensionaleffects ......................... 76
6.5.6 Variance in e withvelocity......................... 76
6.6 Conclusions ..................................... 76
7 Positioning Experiments 79
7.1 Preliminaries..................................... 79
7.1.1 ExperimentalSetup............................. 79
7.1.2 Errormodels ................................ 80
7.1.3 Positioningfigures ............................. 81
7.2 Planningmethods.................................. 82
7.2.1 Exacttwo-tapplans ............................ 82
7.2.2 Conservativetwo-tapplans........................ 82
7.2.3 Multi-tapplans ............................... 84
7.3 Highprecisionpositioning............................. 88
7.3.1 Experiment ................................. 88
7.3.2 SensitivityAnalysis ............................ 90
7.3.3 Discussion.................................. 93
8 Vibratory Manipulation 95
8.1 Vibratorymanipulationinonedimension.................... 96
8.1.1 Intermittenttapping ............................ 96
8.1.2 Continuoustapping ............................ 97
8.1.3 Vibratorymanipulationandballbouncing ............... 98
8.1.4 Roboticjuggling .............................. 101
8.2 Vibratorymanipulationintwodimensions ................... 101
8.2.1 Limitingcaseofintermittenttapping .................. 102
8.2.2 Limitingcaseofcontinuoustapping................... 103
8.3 Comparisonwithpushing............................. 106
8.4 Examplesforthelimitingcases .......................... 107
8.5 Discussion...................................... 108
8.5.1 Limitingcaseofintermittenttapping .................. 108
8.5.2 Continuoustappinglimitingcase .................... 109
8.5.3 Trackingversusfollowingtheobject................... 109
9 Conclusions 111
9.1 Contributions .................................... 111
9.2 Futurework ..................................... 112
A Impact Analysis 115
A.1 Two dimensional rigid-body collisions with friction . . . ........... 115
A.1.1 Equationsofmotion ............................ 115
A.1.2 Impulsespaceconstructs ......................... 116
A.1.3 Impactprocess............................... 117
A.1.4 End of the collision . . .......................... 118

iv
CONTENTS
A.1.5 Impulsesolutions.............................. 118
A.2 Generatingimpulsewithinafrictioncone.................... 119
A.2.1 Generalcase................................. 120
A.2.2 Generalizedcentralimpacts........................ 121
A.3 SpecialCases..................................... 121
A.3.1 Sphericalstriker............................... 123
A.3.2 Sphericalstrikerandcircularobject ................... 123
A.3.3 Longthinrodstriker............................ 123
A.3.4 Longthinrodstrikerwithcircularobject ................ 124

List of Figures
2.1 Notationfortheaxisymmetricinverseslidingproblem ............ 10
2.2 Exampledisplacementfunctions ......................... 12
2.3 Levelcurvesofthedisplacementfunctions ................... 15
2.4 Comparing two points on an xf levelcurve................... 16
2.5 Theangularvelocitytrajectoriescross....................... 16
2.6 Thetranslationalvelocitytrajectoriescross. ................... 17
2.7 Solvingfortheinitialvelocities .......................... 18
2.8 Relationshipbetweenvelocityanddisplacementratios ............ 20
2.9 The x f displacement function for the two-dimensional barbell and a graph
ofitslevelcurves................................... 22
2.10 The f displacement function (net rotation) for the two-dimensional barbell
andagraphofitslevelcurves. .......................... 22
2.11 The y f displacement function for the two-dimensional barbell and a graph
itslevelcurves. ................................... 23
3.1 Notationfortheimpactproblem ......................... 27
3.2 Directedvelocityratiomappings ......................... 30
4.1 Planningforaxisymmetricobjects ........................ 32
4.2 Small time local controllability of axisymmetric objects . ........... 33
4.3 Thereachableconfigurationsforanearlyaxisymmetricobject ........ 35
4.4 States with the same orientation that can reach the goal in one tap . . . . . . 35
4.5 States with a different orientation that can reach the goal in one tap . . . . . 36
4.6 An illustration of the spiral staircase ....................... 37
4.7 Thespanningdistance ............................... 37
4.8 The spiral staircase and reachable displacement cone projected onto the plane. 37
4.9 Intersection between the spiral staircase and the reachable displacement cone 38
5.1 Thefirstspring-loadedtappingdevice. ..................... 42
5.2 Double impact example for the first spring-loaded tapping device. . . . . . 43
5.3 Thepneumatictappingdevice .......................... 44
5.4 Theelectromagnetictappingdevice ....................... 45
5.5 The“wreckingball”tapper ............................ 46
5.6 Thesecondspring-loadedtappingdevice .................... 47
v

vi
LIST OF FIGURES
5.7 Position and velocity profiles for the second spring-loaded tapping device. . 47
6.1 Setupforthesingletapexperiments ....................... 52
6.2 Objectalignmentmethods............................. 53
6.3 Intended sampling of the space of initial conditions for runs A–G of the plexiglasdiskexperiment.
............................... 54
6.4 Netdisplacementsfortheplexiglasdiskexperiment. ............. 56
6.5 InitialvelocitiesforRunsA–Doftheplexiglasdiskexperiment ....... 57
6.6 Initial velocities for Runs E, C, F, and G of the plexiglas disk experiment . . 57
6.7 Plexiglasdiskvelocityprofiles—goodfit.................... 59
6.8 Plexiglasdiskvelocityprofiles—acceptablefit................. 60
6.9 Plexiglasdiskvelocityprofiles—unacceptablefit ............... 61
6.10 Displacements and initial velocities for the aluminum square experiment. . 63
6.11Forceandtorquefunctionsonasquareobject.................. 64
6.12 Calculated velocity profiles for the aluminum square experiment, unscaled
torque......................................... 65
6.13 Comparison of velocity profiles for the aluminum square experiment . . . . 67
6.14 Measured displacements and initial velocities for the aluminum disk experiment..........................................
69
6.15 Comparison of velocity profiles for the aluminum disk experiment . . . . . 71
6.16Spikesinvelocityprofilesduetoobjectoverhang. ............... 74
6.17Curvatureinobjectpaths.............................. 75
6.18Representativetrialsfromthetripoddiskexperiment ............. 77
7.1 Experimentalsetupforpositioningexperiments................. 80
7.2 The error projection and the effect of maximum ellipse radii on the set of
reachablestates.................................... 81
7.3 Theexacttwo-tapplanningmethod ....................... 82
7.4 Exampleofanexacttwo-tapplan. ........................ 83
7.5 Exampleofanexacttwo-tapplanwithreplanning................ 83
7.6 Theconservativetwo-tapplanningmethod................... 83
7.7 Exampleofaconservativetwo-tapplanwithreplanning............ 84
7.8 Example of a conservative two-tap plan with replanning. (Worst case) . . . 85
7.9 Construction of a sequence of cones leading to the goal for multi-tap planning 85
7.10Thefirstfourconesfromthegoal......................... 86
7.11 The sequence of cones leading to the goal for multi-tap planning . . . . . . 86
7.12 The extended reachable displacement cone and sequence of cones leading to
thegoal........................................ 87
7.13Exampleofamulti-tapplan............................. 88
7.14Exampleofamulti-tapplan............................ 89
7.15Highprecisionpositioningtrial.......................... 89
7.16Notationforhighprecisionpositioninganalysis................. 90
7.17 Final position in terms of x f and a ........................ 91
7.18Normoffinalconfigurationjacobian....................... 93

LIST OF FIGURES
vii
8.1 Onedimensionalintermittenttappinglimitingcase .............. 97
8.2 Onedimensioncontinuoustappinglimitingcase................ 98
8.3 Illustration of regular periodic bouncing . . . .................. 99
8.4 Relationship between frequency and amplitude for stable periodic bouncing 100
8.5 Illustration of regular periodic bouncing for intermittent tapping . . . . . . 101
8.6 ConstructionforfindingthenetCOR ...................... 102
8.7 Illustrations of the state space for a sliding rotating axisymmetric object . . 104
8.8 Maximum T forpushingadisk.......................... 107
F
8.9 Illustration of the motion constraints on a disk under pushing and the limitingcasesoftapping.................................
108
8.10Reachingpointsoutsidetheinitialimpactcone................. 109
A.1 Impactgeometryandnotation. .......................... 117

viii
LIST OF FIGURES

List of Tables
6.1 Summary of materials and parameters for single tap experiments . . . . . . 53
6.2 Comparisonofdisplacementsfortheplexiglasdiskexperiment ....... 58
6.3 Comparison of initial velocities from sliding for the plexiglas disk experiment 61
6.4 Comparison of initial velocities from impact for the plexiglas disk experiment 62
6.5 Comparison of displacements for the aluminum square experiment . . . . . 66
6.6 Comparison of velocities from sliding for the aluminum square experiment 66
6.7 Comparison of initial velocities from impact for the aluminum square experiment
......................................... 68
6.8 Comparison of displacements for the aluminum disk experiment . . . . . . 70
6.9 Comparison of initial velocities from sliding for the aluminum disk experiment 72
6.10 Comparison of initial velocities from impact for the aluminum disk experiment 73
6.11Effectofstrikerheight ............................... 73
6.12Momentumcomparisonbeforeandafterimpact ................ 75
6.13Energycomparisonbeforeandafterimpact................... 75
6.14Variationinthecoefficientofrestitution..................... 76
8.1 Motionconstraintsforadiskandaring ..................... 105
A.1 Sliding modes classified by the parameters P d , P q , s ,and r ......... 119
A.2 Generating impact in a friction cone in impulse space, sticking or sliding case 121
A.3 Generating impact in a friction cone in impulse space, reversed sliding or
slidingcase...................................... 122
A.4 Generalizedcentralimpacts............................ 123
ix

x
LIST OF TABLES

Chapter 1
Introduction
Manipulation is the act of moving objects. A manipulation task might be to move a large
boxfromoneplacetoanother,toassembletwoparts,toopenastubborndoor,topush
a stack of papers to a corner of a desk, to dribble a basketball, to shoot a basket, to join
connectors, or to flip pancakes. Even this short list of tasks shows a variety of modes of
manipulation, including pushing, sliding, throwing, and bouncing. Furthermore, it hints
at how people greatly increase the range of tasks they can perform by using different parts
of their arms and hands and by using different manipulation strategies.
The physics and the strategies behind a mode of manipulation are largely independent
of the manipulator. By understanding the essence of different modes of manipulation, we
can enhance the abilities of manipulators such as robotic arms and parts feeding systems.
This thesis is about the use of impulsive forces to manipulate objects, a mode of manipulation
I have termed impulsive manipulation. I broadly define impulsive manipulation
to be any form of manipulation consisting of:
a strike to an object which rapidly imparts some change in its initial velocities, followed
by
free motion of this object subject to forces and constraints in its environment, such as
friction, drag, and collisions.
This definition of impulsive manipulation includes many different forms of manipulation
involving a variety of physical processes and effects. For example, kicking a soccer ball
involves deformation during a kick and aerodynamics of flight, whereas playing billiards
involves rigid body impact, rolling, and sliding. In order to focus my efforts, I have limited
the scope of this thesis to the study of tapping planar rigid objects which then slide on a
support surface.
1.1 Motivations
There are many tasks in which it is required to position an object on a support surface; examples
include aligning masks and silicon wafers during fabrication processes and moving
1

2 CHAPTER 1. INTRODUCTION
parts or assemblies from one workcell to another. Tapping is a viable method to perform
these tasks, and in some instances it may be the preferred method — consider the task of
aligning a vice parallel to the travel of a milling machine bed. Generally, a machinist will
clamp the vice down most of the way and gently tap the vice in order to make minute
changes in its orientation.
Another motivation for the study of tapping is to expand the abilities of manipulators
by transferring the underlying physics and strategies of tapping or other modes of manipulation.
One way in which this can be done is through a minimalist approach to robotics.
Instead of using a complex robot, with many actuators and sensors, we embed knowledge
of the mechanics of a task in a simpler robot, either through incorporating this knowledge
into a planner or through the design of this robot. For example, sufficient knowledge of
the mechanics may make it unnecessary to use sensing or may allow the design of a mechanism
that mechanically produces stable or convergent behavior.
Tapping (and impulsive manipulation in general) is well suited for this minimalist approach
for several reasons. It is a nonprehensile (nongrasping) mode of manipulation and
therefore does not limit the shape and size of objects that can be manipulated. Tapping
can be done using a simple actuator, and it can be very fast because a robot need neither
grasp the object nor track it as it moves. (Unlike the traditional pick-and-place paradigm
of robotic manipulation, moving an object does not have to be limited to the speed or the
workspace of the manipulator.) High interaction forces are inherent in impulsive manipulation.
This can be desirable in many situations and can be limited in others through choice
of materials and impact strength.
One reason that tapping is not in the repertoire of today’s robots is that impact and friction
are complex nonlinear phenomena and are thus more difficult to include in a robot’s
conception of its world; neither are they as predictable as the traditional pick-and-place
paradigm. Furthermore, generating the controlled impact required for manipulation has
not received much attention. Thus, another motivation for this thesis is to gain experience
and insight in impact and friction phenomena and in the design of mechanisms to generate
a controlled impact.
1.1.1 Potential applications
Two potential applications will serve to set the context of this thesis and to indicate some
of its directions.
Micropositioning
Micropositioning of macroscopic objects is difficult because large forces are required to
break static friction, yet small forces or forces applied over a short time duration are required
in order to effect small movements. Tapping is ideal for micropositioning because
an impact breaks static friction instantaneously; the object motion is then determined by
the amount of impulse delivered by the tap.
There are several different ways in which robotic micropositioning through tapping
could be implemented:

1.2. THESIS OUTLINE 3
using a general purpose robot to position a micropositioning tapping device at points
around the object as required
using a number of fixed tapping devices to position an object within a micropositioning
“cell”
by placing a number of tapping devices on the object itself
Because of the variations in impact and friction, these methods require some sort of position
sensing. The experimental work in this thesis is aimed toward the first method and
uses overhead cameras for position measurement.
Tapping can be more accurate than the pick-and-place method for micropositioning
because the act of releasing a grasp can cause object motion. In addition, pick-and-place
positioning is only as as precise as the manipulator. With tapping, there is the potential
of positioning the object more precisely than the manipulator that positions a tapping device;
the positioning accuracy will then be limited by the impulse resolution of the tapping
device. (Hence, the importance of designing specialized tapping devices to optimize performance.)
Parts feeding
Many of the parts feeders in use today are vibrational parts feeders. One particular system
of interest is SONY’s APOS system. This parts feeder uses trays with specially designed
holes. A tray is held at an angle in an APOS machine, which vibrates the tray while pouring
parts over it. The holes are designed so that parts will remain in them during the vibration
only if they are in the correct orientation.
The interaction between parts and these parts trays is complex; friction, impact, sliding,
and rolling all come into play. This interaction includes forms of impulsive manipulation
beyond the scope of this thesis — particularly situations when objects become airborne
in between taps. Currently, these parts trays are designed by hand, guided by experience
and experiments. We hope to eventually apply a general understanding of impulsive manipulation
and other modes of manipulation to analyze this system and automate the design
of these parts trays from CAD models of the parts.
1.2 Thesis outline
This thesis consists of three main parts: analysis of the mechanics, planning, and controllability
of tapping; experiments; and a study of vibratory manipulation.
The first problem this thesis considers is finding a solution to the inverse mechanics of
tapping. The key question is:
Given a start configuration (position and orientation) and a desired goal configuration,
how should an object be tapped in order for it to slide from the start
to the goal?

4 CHAPTER 1. INTRODUCTION
Since the manipulator interacts with the object only at the instant of impact, the mechanics
of tapping can be divided into two subproblems: the inverse sliding problem and the impact
problem, the subject of Chapters 2 and 3 respectively.
The inverse sliding problem is to determine the initial velocities that will allow an object
to achieve the correct displacement (distance and rotation) during the sliding motion;
the impact problem is to determine, if possible, how to generate those initial velocities by
striking the object.
The solution to the inverse mechanics of tapping shows how to plan for a single tap. In
Chapter 4, I extrapolate from this solution to demonstrate a method for planning multiple
taps; this leads naturally to a demonstration of the controllability of tapping.
Experiments have been a large part of this dissertation, and in order to perform these
experiments, I needed some means of generating impact. Designing a device to produce a
controlled repeatable impact turned out to be a difficult problem; Chapter 5 describes the
tapping devices used in this thesis and discuss some of the key issues in their design.
Most of the experiments were designed to determine how well the models used for
analysis reflect actual object behavior. These experiments consisted of many single tap
runs, varying the impact parameters (and thus varying the initial object velocities) between
runs. The results of these experiments in regard to both the impact and sliding models are
described and discussed in Chapter 6. The remainder of the experiments demonstrated
positioning via tapping; the culmination of the experimental effort was a demonstration
that tapping can be used to position an object more precisely than the tapping device itself
is positioned. The results of these experiments and the relevant planning and control issues
are described in Chapter 7.
The final subject of this thesis is vibratory manipulation, which considers the behavior of
a sliding object subject to a regular or servoed striker motion, particularly a high frequency
low amplitude motion. In this sense, vibratory manipulation considers system level issues
whereas the bulk of this thesis concentrates on issues of mechanics and planning. I discuss
and develop this subject further in Chapter 8.
1.3 Assumptions
I have made a number of assumptions in this thesis: the support surface is flat and homogeneous,
and the objects are planar, rigid, and are of known geometry, mass, support
distribution, and material properties. The only forces acting upon an object during free motion
are those due to Coulomb friction. I make some additional assumptions specifically
relating to the impact and friction phenomena.
1.3.1 Impact
The classical model of impact assumes that objects are rigid in the sense that the area near
the contact point which undergoes deformation during impact must be small compared to
thesizeoftheobject.
Using the classical model of impact, a planar collision between two rigid bodies can
be characterized by:

1.4. RELATED WORK 5
the geometry and relative velocities of the bodies
a coefficient of restitution
a coefficient of friction between the striker and the object.
A more detailed discussion of impact is left to Chapter 3.
1.3.2 Friction
In this thesis, I assume that Coulomb friction applies between the object and the support
surface and between the object and the striker. Coulomb friction provides a simple linear
relationship between the normal force and the frictional force on a sliding object. However,
as Feynman [24] notes:
...this lawis theresultofanenormouscomplexityofeventsandisnot,fundamentally,
a simple thing. If we continue to study it more and more, measuring
more and more accurately, the law will continue to become more complicated,
not less. ...the more deeply we study it, and the more accurately we measure,
themorecomplicatedthetruthbecomes...
Despite the underlying complexity of this phenomenon, the Coulomb friction model works
well over a broad range of materials, velocities, and contact situations.
A difficulty in calculating the friction on a body with a finite pressure distribution is
that the actual pressure distribution between a body and a support surface is never known.
The actual pressure distribution is determined by microscopic variations in the surface of
the object and the support surface. Much of the analysis in this thesis assumes a constant
uniform support distribution; the results of the single tap experiments show that this is not
an unreasonable assumption.
1.4 Related Work
There has been little previous work on impulsive manipulation in the context of tapping.
However, this thesis does draw from research in a number of related areas, and in this
section, I have tried to identify the most relevant works in each area.
1.4.1 Impulsive manipulation
The first instance of impulsive manipulation in the robotics literature was work done by
Higuchi [29] who used an electromagnetic coil to deliver an impulsive force for linear
micropositioning. More recently, Yamagata and Higuchi [65] have developed a piezoelectric
device to deliver impulsive forces. This device creates an impulsive force by rapidly
expanding a piezoelectric material which pushes against a reaction mass. They have used
this device for precision alignment of optical sensors (Yamagata and Higuchi [65]) and for
a positioning table for a scanning tunneling microscope which operates in an ultrahigh

6 CHAPTER 1. INTRODUCTION
vacuum (Yamagata et al. [67]). They have also explored the use of thermal expansion instead
of the piezoelectric effect for analogous micro electrical mechanical systems (MEMS)
devices (Yamagata et al. [66]).
Zhu et al. [70] have studied what they call “releasing manipulation” which is based
in part on a publication of some early results of this thesis (Huang et al. [33]). Releasing
manipulation is similar to the subject of this thesis in that it involves an object sliding on
a support surface; it differs in that the initial velocities are not produced by impact but by
smoothly accelerating an object with a manipulator and then releasing it when the required
velocities have been attained. In this paper, Zhu et al. presented the mechanics, simulations,
and experimental results for a rectangular object sliding on a surface. Their experimental
results, like the results of experiments I have since conducted (presented in Chapter 6),
show considerable variation in the angle rotated and distance translated.
This thesis builds upon the work of Voyenli and Eriksen [62] who analyzed the physics
of sliding rotating disks and rings. Their development of the state space behavior of such
an object led to their observation that sliding disks and rings will come to rest only at a
certain ratio of angular to translational velocity. Goyal et al. [28] developed a more general
formulation of this idea and called these special velocity ratios “eigendirections,” referring
to their vector representation in generalized velocity space.
1.4.2 Friction
The study of friction dates back to Leonardo da Vinci in the mid 1400s, although our “modern”
theories of dry friction did not appear until Amontons and Coulomb published their
works on the subject in 1699 and 1781 respectively.
The text by Bowden and Tabor [14] is an introduction to tribology and the mechanisms
of friction. Two other books by Bowden and Tabor ([12] and [13]) present an extensive
study, including many experimental results.
There has been much work in studying Coulomb friction on an object resting on a
support surface. Goyal et al. ([27] and [28]) developed the limit surface, a graphical representation
of the relationship between the velocities of an object and the net force and torque
due to friction. Howe and Cutkosky [32] examined practical approximations to the limit
surface and performed experiments in the context of robotic grasping.
There has also been work in measuring the pressure distribution and center of friction.
Lynch [39] studied estimation of the pressure distribution of an object though quasistatic
pushes. Yoshikawa and Kurisu [68] calculated the center of friction from pushing an object
with a mobile robot.
Armstrong-Hélouvry [6] did a detailed study of friction in his work in controlling a
robotic arm.
1.4.3 Impact
The study of collision between two bodies has been studied since the time of Galileo,
Descartes and Newton, and it was in their time (the mid 1600s) that the first theories of
impact surfaced.

1.4. RELATED WORK 7
The text by Goldsmith [26] is a standard reference on impact with friction; he reviews
the classical methods of analyzing impact and presents analysis of the micro-mechanics of
impact. Bowden and Tabor [13] present analysis and experiments in the micro-mechanics
of impact in their book. A more recent text on impact is by Brach [15]; he discusses a larger
range of phenomena such as tangential and torsional restitution, energy loss, and collision
of planar linkages.
Routh [54] demonstrated a graphical method for analyzing planar impact with friction
which was more recently presented by Wang and Mason [63]. Stronge [57], in addition to
proposing a new restitution hypothesis, gave a brief history of the development of impact
theories.
1.4.4 Nonprehensile manipulation
Nonprehensile (nongrasping) manipulation is an important type of manipulation because
of its versatility. Grippers tend to be designed to manipulate a certain size or shape of
parts. In contrast, nonprehensile modes of manipulation do not need a special end effector
and can be used on a larger class of objects.
Much research has been focused on pushing, which was originally analyzed by Mason
[44]. Many others have done additional work in pushing, including Peshkin and
Sanderson ([50] and [49]), Lynch ([40] and [38]), Alexander and Maddocks [5], Goldberg [25],
Mani and Wilson [42], Akella and Mason [4], Kurisu and Yoshikawa [37], and Pham et
al. [51].
Other forms of nonprehensile manipulation include the whole arm manipulation of
Salisbury [55] and Trinkle et al.([60] and [59]) and the “palmar” manipulation of Erdmann
[22], Zumel [71], Yun [69], and Paljug and Yun [48].
1.4.5 Impact-based dynamic simulation
Related to the study of vibratory manipulation in this thesis is the idea of impact-based
dynamic simulation, as described by Mirtich and Canny [47]. This method of simulation
uses only impact constraints for dynamic modeling. Noncolliding contacts, such as rolling,
sliding, and resting, are modeled as a series of microcollisions. Kinematic constraints, however,
pose some difficulty because of the potentially large number of collisions required to
model such a constraint.
1.4.6 Minimalism in robotics
One theme which underlies this work is that of minimalism in robotics. Canny and Goldberg
([19] and [20]) use the term “RISC robotics” (reduced intricacy in sensing and control).
This is the idea of enabling simpler robots to do more complex tasks by creating more intelligent
planners or “compiling” intelligence into a mechanism compensate for the use of
fewer actuators and sensors.
The walking machines of McGeer [45] and [46] are perhaps the earliest and most
dramatic example of minimalism. These machines have no actuators or sensors; they are
designed so that the mechanism (powered by gravity) produces a stable walking gait.

8 CHAPTER 1. INTRODUCTION
Other examples include the tray tilting of Erdmann and Mason [23] and the parts
orienting of Akella et al.([2] and [3])
1.4.7 Parts feeding
Vibrational parts feeders are related to impulsive manipulation and tapping, although in
these systems objects tend to become airborne between impacts with a vibrating surface.
The most common type of vibratory parts feeder is the bowl feeder which is described
in (Boothroyd et al. [11]). Automated design and configuration of bowl feeders has been
studied by Boothroyd [10], Caine [18], Christiansen et al. [21], and Berkowitz and Canny [8].
The SONY APOS system (Hitakawa [30]) is another type of vibratory parts feeding system,
which has been studied by Krishnasamy [36].
Other work in vibratory parts feeding includes that of Böhringer et al. [9], who oriented
planar parts by taking advantage of the vibrational modes of the support surface, and
Swanson et al. [58], who studied parts orienting using a vibratory juggling motion.
1.4.8 Robotic juggling & ball bouncing
The vibratory manipulation discussed in Chapter 8 is closely related to robotic juggling
and the problem of a bouncing ball on a sinusoidally vibrating table. The most relevant
work in robotic juggling studies stability of single ball (or puck) juggling, which has been
done by Schaal and Atkeson [56], Bühler and Koditschek [16], Rizzi and Koditschek [53]
and Burridge et al. [17].
A ball bouncing on a sinusoidally vibrating table can display bounded chaotic behavior.
The first works to study this problem were by Holmes [31], who did analysis of
approximated mechanics, and Bapat et al. [7] who did analysis and simulation of the approximated
and actual mechanics. Other analytical study has been done by Wiesenfeld and
Tufillaro [64], and some experiments have been performed by Tufillaro and Albano [61].
This work will be discussed in greater detail in Chapter 8.

Chapter 2
The Inverse Sliding Problem
The inverse sliding problem is the central problem of the mechanics of tapping. Given
some initial velocities, it is fairly straightforward to do forward integration to determine
where an object will slide to a stop. This chapter addresses the inverse problem, i.e. given
a desired displacement, determining the required initial velocities. In Chapter 3, I address
the second part of the mechanics: how to generate those initial velocities via impact.
The inverse sliding problem can be formally stated as follows:
A rigid planar object of known geometry, support distribution, and frictional
properties slides on a uniform surface, slowing down and coming to rest only
due to Coulomb friction. Given a desired displacement (translation and rotation),
determine the required initial velocities.
This chapter addresses separately the classes of axisymmetric and nonaxisymmetric
objects. Axisymmetric objects are those objects whose pressure and mass distribution are
functions of radius only. Axisymmetric objects, examples of which include rings, disks,
and annuli, have the properties that:
the net force and torque on the object are independent of orientation, and
the net force on the object is always parallel to the translational velocity.
These properties simplify the equations of motion for axisymmetric objects.
It is the coupling between translation and rotation of a sliding object that makes this
problem difficult. This chapter describes a method for searching for the required initial
velocities for the axisymmetric case and the application of a general numerical root finding
method to the nonaxisymmetric case. This chapter also develops some additional properties
of the dynamics of sliding axisymmetric objects which will be used for planning and
controllability in Chapter 4.
2.1 Axisymmetric case
Since the net force due to friction on an axisymmetric object is always parallel to the translational
velocity, axisymmetric objects will always slide in a straight line. Thus the axisym-
9

10 CHAPTER 2. THE INVERSE SLIDING PROBLEM
y
ω
x
θ
v
θ f
start
x
x f
goal
Figure 2.1: Notation for the axisymmetric inverse sliding problem. The goal configuration
is at a distance x f along the ^x axis and a rotation of f . The object’s position and orientation
during the sliding motion are x and , its velocities v and !.
metric inverse sliding problem need only consider one dimension in translation and one
dimension in rotation.
Figure 2.1 shows the notation used in this section. We place the global coordinate
frame at the center of mass (COM) of the object in the start configuration so that the ^x axis
passes through the COM of the object in the goal configuration. The desired displacement
is represented as a translation of d along the ^x axis and a rotation of about the COM. The
object’s configuration is represented as its position x along the axis and its rotation with
respect to the start configuration. The object’s translational velocity is v, its the rotational
velocity !. The object is of mass M and has a moment of inertia I about the COM.
This section first develops the net force and torque load on a sliding rotating object,
formulates the equations of motion, and then mathematically states the inverse sliding
problem. Unfortunately, the problem cannot be solved directly because there are no analytic
solutions to the equations of motion. However, reasoning about the net force and
torque load and about the velocity profiles leads to a method to search for the desired
initial velocities to achieve a desired displacement.
2.1.1 Force and torque due to friction
Under Coulomb friction, the frictional force at a point on the object directly opposes the
velocity of that point but is independent of the magnitude of the velocity. The net force
and torque due to friction are simply integrals over the support distribution of the object.
The velocity of a point ~r on the object is given by:
The direction of this velocity is:
~u = v^x + !^z ~r (2.1)
^u =
v^x + !^z ~r
jv^x + !^z ~r j = ^x + ! ^z ~r
v
j^x + ! ^z ~r j (2.2)
v
Note that the direction of this velocity is dependent only upon the the ratio of angular
velocity to translational velocity, ! , and the vector ~r.
v
The differential force load due to friction at this point is:
~df = dmg^u (2.3)

2.1. AXISYMMETRIC CASE 11
(The force that acts on the object at this point is then , ~ df.) Since the net force load will lie
along the x axis, we can write:
The net torque load is given by:
T ( ! v )= Z
F ( ! v )= Z ~ df = g Z ^u dm (2.4)
Z
~r df ~ = g
~r ^u dm (2.5)
The net force and torque load are nonlinear functions and generally do not have analytic
forms.
Mason [43] has shown that the net force is strictly monotonic decreasing and the net
torque is strictly monotonic increasing (with respect to ! ). This property will be essential
v
in the following subsections.
Intuitively, we can understand this property from the following argument. The magnitude
of the frictional force at any given point is fixed; its direction is determined by the
ratio ! . For a fixed velocity v, when! =0(i.e. when ! =0), all the frictional forces oppose
translation, so there will be maximum force and zero torque. As ! increases (i.e. as ! v
v v
increases), the frictional forces will increasingly oppose the rotational motion. This results
in increasing torque and decreasing force.
2.1.2 Equations of motion and displacement functions
The equations of motion that govern this system are:
M _v = ,F ( ! v ) (2.6)
I _! = ,T ( ! v ) (2.7)
_x = v (2.8)
_ = ! (2.9)
It is here that the coupling between translation and rotation of a sliding object becomes
apparent.
The total distance and angle can be treated as functions of the initial translational and
rotational velocities (v 0 and ! 0 respectively). I denote these displacement functions by x f
and f . These functions are given by:
x f (v 0 ;! 0 )=
f (v 0 ;! 0 )=
Z
t f
0
Z t f
0
v(t) dt (2.10)
!(t) dt (2.11)
where v(t) and !(t) are solutions to the equations of motion, subject to the conditions:
v(0) = v 0 !(0) = ! 0 (2.12)

12 CHAPTER 2. THE INVERSE SLIDING PROBLEM
50
40
30
wo
20
10
0.4
1
xf 50
0.5
40
0
30
2.0
wo
1.6 20
1.2 0.8
10
vo 0.4
30
20
10
0
1.2 1.6 2.0
0.8 vo
thf
Figure 2.2: Example displacement functions x f and f for a disk of radius 0.05 meters and
mass 0.1 kg; the coefficient of friction is 0.25. v 0 is in meters/sec, ! 0 in radians/sec, x f in
meters, and f in radians. These surfaces were numerically generated.
and
v(t f )=0 !(t f )=0 (2.13)
where t f is the time at which the object comes to rest.
The inverse sliding problem for axisymmetric objects can now be stated as solving:
x f (v 0 ;! 0 ) = d (2.14)
f (v 0 ;! 0 ) = (2.15)
for ! 0 and v 0 given a desired translation d and rotation .
Although x f and f cannot be computed analytically, certain properties can be deduced
which will lead to a method for finding the required ! 0 and v 0 .
Note that the x f function is even with respect to ! 0 and f odd, i.e. x f (v 0 ;! 0 ) =
x f (v 0 ; ,! 0 ) and f (v 0 ;! 0 )=, f (v 0 ; ,! 0 ). I will often consider only positive rotations for
clarity, but the results are symmetric.
2.1.3 Monotonicity of the displacement functions
For the remainder of this section, I assume that the initial rotational velocity ! 0 is nonnegative.
Although analogous properties and methods hold for negative initial rotational
velocities, the notation and discussion becomes cumbersome when handling both cases
simultaneously.
The displacement functions x f (v 0 ;! 0 ) and f (v 0 ;! 0 ) are monotonic in v 0 and ! 0 in
the following sense: if we start at some point (v 0 ;! 0 ) these functions both increase as v 0
increases (with ! 0 held constant), and vice versa. Figure 2.2 shows an example of the
displacement functions.
Intuitively, if the object has a greater initial translational velocity it will slide further.
This means that it will take more time for the object to come to rest, so it will also rotate

2.1. AXISYMMETRIC CASE 13
more. Similarly, if the object has a greater initial angular velocity, it will rotate more and
also slide further.
More formally, we have the following implications concerning two different trajectories.
v 01 >v 02 ! 01 = ! 02 =) x f1 >x f2 (2.16)
v 01 >v 02 ! 01 = ! 02 6=0 =) f1 > f2 (2.17)
v 01 = v 02 6=0 ! 01 >! 02 =) x f1 >x f2 (2.18)
v 01 = v 02 ! 01 >! 02 =) f1 > f2 (2.19)
where x fi and fi is the displacement resulting from the initial velocities ! 0i and v 0i .These
statements can be proven by reasoning about the velocity profiles of these two trajectories.
Theorem 1 If, at some time t, two trajectories satisfy:
then at time t + they will satisfy:
v 1 >v 2 and ! 1 = ! 2 6=0
v 1 >v 2 and ! 1 >! 2
Proof Under the conditions at time t,wehave !1
v 1
< !2
v 2
. From the monotonicity properties
of subsection 2.1.1, this implies that T ( !1
) ! 2 while the condition v 1 >v 2 still holds.
Theorem 2 If, at some time t, two trajectories satisfy:
v 1 = v 2 6=0 and ! 1 >! 2
then at time t + they will satisfy:
v 1 >v 2 and ! 1 >! 2
Proof Under the conditions at time t, wehavej !1
v 1
j > j !2
v 2
j. From the monotonicity properties
of subsection 2.1.1, this implies that F ( !1
v 1
) j _v 2 j, and by the definition of the derivative and of limits, this implies
that at time t + we will have v 1 ! 2 still holds.
Theorem 3 If at some time t 0 we have:
v 1 >v 2 ! 1 >! 2
then for all t>t 0 the following condition will hold:
v 1 v 2 ! 1 ! 2

14 CHAPTER 2. THE INVERSE SLIDING PROBLEM
Proof The only way that this condition could be violated is if v 1 (t) crossed v 2 (t) or ! 1 (t)
crossed ! 2 (t). Both of these crossings cannot happen simultaneously because the derivatives
_v and _! are uniquely defined by the equations of motion. However it could be possible
for v 1 (t) to cross v 2 (t) while ! 1 (t) >! 2 (t), orviceversa.
But in order for v 1 (t) to cross v 2 (t), atsometimet we must have v 1 = v 2 . Then by
Theorem 2, at time t + we will have v 1 >v 2 .Inorderfor !1
v 1
to cross ! 2 (t), atsometimet
we must have ! 1 = ! 2 . Then by Theorem 1, at time t + we will have ! 1 >! 2 . Thus the
condition will not be violated.
We are now in a position to prove the implications from the beginning of this subsection.
The left hand sides Equations 2.16 and 2.17 are the condition for Theorem 1, so for
some period of time after t =0, we will have v 1 >v 2 and ! 1 >! 2 . By Theorem 3, we will
have the condition v 1 v 2 and ! 1 ! 2 for the remaining time. Together, these imply:
which simplifies to:
Z t f 1
0
Z t f 1
0
v 1 (t) dt >
! 1 (t) dt >
Z t f 2
0
Z t f 2
0
v 2 (t) dt (2.20)
! 2 (t) dt (2.21)
x f (v 01 ;! 01 ) >x f (v 02 ;! 02 ) (2.22)
f (v 01 ;! 01 ) > f (v 02 ;! 02 ) (2.23)
Similar reasoning can be employed to prove equations 2.18 and 2.19.
2.1.4 Level curves of displacement functions
We would like to solve for the required initial condition by intersecting the level curve of x f
corresponding to a translation d and the level curve of f corresponding to a rotation .
However, since there is no analytic form for the displacement functions, there is no analytic
form for their level curves. The monotonicity properties of the displacement functions can
be used to determine some properties of these level curves which will lead to a method for
finding the solution; this solution is unique and always exists.
Trivial cases of the displacement functions
The form of the displacement functions along the v 0 and ! 0 axes can easily be determined.
In these cases, the coupling between translation and rotation disappears, the problem is
reduced to elementary physics, and we get:
x f (0;! 0 ) = 0 (2.24)
f (v 0 ; 0) = 0 (2.25)
x f (v 0 ; 0) =
f (0;! 0 ) =
1
2 Mv2 0
Mg
1
2 I!2 0
g R j~r j dm
(2.26)
(2.27)

2.1. AXISYMMETRIC CASE 15
50
xf
50
thf
40
40
wo
30
wo
30
20
20
10
10
0.4 0.8 1.2 1.6 2.0
vo
0.4 0.8 1.2 1.6 2.0
vo
Figure 2.3: Level curves of the displacement functions x f and f from Figure 2.2.
Monotonicity of the level curves
To find a level curve of x f at a height d, we start on the v 0 axis. There, we can solve
d = x f (v 0d ; 0) for v 0d using Equation 2.26. Imagine following the level curve of x f using a
zigzag motion, repeatedly increasing ! 0 on the first step and then adjusting v 0 on the next.
If we take a step upwards, increasing ! 0 , then we leave the level curve; by Equation
2.16, we know that x f increases as we move upwards, parallel to the positive ! 0 axis.
In order to get back on the level curve, we must adjust the v 0 coordinate. By Equation 2.17
we know that x f increases parallel to the positive v 0 axis. Since we need to go downhill to
the level curve, we decrease the v 0 coordinate. This implies that the level curve is monotonic
decreasing — as its ! 0 coordinate increases, its v 0 coordinate decreases.
A level curve of x f then starts at some point on the v 0 axis and moves upwards and
to the left, asymptotically approaching the ! 0 axis. This level curve must lie between the
lines v 0 = v 0d and v 0 =0.
Similar reasoning can be employed to show that the level curve of f at a height
begins at a point ! 0 on the ! 0 axis (where ! 0 is the solution to = f (0;! 0 ) using
Equation 2.27). This curve is monotonic decreasing as it asymptotically approaches the v 0
axis, and the curve must lie between the lines ! 0 = ! 0 and ! 0 =0.
Figure 2.3 shows examples of the level curves of x f and f .
Monotonicity along the level curves
A level curve of x f and a level curve of f must intersect somewhere within the region
bounded by the v 0 and ! 0 axes and the lines v 0 = v 0d and ! 0 = ! 0 , but their monotonicity
properties do not guarantee a unique intersection point. However, the uniqueness of the
intersection can be proven by showing that f is monotonic along a x f level curve and vice
versa.
Consider two different points on a level curve of x f . (See Figure 2.4.) Since the level

16 CHAPTER 2. THE INVERSE SLIDING PROBLEM
x level curve
f
ω 0
2
1
v 0
Figure 2.4: By comparing velocity trajectories for two sets of initial conditions corresponding
to two points on a level curve of x f , we can show that f increases along an x f level
curve.
v
1
2
ω
2
1
t
t
Figure 2.5: The angular velocity trajectories cross.
curve is monotonic decreasing, we can assume:
v 01 > v 02 (2.28)
! 01 < ! 02 (2.29)
and therefore !0 1
v 01
< !0 2
v 02
.Aslongas !1
v 1
< !2
v 2
, then we will have F 1 >F 2 and T 1

2.1. AXISYMMETRIC CASE 17
v
1
2
ω
2
1
t
t
Figure 2.6: The translational velocity trajectories cross.
If ! 1 and ! 2 cross (Figure 2.5), then we will have v 1 v 2 for the entire trajectory,
implying that x f1 >x f2 which violates our assumption that the initial conditions were on
a level curve of x f .
Consequently, v 1 and v 2 must cross (Figure 2.6), which results in the condition ! 1 ! 2
for the entire trajectory, so f1 < f2 . Therefore, f increases monotonically along level
curves of x f . (Since by assumption the two points are from a level curve of x f , the net area
between the two translational velocity profiles will be zero.)
Similar reasoning can be applied to show that x f increases monotonically along level
curves of f . These properties imply that any given level curve of x f and any given level
curve of f will intersect exactly once.
2.1.5 Solving for initial velocities
The level curve of f for any given and the level curve of x f for any given d are guaranteed
to have a unique intersection point whose coordinates will correspond to the required
initial velocities to achieve that displacement. The intersection point is also bounded, but
the question still remains of how to determine the coordinates of this point.
The only way to get information about the location of the level curves is to do a forward
integration on some initial conditions and compare the resulting displacement to the
desired displacement. If a displacement component (either translation or rotation) is less
than that of the desired displacement, then that point is “below” the corresponding level
curve; if it is greater than the corresponding displacement component, then it is “above”
that level curve.
A variation on bisection can be used to zoom in on the coordinates of the intersection
point. Starting with the rectangular region that bounds the intersection point, we integrate
the initial conditions corresponding to the center of the rectangle. This divides the rectangle
into four smaller rectangles. Based upon whether the center is above or below the x f
and f level curves, we can eliminate at least one and possibly three of the newly formed
subrectangles. Continuing in this manner, we can eliminate regions of the bounding rectangle
by repeatedly subdividing rectangles and integrating points, eventually zooming in
on the intersection point to any desired accuracy. Figure 2.7 illustrates this process.

18 CHAPTER 2. THE INVERSE SLIDING PROBLEM
ωo
ωo α
vo
d
vo
Figure 2.7: Solving for the initial velocities. By integrating points and determining whether
they lie above or below the level curves, we can eliminate portions of the bounding rectangle
and zoom in on the intersection point. The initial velocities corresponding to the
intersection point are those that will produce the a translation d and rotation .
Implementation notes
I have implemented this procedure in C on a UNIX workstation. The underlying forward
integration routines use Gaussian quadrature integration to numerically integrate the net
force and torque due to friction. The equations of motion are integrated using a fixed step
fourth order Runge-Kutta integration. My integration routines have drawn from Numerical
Recipes in C [52], but I have written my own because of conditions unique to this problem,
namely that once the object comes to rest, friction no longer acts upon it.
For function integration, I experimented with less sophisticated routines (than Gaussian
quadrature) including trapezoidal, Simpson, and Romberg integration. I have found
that Gaussian quadrature integration gives virtually identical results as the other methods
with far fewer function evaluations.
For integration of the equations of motion, I have experimented with more sophisticated
routines (than fixed-step fourth order Runge-Kutta), such as adaptive step-size fifth
order Runge-Kutta and the Bulirsch-Stoer method. The fixed-step Runge-Kutta will overshoot
the velocities, i.e. will improperly let the velocities change sign, so I modified the
integration routine to back up and take smaller steps. The adaptive stepsize Runge-Kutta
and the Bulirsch-Stoer methods never overshot; however, the step size became so small towards
the end that these methods used far more derivative evaluations than the fixed-step
Runge-Kutta. Again, the resulting answers were virtually identical.
A substantial amount of code is required to keep track of all the squares, points, results
of integration, and whether a square or point has been eliminated, but most of the running
time is occupied by numerical integration. My implementation takes approximately 3 seconds
to run on a Sun SPARC 5 using an accuracy tolerance for the resulting displacement
that is beyond the accuracy of the high speed measurement device I have used for the single
tap experiments. The linear position tolerance is set to 0.05 mm, the angular position
tolerance to 0.057 degrees. It typically takes from 7 to 10 iterations to achieve this degree
of accuracy. For an experiment with 70 trials, the average number of iterations was 8.6 and
required integrating an average of 26 points.

2.1. AXISYMMETRIC CASE 19
Convergence of this method would be linear like bisection if all but one rectangle
was eliminated at each iteration. However this is not quite the case. For the previously
mentioned experiment, there was an average of 1.2 rectangles at the end of the last iteration
(most often 1 but sometimes 2).
2.1.6 Additional axisymmetric properties
In this subsection, I develop some additional properties of the dynamics of a sliding axisymmetric
object; these properties will be used for planning and controllability in Chapter
4.
Trajectory scaling
We can show through straightforward use of the chain rule, that if v(t) and !(t) are solutions
to the equations of motion, then so are kv( t k ) and k!( t ). If the first pair of trajectories
k
produce a displacement of x f and f , then the second produce a displacement of k 2 x f and
k 2 f .
One can also think about this property in terms of the vector field:
(v; !) 7! ( ,F ( ! v )
M ; ,T ( ! v )
) (2.34)
I
Since this field is dependent only upon the value of ! (i.e. the “angle” of the velocities),
v
the evolution of this system is not dependent upon the amount of energy of the initial
conditions but simply the ratio ! . Thus, paths in state space can be scaled.
v
One implication of trajectory scaling is that a line through the origin in the space of
initial velocities (constant !0
v 0
) can be mapped to a line through the origin in displacement
space (constant xf
f
).
Relationship between initial velocity and displacement ratios
We can draw upon the monotonicity along level curves (i.e. that x f increases along a f
and vice versa) and from the trajectory scaling property to show that xf
f
is a monotonic
decreasing function of !0
v 0
.
Consider a level curve of f as shown in Figure 2.8. Since x f monotonically increases
along this level curve, the ratio xf
f
the f level curve can be parameterized as a function of !0
v 0
will intersect the level curve exactly once. The value of !0
v 0
also monotonically increases. Because of monotonicity,
— every line from the origin
monotonically decreases along
(because of
this level curve. Since the ratio xf
f
remains constant along a line of constant !0
v 0
trajectory scaling), we can conclude that xf
f
monotonically decreases as a function of !0
v 0
.
Mapping initial velocity cones to displacement cones
The trajectory scaling property showed that a line through the origin in initial velocity
space maps to a line through the origin in displacement space. Now, with the monotonicity

20 CHAPTER 2. THE INVERSE SLIDING PROBLEM
ω 0
v 0
x increases
f
ω 0
v 0
decreases
θ level curve
f
Figure 2.8: Relationship between velocity and displacement ratios. We consider a level
curve of f to show that xf
f
is a monotonic decreasing function of !0
v 0
.Sincex f increases
along a f level curve, the value of xf
f
increases along this level curve; the value of !0
v 0
decreases along a level curve.
relationship between the initial velocity ratio and the displacement ratio, we can show that
cones map from one space to the other.
The cones we consider are continuous ranges of rays starting at the origin. A cone in
initial velocity space consists of a set of rays at the origin with a continuous range of slopes.
From the monotonicity property between initial velocity ratios and displacement ratios,
we know that this continuous range of slopes (values of !0
v 0
) maps to a continuous range of
slopes (values of xf
f
) in displacement space. Furthermore, the cones can be characterized
by their maximum and minimum slope, and the mapping between cones has a unique
inverse.
2.2 Nonaxisymmetric case
For nonaxisymmetric objects undergoing translation and rotation, the net force due to friction
will generally not be in the same direction as the net translational velocity. The underlying
reason for this is that Coulomb friction is not dependent upon velocity magnitude.
Consider the velocity vectors at every point on the object. The components of these vectors
normal to the net translational velocity sum to zero; they do not after the vectors have been
normalized.
Since these objects do not move in a straight line, the equations of motion now have
six state variables:
M ~v _ = , F ~ ( ! ; ^v; )
j~vj
(2.35)
I _! = ,T ( ! ; ^v; )
j~vj
(2.36)
_~x = ~v (2.37)
_ = ! (2.38)

2.2. NONAXISYMMETRIC CASE 21
Also note that the force and torque load are now functions of the direction of the translational
velocity ^v and the orientation of the object in addition to the velocity ratio !
j~vj .
In this section, I will discuss a general solution to the nonaxisymmetric case, show an
example of how the displacement functions (x f , y f ,and f ) are different for nonaxisymmetric
objects, and suggest a practical solution for this case of the inverse sliding problem.
2.2.1 General solution
Although the force and torque functions, for a given value of display the same monotonicity
properties (with respect to
j~vj ! ) as in the axisymmetric case, these functions change
shape with variation in . Thus the rotation of the object precludes the use of these monotonicity
properties in the same way that was used to find a solution for the axisymmetric
case.
A general solution to the nonaxisymmetric case must then rely upon general purpose
numerical methods. This problem is related to two point boundary value problems —
problems in which a solution to a set of differential equations must be found that satisfies
constraints at two boundaries. In typical two point boundary value problems, the two
boundaries are defined in terms of time, and the constraints at the boundaries are upon the
state variables. For the nonaxisymmetric inverse sliding problem, the second boundary
is determined by the state variables (i.e. when the object comes to rest) and the value at
the second boundary is an integral over the trajectory (i.e. the object displacement). This
problem lends itself naturally to the shooting method.
Applying the shooting method to the nonaxisymmetric case is equivalent to performing
multidimensional root finding on the displacement functions. Multidimensional root
finding is a difficult problem and is best solved when one has some idea of where a solution
lies. Various methods are generally based on the Newton-Raphson method — the jacobian
is used to form a set of linear equations to solve for a step which brings all functions closer
to zero simultaneously. In the shooting method, the Jacobian is estimated numerically.
Neither existence nor uniqueness of a solution is guaranteed in this case.
2.2.2 An example
Although the prospects for a simple solution to the nonaxisymmetric case look bleak, examination
of a few objects suggests that the problem is not quite as bad as it seems.
This subsection examines the level curves of the displacement functions for a twodimensional
barbell — two point masses on the end of a massless rigid rod. In this example,
the rod is 0.1 meters long and each point mass is 0.1 kg; the coefficient of friction is 0.2.
Figures 2.9, 2.11, and 2.10 show the displacement function and level curves for the x f , y f ,
and f respectively. These functions were computed for a translational velocity along the
positive x axis and with the barbell initially aligned with the x axis.
Note that the x f and f displacement functions are smooth, though there are of ripples
in these surfaces. The level curves of these functions all appear to be monotonic. The y f
displacement function is quite different — it has a number of distinct peaks.

22 CHAPTER 2. THE INVERSE SLIDING PROBLEM
1200
xf level curves
xf (m)
2
1.5
1
0.5
1200 0
800
400
wo (deg/sec)
0
0
1
2
vo (m/sec)
3
initial rot. velocity (deg/sec)
1000
800
600
400
200
0
0 0.5 1 1.5 2 2.5
initial trans. velocity (m/sec)
Figure 2.9: The x f displacement function for the two-dimensional barbell and a graph of
its level curves. The level curves are at 6 cm intervals.
1200
thf level curves
thf (deg)
1000
500
1200 0
800
400
wo (deg/sec)
0
0
1
2
vo (m/sec)
3
initial rot. velocity (deg/sec)
1000
800
600
400
200
0
0 0.5 1 1.5 2 2.5
initial trans. velocity (m/sec)
Figure 2.10: The f displacement function (net rotation) for the two-dimensional barbell
and a graph of its level curves. The level curves are at 13 degree intervals.

2.2. NONAXISYMMETRIC CASE 23
yf (m)
x 10 −3
20
10
0
1200
800
400
wo (deg/sec)
0
0
1
2
vo (m/sec)
3
initial rot. velocity (deg/sec)
1200
1000
800
600
400
200
yf level curves
0
0 0.5 1 1.5 2 2.5
initial trans. velocity (m/sec)
Figure 2.11: The y f displacement function for the two-dimensional barbell and a graph its
level curves. The level curves are at 1.2 mm intervals.
2.2.3 A practical solution
The monotonicity of the level curves of the x f and f displacement functions permits the
use of the axisymmetric solution to solve for the initial velocities for a given translation
along the x axis and for a given rotation. However, this assumes that the initial translational
velocity lies on the x axis. There will also be some translation along the y axis.
This suggests the use of a one dimensional numerical root finding method in which
the independent variable is the direction of the velocity. The required translational displacement
can be projected onto the axis defined by the direction of the initial translational
velocity, and the axisymmetric solution can be used to solve for this projected translation
and the desired rotation. The resulting initial velocities determine the amount of translation
normal to the initial velocity direction. The difference between this actual normal
translation and the required normal translation to reach the goal is then taken as the value
of the function whose root we wish to find.
Alternatively, the translation normal to the initial translational velocity can simply be
ignored. In the examples I have tried, the normal translation is usually much smaller than
the translation in the direction of the initial translational velocity. For the two-dimensional
barbell of the previous subsection, y f can be nearly 32% of x f (for smaller displacements),
is 0.0261, which corresponds to a deviation of 1.5 degrees
from the direction of the initial translational velocity — less than the typical errors due to
variations in impact and friction, as the single tap experiments of Chapter 6 show. Deviations
in direction of this magnitude can effectively be dealt with using planning and
feedback control strategies such as those used in the positioning experiments of Chapter 7.
but the average value of of jyfj
x f

24 CHAPTER 2. THE INVERSE SLIDING PROBLEM

Chapter 3
The Impact Problem
Once the inverse sliding problem has been solved, determining the required initial velocities
to achieve a desired displacement, the question that remains is how to generate those
initial velocities (if possible) via impact. This is the impact problem.
The solution to this problem must consider not only the mechanics of impact but also
the shape of the object. This chapter begins with a discussion of the mechanics of impact
and different impact models, but most of the details concerning impact have been left to
Appendix A in order that this chapter can focus on the issues regarding object shape.
At any given contact point on an object, there will be some range of directions in which
translational velocity can be produced by an impact. Each of these directions will have a
different lever arm and will thus produce a proportionately different amount of rotational
velocity. In order to produce a given rotational and translational velocity, the boundary of
the object must be searched for a point which has appropriate lever arm for a translational
velocity in the desired direction. This chapter describes a method to represent all velocities
that can be generated by striking a given object, and how the boundary search can be
conducted in terms of this representation.
3.1 Impact models
The collision between two rigid bodies (assumed to contact at a single point) is analyzed
through a progression of phases.
Once the two objects have made contact, the compression phase begins; in this phase,
small local (elastic and plastic) deformations occur which gradually deliver impulse to
the objects until their relative normal velocity reaches zero. Then the restitution phase
begins, and the “decompression” of the contact area (due to elastic deformation) adds more
impulse to the bodies, propelling them apart, until the collision ends.
Impact is characterized by a coefficient of restitution e which, at the simplest level,
represents the amount of elasticity of the collision. When e =0, the collision is perfectly inelastic
(or plastic); when e =1, the collision is perfectly elastic. This coefficient is generally
assumed to be between 0 and 1 and is largely a property of the different materials involved
in the collision. Although it is often presumed to be a constant, the coefficient of restitution
25

26 CHAPTER 3. THE IMPACT PROBLEM
does vary with the velocity of the impact.
Collision was first studied for direct central impact (i.e. the COMs and the contact normal
lie on the same line, and there is no relative tangential velocity). Early experiments
typically involved the collision of two spheres. Newton’s law (for impact) defines the coefficient
of restitution to be , v, v+
, the ratio of the (normal) velocity after impact to that before
impact. This law is undisputed for direct central impact, and all other proposed restitution
hypotheses simplify to Newton’s law under these conditions.
If the objects collide with some tangential velocity (in addition to normal velocity),
then there will be tangential impulse due to some phenomena; this also affects the collision
process. Most often, Coulomb friction is taken to apply at the contact, so some small relative
motion during a “sliding” mode of the collision is postulated. Other effects, such as
tangential and torsional restitution, are sometimes held to apply instead.
Under Coulomb friction, the tangential impulse will oppose the direction of the relative
sliding motion between the two objects, and the tangential impulse is related to the
normal impulse by a coefficient of friction. If enough tangential impulse is delivered to
the objects that the relative sliding velocity becomes zero, then the collision enters either a
“sticking” mode, in which the relative tangential velocity remains zero, or a “reversed sliding”
mode in which the objects will slide in the opposite direction. Whether this transition
to “sticking” or “reversed sliding” occurs in the compression or restitution phase can make
a difference in the outcome of the collision.
Newton’s law has been shown to inappropriately yield energy increases under collisions
in which enter a sticking or reversed sliding mode. An alternative to Newton’s
law is Poisson’s hypothesis. Poisson defined the coefficient of restitution as the ratio of
the normal impulse delivered during restitution to the normal impulse delivered during
compression. Poisson’s hypothesis and Newton’s law yield identical results for impacts in
which there is no slip or in which slip continues through the entire impact.
Stronge [57] has proposed the internal dissipation hypothesis in which the square of
the coefficient of restitution is defined to be the ratio of elastic strain energy released during
restitution to the energy absorbed by deformation during compression. This hypothesis is
also identical to Newton’s law for impacts in which there is no slip or in which slip continues
through the entire impact. Stronge shows that Poisson’s hypothesis dissipates too
much energy for perfectly elastic collisions involving a reversed sliding mode. The internal
dissipation hypothesis results in a collision that terminates at some point in between where
Poisson’s hypothesis and Newton’s law would terminate the collision.
In this thesis, I have adopted Poisson’s hypothesis for several reasons. Since I will be
applying this analysis to physical experiments, there will be no materials with a coefficient
of restitution of 1. Poisson’s hypothesis also lends itself more easily to the analysis of this
thesis than Stronge’s hypothesis, though not quite as easily as Newton’s law would.
Although the choice of a restitution hypothesis affects the particular striker parameters
to achieve a given impulse, it does not (at least among these three hypotheses) affect the
range of impulses which can be achieved via impact at a given contact point.
Routh [54] presented a graphical method for analyzing frictional impact that was more
recently presented in Wang and Mason [63]. This method traces the progress of the impact
in impulse space, using various lines to represent the relative accrual of normal and tan-

3.2. IMPACT PROBLEM CONSTRUCTS 27
−1
tan µ r
r
v so
striker
β
Figure 3.1: Notation for the impact problem. A striker with velocity v s0 at an angle of
incidence with respect to the contact normal collides with the object at a point ~r. By
varying the values of v s0 and , the collision can deliver any impulse within a friction cone
in impulse space. The resulting translational velocity will be in the same direction as the
impulse, so it will lie within the shaded friction cone in the contact frame. (Coefficient
of friction between the striker and the object is denoted by r .) The resulting rotational
velocity will be determined by the impulse and the vector ~r.
gential impulse in different impact modes as well as to represent constraints such as the
boundary between the compression and restitution phases. Wang and Mason present results
for Newton’s and Poisson’s laws of restitution.
In Appendix A, I have summarized Wang and Mason’s results and use them as a
starting point for showing some properties of impact.
3.2 Impact problem constructs
From Appendix A, this chapter takes the following result as a starting point:
For any contact point on the object, an impact can generate any impulse within
a friction cone in impulse space, i.e. any impulse in which the tangential component
divided by the normal component is less than or equal to the coefficient
of friction between the striker and the object.
This is illustrated in Figure 3.1. The details of determining the actual striker parameters to
generate a particular impact are left to the appendix.
This result comes with two assumptions about the striker:
The striker has no rotational velocity; thus there are two parameters — the initial
velocity of the striker and the angle of incidence at which it collides with the object.
Either the striker’s COM always lies on the contact normal or the striker has infinite
mass or angular inertia.
These issues are discussed in greater depth in Appendix A.

28 CHAPTER 3. THE IMPACT PROBLEM
3.2.1 From impulse to velocities
The impulse generated through impact is related to the initial velocities of the object by:
M~v 0 = ~P (3.1)
I! 0 =(~r ~ P ) z (3.2)
where ~r is the vector from the COM to the contact point. The direction of the impulse
determines the direction of the linear velocity, and the contact point determines the lever
arm with which this impulse acts on the object.
Note that ~v 0 and ! 0 are both proportional to the magnitude of the impulse. Since the
constraint on possible impulses is on the direction of the impulse, not the magnitude, it
makes sense to consider the ratio of velocities:
! 0
j~v 0 j = M I (~r ^P ) z (3.3)
which is a function of the direction of the impulse and the contact point.
At a given contact point, a continuous range of ~P (such as all the vectors in a friction
cone at the contact point) will result in a continuous range of !0
j~v 0j
. However, each of these
velocity ratios pertains to a velocity in a different direction. Since we generally want to
produce a velocity in a given direction, we must consider other contact points that can
produce velocity in this direction; these other points will generally have different !0
ratios
j~v 0j
for translational velocity in the desired direction.
3.2.2 Directed velocity ratio sets & impact cones
In order to determine the set of velocity ratios achievable in a given velocity direction, we
must search the boundary of the object to find all points at which an impact can produce
a translational velocity in that direction. This subsection shows how this search can be
“precompiled” into a mapping which I denote by K.
Let the boundary of the object be parameterized by , anddefineS() to be the set of
all directions in which impulse can be generated at the point , i.e.thesetofallvectorsin
the friction cone at the contact point. The set of velocity ratios for translational velocity in
the direction ^v 0 is given by:
K(^v 0 )= M
I (~r() ^v 0) z
9 ^v 0 2S()
(3.4)
The set K(^v 0 ) for smooth convex objects will consist of a continuous range of !0
j~v 0j
for any
given ^v 0 . If there are any points on the boundary at which the object should not (or cannot)
be tapped, they can simply be excluded from consideration when forming the K mapping.
When K(^v 0 ) consists of a continuous range of possible velocity ratios, it is convenient
to represent this set as an impact cone in velocity space (the ! 0 v 0 plane for translational
velocity in a particular direction). This cone consists of all vectors for which !0
2K(^v j~v 0j 0)
and thus contains all points in velocity space that can be reached in a single tap.

3.3. EXAMPLES 29
3.2.3 From velocities to impulse
Given some desired initial velocities ! 0 and ~v 0 , the condition:
! 0
j~v 0 j 2K(^v 0) (3.5)
must be satisfied in order to generate those velocities via impact. The impulse that will
generate these velocities is ~P = M~v 0 , however, the contact at which to apply this impulse
must still be determined. This will require a search of the the boundary of the object for a
point which can produce the proper !0
j~v 0j
in the right direction. For simple boundary shapes,
it may be feasible to construct the contact point mapping:
B( ! 0
j~v 0 j ; ^v !
0)=
0
j~v 0 j = M R j~r() ^v 0j
(3.6)
Given a contact point and the required impulse, the methods in Appendix A can be used
to determine the striker parameters v s0 and .
3.3 Examples
This section develops the mapping used to solve the impact problem for several sample
objects. Illustrations of these objects and the sets K(^v 0 ) are shown in Figure 3.2.
3.3.1 Circular object
For a circular object, the set K is independent of ^v 0 and is given by:
K(^v 0 )=
", MR
p r
; MR
I 1+
2
r
p r
I 1+
2
r
#
(3.7)
where r is the coefficient of friction between the striker and the object. If we parameterize
the boundary by an angle from some reference axis and the direction of ^v 0 by the angle
from the same axis, then we can define the contact point mapping:
3.3.2 Square object
sin,1 B( ! 0
j~v 0 j ;)= + + !0
j~v 0 j
I
MR
(3.8)
K() =8><
>:
,
MR
sin( 3 I 4
,
MR
sin( 5 I 4
,
MR
sin( 7 I 4
,
MR
sin( 9 I 4
) MR
, ) ; sin( 3 I
+ jjtan ,1
4 r
) MR
, ) ; sin( I
+ j , 4 2 jtan,1 r
) MR
, ) ; sin( , I
4
, ) ;
MR
I
sin( , 3 4 )
j , j tan ,1 r
j , 3 2 jtan,1 r
; otherwise
(3.9)

30 CHAPTER 3. THE IMPACT PROBLEM
Wo/|Vo|
10
7.5
5
2.5
-2.5
-5
-7.5
-10
Wo/|Vo|
Pi/2 Pi 3Pi/2 22 Pi Pi theta
η
v
40
20
-20
Pi/2 Pi 3Pi/2 22 Pi Pi theta
η
v
-40
Figure 3.2: Directed velocity ratio mappings. These plots show the mapping K, which
contains all possible velocity ratios that can be achieved by impact as a function of the
direction of the resulting translational velocity (parameterized by orientation .

Chapter 4
Planning & Controllability
The previous two chapters focused separately on sliding and impact in order to solve the
two parts of the problem for planning a single tap. In this chapter, those results are combined
in order to address planning for multi-tap plans. This will naturally lead to a discussion
of the controllability of tapping that serves to precisely characterize this mode of
manipulation.
This chapter addresses only objects with an axisymmetric support distribution. Since
there seems to be no simple characterization of the possible displacements for objects with
nonaxisymmetric support distributions, these objects do not lend themselves to multi-tap
planning. Even with axisymmetric objects, it does not make sense to extrapolate too many
steps into the future because of the inherent sensitivity of impact and of friction on a dynamically
sliding object.
The two parts of this chapter address axisymmetric and nearly axisymmetric objects
respectively. The former class considers circular objects which obey the axisymmetric mechanics.
The latter class is aimed at objects whose sliding motion is reasonably modeled
with the axisymmetric mechanics but whose boundary is not circular; it is simply the combination
of a more general approach to impact with the axisymmetric sliding mechanics.
For each of these cases, the backprojection of the set of possible displacements is constructed
and used to show that any configuration can be reached in at most two taps; then
the controllability of the respective class of objects is discussed.
4.1 Axisymmetric case
The main construct used for planning under the axisymmetric mechanics is the reachable
displacement cone. This cone exists in displacement space and contains all the displacements
that can be reached in a single tap.
In Section 3.2.2, the impact cone was introduced as a representation of all possible initial
velocities that can be produced by impact in a given translational velocity direction.
This cone consists of a continuous range of rays in initial velocity space. In Section 2.1.6 it
was shown that under the axisymmetric sliding mechanics, a cone in the space of initial velocities
can be mapped to a cone in displacement space. Putting these two results together
31

32 CHAPTER 4. PLANNING & CONTROLLABILITY
θ
goal
start
x
Figure 4.1: Planning for axisymmetric objects. The reachable displacement cones for each
direction are placed at the start configuration, the backprojections at the goal configuration.
(For circular axisymmetric objects, these cones are identical.) The region of intersection is
the set of all points that may be used as an intermediate configuration in getting from the
start to the goal in two taps.
yields the reachable displacement cone.
The impact cone for a circular object is symmetric about the v 0 axis (see Section 3.3.1).
Since the v 0 axis maps to the x f axis (pure translational velocity results in a pure translation),
the reachable displacement cone will be symmetric about the x f axis in displacement
space. Because the object is circular, any displacement within this cone can be achieved in
any given translational direction.
The backprojection of the reachable displacement cone, i.e. the set of states from which
the current state can be reached with one tap, is simply the reflection of the reachable displacement
cone through the current state. Because of the symmetry of the reachable displacement
cone about the x axis for this case, the backprojection for tapping in one direction
is the same as the reachable displacement cone for tapping in the opposite direction.
4.1.1 Planning
Since the reachable displacement cone is invariant with respect to orientation, the problem
of positioning an axisymmetric object in the plane can be reduced to positioning the object
along the line connecting the COMs in the start and goal configurations.
A single tap will be able to produce the desired translation, but may not be able to
produce enough rotation. Consider the problem in displacement space; assume the origin
corresponds to the start state and place the reachable displacement cone for both translation
directions at the origin. Place the backprojections of the reachable displacement cones
for both directions at the goal state. The intersection of these pairs of cones is the set of all
states that can be used as an intermediate state to get from the start to the goal state in two
taps. These constructs are illustrated in Figure 4.1.
It is also not difficult to see that the Minkowsky sum of a reachable displacement cone
for translation in one direction with that for translation in the opposite direction covers the

4.2. NEARLY AXISYMMETRIC CASE 33
θ
x
Figure 4.2: The set of configurations reachable in two taps without leaving a neighborhood
of the start configuration includes a (smaller) neighborhood of the start configuration, so
circular axisymmetric objects are small time locally controllable.
space of possible displacements.
4.1.2 Controllability
We have adopted a number of definitions from nonlinear control theory in order to be
precise about the capabilities of modes of manipulation. A system is controllable if it can
reach any goal configuration from any start configuration. A system is small time locally
controllable if, for any given neighborhood about the start configuration, the system can
reach a neighborhood of the start configuration without leaving the given neighborhood.
For our purposes, the “small time” part of small time local controllability does not
refer to time; our interest in this property is in its characterization of the maneuverability
of an object. A mode of manipulation which is small time locally controllable can follow
any path arbitrarily closely.
An axisymmetric object is obviously controllable since it has been shown that any
configuration can be reached with at most two taps. For small time local controllability,
consider the set of states that can be reached without leaving a given neighborhood by
tapping once in the positive x direction and then tapping once in the negative x direction.
As illustrated in Figure 4.2, this set covers the central portion of the given neighborhood,
which includes a smaller neighborhood of the start state. Since this can be done in any
translation direction, a circular axisymmetric object is small time locally controllable.
4.2 Nearly axisymmetric case
For objects that do not have a circular boundary, the K mapping, relating ^v 0 to a set of
! 0
j~v 0j
, is more complex. As in the axisymmetric case (since we assume axisymmetric sliding
mechanics for this object), a range of !0
j~v 0j
can be mapped to a reachable displacement cone.
However, this cone will be different for different ^v 0 , and in general, it will not be symmetric
about the x axis.

34 CHAPTER 4. PLANNING & CONTROLLABILITY
In order to fully appreciate the possible displacements for such an object, we would
need to create a rather complex construct: take a graph of K (such as those in Figure 3.2) and
convert the !0
f
j~v 0j
rangeineachverticalslicetoa
j~x f j
range. Wrap the resulting graph around
a unit cylinder along the axis in displacement space. The set of possible displacements
consists of all points along all rays emanating from the origin and passing through a point
of this cylindrical graph. In addition to having a complicated shape, this set will rotate as
the object rotates (as it moves up and down in the direction in displacement space).
The ability to reach any configuration with at most two taps can be shown with a
much more restrictive set of actions — tapping the object in a single translational velocity
direction such that the impact cone for this direction includes both positive and negative
values of ! 0 . This will involve tapping the object at a range of contact points in order
toproducearangeof !0
j~v 0j
in the given velocity direction. This restriction will make the
displacement space constructs considerably simpler and easier to manipulate.
4.2.1 Planning
The approach taken to planning is essentially the same as that for the axisymmetric case.
First the set of all configurations that can reach the goal in one tap is constructed; I refer
to this construct as the “spiral staircase.” The spiral staircase is then intersected with the
reachable displacement cone at the start configuration. The result is a set of intermediate
points which may be used to reach the goal configuration in two taps.
Constructing the spiral staircase
Since the object may only be tapped in one direction (relative to its orientation), the orientation
of the object and its two dimensional position must now be taken into account.
Figure 4.3 shows a reachable displacement cone in the three dimensional displacement
space. This cone is in a plane defined by the direction in which the object can be tapped
and the axis. This cone may not be symmetric because of impact geometry.
Note that if the object is tapped and there is some net rotation, then the object will
point in a different direction. The reachable displacement cone for the next tap will also
point in that direction (and its origin will be higher or lower in the direction).
Assume that the origin of the coordinate system is placed at the goal configuration
and that the x axis is aligned with the direction of tapping of the object, thus defining the
zero orientation.
The spiral staircase will be defined for values of between 0 and ,2. A given
slice of the spiral staircase contains all points at that orientation that can reach the goal
configuration in one tap. First consider the =0slice; the states with the same orientation
as the goal that can reach the goal in a single tap are simply those directly behind the
goal, as illustrated in Figure 4.4. These states form a line from the goal configuration off to
infinity in the xy plane.
Now consider the = , slice. Objects in configurations corresponding to this plane
4
are oriented , relative to the goal configuration, so the reachable displacement cone is
4
pointed in that direction. The set of states in this slice which can reach the goal configuration
also lie on a line through the axis. As illustrated in Figure 4.5, the closest point is

4.2. NEARLY AXISYMMETRIC CASE 35
θ
x
y
Figure 4.3: The set of reachable configurations for a nearly axisymmetric object which can
only be tapped in one direction. The reachable displacement cone lies in a plane parallel to
the axis and in the direction in which the object can be tapped. The arrow in the xy plane
represents the object and its orientation.
θ
x
y
Figure 4.4: The set of states with the same orientation as the goal configuration that can
reach the goal configuration in a single tap are those directly behind the goal. The reachable
displacement cone is shown for one of those points.

36 CHAPTER 4. PLANNING & CONTROLLABILITY
θ
x
y
∆θ
∆θ
Figure 4.5: The set of states with a different orientation than the goal configuration that can
reach the goal in a single tap fall on a line through the origin in a plane parallel to the xy
plane. Here, the reachable displacement cone is shown for the closest of these points in one
slice; all points behind it can also reach the goal in one tap.
the one for which the goal state lies on the upper edge of the reachable displacement cone.
The rest of the points in this slide lie on the line directly behind this point.
The spiral staircase can be mathematically described by the points:
(x; y; ) =(k cos ;k sin ;) for 2 [0; ,2];k2 [,s;,1) (4.1)
where s is the slope of the upper edge of the reachable displacement cone. This set forms a
surface that spirals around the axis. See Figure 4.6 for an illustration.
Solving for intermediate states
Consider where the reachable displacement cone at the start state encompasses all orientations
from ,2 to 0. This will happen at a certain distance from the start state which I will
refer to as the spanning distance. (See Figure 4.7.)
The spiral staircase will sweep through all points (x; y) at some orientation except in a
vicinity of the goal state (because of the varying offset of the cone from the axis).
Consider projecting both the spiral staircase about the goal configuration and the
reachable displacement cone from the start configuration onto the xy plane as illustrated in
Figure 4.8. The spiral staircase covers the entire plane except for a region enclosed by a spiral
near the goal configuration. The reachable displacement cone becomes a ray originating
from the start configuration.
The reachable displacement cone and the spiral staircase must intersect at all points
(x; y) along the collapsed reachable displacement cone which are further away from the
start configuration than the spanning distance and which lie on the collapsed spiral staircase.
The value of the actual intersection point is determined by the spiral staircase. There
may be points closer to the start configuration that are also in this intersection, but the determining
these intersection points is more difficult. Figure 4.9 shows an illustration of a
spiral staircase and the reachable displacement cone in three dimensions.

4.2. NEARLY AXISYMMETRIC CASE 37
Figure 4.6: An illustration of the spiral staircase, the set of states with orientations from
,2 to 0 which can reach the goal state in a single tap. Each ray in the spiral staircase
extends to infinity, though this is not shown here.
θ
0
-2π
spanning
distance
Figure 4.7: The reachable displacement cone will span all orientations from ,2 to 0 beyond
some minimum distance (the spanning distance) from the cone origin.
spanning
distance
start
goal
Figure 4.8: The spiral staircase about the goal configuration and the reachable displacement
cone from the start configuration are shown here projected into the xy plane. These constructs
must intersect at some value for all (x; y) points along the reachable displacement
cone whose distance from the start position is greater than the spanning distance.

38 CHAPTER 4. PLANNING & CONTROLLABILITY
Figure 4.9: An illustration of the three dimensional intersection between the spiral staircase
and the reachable displacement cone. As this illustration hints, there may be intersection
points closer to the start configuration than the spanning distance.
Any of the points in this intersection may be used as an intermediate point to reach
the goal configuration in two taps.
4.2.2 Controllability
This system is also obviously controllable, since any goal configuration can be reached with
at most two taps. However, with a single translation direction, it is not small time locally
controllable.
Lynch [40] showed that a pushed object is small time locally controllable if the set
of generalized velocity directions along which the object can be pushed positively span a
great circle of the velocity sphere which is not the ! =0plane. Similarly, tapping a nearly
axisymmetric object is small time locally controllable if the possible displacements directions
positively span a great circle of the unit sphere in displacement space that does not
lie in the =0plane. This condition could be satisfied with as few as two contact points or
two translation directions. This condition determines the number of fixed actuators (fixed
either relative to the object or relative to the world) that are required in order to position
an object arbitrarily. The actual number required will depend on the object geometry and
thetypeofactuator.

Chapter 5
Generating Impact
Designing devices to generate impact has been an important part of the experimental effort
of this thesis. In the course of this experimental work, I have used a variety of different
approaches and devices for tapping objects.
This chapter first discusses the background and philosophy behind our efforts in tapping
device design. Through consideration of existing devices to generate impact and preliminary
experiments, I arrived at the goal of designing a specialized tapping device that
creates a single controlled repeatable impact. The remainder of the chapter presents the
designs of the five tapping devices used in this thesis, relates experimental observations of
their performance, and discusses the issues I have found to be important in tapping device
design.
5.1 Background and philosophy
Impact is a fairly common phenomena, used in many different ways. The designs and
design criteria for the tapping devices used in this thesis were influenced by the methods of
generating impact and applications of impact from many different fields, so a brief survey
is presented here.
Coarse uses of impact can be found in fields such as mining (pneumatic or percussive
drills, stopers, and drifters), destruction (jackhammers and wrecking balls), and construction
(pile drivers and hammer drills). In most of these areas, variations in the impulse
delivered by an impact is not critical; often the objective is simply to deliver the largest
possible impact in order to crush materials.
Many sports and games involve impact, including golf, baseball, basketball, racquet
sports, soccer, cricket, football, croquet, and billiards. In these activities, the objective of the
impact is some combination between delivering a large impact and exerting fine control
over the impact. Generally it is desired to control the direction of an object in addition to
its velocities.
Many musical instruments involve impact, e.g. the xylophone, the piano, and most
percussive instruments. Here the objective is to exert fine control over the strength and
contact point of the impact in order to excite vibratory modes of some resonator.
39

40 CHAPTER 5. GENERATING IMPACT
From this brief summary, it appears that mechanical means of propelling free flying
masses towards objects to generate impact are only used for generating large impacts, and
compliant linkages (mostly human limbs, possibly with some tool) are always used when
precisely controlled impacts are required.
One possible exception is the action of a piano. Although a person is used to press
the keys, the impact is generated by a linkage that essentially throws the hammer (which
rotates about a pivot) at the strings and catches it after it rebounds. Another such device is
the doorbell; it produces a controlled impact using an electromagnetic coil to accelerate a
mass on a spring.
5.1.1 Methods of generating motion and impact
Although impact is usually generated by propelling a free rigid mass at some object, there
are other alternatives, such as electromagnetic forces, using a burst of air, striking an object
directly with some linkage, or using a reaction mass. In regard to the subject of this thesis,
the key criterion is whether initial velocities can be imparted to an object in a sufficiently
short time, thus minimizing the interaction of the forces that produce these initial velocities
with the forces that govern the dynamics of the object motion. Such interaction complicates
analysis of the system.
One implicit focus of this work is on robotic or mechanical implementation for manipulating
parts, materials, or assemblies. The primary purpose of these items is generally
not “to be manipulated”, so they may have limitations on how they can be fitted for manipulation
processes. Consequently, I have pursued the most general approach to creating
impulse: collision between a striker and an object.
Striking an object directly with some linkage is undesirable because transmissions to
create motion of a linkage are generally stiff; also, impact tends to destroy bearings. This
brings us back to the most common method of creating impulse: propelling a striker at an
object.
This leads to the question of how motion is generated. Four basic categories of methods
to generate motion are:
electromagnetic forces (electric motors, solenoids, and such),
gravitational forces (falling objects),
mechanical processes ( springs and compressed substances), and
chemical/electrical/mechanical processes (explosions, muscles, piezoelectrics).
In the course of this experimental work, I have explored the first three categories.
5.1.2 Other impact generating devices for manipulation
Higuchi [29] used an electromagnetic coil to induce eddy currents in a conductive plate
in order to generate impulse. A condenser was used to store electrical energy and was
discharged by connecting it to the electromagnetic coil. By varying the amount of energy
stored in the condenser and by varying the capacitance of the condenser, Higuchi was

5.1. BACKGROUND AND PHILOSOPHY 41
able to generate one dimensional displacements of an object with a ratio of minimum to
maximum displacement of approximately 1:25 1 , which corresponds to an energy ratio of
1:5.
Yamagata and Higuchi [65] created an impact generating device, partly based on
Higuchi’s earlier work, that uses a piezoelectric element as an actuator. This device consists
of two masses, a main body which rests on a support surface and a reaction mass which
is attached to a side of the main body by the piezoelectric element (and does not rest on
the support surface). Rapid expansion of the piezoelectric element delivers an impulse to
the main body of the motion mechanism. Slow contraction of the piezoelectric brings the
reaction mass back to its starting position without producing forces on the main body that
exceed static friction. Yamagata and Higuchi show results from an experiment which produced
displacements of 4nm and from an experiment which produced displacements of
2m to5m; however, it is not clear whether these experiments were done with the same
object.
5.1.3 Preliminary tapping experiments
Preliminary experiments with tapping suggested the use of specialized tapping devices
and helped identify several key criteria in their design.
These preliminary experiments used an Adept 550 robot to generate impact. At this
time, I was using plywood parts sliding on a plywood surface. The striker was a small
plywood disk held in the gripper of the robot. In order to generate impact, the Adept was
commanded to accelerate to a specified velocity, striking the object with the small disk.
By videotaping the impact, I found that this method did not work well for striker
velocities below 400 mm/sec. For these velocities, the robot appeared to push the object
for a significant distance. For velocities greater than 400 mm/sec, the impact was “clean”,
but resulted in large displacements.
There was significant variation in the displacement of the object, though it was not
clear how much of this was due to variations in the impact and how much to variations in
the sliding. Other preliminary experiments in which the striker velocity could be measured
(done with the first spring-loaded tapping device described later in this chapter) revealed
that there was significant variation in both the impact and sliding process.
5.1.4 Design goals, criteria, and constraints
The goal in designing a tapping device for impulsive manipulation is to create a repeatable
device that generates a single clean controllable impact.
A clean impact is important to minimize the interaction between the dynamics of the
striker and the dynamics of the object. Essentially, this means making the collision time as
short as possible so that the striker does not end up pushing the object during an extended
collision.
Multiple impacts simply complicate the impact process, making it less amenable to
analysis and adding more variation to the resulting object velocities.
1 I have estimated a ratio of minimum to maximum displacement of 1:25 from the graphs of (Higuchi [29]).
The resulting energy ratio of 1:5 roughly corresponds to his graphs.

42 CHAPTER 5. GENERATING IMPACT
motor
Figure 5.1: The first spring-loaded tapping device.
solenoid
It is best to have a dynamic range as large as possible. For positioning tasks, one
would like to reach the vicinity of the goal configuration with a few large taps and then
use very small taps to achieve a high positioning accuracy. My goal was to achieve a ratio
between 1:50 and 1:100 in pure translation (or in energy), which would correspond to a
ratio between 1:7 and 1:10 in striker velocity. It was required that the tapping device should
be able to produce displacements from 1 mm to 15 mm for the set of objects I was using.
The tapping devices used in this work have only achieved translation ratios in the range
of 1:15 to 1:30. The difficulty is in determining the smallest impact that provides consistent
results. The impact can always be made smaller (thus increasing the translation ratio) at
the cost of repeatability.
The objects used in the experimental work of this thesis were short flat objects, so it
was essential that the tapper tip was low enough to the surface to tap these objects.
5.2 The tapping devices
This section describes the tapping devices designed and built in the course of this thesis in
chronological order.
5.2.1 First spring-loaded tapping device
The first spring-loaded tapping device, illustrated in Figure 5.1, is a two stage device. The
first stage is an aluminum rod with a moderately stiff spring; the second stage is wooden
dowel with a fairly loose spring. A motor and leadscrew move a translation stage back
and forth; a solenoid-actuated latch on this translation stage draws the aluminum rod back
and releases it. The size of the tap is adjusted by varying the length of time the motor is
turned on when drawing back the striker. This tapping device was designed to be operated
completely under robot control, so it was important to have some automatic way to reset
the device.
This device uses two stages (a “power” stage and a tapper tip) because of concerns
about the force of a spring acting on the object during and after impact. The energy from

5.2. THE TAPPING DEVICES 43
28
striker stage & object positions
2000
striker stage velocity
position (mm)
26
24
22
object
stage 2
stage 1
velocities (mm/sec)
1500
1000
500
0
20
0 0.01 0.02 0.03 0.04
time (sec)
−500
0 0.01 0.02 0.03 0.04
time (sec)
Figure 5.2: Double impact example for the first spring-loaded tapping device.
the power spring is transferred to the tapper tip, and the spring on the tapper tip was
intended to be just strong enough to return the tip to its starting position.
The choice of wood for the tapper tip may not have been a good decision; on the
other hand, it did not seem to be a bad choice either. It was originally picked to match
the material of the object. However, wood has slightly unusual friction and deformation
properties due to its fibrous structure.
There are greater difficulties with this device, though. The tapper tip is not exactly
spherical, so contact normals differed from what was expected. There is some misalignment
between the axes of the two stages, which made calibration of the tapper more difficult,
and there is perhaps too much play in the in the stages, particularly in the tapper
tip. These factors are undoubtedly responsible for most of the variation in impact in the
experiments with this device.
In later experiments using a high speed measurement system, I found that this device
does often produce double impacts. Figure 5.2 shows one such occurrence.
5.2.2 Pneumatic tapping device
The next tapping device, illustrated in Figure 5.3, was pneumatic. Pressurized air propels
a striker along a tube until it strikes the object. A solenoid-actuated valve midway down
the tube remains open in order to relieve pressure behind the striker, so that air pressure
might not continue to exert force on the striker during and after the collision. The striker
is then free to rebound. Both the length of a burst of air (controlled by an external solenoid
valve) and the air pressure can be adjusted to affect the striker velocity. In order to reset
the device, the valve is closed, and a venturi generator creates a partial vacuum behind the
projectile.
In practice, the burst of air was very short, in part because the air should be turned off

44 CHAPTER 5. GENERATING IMPACT
6"
detail of rear block
(rear view, 1.5X)
3.25"
air for
vacuum
air for
propulsion
Figure 5.3: The pneumatic tapping device. External solenoid valves control the flow of air
to the two ports on this device. One port provides air to propel the tapper tip; the other
produces a partial vacuum using a venturi generator to pull the tapper tip back.
by the time the the striker passes the relief valve. The pulse sent to the external solenoid
valve was approximately 5 ms, and the range of air pressure used was from 20 to 40 psi.
It was easy to generate a range of impulses by varying the air pressure; there was a lesser
degree of control through varying the length of the burst of air. A longer tube would
expand the range of velocities achievable by varying the length of the burst of air; however,
there would also be greater friction as the striker slides to the front of the tube.
This device has the advantage of being much more compact than the first springloaded
tapper, although it requires external “plumbing” and hardware. One disadvantage
of this device is that the steel striker pits the aluminum objects used, possibly invalidating
the impact model.
5.2.3 Electromagnetic tapping device
The electromagnetic tapping device is illustrated in figure 5.4. This device is a brass tube
with five coils. The striker is a short section of steel shaft which is free to move in the tube.
Coils are turned on in sequence in order to accelerate the projectile as well as to draw the
projectile back to the end of the tube. The front four coils are used for accelerating the
striker; the back-most coil serves to position the striker in its starting position.
There is no feedback on the position of the striker within the tube, so the coils must be
sequenced open loop. The timing of the coils must be adjusted so that as the striker passes

5.2. THE TAPPING DEVICES 45
Figure 5.4: The electromagnetic tapping device. Five coils are used to accelerate and draw
back a steel striker inside a brass tube.
a coil, that coil is turned off so that it pulls the striker backwards as little as possible.
5.2.4 Wrecking ball tapper
The first three tapping devices were designed to be end effectors of a robot and operated
under robot control. As I began a more detailed investigation of the models and mechanics
used in this thesis (the results of which are in Chapter 6), it became clear that I needed a
simpler more repeatable device for generating impact. Appropriateness for use as a robot
end effector was not the primary concern.
The resulting device, which I refer to as “the wrecking ball”, is illustrated in Figure 5.5.
It consists of a hard steel ball bearing suspended from above. It is drawn back with a
thread to a bracket; the velocity and angle of incidence of the wrecking ball can be varied
by adjusting this bracket. A loop of the draw string is passed through a hole in the bracket
and a metal pin used to hold it there until release.
The advantages of this device are its simplicity, its repeatability, and the ability to
easily and accurately vary its angle of incidence. The wrecking ball is spherical, so there
is no problem with variation in the contact normal. Although there is some disturbance
due to the release of the wrecking ball, the variation in initial object velocities produced by
this tapping device was far less than with the previous tapping devices. Furthermore, it is
possible to calculate the initial velocity of the striker based on measurements of its change
in height.
This device was only used to tap objects at the edge of the support surface because the
wrecking ball is too large to clear the surface and strike the side of an object. The wrecking
ball does slightly dent aluminum objects, but not nearly as much as the pneumatic tapper,
probably because of the much larger radius of the wrecking ball.
5.2.5 Second spring-loaded tapping device
The final tapping device is illustrated in Figure 5.6. This spring-loaded device consists
of a single stage and was designed to be reconfigurable so that minor modifications and
additions could easily be made. It was intended to be used as an end effector, although
not under automatic control. The striker is a rod which is pushed back and secured with
a simple latch. A stop can be manually adjusted in order to vary the compression of the
spring.
Taking some lessons from the design of previous tapping devices, this device was built
with much greater precision and features linear ball bearings. The striker is more massive

46 CHAPTER 5. GENERATING IMPACT
18"
Figure 5.5: The “wrecking ball” tapper. An adjustable bracket (both in angle and in length)
is used to pull back a steel ball bearing suspended from above by a thread.
that that of previous designs (113.8 g versus 5.5 g for the first spring-loaded tapper and
29.3 g for the wrecking ball). The tip of the striker is a softened steel ball bearing, so that
the contact normals would be reasonably precise.
This device has proved to be very repeatable, and makes a nice sound upon impact.
There is a limit on the angle of incidence — beyond 15 to 20 degrees, the impact excites
some vibrational mode of the rod which makes an unpleasant sound and produces a much
smaller impulse.
When calibrating this tapping device, it is important to leave some distance between
the points at which the spring stops acting on the striker and the striker collides with the
object. This distance should be minimized because there is friction in the linear bearings
that then slows the striker. However, if this distance is too short, it is possible for the striker
to rebound after the collision, bounce against the spring, and strike the object again. This
has not been a problem in practice.
Most likely due to the ball bearings, this striker seems to be modeled well under the
assumption that it has infinite angular inertia. The position and velocity profiles for this
striker, an example of which are shown in Figure 5.7, show a particularly clean impact.
5.3 Issues in tapping device design
This section summarizes the issues I have found to be important through background reading
and through the design of the tapping devices used in this thesis.
Release mechanisms
For any device which is powered by stored mechanical potential energy, the design of
a release mechanism will influence the repeatability of the device. In these devices, the
striker is subject to some compression or gravitational forces but is prevented from moving
by some (usually kinematic) constraint. The goal of a release mechanism is to remove this
constraint quickly while minimizing the effect of (and variations in the effect of) the release
upon the object.
For kinematic constraints, this description suggests a number of standard solutions.

5.3. ISSUES IN TAPPING DEVICE DESIGN 47
15"
top
side
11 mm dia.
collar
foam
bottom
Figure 5.6: The second spring-loaded tapping device. The strength of the impact can be
adjusted by moving the spring backstop. The striker rod is cocked and released by hand,
using a simple latch.
25
striker position
600
striker velocity
position (mm)
20
15
10
5
0
velocity (mm/sec)
400
200
0
−5
0 0.1 0.2 0.3 0.4
time (sec)
−200
0 0.1 0.2 0.3 0.4
time (sec)
Figure 5.7: Position and velocity profiles for the second spring-loaded tapping device.

48 CHAPTER 5. GENERATING IMPACT
Quick release and minimizing the effect of the release suggest using two sharp edges sliding
against each other. Of course, sharp edges are prone to wear, so using hard materials
will improve the repeatability over time. Components of a release mechanism should be
stiff to reduce variations due to the speed of the release. Clearance between parts should be
reduced where appropriate to avoid unwanted motion. Friction can be reduced through
choice of materials or use of lubricants. Finally, conditions where jamming or wedging
might occur should be avoided. More novel release mechanisms might involve electrorheological
fluids or piezoelectric elements.
Another type of constraint is due to some larger force, such as electromagnetic force,
keeping the striker immobilized. I considered using an electromagnet for the wrecking ball
release, but I did not pursue this because I feared gradual magnetization of the parts would
affect repeatability.
Calibration
There are two types of calibration required for a tapping device: geometric calibration and
striker velocity calibration.
When the tapping device will be used as a robotic end effector, kinematic calibration is
much easier if it is constructed so that the axis of the striker lies on one of the axes of the tool
frame. One related design issues is whether the striker should always contact the object at
the same offset from the tapping device. (Because the second spring-loaded tapping device
uses an adjustable back stop, the spring stops acting on the striker at different points for
different settings.)
Determining the relationship between the parameters of the tapping device and the
striker velocity is difficult to do without some means of measuring the striker velocity directly.
For one dimensional impact, the striker velocity and coefficient of restitution cannot
be distinguished only from measurements of object translation. One alternative is to use
some method of generating motion that produces velocities that can easily be calculated,
such as a linear spring or a pendulum.
Dynamic range
As mentioned Section 5.1.4, it is desirable to have as large a dynamic range as possible
in order to do positioning more efficiently. In addition, a large dynamic range aids in the
ability to manipulate objects of different mass on the same scale of displacements (because
object velocities will be inversely proportional to object mass).
Striker tip material and shape
The material of the tip of the tapper in relation to the object material is another factor
to consider. The friction between the striker tip and the object will affect the amount of
tangential impulse that can be delivered to the object. The relative hardness of the two
materials will affect the amount of plastic deformation in the impact, thus affecting the
coefficient of restitution.

5.3. ISSUES IN TAPPING DEVICE DESIGN 49
The shape of the striker tip is important, in part because of how it affects local deformations
of the object during the collision, but primarily because impact is sensitive to the
direction of the contact normal. When the angle of incidence of the striker tip is changed,
a spherical tip is the simplest to compensate for.
Striker Mass
The mass of the striker in relation to the mass of the objects being manipulated is also
important because it affects the striker velocities required. A light striker will have to be
accelerated to higher velocities to carry the same momentum as a more massive striker.
Inefficiency of impact
Impact can be inefficient due to energy losses through deformation. Small objects can generally
be treated as ideal rigid objects, but larger objects show losses from the shock wave
of an impact traveling through the object. This also has implications in the design of multistage
tapping devices.
Duration of impact
In order to avoid having the forces that accelerate the tapper tip also act on the object
during an extended impact, it is desirable for the tapper tip to be unconstrained in at least
one dimension at impact, i.e. the tapper tip should be free to rebound. However, this can
potentially cause problems with double impacts, another phenomenon which complicates
the impact process.
Method of acceleration
Of the various methods of generating striker motion, the spring-loaded method has been
the easiest to build and use; however, there does not seem to be any inherent reason why
any one method is better than any other.

50 CHAPTER 5. GENERATING IMPACT

Chapter 6
Single Tap Experiments
In order to determine how well the models used in the analysis predict object behavior, I
performed a number of single tap experiments. These experiments involved a variety of
materials, object shapes, and tapping devices. This chapter presents three experiments that
illustrate the development of how the models were adapted to reflect experimental results.
Each experiment consists of a number of runs at different tapper settings, and each
run consists of a number of trials in order to determine the amount of variation for a given
setting. Object trajectories and sometimes striker trajectories were recorded using a high
speed position measurement system. By calculating initial velocities from these measurements,
the effect of tapping device parameters on initial velocities (the impact model) and
the effect of initial velocities on resulting displacements (the sliding model) can be evaluated
separately.
There are two major deviations from the assumptions made for the models used in this
thesis. The first is that the support distribution of the objects used in these experiments
is not uniform. Because the actual support distribution depends upon the microscopic
variations in the surfaces in contact, the support distribution is unknown. My hypothesis
was that the actual support distribution would be close enough to uniform that the sliding
model would still be applicable. The second deviation is that objects are three dimensional,
not planar. This affects both the sliding model (because it causes a bias in the support distribution
during sliding) and the impact model (because a strike above or below the COM
will produce a moment not parallel to the z axis). My hypothesis was that the objects used
were short enough (close enough to planar) that there would be no significant deviations
from the impact model.
The first experiment, the plexiglas disk experiment, showed reasonable agreement between
the (unmodified) sliding model and the experimental results but some discrepancy
with the impact model. The second experiment, the aluminum square experiment, showed
reasonable agreement between the impact model and the experimental results but required
a torque scaling factor for the sliding model to reasonably predict object motion. The third
experiment, the aluminum disk experiment, also required a torque scaling factor for the
sliding model to match experimental results and showed some discrepancy with the impact
model.
51

52 CHAPTER 6. SINGLE TAP EXPERIMENTS
OptoTrak
OptoTrak
tapping
device
tapping
device
Figure 6.1: Setup for the single tap experiments. The OptoTrak measures the position of
infrared LEDs attached to the object and the support surface.
6.1 Experimental materials, setup, and procedures
Figure 6.1 illustrates the experimental setup used in these experiments. An OptoTrak measurement
system looks downward upon the experimental setup to measure the positions
of infrared LEDs attached to the object and the support surface. Two LEDs are used to
measure the position and orientation of the object, three LEDs for the plane of the support
surface. A LED can be attached to the striker (for the spring-loaded tapping devices) in
order to measure the striker velocity. The wires from these LEDs can exert small forces on
the object and the striker, so they are carefully arranged and suspended in order to affect
object and striker motion as little as possible.
The support surface LEDs are measured at the beginning of each run in order to determine
a local frame. Each run generally consists of 10 trials. During analysis, the object
positions are projected onto the surface plane to measure the direction of the object motion.
This projection was also used to check for motion normal to the surface of the plane,
though none greater than the accuracy of the OptoTrak was observed.
The support surfaces were also equipped with some method of aligning the object at
the same starting point; the two methods used are illustrated in Figure 6.2. The plexiglas
disk on the aluminum surface was pushed against two pins that were mounted on the
surface; a spacer between the pins and the disk was removed after alignment to keep the
disk from actually touching the pins. For the other experiments, a removable guide with a
number of pins was used.
The angle of incidence of the striker could be adjusted easily and fairly precisely with
the wrecking ball tapping device. However, with the second spring-loaded tapping device,
changing the angle of incidence required re-orienting both the tapping device and the support
surface. The measurements of the angle of incidence for the wrecking ball are therefore
more accurate than those for the second spring-loaded tapping device. On the other hand,
the striker velocity of the second spring-loaded tapping device could be measured directly

6.1. EXPERIMENTAL MATERIALS, SETUP, AND PROCEDURES 53
Figure 6.2: Object alignment methods. One support surface was equipped with two pins
that were used to place disks in the same start configuration. Other surfaces used a removable
guide which also used several pins for object alignment.
Object shape disk square disk
Object material plexiglas aluminum aluminum
Object mass 52.5 g 205.1 g 245.5 g
Object size r=49mm s = 76.2 mm r=48mm
Surface material aluminum aluminum formica
0.19 0.21 0.17
striker WB WB S2
e 0.54 0.76 0.86
Table 6.1: Summary of materials and parameters for single tap experiments. WB refers to
the wrecking ball tapper, and S2 refers to the second spring-loaded tapper.
by the OptoTrak system, whereas the velocity of the wrecking ball was estimated based on
its change in height.
Table 6.1 shows the combinations of object, surface, and striker used for the experiments
in this chapter.
6.1.1 Measurement issues
The OptoTrak was used to measure the position of the object at 500 Hz. The stated position
accuracy of the system is 0.1 mm. The “baseline” of the two markers on the object ranged
from 62.9 mm to 83.4 mm, which translates to an angular accuracy ranging from 0.18 to
0.14 degrees. With data measured at 500 Hz, this could lead to errors in the calculated
translational velocity of 100 mm/sec and in the calculated rotational velocity from 180
to 140 deg/sec (using a first order difference to compute velocities).
In practice, the data are better behaved than these errors might suggest, but there can
still be significant noise in velocity profiles, particularly in angular velocity during small
rotations.
The analysis of these data requires measurement of the initial velocities of the object.
For the experiments presented in this chapter, the initial velocities were taken as the maxima
of the velocity profiles (calculated by a first order difference), with the exception of the

54 CHAPTER 6. SINGLE TAP EXPERIMENTS
ω 0
E
D
C
F, G
B
A
v 0
Figure 6.3: Intended sampling of the space of initial conditions for runs A–G of the plexiglas
disk experiment.
plexiglas disk experiment.
In the plexiglas disk experiment, there was some slight problem with object overhang
(discussed in Section 6.5) which resulted in a small spike in the rotational velocity profiles.
The initial translational velocities were measured automatically, but the initial rotational
velocities were estimated manually from the calculated rotational velocity profiles.
6.1.2 Measuring material parameters
The question at hand is how well the model predicts the experimental data; however, before
the model can be used, some of the data are required to determine material parameters:
the coefficient of friction between the object and the support surface
e the coefficient of restitution between the striker and the object
r the coefficient of friction between the striker and the object
The values of and e are determined using data from pure translational trials. The slopes
of lines fit to the translational velocity profiles were averaged in order to determine . The
calculated or measured striker velocity and the measured initial translational velocity are
used to determine e.
There is no simple way to measure the value of r directly with the object and striker;
however, r should be close to the coefficient of friction between the object and aluminum
— around 0.15 to 0.20 for the object materials used here. For the circular objects and the
tapping devices used in this chapter, the impact is not directly dependent upon the value
of r . So long as the impulse ratio Px
P y
is less than r then the value of r does not affect the
impact process.
6.2 Plexiglas disk experiment
This experiment used the wrecking ball to tap a plexiglas disk sliding on an aluminum
surface. There were seven runs (lettered A–G), each of which contained ten trials. The
settings of the tapping device were determined in an attempt to sample the space of initial
velocities as illustrated in Figure 6.3. Although the tapper parameters were adjusted to
try to achieve the desired initial velocities, there is enough variation in initial velocities

6.2. PLEXIGLAS DISK EXPERIMENT 55
that it was difficult to achieve these velocities exactly. These nominal initial velocities were
chosen in order to experimentally test the monotonicity properties described in Chapter 2.
Figure 6.4 shows the net displacements for this experiment as well as the average
displacement and standard deviation for each run. (Note that run C is included in both
graphs and tables.) There is a significant amount of variation most runs due to variations
in both impact and sliding.
6.2.1 Evaluation of the sliding model
This subsection examines how well the sliding mechanics model the motion of the disk.
First, the results of the monotonicity test are discussed, and then a general evaluation of
the sliding model is done by a comparison of the actual and predicted displacements and
by a comparison of the actual and predicted velocity profiles.
Montonicity conclusions
Figures 6.5 and 6.6 show graphs of the level curves of x f and f over the relevant space
of initial velocities. Examining these graphs reveals that over the velocity ranges of this
experiment, the expected monotonicity effect in x f would be less than 0.5 mm, in f it
would be about 2 deg. However, the noise in the measured translational displacements is
generally greater than the size of this effect. (See Figure 6.4.) For rotational displacements,
the monotonicity effect can be observed for differences in translational velocity of more
than 40–50 mm/sec.
Comparison of displacements
Figures 6.5 and 6.6 show graphs of the measured initial velocities in addition to plots of
the level curves of x f and f . These graphs show a large variation in the initial velocities
of the object — variation due to the impact process. A comparison of the initial velocities
graph with the graph of the level curves shows that even if the sliding model were perfect,
we would expect a lot of variation in the object displacement because of these variations in
initial velocities.
In order to evaluate the sliding model independently of the impact process, the actual
displacement of the object must be compared with the predicted displacement given the
measured initial velocities. It is this difference that will show the accuracy of the sliding
model.
Table 6.2 shows the results of this comparison. In the worst case, the average translation
difference is 5.5% of the average measured distance. It is not as meaningful to make
a similar comparison for rotation since some of the rotations are rather small. For runs
C–G, the average rotation difference ranges from 4.8% to 26.8% of the average measured
rotation.
Comparison of velocity profiles
The solution to the inverse sliding problem described in Chapter 2 can be used to calculate
what the initial velocities must have been to produce a given displacement. By comparing

56 CHAPTER 6. SINGLE TAP EXPERIMENTS
rotation (deg)
8
7
6
5
4
3
2
1
0
−1
Net Displacements, runs A−D
C
−2
23 24 25 26 27 28 29 30
translation (mm)
A
D
B
rotation (deg)
11
10
9
8
7
6
5
4
3
E
Net Displacements, runs E, C, F, G
C
2
15 20 25 30 35 40 45 50 55 60
translation (mm)
Run A B C D
x x x x x x x x
Angle of incidence (deg) 0 10 20 30
Calculated striker velocity (mm/sec) 543 560 594 642
Average distance (mm) 27.8 1.5 26.8 1.2 25.4 1.2 27.2 1.0
Average rotation (deg) 0.9 0.3 -2.3 0.6 -4.1 0.9 -5.9 1.1
Run E C F G
x x x x x x x x
Angle of incidence (deg) 30 20 10 10
Striker velocity (mm/sec) 578 594 700 754
Average distance (mm) 20.9 0.9 25.4 1.2 44.4 2.5 52.9 1.5
Average rotation (deg) -4.1 0.6 -4.1 0.9 -6.2 1.6 -7.7 1.7
Figure 6.4: Net displacements for the plexiglas disk experiment.
F
G

6.2. PLEXIGLAS DISK EXPERIMENT 57
80
Initial velocities, runs A−D
80
Level curves of X_f (mm)
rotational velocity (deg/sec)
70
60
50
40
30
20
C
B
10
A
0
280 285 290 295 300 305 310 315 320
translational velocity (mm/sec)
D
intial rot. velocity (deg/sec)
70
60
50
40
30
20
10
23 24 25 26 27 28
0
280 285 290 295 300 305 310 315 320
initial trans. velocity (mm/sec)
Figure 6.5: Initial velocities for Runs A–D of the plexiglas disk experiment. Also shown
is a graph of level curves of x f over the same range of initial conditions; this gives some
indication of how much variation in translation is expected due to the variation in initial
velocities.
60
Initial velocities, runs E, C, F, G
60
Level curves of Theta_f (deg)
rotational velocity (deg/sec)
55
50
45
40
35
30
25
E
C
F
G
intial rot. velocity (deg/sec)
55
50
45
40
35
30
25
3
4
5
6
7
8
20
260 280 300 320 340 360 380 400 420 440
translational velocity (mm/sec)
20
260 280 300 320 340 360 380 400 420 440
initial trans. velocity (mm/sec)
Figure 6.6: Initial velocities for Runs E, C, F, and G of the plexiglas disk experiment. Also
shown is a graph of the level curves of f over the same range of initial velocities; this
gives some indication of how much variation in f is expected due to variation in initial
velocities.

58 CHAPTER 6. SINGLE TAP EXPERIMENTS
A B C D E F G
Average measured translation 27.8 26.8 25.4 27.2 20.9 44.4 52.9
standard deviation 1.53 1.17 1.18 0.96 0.93 2.48 1.47
Average translation difference 0.2 -1.1 -1.4 -0.4 -1.1 -0.2 1.3
standard deviation 2.4 0.8 0.9 0.9 0.6 2.2 1.8
Average measured rotation -0.8 2.3 4.1 5.9 4.1 6.2 7.7
standard deviation 0.43 0.62 0.88 1.15 0.61 1.57 1.67
Average rotation difference 0.9 0.6 1.1 1.0 0.7 -0.3 -0.6
standard deviation 0.3 0.6 0.6 0.3 0.3 0.9 0.6
Table 6.2: Comparison of displacements for the plexiglas disk experiment. The calculated
displacements were the result of a forward simulation based on the measured initial velocities.
This table shows the average difference between the computed displacement and the
measured displacement for each run.
the velocity profiles based upon these calculated velocities with the actual velocity profiles,
the fidelity of the sliding model to the actual object behavior can be evaluated.
This is perhaps the truest test of whether the sliding model accurately reflects the object
behavior, and it avoids any error in measuring initial velocities that would affect the
comparison of displacements.
The translational velocity profiles all fit fairly well, but there is some variation in the
rotational velocity profiles. Some profiles fit very well (Figure 6.7), most are acceptable
(Figure 6.8), and some fit poorly (Figure 6.9). The angular velocity profiles are noisy enough
that there is no meaningful statistical test of the goodness-of-fit.
A quantitative measure is the difference between the calculated initial velocities and
the measured initial velocities; however this does not provide as much information about
how well the sliding mechanics model the object behavior. Like for the displacement comparison,
this measure is computed for each trial in order to discount the variation due to
the impact process; the results of this comparison are shown in Table 6.3. The average difference
in initial translational velocity ranges from 2.4% to 7.5% of the average measured
initial translational velocity; for initial rotational velocity, the range is 5.1% to 19.0%.
6.2.2 Evaluation of the impact model
Table 6.4 shows a comparison between the measured initial velocities and the calculated
initial velocities, based on the calculated striker velocity.
The translational velocities match very closely. Two of the predicted initial rotational
velocities are very close, two are much too high, and two are much too low. The initial rotational
velocity comes entirely from the tangential impulse, so in two cases the tangential
impulse is about right, in two cases it is too high, and in two cases it is too low. However,
the measured angle of the translational velocity (relative to the contact normal) is not as
large as the model predicts, implying that there is more tangential impulse than what the
model predicts for all but one trial.

6.2. PLEXIGLAS DISK EXPERIMENT 59
400
Run B, trial 6
60
Run B, trial 6
trans. velocity (mm/sec)
300
200
100
0
rot. velocity (deg/sec)
40
20
0
−100
0 0.05 0.1 0.15 0.2 0.25
time (seconds)
500
Run G, trial 2
−20
0 0.05 0.1 0.15 0.2 0.25
time (seconds)
80
Run G, trial 2
trans. velocity (mm/sec)
400
300
200
100
0
rot. velocity (deg/sec)
60
40
20
0
−100
0 0.1 0.2 0.3
time (seconds)
−20
0 0.1 0.2 0.3
time (seconds)
Figure 6.7: Plexiglas disk velocity profiles, measured and calculated. These examples are
representative of trials that showed a good fit between the predicted and measured velocity
profiles. The predicted velocity profiles were calculated based on measured displacements.

60 CHAPTER 6. SINGLE TAP EXPERIMENTS
400
Run C, trial 6
80
Run C, trial 6
trans. velocity (mm/sec)
300
200
100
0
rot. velocity (deg/sec)
60
40
20
0
−100
0 0.05 0.1 0.15 0.2 0.25
time (seconds)
400
Run D, trial 1
−20
0 0.05 0.1 0.15 0.2 0.25
time (seconds)
100
Run D, trial 1
trans. velocity (mm/sec)
300
200
100
0
rot. velocity (deg/sec)
80
60
40
20
0
−100
0 0.05 0.1 0.15 0.2 0.25
time (seconds)
300
Run E, trial 3
−20
0 0.05 0.1 0.15 0.2 0.25
time (seconds)
100
Run E, trial 3
trans. velocity (mm/sec)
200
100
0
rot. velocity (deg/sec)
80
60
40
20
0
−100
0 0.05 0.1 0.15 0.2 0.25
time (seconds)
−20
0 0.05 0.1 0.15 0.2 0.25
time (seconds)
Figure 6.8: Plexiglas disk velocity profiles, measured and predicted. These examples are
representative of the bulk of the trials, which showed an acceptable fit between the measured
and predicted velocity profiles. The predicted velocity profiles were calculated based
on measured displacements.

6.2. PLEXIGLAS DISK EXPERIMENT 61
400
Run C, trial 10
80
Run C, trial 10
trans. velocity (mm/sec)
300
200
100
0
rot. velocity (deg/sec)
60
40
20
0
−100
0 0.05 0.1 0.15 0.2 0.25
time (seconds)
−20
0 0.05 0.1 0.15 0.2 0.25
time (seconds)
Figure 6.9: Plexiglas disk velocity profiles, measured and predicted. These trials are representative
of the few trials which did not show an acceptable fit between measured and
predicted velocity profiles. The predicted velocity profiles were calculated based on measured
displacements.
A B C D E F G
Average measured v 0 300 302 301 306 272 387 413
standard deviation 6.2 3.8 11.6 6.3 5.7 6.7 7.6
Average v 0 difference -21.9 -13.5 -6.3 -12.0 -6.5 -19.0 -30.8
standard deviation 11.4 8.2 6.4 6.3 6.0 10.3 9.6
difference percentage -7.3 -4.5 -2.1 -3.9 -2.4 -4.9 -7.5
Average measured ! 0 -24.7 -46.2 -60.0 -47.2 -40.7 -44.9
standard deviation 8.4 5.5 11.6 5.9 6.7 10.3
Average ! 0 difference -7.5 -4.5 -8.8 -8.1 -6.2 2.1 3.5
standard deviation 2.7 4.6 5.2 3.4 2.8 5.7 3.5
difference percentage 18.2 19.0 13.5 13.2 -5.1 -7.8
Table 6.3: Comparison of calculated (from sliding) and measured initial velocities for the
plexiglas disk experiment. The calculated initial velocities were determined by applying
the inverse sliding problem solution to the measured displacements. This table shows the
average velocity difference for each run.

62 CHAPTER 6. SINGLE TAP EXPERIMENTS
A B C D E F G
Angle of incidence 0 10 20 30 30 10 10
Calculated striker velocity 543 560 594 642 578 700 754
Predicted v 0 (mm/sec) 299 304 309 309 278 380 410
Average measured v 0 300 302 301 306 272 387 413
standard deviation 6.2 3.8 11.6 6.3 5.7 6.7 7.6
Predicted ! 0 (deg/sec) 0.0 24.5 51.1 80.8 72.7 30.6 33.0
Average measured ! 0 24.7 46.2 60.0 47.2 40.7 44.9
standard deviation 8.4 5.5 11.6 5.9 6.7 10.3
Predicted direction (deg) 0 2.0 4.1 6.4 6.4 2.0 2.0
Actual direction 0 3.2 5.2 6.7 6.2 3.4 3.2
standard deviation 0.4 0.4 0.4 1.1 0.8 0.3 0.3
Table 6.4: Comparison of initial velocities from impact for the plexiglas disk experiment.
This table compares the predicted initial velocities and velocity direction (based on striker
velocity, coefficient of restitution, and angle of incidence) with the measured values.
6.3 Aluminum square experiment
This experiment used the wrecking ball tapping device to tap an aluminum square sliding
on an aluminum surface. This experiment consisted of five runs (lettered A–E). The rotational
velocity profiles are considerably less noisy than in the plexiglas disk experiment;
the primary reason for this is that the rotations are greater (in turn due to the fact that an
impulse can have a greater lever arm due to the geometry of the square). The resulting
displacements and the measured initial velocities are shown in Figure 6.10.
I hypothesized that a “nearly” axisymmetric object such as a square would be modeled
reasonably well by the axisymmetric mechanics. In this sense, this experiment was a
combination of the axisymmetric mechanics with a more general impact problem.
A key issue is what size disk to use to model the square. Without examining the force
andtorquefunctions,Idecideduponusingthediskthathadthesametorqueasthesquare
for pure rotation. Setting the expressions for this torque on a disk of radius R and a square
of side s equal to one another, we get:
2
"2
3 gRM = 1 12 gsM p
2 , log
p
2s , s
2
!
+ log
p
2s + s
Given s we can then solve for R. For the square used in this experiment, s =76:2 mm and
R =43:7 mm (87:4 mm in diameter).
Figure 6.11 shows the force and torque functions for the square and for the disk approximation.
The relevant range of ! in simulation is generally from 15 to 20, and the
v
difference between the real force and torque functions and those of the equivalent disk are
fairly small in this range.
However, the sliding model does not adequately reflect the motion of the square. In
particular, the predicted rotational velocity profiles are qualitatively different than the ac-
2
!#
(6.1)

6.3. ALUMINUM SQUARE EXPERIMENT 63
rotation (deg)
20
15
10
5
D
C
Displacements, runs A−E
B
E
rotational velocity (deg/sec)
A
50
A
0
11 12 13 14 15 16 17 18
0
180 200 220 240 260 280
translation (mm)
translational velocity (mm/sec)
Run A B C D E
x x x x x x x x x x
Angle of incidence 0 0 -20 20 0
Calculated striker velocity 1.05 1.05 1.05 1.05 1.19
Average distance (mm) 14.7 0.9 13.8 1.1 12.3 0.7 12.8 0.4 16.6 0.5
Average rotation (deg) -1.1 0.5 -13.0 1.2 -9.9 0.4 -14.8 0.4 -16.8 0.5
350
300
250
200
150
100
Initial velocities, runs A−E
Figure 6.10: Displacements and initial velocities for the aluminum square experiment.
D
C
B
E
tual velocity profiles; the predicted rotation is significantly higher than the actual rotation.
Figure 6.12 shows the measured and predicted velocity profiles for two typical trials.
It turns out that scaling the torque by a factor of 2.17 enables the model to reasonably
predict the object motion. The use of a scaling factor was suggested by examining a plot
of the measured torque versus the measured ! ratio (all calculated from position data
v
and consequently quite noisy). The actual value of this scale factor was computed by an
optimization, using the sum of the squares of displacement differences for each run and
the sum of the squares of velocity differences for each run as objective functions. (The
displacement metric actually produced an optimal scaling factor of 2.14, the velocity metric
2.20.) The remainder of this section uses scaled torque for both inverse sliding problem
calculations and for trajectory simulation.
6.3.1 Evaluation of the sliding model
As in the plexiglas disk experiment, the differences between displacements and between
initial velocities are used as a quantitative measure of how well the sliding model reflects
object motion. However, since the objective function for optimizing the torque scaling is
based on these differences, perhaps the most convincing argument of the suitability of the

64 CHAPTER 6. SINGLE TAP EXPERIMENTS
0.44
force on square
0.015
torque on square
0.42
Newtons
0.4
0.38
0.36
real
disk
Nm
0.01
0.005
scaled
real
0.34
disk
0.32
0 5 10 15 20 25
w/v
0
0 5 10 15 20 25
w/v
Figure 6.11: Force and torque functions on a square object. The solid line is the real function;
the dashed line is the function for the equivalent disk. In the torque plot, the dotted line is
the scaled torque which is used later in this section.
sliding model is a comparison of velocity profiles.
Table 6.5 shows the comparison between displacements (measured translation versus
calculated translation based on measured initial velocities). The difference in translation
ranges from 1.6% to 11.6% of the average measured translation, the difference in rotation
ranges from 2.9% to 6.6% of the average measured rotation.
Figure 6.13 shows for three representative trials the measured velocity profiles and
the predicted velocity profiles based on the initial velocities calculated from the measured
displacements using the inverse sliding problem solution. The match between measured
and calculated profiles for both translation and rotation seems good.
Table 6.6 shows the comparison between initial velocities (measured velocities versus
calculated). The difference in initial translational velocity ranges from 1.0% to 6.2% of the
average measured initial velocity, the difference in rotation ranges from 1.2% to 5.3% of the
average measured initial rotational velocity.
6.3.2 Evaluation of the impact model
These runs in this experiment used two different contact points along the edge of the square
as well as different angles of incidence of the striker. Table 6.7 shows the predicted initial
velocities which are calculated from the coefficient of restitution, the striking geometry,
and the estimated striker velocity. For comparison, the table also shows the measured
initial velocities.

6.3. ALUMINUM SQUARE EXPERIMENT 65
250
Run B, trial 3
50
Run B, trial 3
trans. velocity (mm/sec)
200
150
100
50
0
rot. velocity (deg/sec)
0
−50
−100
−150
−200
−50
0 0.05 0.1 0.15 0.2 0.25
time (seconds)
300
Run E, trial 10
−250
0 0.05 0.1 0.15 0.2 0.25
time (seconds)
100
Run E, trial 10
trans. velocity (mm/sec)
200
100
0
rot. velocity (deg/sec)
0
−100
−200
−100
0 0.05 0.1 0.15 0.2 0.25
time (seconds)
−300
0 0.05 0.1 0.15 0.2 0.25
time (seconds)
Figure 6.12: Measured and calculated velocity profiles for the aluminum square experiment
using the unscaled torque. Although the translational velocity matches well, the rotational
velocity is qualitatively different.

66 CHAPTER 6. SINGLE TAP EXPERIMENTS
A B C D E
Average measured translation 14.7 13.8 12.3 12.8 16.6
standard deviation 0.9 1.1 0.7 0.4 0.5
Calculated translation 13.0 12.8 12.1 11.8 17.0
standard deviation 1.1 1.0 0.7 0.6 0.8
Translation difference 1.7 1.0 0.2 0.9 -0.4
standard deviation 0.5 0.3 0.5 0.4 0.4
Translation diff. percentage 11.6 7.2 1.6 7.4 2.4
Average measured rotation -1.1 -13.0 -9.9 -14.8 -16.8
standard deviation 0.5 1.2 0.4 0.4 0.5
Calculated rotation 0 -12.6 -10.4 -13.8 -17.5
standard deviation 0 1.1 0.6 0.6 0.9
Rotation difference -1.1 -0.4 0.4 -1.0 0.7
standard deviation 0.5 0.5 0.6 0.5 0.5
rotation diff. percentage 2.9 3.3 6.6 4.1
Table 6.5: Comparison of displacements for the aluminum square experiment. The calculated
displacements were determined from a forward simulation using the measured
initial velocities (and the scaled torque). This table shows the average difference for each
run and its percentage with respect to the average measured values.
A B C D E
Average measured v 0 231 220 217 207 253
standard deviation 10.4 8.3 6.5 5.4 6.0
Average v 0 difference -14.4 -9.3 -2.7 -8.5 2.5
standard deviation 4.8 2.6 3.8 3.7 2.5
difference percentage -6.2 -4.2 -1.2 -4.1 1.0
Average measured ! 0 -241 -202 -284 -292
standard deviation 12.8 9.8 7.5 11.4
Average ! 0 difference 19.7 -2.9 -10.8 7.7 -9.0
standard deviation 8.5 6.5 8.7 6.6 6.5
difference percentage 1.2 5.3 -2.7 3.1
Table 6.6: Comparison of calculated (from sliding) and measured initial velocities. The calculated
initial velocities were determined by applying the inverse sliding problem solution
(with scaled torque) to the measured displacements.

6.3. ALUMINUM SQUARE EXPERIMENT 67
250
Run B, trial 3
50
Run B, trial 3
trans. velocity (mm/sec)
200
150
100
50
0
rot. velocity (deg/sec)
0
−50
−100
−150
−200
−50
0 0.05 0.1 0.15 0.2 0.25
time (seconds)
250
Run D, trial 5
−250
0 0.05 0.1 0.15 0.2 0.25
time (seconds)
100
Run D, trial 5
trans. velocity (mm/sec)
200
150
100
50
0
rot. velocity (deg/sec)
0
−100
−200
−50
0 0.05 0.1 0.15 0.2 0.25
time (seconds)
300
Run E, trial 10
−300
0 0.05 0.1 0.15 0.2 0.25
time (seconds)
100
Run E, trial 10
trans. velocity (mm/sec)
200
100
0
rot. velocity (deg/sec)
0
−100
−200
−100
0 0.05 0.1 0.15 0.2 0.25
time (seconds)
−300
0 0.05 0.1 0.15 0.2 0.25
time (seconds)
Figure 6.13: Measured and predicted velocity profiles for the aluminum square experiment.
The predicted velocity profiles were computed using the inverse sliding problem solution
(with the scaled torque) and the measured displacements.

68 CHAPTER 6. SINGLE TAP EXPERIMENTS
A B C D E
Striker velocity 1.05 1.05 1.05 1.05 1.19
Strike point 0 -0.506 -0.506 -0.506 -0.506
Angle of incidence 0 0 -20 20 0
Predicted translational velocity (mm/sec) 231 221 206 211 251
Measured translational velocity 231 220 217 207 253
standard deviation 10.3 8.3 6.5 5.4 6.0
Predicted rotational velocity (deg/sec) 0 -266 -218 -283 -302
Measured rotational velocity -241 -202 -284 -292
standard deviation 12.8 9.8 7.5 11.4
Predicted velocity direction (deg) 0 1.6 -2.1 5.2 1.6
Measured velocity direction 0 6.0 -0.1 14.5 6.8
standard deviation 0.5 1.0 0.8 0.5 0.7
Table 6.7: Comparison of initial velocities from impact for the aluminum square experiment.
The predicted initial velocities and velocity direction were calculated from the impact
geometry, the striker velocity, and angle of incidence.
6.4 Aluminum disk experiment
This experiment used the second spring-loaded tapping device to tap an aluminum disk
sliding on a formica surface. This experiment consisted of four runs (lettered A–D), testing
a few settings of the tapping device and angle of incidence. The displacements for this
experiment and the measured initial velocitieis are shown in Figure 6.14.
The primary purpose of this experiment was to determine material parameters of this
disk and surface in order to perform positioning experiments, the subject of Chapter 7.
There were several pure translation ( = 0) trials at different tapper settings in order to
determine a coefficient of restitution. These are not discussed here but are mentioned in
Section 6.5.6 in conjunction with observations on the dependence of the coefficient of restitution
on velocity.
The unmodified axisymmetric mechanics did not model the motion of this object very
well, particularly with respect to rotational motions. Like the aluminum square, measured
rotations were much less than predicted. Expanding upon the approach to the aluminum
square experiment, I did an optimization on the coefficient of friction (between the object
and support surface) first and then on the torque scaling factor.
The objective function for the coefficient of friction optimization was computed by taking
the difference between the measured initial velocity and the calculated initial velocity
(based on measured displacement), averaging the differences over each run, and summing
and squaring the average differences for each runs. The resulting coefficient of friction
was 0.17, slightly lower than what I had originally estimated from the pure translational
velocity profiles. The torque scaling factor was optimized next, using a similar objective
function based on initial rotational velocity, and resulted in a scale factor of 3.64.

6.4. ALUMINUM DISK EXPERIMENT 69
rotation (deg)
8
6
4
B
Displacements, runs A−D
C
D
rotational velocity (deg/sec)
40
2
20
A
A
0
10 15 20 25 30 35 40
0
200 250 300 350 400
translation (mm)
translational velocity (mm/sec)
Run A B C D
x x x x x x x x
Angle of incidence (deg) 0 14.5 7 7
Striker velocity (mm/sec) 419 419 431 632
Average distance (mm) 17.4 0.3 15.5 0.9 18.7 1.1 37.1 0.7
Average rotation (deg) 0.5 0.1 4.7 0.7 4.2 0.4 7.6 0.3
120
100
80
60
B
Initial velocities, runs A−D
Figure 6.14: Measured displacements and initial velocities for the aluminum disk experiment.
C
D

70 CHAPTER 6. SINGLE TAP EXPERIMENTS
A B C D
Average measured translation 17 16 19 37
standard deviation 0.3 0.9 1.1 0.7
Calculated translation 18 14 19 40
standard deviation 0.4 0.9 1.6 1.1
Translation difference -0.5 1.7 0.2 -2.7
standard deviation 0.2 0.3 0.6 1.0
Translation diff. percentage -2.6 10.8 1.2 -7.2
Average measured rotation 0.5 4.7 4.2 7.6
standard deviation 0.1 0.7 0.4 0.3
Calculated rotation 0.0 4.1 4.4 8.2
standard deviation 0.0 0.7 0.8 0.3
Rotation difference 0.5 0.6 -0.1 -0.6
standard deviation 0.1 0.4 0.6 0.2
rotation diff. percentage 12.3 -3.2 -7.8
Table 6.8: Comparison of displacements for the aluminum disk experiment. The calculated
displacements were calculated from the measured initial velocities using a forward
simulation.
6.4.1 Evaluation of the sliding model
As in the previous experiments, a comparison between displacements and between initial
velocities is used for some quantitative measure of how well the model reflects the object
behavior. Table 6.8 shows the comparison between the measured displacement and the
displacement calculated from the measured initial velocities. The average difference in
translation ranges from 1.2% to 10.8% of the average measured translation, and the average
difference in rotation ranges from 3.2% to 12.3% of the average measured rotation.
Figure 6.15 shows a number of representative measured and calculated translational
and rotation trajectories.
6.4.2 Evaluation of the impact model
The impact model predicted translational velocities well, but the predicted rotational velocities
were 25% to 50% of the measured initial rotational velocities. In order to have
a usable model of impact for positioning experiments, I took the form of the model (i.e.
the relationship between striker parameters and the resulting impulse, based on parameters
computed from striker and object mass and geometry) and estimated the parameters
based on experimental data.
Specifically in terms of the impact model, I estimated the parameters B 1 and B 2 which
are the ratios between relative object velocity and the resulting impulse for the tangential
and normal directions respectively. Whereas B 2 is fairly consistent over the four runs, B 1
varies considerably, particularly for Run B. The average B 2 results in underestimates of the
initial rotational velocity for Runs C and D and an overestimate for Run B, however the

6.4. ALUMINUM DISK EXPERIMENT 71
250
Run B, trial 7
100
Run B, trial 7
trans. velocity (mm/sec)
200
150
100
50
0
rot. velocity (deg/sec)
80
60
40
20
0
−50
0 0.1 0.2 0.3
time (seconds)
250
Run C, trial 7
−20
0 0.1 0.2 0.3
time (seconds)
100
Run C, trial 7
trans. velocity (mm/sec)
200
150
100
50
0
rot. velocity (deg/sec)
80
60
40
20
0
−50
0 0.1 0.2 0.3
time (seconds)
400
Run D, trial 7
−20
0 0.1 0.2 0.3
time (seconds)
150
Run D, trial 7
trans. velocity (mm/sec)
300
200
100
0
rot. velocity (deg/sec)
100
50
0
−100
0 0.1 0.2 0.3 0.4
time (seconds)
−50
0 0.1 0.2 0.3 0.4
time (seconds)
Figure 6.15: Comparison of predicted and measured velocity profiles for the aluminum
disk experiment. The predicted velocity profiles were computed based on initial velocities
from the inverse sliding problem solution applied to the measured displacements.

72 CHAPTER 6. SINGLE TAP EXPERIMENTS
A B C D
Average measured v 0 244 214 248 363
standard deviation 2.7 7.0 10.5 5.1
Average calculated v 0 241 226 249 351
standard deviation 2.1 6.2 7.2 3.1
Average v 0 difference 3.5 -12.7 -1.5 12.4
standard deviation 1.4 2.8 3.9 4.6
difference percentage 1.4 -5.9 -0.6 3.4
Average measured ! 0 90.9 83.0 106.4
standard deviation 13.1 12.9 2.9
Average calculated ! 0 9.1 97.7 80.1 102.2
standard deviation 1.2 12.4 5.0 3.1
Average ! 0 difference -9.1 -6.8 2.9 4.2
standard deviation 1.2 9.2 11.9 1.7
difference percentage -7.5 3.5 4.0
Table 6.9: Comparison of initial velocities from sliding for the aluminum disk experiment.
The calculated velocities were determined by applying the solution to the inverse sliding
problem to the measured displacements.
predicted values are closer to the measured values than they were before.
Table 6.10 shows these predicted velocities (based on angle of incidence and measured
striker velocities) and the measured object velocities.
6.5 Experimental observations
6.5.1 Striking height
Since the striker may contact the object at a point above, below, or at the same height above
the support surface as the COM, the impact may produce some moment about the x or y
axis (i.e. a moment that would press one side of the object into the surface and lift the
other side up). A few pure translational runs in the plexiglas disk experiment were taken
in order to determine how much the striking height affects object motion.
This experiment used the wrecking ball tapper which was adjusted in between runs so
that for one run, it struck the disk at approximately 1 of the object height above the surface,
4
onerunattheleveloftheCOM, and one run near the top of the object.
Since the wrecking ball delivered a different impulse for each of these three runs, it
is not informative to examine the resulting displacements or initial velocities. Instead, I
present a comparison of the coefficients of friction of these trials. This serves to lump all
effects of the three dimensional behavior into the coefficient of friction.
As Table 6.11 shows, the coefficient of friction was slightly higher for the run in which
the striker struck the object above the COM and slightly lower for the run in which the
contact point was below the COM. However, the differences between the coefficients of

6.5. EXPERIMENTAL OBSERVATIONS 73
A B C D
Striker velocity 419 419 431 632
Angle of incidence 0 -14.5 -7 -7
Predicted translational velocity (mm/sec) 247 240 252 370
Measured translational velocity 244 214 248 363
standard deviation 2.7 7.0 10.5 5.1
Predicted rotational velocity (deg/sec) 0 125 63 92
Measured rotational velocity 90.9 83.0 106.4
standard deviation 13.1 12.9 2.9
Predicted velocity direction (deg) 0 4.9 2.3 2.3
Measured velocity direction 0 12.6 7.0 7.4
standard deviation 0.2 1.2 0.1 0.1
Table 6.10: Comparison of initial velocities from impact for the aluminum disk experiment.
The predicted initial velocities were calculated based upon the striker velocity and angle
of incidence and the coefficient of restitution.
Below COM 0.1778 0.0058
At COM 0.1818 0.0110
Above COM 0.1888 0.0088
Table 6.11: Effect of striker height. This table shows the measured coefficient of friction
from runs where the contact point of the striker was varied with respect to the level of the
COM.
friction are less than the standard deviations of the data. The effect of striker height appears
to be small but noticeable.
6.5.2 Object overhang
In some of the first experiments with the wrecking ball tapper, the initial object position was
such that a small portion of the object extended beyond the edge of the support surface.
This was done in order to prevent the possibility of the wrecking ball colliding with the
edge of the support surface and thereby affecting the impact.
I noticed a slight spike followed by a slight dip in many of the velocity profiles, which
I attributed to measurement error and treated as a slight annoyance in measuring initial
velocities. This effect can be observed in the measured velocity profiles for the plexiglas
disk experiment presented in Section 6.2.
An experiment with the aluminum square produced much more dramatic spikes in
the velocity profiles, as shown in Figure 6.16. In fact, a change of resistance could be felt as
one pushed the the object over the edge of the support surface. A later experiment with the
object starting entirely on the support surface (with the wrecking ball carefully adjusted so
that it would not strike the edge of the support surface) showed no such spike.
In the original aluminum square experiment, I expect that this effect was due to the

74 CHAPTER 6. SINGLE TAP EXPERIMENTS
250
Run B, trial 2
200
Run B, trial 2
trans. velocity (mm/sec)
200
150
100
50
0
rot. velocity (deg/sec)
150
100
50
0
−50
0 0.05 0.1 0.15 0.2 0.25
time (seconds)
−50
0 0.05 0.1 0.15 0.2 0.25
time (seconds)
Figure 6.16: Spikes in velocity profiles due to object overhang.
topography of the surface of the object. Perhaps some “lip” of the object was catching on
the edge of the support surface. To some lesser extent, there are probably other effects,
such as some aerodynamic effect, in this and in other experiments.
6.5.3 Object path
The path of sliding objects tends to curve, as shown in Figure 6.17. The likely explanation
for this is that the COM of the object is above the plane in which friction acts on the
object. This produces a moment about the COM that pushes the front end of the disk downward,
thus increasing the effect of the friction that opposes rotation. A object with positive
rotation then tends to curve towards the right.
6.5.4 Breaking static friction
For the aluminum disk experiment, a number of pure translational trials in which the object
motion and the striker motion were measured allow calculation of striker velocity before
and after impact. Taken with the initial object velocity, the momentum and energy before
and after the impact can be compared.
Table 6.12 shows the momentum comparison. Some momentum is lost during the impact,
presumably due to breaking static friction. This does not seem to be a fixed amount or
fixed percentage of momentum, but one might expect that breaking static friction involves
exerting some force over a distance, resulting in a fixed energy loss instead.
Table 6.13 shows the energy lost during collision. Of course, most of this energy loss is
due to plastic deformation during impact. It is difficult to say how much energy is used in
breaking static friction of whether this energy is constant because it is not easy to determine
both the energy loss and the coefficient of restitution from measured velocities.

6.5. EXPERIMENTAL OBSERVATIONS 75
1
Object paths, aluminum disk experiment, Run A
(mm)
0.8
0.6
0.4
0.2
0
0 2 4 6 8 10 12 14 16 18 20
(mm)
Figure 6.17: Curvature in object paths. This graph shows object paths for Run A of the
aluminum disk experiment. Ideally, these paths should be straight lines. Instead, the path
curves slightly, likely due to the slight positive rotation observed in this run.
Run Pbefore Pafter P percentage
hi 0.0737 0.0684 0.0054 7.3
0.0006 0.0012
md 0.0476 0.0457 0.0020 4.2
0.0009 0.0012
lo 0.0229 0.0212 0.0017 7.4
0.0011 0.0014
Table 6.12: Momentum comparison. The momentum before and after impact for the aluminum
disk experiment is compared. All momenta are in kg m/sec, and the percentages
are in relation to the momentum before the impact.
Run Ebefore Eafter E percentage
hi 0.0239 0.0192 0.0045 18.3
0.0004 0.0004
md 0.0100 0.0088 0.0011 11.0
0.0004 0.0003
lo 0.0023 0.0021 0.0002 8.7
0.0002 0.0002
Table 6.13: Energy comparison. This table compares the kinetic energy of the object and
striker before and after collision.

76 CHAPTER 6. SINGLE TAP EXPERIMENTS
Run v s0 (mm/sec) e
hi 648 0.814 0.025
md 419 0.893 0.039
lo 202 0.887 0.047
Table 6.14: Variation in the coefficient of restitution. This table shows the average computed
coefficient of restitution for three trials using different striker velocities. In accordance
with typical variation, the coefficient of restitution is the lowest for the trial with the
highest striker velocity.
6.5.5 Three dimensional effects
One way to know the exact support distribution is to use three points of support. I did
an experiment using a disk which had been fitted with three rounded “feet” spaced at
120 degree intervals at a common radius. However, this seemed to only exacerbate three
dimensional effects, perhaps because the COM of the object was higher. The measured
trajectories, as shown by representative trials in Figure 6.18, did not match the predicted
trajectories. In particular, the predicted trajectories did not reverse direction or stop and
start. My hypothesis is that the disk is actually skittering — there is some combination of
translation and rotation about one of the feet.
6.5.6 Variance in e with velocity
The coefficient of restitution does have some dependence upon the velocity of impact because
of varying amounts of plastic deformation with the energy of the impact. Table 6.14
shows the calculated coefficient of restitution for each of three pure translational runs conducted
during the aluminum disk experiment. Despite variance in the average, it is clear
that the run with highest velocity impact has the lowest coefficient of restitution.
6.6 Conclusions
These experiments have shown that the sliding model (modified, if necessary) can predict
object motion within 5–10% for translations and 10–25% for for rotations. The impact
model generally predicts initial translational velocities to within a few percent but may be
anywhere from a few percent to 33% off in predicted rotational velocity.
The addition of a torque scaling factor is somewhat distressing because there is no
clear explanation why it is required. In addition, this factor has varied widely — no scaling
factor was required for the plexiglas disk experiment; a factor of 2.17 was used for the
aluminum square experiment, a factor of 3.64 for the aluminum disk experiment. These
factors are too large to be accounted for solely by a nonuniform support distribution; the
maximum amount of torque is produced by having all points of support as far from the
COM as possible, and this would only increase the torque by a factor of 2 or less. The three
dimensional extent of objects is another possible cause, but the fact that the COM of the
object is not in the plane likely has a small effect on the pressure distribution. It seems

6.6. CONCLUSIONS 77
2
orientation
50
angular velocity
(degrees)
1.5
1
0.5
(deg/sec)
40
30
20
10
0
0
0 0.05 0.1 0.15 0.2 0.25
time (sec)
0
−0.5
orientation
−10
0 0.05 0.1 0.15 0.2 0.25
time (seconds)
20
0
angular velocity
(degrees)
−1
−1.5
−2
−2.5
(deg/sec)
−20
−40
−60
−3
0 0.05 0.1 0.15 0.2 0.25
time (sec)
−80
0 0.05 0.1 0.15 0.2 0.25
time (seconds)
Figure 6.18: Representative trials from the tripod disk experiment. Note how the rotation
slows down and resumes in the first example and how it reverses direction in the second.

78 CHAPTER 6. SINGLE TAP EXPERIMENTS
plausible that frictional torque is caused by some mechanism that differs from classical
mechanics since Coulomb friction is an approximation to a complex nonlinear phenomena.
Additional experiments with more precise objects and surfaces and with faster and more
accurate measurements would be required to investigate this matter further.
The impact model seems much better at predicting rotational velocities when they are
produced by normal impulse acting at some lever arm, as was the case for the aluminum
square experiment, instead of by the tangential impulse which is due to friction, as in the
disk experiments. The impact model does not predict the direction of the impulse well; this
also bears some relationship to the amount of tangential impulse. In fact, the measured
initial velocities are usually not consistent with an impulse that can be produced at the
contact point, indicating that there may be other processes at work.
Despite these shortcomings, it still appears that the sliding and impact models are capable
of predicting object motion well enough to be used for positioning experiments. Even
if the models perfectly predicted the averages for each trial, there is still considerable variation
due to the physical phenomena. This error, along with any error due to the models,
will have to be addressed by planning and feedback strategies in order to do positioning
experiments.
The experiments in this chapter were done with reasonable care. While it is possible
to more carefully fabricate objects, use smoother and more homogeneous support surfaces,
and build more precise tapping devices, i.e. make the world conform better to the assumptions
of the models used, it is not clear that we want to. In order for manipulation by
tapping to be generally and easily applicable, it must address the factors that deviate from
model assumptions but are common in real world applications — three dimensional parts,
variation in the coefficient of restitution, variations in striking height, etc.

Chapter 7
Positioning Experiments
In order to apply the mechanics and the planning approach described in this thesis towards
positioning an object, a number of practical issues must be addressed. The planning
methods described in this chapter explicitly consider the range of displacements that the
tapping device can produce as well as the displacement errors due to variations in impact
and friction. Chapter 4 demonstrated that axisymmetric objects can reach any configuration
within two taps but suggested no criteria for choosing among the infinite number of
possible subgoals. Furthermore, it may be desirable to use more than two taps to reach a
goal.
The planning methods used in this chapter can be viewed as feedback control strategies
— given the discrepancy between the object’s configuration and the goal configuration,
they determine the first of a sequence of taps that will take the object to the goal.
These planning methods show a progression in how the error model is incorporated, and
each has a different underlying assumption about the relative importance of minimizing
the number of taps, minimizing the length of each tap, and producing a robust plan.
The first half of this chapter describes three experiments in positioning a circular object.
The first experiment uses two tap plans under the assumption that there are no errors,
the second uses a simple error model to plan two tap plans, and the third uses a slightly
more sophisticated error model to plan multi-tap plans. The second half of this chapter
describes an experiment and analysis in support of the conjecture that tapping can be used
to position an object more precisely than the manipulator can position the tapping device.
7.1 Preliminaries
7.1.1 Experimental Setup
As illustrated in Figure 7.1, the positioning experiments were done using an Adept 550
(SCARA) robot to position the second spring-loaded tapping device relative to the object.
An overhead camera was used to determine the object position and orientation by locating
a black circular target on the object which has a white stripe across its diameter; the
accuracy of the vision system is between 0.5 mm and 1.0 mm in position.
79

80 CHAPTER 7. POSITIONING EXPERIMENTS
Figure 7.1: Experimental setup for positioning experiments.
The object was the same aluminum disk used in the single tap experiments of Chapter
6. For this object, the second spring-loaded tapping device can produce a minimum
translation of 1.7 mm, and I have set the maximum translation to 35 mm in order to allow
a full rotation range over the entire translation range. This disk slides on a white formica
surface (the surface is a formica-covered particle board bookshelf). By default, the Adept
taps the object from its current location directly towards the goal; in the third experiment,
however, there was the option to tap the object away from the goal (in a direction 180
degrees from the goal).
The task for all the experiments in this chapter was to move the object 10 mm with
a rotation of 10 degrees; this is approximately three times as much rotation as can be accomplished
in a 10 mm translation. The desired positioning accuracy was 1 degree in
orientation and 1 mm in position.
7.1.2 Error models
I have assumed that the error model is linear although this is difficult to show conclusively
from the single tap experiments of the previous chapter. The error model relates the
displacement of a tap to the radii of an error ellipse in configuration space:
rx
r
=
exx
e x
e x
e
xf
j f j
In general, this error matrix has full rank and would represent a projection like that illustrated
in the leftmost graph of Figure 7.2, reflecting the fact that a translational displacement
primarily results in translational error but will also produce some rotational error,
and likewise with a rotational displacement.
The consequence of this projection is that the set of states which have an error ellipsoid
smaller than some given dimensions in rotation and translation forms a nonconvex set. The
process of intersecting the reachable displacement cone with the acceptable error ellipsoid
dimensions is also shown in Figure 7.2. Since this shape is unwieldy to use in planning, I
first considered a conservative kite-shaped approximation to this set. However even this
approximated shape was awkward to use for planning, so I turned to simpler error models.
(7.1)

7.1. PRELIMINARIES 81
r
θ
θ
f
r
θ
θ
f
x
f
max r
θ
r
x
max r
x
r
x
x
f
Figure 7.2: The error projection and the effect of maximum ellipse radii on the set of reachable
states.
One simplification is to assume that the error matrix is diagonal — r x and r are independent
and proportional to x f and f respectively. However, the single tap experiments
showed that variation in rotation is much greater than that in translation, so in the interest
of simplicity, I have considered only rotational error, which collapses the error ellipsoid to
a line. For the second experiment (conservative two-tap plans), I have assumed an error
matrix of the form:
0 0
(7.2)
0 e
The drawback of this error matrix is that a pure translation, no matter how far, will have
zero rotational error. For the third experiment (multi-tap plans), I have made the more
conservative assumption that the radius of the error ellipsoid is constant for a given
translational displacement and have set the error matrix:
0 0
(7.3)
e x 0
to reflect the maximum radius for all possible rotations.
The planning methods in this chapter do not explicitly consider error in the direction
of the tap. Since the object is circular and can thus be tapped in any direction, errors in
direction simply contribute to errors in distance to the goal. (Recall that these experiments
are done so that the object is always tapped directly towards (or away from) the goal.)
7.1.3 Positioning figures
The figures in this chapter used to graphically illustrate positioning experiments are of two
types: cartesian space and configuration space. In both cases, the start position is marked
with a black dot labeled with the letter S. For each tap, the intended tap is shown by a
dotted line with a gray dot at the end. The actual tap is shown with a black line with a
black dot at the end, labeled with the tap number.
In cartesian space plots, the x and y axes are scaled differently in order to clearly show
the path of the object; displacements in the y direction are typically much smaller than that
in the x direction. These plots are always oriented so that the goal lies directly to the right
of the start configuration.

82 CHAPTER 7. POSITIONING EXPERIMENTS
θ
goal
subgoal
start
x
Figure 7.3: The exact two-tap planning method takes the closest feasible subgoal.
In configuration space plots, the error ellipsoid is shown about the intended goal configuration
in a dotted line. Although the error ellipsoid has been collapsed to a line for the
second and third planning strategies, the error ellipse is drawn with some radius in the x
dimension. The configuration space plots show the object position projected onto the line
connecting the start to the goal configuration, so the actual translational displacement of
taps may be slightly larger than these plots would indicate.
7.2 Planning methods
7.2.1 Exact two-tap plans
The first set of experiments assumed that there was no error in actuation, so the closest
subgoal was used in formulate a two-tap plan, as illustrated in Figure 7.3. For a given
object state, if the goal can be reached with one tap, then that tap is executed; if not, the
first tap of a two-tap plan is executed. This rule is repeated until the goal is reached.
The drawback of this strategy is that if there is any error in the first tap, the object may
not be within the backprojection from the goal configuration; replanning is then required.
On the other hand, this strategy has the benefit of being very simple.
Figure 7.4 is the best example of this planning method. Although the first tap was
slightly off in direction, the object was still within the backprojection of the goal, and the
second tap brought it to the goal configuration. Not all runs were quite so good; Figure 7.5
shows a more typical run. Here, replanning was done for Taps 2, 3, and 5 (although after
Tap 5, the object was close enough to the goal to terminate the run).
Of the three methods presented in this chapter, this seemed to be the most robust,
consistently taking 5 or fewer taps to position the object.
7.2.2 Conservative two-tap plans
In order to improve upon the exact two-tap planning strategy, the conservative two-tap
planning strategy takes into account the error of the first tap. As shown in Figure 7.6, this

7.2. PLANNING METHODS 83
th (deg)
12.5
2
10.0
1 7.5
5.0
S
2.5
2
S
x (mm) 0.0
0 5 10 15
0 5 10 15
Figure 7.4: Example of an exact two-tap plan.
1
x (mm)
y (mm)
0.0
-0.5
5
4
th (deg)
3
3
10.0
2
7.5
1
5.0 S
5
2.5
2
S
x (mm) 0.0
4
0 5 10 15 20
0 5 10 15 20
Figure 7.5: Example of an exact two-tap plan with replanning.
y (mm)
1 1.5
1.0
0.5
0.0
-0.5
x (mm)
θ
goal
subgoal
start
x
Figure 7.6: The conservative two-tap planning method makes sure the error ellipsoid for
the first tap completely fits within the space of feasible subgoals.

84 CHAPTER 7. POSITIONING EXPERIMENTS
4
3
2
S
0 5 10 15 20 25
1
th (deg)
10.0
7.5
5.0
2.5
0.0
x (mm)
2
S
4
3
0 5 10 15 20 25
y (mm)
1.0
1
0.5
0.0
-0.5
-1.0
x (mm)
Figure 7.7: Example of a conservative two-tap plan with replanning.
strategy aims for a point in the set of possible subgoals that completely contains an error
ellipsoid. As discussed in Section 7.1.2, the error model for this experiment results in error
ellipses that have zero radius in the x direction. This strategy guarantees (subject to the
assumptions inherent in this error model) that after the first tap, the object will be in a state
that can directly reach the goal configuration. Like the exact two-tap planning strategy, if
the goal can be reached with a single tap, that tap is executed; otherwise, the first tap of the
two-tap plan is executed, and this process is repeated until the goal is reached.
Surprisingly, this strategy did not perform quite as well as the exact two-tap plans.
Figure 7.7 shows one of the better runs, which only required replanning for Tap 3. Most
runs took about 5 taps.
The worst case run for this strategy is shown in Figure 7.8. This points to the problem
with my implementation of this strategy — the tapping accuracy is not as good as the error
model; all of the taps in this trial over-rotated. Generally, the first tap of a two-tap plan
did bring the object within the backprojection from the goal, but after the next tap, another
two-tap plan had to be created. Although the object was repeatedly tapped back and forth,
itdidgetcloserandclosertothegoalconfiguration.
7.2.3 Multi-tap plans
One drawback of the first two strategies is that the final tap to reach the goal is often a large
tap. Since the error scales with the displacement, a better solution is to have a sequence of
taps that decrease in size as the object approaches the goal. In addition, every tap should
be guaranteed (again, within the assumptions of the error model) to leave the object in a
state from which it can reach the next subgoal.
As discussed in Section 7.1.2, the error model for this experiment also results in error
ellipses with zero radius in the x direction.
Planning method
Consider the set of states which can reach the goal in one tap subject to a maximum bound
on the error (in ) and to the minimum displacement that can be produced by the tapping

7.2. PLANNING METHODS 85
5 3
9 7
10
4
8
11
6
2
S
0 5 10 15 20 25
1
x (mm)
th (deg)
12.5
10.0
7.5
5.0
2.5
0.0
S
2
6
48
10 11
9
7 5
0 5 10 15 20 25
3
1
x (mm)
y (mm)
0.5
0.0
-0.5
-1.0
-1.5
Figure 7.8: Example of a conservative two-tap plan with replanning. (Worst case)
device. This set, which I will refer to as Cone 1, forms a symmetric trapezoidal region, as
illustrated in Figure 7.9.
The set of states that can reach Cone 1 in a single tap, subject to the same constraints,
is the union of two symmetric trapezoidal regions. This second cone may overlap the first,
depending upon the choice of the acceptable error in and the minimum displacement.
Since any state in Cone 2 that is also in Cone 1 should aim directly for the goal, I consider
Cone 2 to be the set of states outside of Cone 1. This second cone can in turn be backprojected
to form Cone 3. Figure 7.9 illustrates the process of backprojecting these cones, and
Figure 7.10 shows the first four cones together in relationship to each other and the goal.
This sequence of cones is fixed relative to the goal, and by definition, once the object
is within one cone, it can always reach the next cone towards the goal. The question recone
1
cone 2
cone 3
goal
cone 1
cone 2
Figure 7.9: Construction of a sequence of cones leading to the goal for multi-tap planning.
The backprojection of a cone can be found by examining the backprojection using the
smallest tap and that using the largest tap. The backprojected cone will include all states
in between. The lightly shaded region is the part of the backprojection that overlaps with
the cone.

86 CHAPTER 7. POSITIONING EXPERIMENTS
goal
cone 1
cone 2
cone 3
cone 4
Figure 7.10: The first four cones from the goal to the right. Cones 3 and 4 and the second
half of Cone 2 are vertical slices of the same cone.
10
th
0
-10
0 51015
-50
0
x
50
r_th
Figure 7.11: The sequence of cones leading to the goal for multi-tap planning. The cones
shrink with increasing error ellipsoid radius r .

7.2. PLANNING METHODS 87
15
10
5
0
-5
-10 th
-15
-20
-25
-30
-80 -60
-40
x
-20
0
151010
5
0
r_th
Figure 7.12: The intersection of the extended reachable displacement cone and the sequence
of cones leading to the goal is the set of possible subgoals for the first tap.
maining is how to get from an arbitrary start configuration to one of these cones; again, the
error ellipsoid from that tap should fit entirely within the entry cone.
The size of the error ellipsoid, by assumption, varies with the translational displacement
of the object. In order to determine where the object can properly enter a cone, the
radius of the error ellipsoid r must be considered as another dimension to this problem.
Drawing upon principles used in the construction of configuration spaces, we reduce
the size of the sequence of cones to the goal in the direction as r increases. This extension
along the r dimension is shown in Figure 7.11. The reachable displacement cone from the
start configuration, which normally exists in the x- plane will now extend linearly into the
r dimension. These constructs are shown together in Figure 7.12.
There is often a set of points in the intersection of these constructs to choose as a subgoal.
I have made particular choices for this experiment: when entering a cone from a point
not on any cone, I chose the point closest to the goal orientation and then the point closest
to the start configuration; when transitioning from one cone to the next, I chose the point
closest to the goal orientation and then the point closest to the middle of the cone.
Experiment
The multi-tap planning experiments were the most consistent of all the experiments in this
chapter. With one exception, the plans all took 5 taps and required no replanning. A typical
run is shown in Figure 7.13. From the start configuration, the object reached Cone 4 in the
first tap, and progressed to the next cone closer to the goal for each subsequent tap.
One run which hinted at the drawbacks of this planning method was the first run,
which used a lower limit (0.5 degrees instead of 1.0 degrees) on the allowable orientation
error at the goal; this made all the cones narrower in this run than in the others. This run
is shown in Figure 7.14. Replanning occurred after Tap 3, taking the object away from the
goal in order to get back into the sequence of cones. In Tap 8 (from Cone 1), the object
rotated too much, leaving it outside the sequence of cones. More taps were required to get
it back into a cone. This is, of course, a failure of the error model to accurately reflect the

88 CHAPTER 7. POSITIONING EXPERIMENTS
1
4
2 3
S
-25 -20 -15 -10 -5 0 5 10
S 4
3
1 2
-25 -20 -15 -10 -5 0 5 10
5
x (mm)
5
x (mm)
th (deg)
10.0
7.5
5.0
2.5
0.0
y (mm)
0.0
-0.5
-1.0
Figure 7.13: Example of a multi-tap plan.
behavior of the object, but it underscores the tendency of this model to quickly correct any
slight deviations from the planned sequence of taps.
7.3 High precision positioning
Early in the course of this work, I conjectured that tapping could be used to position an
object more precisely than a manipulator can position a tapping device relative to the object.
The intuition was that the amount of impulse delivered to an object is not too sensitive
to the distance of the tapping device from the object. However, variations in the tapping
device position and orientation also affect the contact normal.
This section describes an experiment to test this conjecture and presents some basic
analysis to support it.
7.3.1 Experiment
In this experiment, the exact two-tap planning method was used, and the requested position
and orientation of the tapping device was then rounded with respect to the global
frame. The first experiment rounded positions to the nearest 10 mm and orientations to
the nearest 5 degrees. The second experiment rounded positions to the nearest 15 mm and
orientations to the nearest 10 degrees. In all trials, the goal was reached (within 1 mm and
1 degree); a typical trial took 5–6 taps to reach the goal. Figure 7.15 shows a typical trial
from the second experiment.

7.3. HIGH PRECISION POSITIONING 89
1
3 8
4
7
5 6
2
9
S
-25 -20 -15 -10 -5 0 5 10 15
th (deg)
10.0
7.5
5.0
2.5
0.0
x (mm)
1 2
3
S
4 5
6
7
9
8
10 11 x (mm)
11
10
y (mm)
1.0
0.5
0.0
-0.5
-1.0
-25 -20 -15 -10 -5 0 5 10 15
Figure 7.14: Example of a multi-tap plan. This run was ended prematurely.
5
4
2
S 1
0 5 10 15 20
6
3
3
th (deg)
10.0
5
1
7.5
5.0S
6
2.5
0.0
x (mm)
2 4
0 5 10 15 20
y (mm)
4.0
3.0
2.0
1.0
0.0
-1.0
x (mm)
Figure 7.15: A high precision positioning trial. The desired tapper position and orientation
were rounded to the nearest 15 mm in position and nearest 10 degrees in orientation
(relative to the global frame).

90 CHAPTER 7. POSITIONING EXPERIMENTS
x
β − υ
υ
y
(i,j) striker offset
φ
P
x
j
nominal contact frame
φ
0
y
P
i
β
Figure 7.16: Notation for high precision positioning analysis. The striker nominally contacts
the object at an angle of incidence of 0 relative to the contact normal and produces
an impulse at an angle of 0 . If the striker is offset by a displacement (x; y) relative to
the striker COM at the nominal contact position, the contact frame is rotates, the angle of
incidence changes, and the angle of the resultant impulse changes.
7.3.2 Sensitivity Analysis
This subsection examines the sensitivity of a single tap to variations in the striker parameters
by evaluating the jacobian for a nominal tap.
Figure 7.16 shows the effect of displacing the striker with respect to the nominal contact
point and describes the notation used in this subsection. The contact frame is rotated
by an angle:
j , i tan
(7.4)
R + r
where R is the radius of the object and r the radius of the striker tip. One important quantity
is the change of the direction of the impulse relative to its nominal direction, given
by:
a = + , 0 (7.5)
A tap will produce some distance x f in the direction a relative to the nominal direction
and some rotation f . As shown in Figure 7.17, these parameters are related to the final
configuration of the object (x; y; ) by:
x = x f cos a (7.6)
y = x f sin a (7.7)
= f (7.8)
The sensitivity of this system to striker variations is characterized (to first order) by

7.3. HIGH PRECISION POSITIONING 91
y
(x,y)
x
a
x f
Figure 7.17: Final position in terms of distance x f and angle a relative to the nominal
translation direction.
the jacobian:
2
4 @x
@y
@
3
5 =
2 6 4
@x
@i
@y
@i
@
@i
@x
@j
@y
@j
@
@j
@x
@
@y
@
@
@
@x
@v s 0
@y
@v s 0
@
@v s 0
3
75 2 64 @i
@j
@
@v s0
3
75 (7.9)
where the striker displacement is (i; j), itsangleofincidence, and its velocity v s0 .
Derivation of the jacobian
The jacobian can be decomposed into several parts, the first of which relates the final position
(x; y; ) to the displacement (x f ; f ) and its direction a, based on the derivatives of
Equations 7.6–7.8 evaluated at a =0:
2 3 2 3
4 @x
@y 5 =
@
4
1 0 0
0 0 5 2 3
4 @x f
x 180 f @ f
5 (7.10)
0 1 0 @a
where @a is assumed to be in degrees.
The change in displacement and its direction are related to changes in the striker parameters
by:
2
4 @x f
@ f
@a
3
5 =
2 6 4
@x f
@i
@ f
@i
@a
@i
@x f
@j
@ f
@j
@a
@j
@x f
@
@ f
@
@a
@
@x f
@v s 0
@ f
@v s 0
@a
@v s 0
3
75 2 64 @i
@j
@
@v s0
The first two rows of this jacobian can be broken down into four components:
@xf
"
@x f @x f
@v
= 0 @! 0
#"
@v 0 @v 0
@P x @P y
#"
@P x @P x
@S 0 @C 0
#"
@S 0 @S 0 @S 0
@i @j @
@ f
@ f
@v 0
@ f
@! 0
@! 0
@P x
@! 0
@P y
@P y
@S 0
@P y
@C 0
@C 0
@i
@C 0
@j
3
75 (7.11)
@C 0
@
@S 0
@v s 0
@C 0
@v s 0
# 2 64 @i
@j
@
(7.12)
The first component relates the change in initial velocities to the change in displacement;
these are the derivatives of the displacement functions of Chapter 2. There appears to be
no analytic expression for these derivatives, so this jacobian is numerically estimated. The
@v s0
3
75

92 CHAPTER 7. POSITIONING EXPERIMENTS
other components can be calculated from the following formulas:
qP 2 x + P 2 y
v 0 =
M
! 0 = P xR
I
(7.13)
(7.14)
P x = S 0
B 1
(7.15)
P y = (1 + e)C 0
B 2
(7.16)
S 0 = (v s0 , k x ) sin( , )
v s0
(7.17)
C 0 = (v s0 , k x ) cos( , )
v s0
(7.18)
Since the second spring loaded tapping device is used for tapping an aluminum disk, Equations
7.18 and 7.18 assume a generalized central impact and a striker of infinite angular
inertia. The constant k in these equations models the deceleration of the striker due to
friction as it travels towards the object.
The last row of the jacobian in Equation 7.11 is obtained by differentiating Equation 7.5.
For the assumed mode of impact, the angle of the momentum is related to the angle of
incidence , (see Figure 7.16) by:
The resulting derivatives are:
@a
@ =
@a
@x = , tan
R + r
@a
@y = 1
R + r
x
(R + r) cos 2 +
2
tan =
B 2
(1 + e)B 1
tan( , ) (7.19)
5 (7.20)
cos
B 2 2
( , )
3
5 (7.21)
cos 2 B 2
( , )
1
1
4 1 ,
B 2
sin 2 (1+e)B 1
( , ) +
2
(1+e)B1
1
4 1 ,
B 2
sin 2 (1+e)B 1
( , ) + (1+e)B1
1+ x
R + r
1
cos 2
B 2
(1+e)B 1
sin 2 ( , ) + (1+e)B1
B 2
cos 2 ( , )
(7.22)
3
Evaluation of the jacobian
2
4 @x
@y
@
3 2
5 = 4
,0:0426 0:26 0:243 0:0764
,9:39 10 ,6 ,0:000052 0:352 0
,0:00434 0:0507 0:0473 0:0115
3
5 2 64 @i
@j
@
@v s0
The jacobian of Equation 7.9 can now be assembled. For a 20 mm, 6 degree tap, we have:
75
(7.23)
3

7.3. HIGH PRECISION POSITIONING 93
10.
6.7
thf
3.4
0
5
10
20
15
xf
25
30
1
0.8
0.6 ||J||
0.4
0.2
0
35
Figure 7.18: Norm of the final configuration jacobian as a function of displacement.
2
andfora10mm,3degreetap:
4 @x
@y
@
3 2
5 = 4
,0:0689 0:135 0:126 0:0562
,4:7 10 ,6 ,0:000026 0:176 0
,0:0088 0:0251 0:0234 0:00803
3
5 2 64 @i
@j
@
@v s0
3
75 (7.24)
Multiplying these jacobians by the striker error vector [7.5 7.5 5 0] T (which corresponds
to the 15 mm, 10 degree rounding done in the experiment) yields [2.85 1.76 0.584] T for
the 20 mm, 6 degree tap and [1.13 0.879 0.239] T for the 10 mm, 3 degree tap. For these taps
(ignoring error in the striker velocity), the maximum error in [@x @y @] T that results from
the maximum error in [@i @j @] T is smaller in each dimension. This offers some analytic
support for the conjecture that tapping can position more precisely than the manipulator.
One way to characterize the jacobian matrices is to find its norm. The norm of the 20
mm, 6 degree jacobian is 0.468, that of the 10 mm, 3 degree jacobian is 0.246. Figure 7.18
shows a plot of the jacobian norms for a range of displacements. From this graph, it appears
that smaller taps have smaller norms (and are thus less sensitive to striker parameter
errors) and that sufficiently large taps will have norms greater than 1. Intuitively, the latter
observation is due to the fact that small angular errors will result in large errors in final
position for large taps.
7.3.3 Discussion
For positioning tasks, we can generally express the linearized relationship between manipulator
positioning error @u and change in object configuration @x by a matrix A:
@x = A@u (7.25)
For pick and place manipulation, the A matrix is the identity, so errors in the object position
are the same magnitude as error in the manipulator. With tapping, we have seen that the

94 CHAPTER 7. POSITIONING EXPERIMENTS
A matrix can have a norm less than one. Under this condition, @x will always be smaller
than @u.
In the preceding analysis, there is an implicit weighting of the different dimensions of
the vectors @u and @x. In particular, one degree, one millimeter, and one millimeter per
second are taken to be equivalent. The previous subsection showed that for the apparatus
of this chapter’s experiments, there is a broad range of taps which have norm less than
one. These taps result in an error in object configuration that is smaller than the error in
the manipulator configuration.
Note that this analysis pertains to a single tap, which is limited in the set of configurations
it can reach. This analysis is applicable to a proof of the stability of tapping which
must address multiple-tap planning and control strategies. It is not difficult to define a
metric that measures distance to the goal and to show that a planning strategy monotonically
reduces this distance. The sensitivity analysis for a single tap will determine limits
on acceptable manipulator error so that the system is still guaranteed to make progress
towards the goal and thus maintain stability. The difficulties in such a proof come from
careful definition of the distance metric and consideration of the acceptable positioning
accuracy and the minimum and maximum possible taps.

Chapter 8
Vibratory Manipulation
Vibratory manipulation is a variant of impulsive manipulation. I define vibratory manipulation
to be any manipulation process involving repeated impacts due to a striker which
follows some regular or servoed periodic motion, generally at a high frequency and low
amplitude. Whereas impulsive manipulation primarily considers the mechanics of a single
tap or a sequence of single taps, vibratory manipulation is concerned with object behavior
due to repeated impacts — a more systems-oriented view. My primary interest is in stable
periodic behaviors of an object that result from such excitation.
This chapter considers vibratory manipulation in the context of a part that slides on
a support surface in between impacts. There are many different scenarios for this sort of
manipulation that we can consider; these include:
Open-loop control of a pusher with superimposed vibration. A pusher with a vibrating tip
follows a predetermined trajectory while a striker tip executes some periodic motion.
The actual object excitation is a high frequency, low amplitude sequence of collisions
between the striker and the object.
Feedback control of a pusher with superimposed vibration. Same as above except that the
gross motion of the pusher is determined by some feedback control law.
Feedback control of impulses. The object is sensed before every impact, and a control
law determines an appropriate impulse to drive the object towards some goal configuration.
Open-loop parts transfer. A sequence of impacts is planned to move an object from start
to goal assuming ideal behavior of the slider and perfect knowledge of all relevant
parameters. The shape of the striker and the details of the striking motion can be
designed so that small errors give rise to restoring impulses.
There are two types of tapping that result under this vibratory manipulation: intermittent
tapping, where the object comes to rest in between taps, and continuous tapping, where
the object is always tapped again before it comes to rest.
This chapter first develops the idea of vibratory manipulation in one dimension. Vibratory
manipulation in one dimension that results in continuous tapping is closely related
95

96 CHAPTER 8. VIBRATORY MANIPULATION
to bouncing a ball on a sinusoidally vibrating table and to robotic juggling. Results from
work in these areas is reviewed and adapted to vibratory manipulation in order to determine
the conditions for stable periodic motion.
The two dimensional case of vibratory manipulation is more complicated; it is related
to the control strategies used for positioning experiments in Chapter 7. These strategies
specify a striker behavior for repeated intermittent tapping; however they involve replanning
at each step instead of a regular or servoed behavior.
The basis of vibratory manipulation in two dimensions is in the limiting cases of impulsive
manipulation. These limiting cases examine the object behavior in the ideal case
when the number of taps approaches infinity and the size of each tap approaches zero.
This chapter presents a detailed analysis of the limiting cases for axisymmetric objects and
examines the relationship between the limiting cases and pushing, concluding with a few
examples.
8.1 Vibratory manipulation in one dimension
Suppose a given task is to move an object a distance d. A single tap could be used to give
the object enough velocity so that it will slide and come to rest after traveling a distance
d. Alternatively, several taps could be used to cover the distance. We could let the object
come to rest after each tap, or we could deliver the next tap before the object stops.
This gives rise to a number of different ways in which the relationship between the
impulse of a tap and the period between taps may be defined. Regardless of the definition
used, I take the limiting case to be the limiting behavior as the impulse delivered by a
single tap approaches zero (and the number of taps approaches infinity). The first two
subsections describe the limiting cases of intermittent tapping and continuous tapping.
These limiting cases are related to the ideal case of vibratory manipulation — they
presume a sequence of identical taps without error and describe how the mechanics change
as the size of a tap decreases. The last two subsections address the more applied problem
of how errors and striker motion interact with the mechanics to affect the resulting object
behavior.
8.1.1 Intermittent tapping
Suppose we are able to move the object a distance d with a single tap. Consider the tap
with 1 of the impulse, producing 1 of the initial velocity. The deceleration due to friction
n n
will be the same, so the distance covered by this tap will be d . It will require n such taps
n
(letting the object come to rest after each one) to move the object a distance d. See Figure 8.1
for an illustration.
The initial velocity required to move the object a distance d is n
v 0 =r2gd
n
(8.1)

8.1. VIBRATORY MANIPULATION IN ONE DIMENSION 97
1 tap
velocity
2 taps
time
n taps
Figure 8.1: One dimensional intermittent tapping. We want to move the object a given
distance using one or more taps, letting the object come to rest after each tap. As the number
of taps increases, the average velocity approaches zero and the total time approaches
infinity.
and the time required for the object to come to rest is
t tap =s
2d
gn
(8.2)
Note that the total amount of time for n taps is
t =s2dn
g
(8.3)
and that the average velocity is
v =r
gd
(8.4)
2n
Under this definition of the limiting case, as the number of taps n approaches infinity and
the initial velocity of each tap approaches zero, the total time required to move the object a
distance d approaches infinity, and the average velocity approaches zero.
8.1.2 Continuous tapping
Although the limiting case for intermittent tapping has useful properties, it would also be
useful to have a limiting case which converges to a nonzero average velocity. This can be
done by tapping the object again before it has come to rest. This limiting case appears to
approach a quasistatic equilibrium in the sense that a constant velocity implies that the
frictional forces are balanced by the “driving forces”.
There are a number of ways in which the relationship between the impulse of a tap and
the period between taps can be defined. The definition used will affect how the velocity
converges. Figure 8.2 shows a few examples.

98 CHAPTER 8. VIBRATORY MANIPULATION
v
(a) (b) (c)
2 taps
t
3 taps
n taps
Figure 8.2: Velocity profiles for possible definitions of the limiting case of continuous tapping
for one dimension. In all cases, the object is tapped before it has come to rest. In Figure
(a), the average velocity remains constant in the limit, whereas in Figures (b) and (c),
the average velocity approaches a limiting value from above and from below respectively.
This approach to the limiting case for continuous tapping departs somewhat from the
stated task of moving an object a distance d, which implicitly assumes that the object starts
and finishes at rest. Although it is possible to address the starting and stopping issues
in this definition, I am primarily interested in characterizing the steady state (periodic)
behaviors that can be achieve in the limit.
8.1.3 Vibratory manipulation and ball bouncing
Suppose that the striker moves with some periodic vibration superimposed on a constant
velocity and that this constant velocity is sufficiently high that the object does not come
to rest in between impacts. When viewed in a frame moving at that constant velocity (see
Figure 8.3), this system is equivalent to a ball bouncing on a vibrating table — between
impacts, the object is subject to a constant acceleration produced by friction, whereas the
ball is subject to constant acceleration produced by gravity.
The problem of a ball bouncing on a sinusoidally vibrating table has been studied
both in theory and experimentally. By applying results from the bouncing ball problem
to one dimensional continuous tapping, the stability of object behaviors can be related to
parameters of a sinusoidal striker vibration.
Holmes [31] studied the bouncing ball problem under the assumption that the vibration
of the table is small compared to the displacement of the ball. Under this assumption,
the velocity of the ball before an impact is the same as its velocity after the previous impact.
Bapat et al. [7] examined and compared the approximated mechanics and the exact
mechanics. In this chapter, I adopt most conventions and notation of Bapat et al.
The motion of the table is given by:
A sin !t (8.5)
and its velocity by:
A! cos !t (8.6)

8.1. VIBRATORY MANIPULATION IN ONE DIMENSION 99
Figure 8.3: An illustration of regular periodic bouncing. The table moves with a sinusoidal
vibration, and the object follows a parabolic trajectory while in flight. This same illustration
applies to tapping an object using a striker with a sinusoidal vibration superimposed on a
constant velocity (when viewed in the frame of that constant velocity).
where ! here is 2 times the frequency of the table oscillation. The velocity of the object
just before or after a collision is denoted by V .
By reasoning about the return map generated by the approximated mechanics ( mapping
the velocity of the ball (just before an impact) and the phase of the impact (relative to
the table oscillation) from one impact to the next ), Holmes showed that for a coefficient of
restitution less than 1, the trajectory of the ball will be bounded. Given the frequency and
amplitude of the oscillation, the velocity of the ball just before an impact (or just after the
previous impact) will eventually satisfy
(1 + e)
jV j < A! (8.7)
(1 , e)
Fixed points of the return map correspond to different modes of stable periodic bouncing.
The simplest mode of object behavior is what Holmes called period one motions and Bapat
et al. called (n,1) motions (a collision every n cycles of the table with a collision pattern
of length 1). For a given n and !, stability analysis of the fixed points showed that in order
to maintain stable periodic bouncing the amplitude of the table must satisfy:
ng (1 , e)
! 2 (1 + e)

100 CHAPTER 8. VIBRATORY MANIPULATION
10
8
Amplitude (mm)
6
4
2
0
2 4 6 8 10 12 14
frequency (Hz)
Figure 8.4: Relationship between frequency and amplitude of a sinusoidally vibrating
striker for stable periodic bouncing (period one, n =1bouncing). Here we have assumed
e =0:8 and =0:25. Increasing the amplitude beyond the range shown here will result in
period doubling and will eventually lead to chaotic behavior. Using an amplitude below
this range will likely result in the object coming to rest against the striker and being pushed
along.
An example
Suppose we take n =1, the coefficient of restitution e =0:8, and the coefficient of friction
=0:25. From equation 8.8, we find that the striker amplitude must satisfy
0:856
! 2 v O ) in order to ensure that the object does not come to rest before it
is struck again. In fact, some tolerance will be required since the mechanics used here are
approximate.
Under continuous motion, an object being “pushed” by a sinusoidally vibrating striker
moving at a constant average velocity v might appear as shown in Figure 8.3 (in the frame
of reference moving at velocity v).

8.2. VIBRATORY MANIPULATION IN TWO DIMENSIONS 101
Figure 8.5: Illustration of an object bouncing off a sinusoidally vibrating striker under intermittent
tapping. This sinusoidal vibration is superimposed on a constant velocity v,
and we view the system from a frame moving at that velocity. Here, the object follows a
parabolic trajectory until it comes to rest relative to the surface; in the moving frame, it
appears to be “falling” with a velocity ,v.
Intermittent bouncing
If the object does come to rest in between impacts, then in the frame moving at the average
velocity of the striker, the object will appear to stop “falling” along a parabolic path
and move at a constant velocity ,v, as illustrated in Figure 8.5. Although not nearly as
amenable to analysis as continuous bouncing, we should still be able to achieve stable periodic
impacts with intermittent motion of the object.
8.1.4 Robotic juggling
Schaal and Atkeson [56] have implemented paddle jugging with a “trampoline-like” racket
and a ping-pong ball. They observed that negative acceleration of the paddle trajectory
at impact seems to provide stability. They calculated the basin of attraction for periodic
juggling to be 0.257 the area of the viable space of initial conditions. However, for a number
of reasons (including mechanical variability and air resistance) the basin of attraction was
significantly larger. In fact, they were unable to avoid getting a periodic (mostly period
one) juggling pattern.
Bühler and Koditschek [16] demonstrated a system that juggles a puck on an inclined
plane using a bat. In order to achieve stable juggling, they developed a “mirror” control
law which servos the bat to nonlinearly reflect the position of the puck. Using this sort of
simple control law for vibratory manipulation is appealing but sensing the object would
be difficult for high frequency, low amplitude motions.
8.2 Vibratory manipulation in two dimensions
The two dimensional case is much more complicated than the one dimensional case because
of the complexity of the mechanics (which is in turn due to the strong coupling between
translational and rotational motion). This section addresses vibratory manipulation
for axisymmetric objects under the idealized assumption that the object is manipulated by
a sequence of identical taps with no error. This section does not address how errors and

102 CHAPTER 8. VIBRATORY MANIPULATION
x f /2
Net COR
θ f
x f
2 tan (θ
f/2)
x f
θ f
Figure 8.6: Construction for finding the net COR for a tap where the object moves a distance
x f and turns an angle f .
striker motion interact with the mechanics to affect stable object behavior.
Since all impacts are assumed to be identical, this section examines the constraints on
object motion in the limit, i.e. as the impulse of a single tap approaches zero. This section
addresses the limiting case of the axisymmetric mechanics for intermittent tapping and for
continuous tapping.
The two limiting cases can be characterized by the locus of centers of rotation (CORs)
that they can produce. As illustrated in Figure 8.6, any planar displacement can be represented
as a pure rotation about some point in the plane. The locus of CORscanbeused
to compare two different modes of manipulation; it is also sufficient information to create
plans for that mode of manipulation
This section assumes that the object boundary is circular and develops the limiting
cases for positive rotations; the constraints on object motion for negative rotations will be
symmetric.
8.2.1 Limiting case of intermittent tapping
For intermittent tapping, the object undergoes a sequence of identical impacts, always coming
to rest before the next impact. In state space, the object repeatedly follows a path starting
at the initial velocities and ending at the origin (zero velocities). I take the limiting
case to be the resultant behavior as the impulse approaches zero but the ratio of the initial
velocities remains the same.
Net COR in the limit
Given the distance traveled and the angle turned, the net COR for the motion can be determined
with a geometric construction as illustrated in Figure 8.6. The location of the net
COR is:
~x f
2 + ^z ~x f
2 tan f 2
(8.12)

8.2. VIBRATORY MANIPULATION IN TWO DIMENSIONS 103
Since the impulse of a tap approaches zero by scaling the initial velocities, the trajectory
scaling of Section 2.1.6 (scaling the velocities by k) is used to take the limit:
k 2 ~x f
lim
k!0 2
+ ^z k2 ~x f
2 tan k2 f
2
= lim
k!0
^z k 2 ~x f
2 tan k2 f
2
(8.13)
Applying l’Hôpital’s rule (differentiating both numerator and denominator with respect to
k), yields:
^z ~x f
lim
= ^z ~x f
(8.14)
k!0
f sec 2 k2 f f
2
Thus, as the impulse delivered by a tap approaches zero, the distance from the center of
mass (COM) to the net COR approaches xf
f
.
Locus of CORs
This limiting case can be characterized by the locus of possible CORs. This represents all
possible motions of the object for a given initial velocity direction.
The trajectory scaling of Section 2.1.6 shows that each value of !0
v 0
corresponds to a
single value of xf
f
, regardless of the magnitude of the initial velocities or the resulting
displacements. It was also shown in Section 2.1.6 that xf
f
is a continuous monotonic decreasing
function of !0
v 0
. Therefore the minimum and maximum values of !0
v 0
that can be
achieved via an impact correspond to the maximum and minimum values of xf
f
for the
possible resulting displacements. These values of xf
f
in turn determine the closest and
furthest CORs possible for this limiting case.
The range of possible !0
v 0
that can be produced by impact was introduced in Section
3.2.2 as impact cones (for the case where the set of possible velocity ratios does not
change with velocity direction). The impact cone for a circular object can be expressed as:
p ! 0
rRM
v 0 I 1+ 2 r
(8.15)
The minimum value of !0
v 0
is 0, corresponding to pure translational velocity (which results
in a pure translation of the object: xf
f
= 1); the maximum value is given by the equality
condition of Equation 8.15. Hence, the range of CORs for the limiting case of intermittent
tapping will be from infinity to the closest COR. Letc min = xf
for the maximum value of
! 0
v 0
that satisfies Equation 8.15. Applying this range of xf
f
to Equation 8.14 determines the
locus of all possible CORs:
fc^z ^v 0 j c 2 [c min ; 1]g (8.16)
This locus consists of the entire plane except for an open disk of radius c min centered at the
object COM.
8.2.2 Limiting case of continuous tapping
For continuous tapping, the object undergoes a sequence of regularly spaced impacts which
always strike the object before it comes to rest. In state space, the object repeatedly follows
f

104 CHAPTER 8. VIBRATORY MANIPULATION
w
(a) (b) (c)
ω
stable
Negative Impact Cone
ω stable
eigendirection
eigendirection
Negative
Impact Cone
v
v
stable
eigendirection
v
Figure 8.7: Illustrations of the state space for a sliding rotating axisymmetric object; the
stable eigendirection is always the dashed-dotted line. Figure (a) shows a typical vector
field, which is a function of only ! . Figure (b) illustrates the case where the initial slope of
v
the path in state space aligns with the negative impact cone above the stable eigendirection.
In this case, the trajectory curves in to the cone. Figure (c) illustrates the case where this
alignment occurs below the eigendirection and the trajectory curves away from the cone.
a path starting at its initial velocities and continues until the object is tapped again; this
“kicks” the state back to the original initial velocities. I take the limiting case to be the
resulting behavior of the object as the amount of time before the next impact approaches
zero.
The same expression (Equation 8.12) for the location of the net COR is used for this
limiting case; however, the limit is now taken with respect to time between impacts t f .As
t f decreases, the displacements ~x f and f both approach zero:
lim
t f !0
~x f
2 + ^z ~x f
2 tan f 2
= lim
tf!0
^z ~x f
2 tan f 2
(8.17)
Assume for the moment that this limit exists. Applying l’Hôpital’s rule (differentiating
numerator and denominator with respect to t f ), yields:
lim
t f!0
^z d
dt f
sec 2 f
~x f
d
dt f
f
= lim
t f!0
^z d
dt f
d
dt f
R
t f
R t f
~vdt 0
t f!0
0 !dt = lim
^z ~v(t f )
!(t f )
= ^z ~v 0
! 0
(8.18)
Thus, the COR in the limit is simply the initial (instantaneous) COR.
The remainder of this subsection examines the conditions under which this limit exists,
and this will determine the locus of CORs possible under this limiting case.
Impact constraint for moving objects
Appendix A demonstrates that it is possible to generate any impact within a friction cone
in impulse space by striking an object at rest. For a given impulse in the friction cone,
we can compute the striker velocity and contact point required to achieve that change in
velocity. Not surprisingly, so long as the same relative velocities between the striker and
the object are recreated, the same change in velocity can be achieved. Thus the impact cone

8.2. VIBRATORY MANIPULATION IN TWO DIMENSIONS 105
!
v
Disk
COR
!
v
Ring
COR
Intermittent Tapping LC 0.080 0.208
Continuous Tapping LC/Pushing 16.395 0.061 8.951 0.112
Impact Cone 9.704 0.103 4.851 0.206
Eigendirection 30.798 0.032 20.000 0.050
Radius 0.050 0.050
Table 8.1: Motion constraints for a disk and a ring for both limiting cases (LCs). The distance
to the COR given for the impact cone, limiting cases, and pushing is the distance to
the closest COR. The distance to the COR corresponding to the stable eigendirection and
the radius of the object are given for comparison. Quasistatic pushing has the same motion
constraints as the limiting case of continuous tapping. These results were numerically
generated, and assume that = r =0:25.
for a circular object (Equation 8.15) can be restated in terms of change in velocities (for
positive and negative changes in rotational velocity):
j!j
v rRM
I p 1+ 2 r
(8.19)
The result is that we can move the impact cone to any point in state space.
This section will primarily make use of the negative impact cone, whichisthesetof
states that can reach the current state with a single impact. Since the impact cone is invariant
with respect to its position in state space, the negative impact cone is simply the
reflection of the impact cone through the current state.
Existence of the limit & locus of CORs
The limit taken in Equations 8.17 and 8.18 exists if and only if the path of the object in state
space passes through the negative impact cone (at the initial velocities) for a nonzero period
of time. In order to determine which initial conditions satisfy this property, the properties
of paths in state space must be examined. The eigendirections of Goyal et al. [27] serve to
guide this analysis.
There are a number of properties of paths in state space for a sliding axisymmetric
object. They are stated here in terms of the vector field (_v; _!) =(,F=M;,T=I) over the
state space.
as ! v increases, the orientation of (_v; _!) monotonically increases from to 3 2 .
(_v; _!) points towards the origin only at values of ! corresponding to eigendirections;
v
most objects have a single stable eigendirection that is neither pure translation nor
rotation.
at all points below the stable eigendirection, (_v; _!) points above the origin, drawing
the system towards the stable eigendirection.

106 CHAPTER 8. VIBRATORY MANIPULATION
at all points above the stable eigendirection, (_v; _!) points below the origin, drawing
the system towards the stable eigendirection.
These properties are illustrated in Figure 8.7a.
If the negative impact cone is placed on the line corresponding to ! = 0(a COR at
v
infinity), the path in state space will lie entirely inside the cone. As the value of ! is increased,
the vector (_v; _!) will rotate counterclockwise. The path of the object in state space
v
will continue to pass through the cone until this vector aligns with the bottom edge of the
cone. At that point:
F ( ! v )
T ( ! v ) = R
(8.20)
q1 + 1 2 r
Let 1=c min be the value of !0
v 0
which satisfies this equation.
If this alignment occurs above the stable eigendirection then the path will curve into
thecone(seeFigure8.7b),sothelocusofpossibleCORsis
fc^z ^v 0 j c 2 [c min ; 1]g (8.21)
If this alignment occurs below the stable eigendirection then the path will curve away from
thecone(seeFigure8.7c),sothelocusofpossibleCORsis
fc^z ^v 0 j c 2 (c min ; 1]g (8.22)
These loci cover the plane except for an open or closed disk (depending upon when alignmentoccurs)ofradiusc
min centered at the COM of the object.
8.3 Comparison with pushing
For comparison to the limiting cases of impulsive manipulation, this section develops the
constraints on the motion of a pushed object.
The pushing force exerted on the object must lie within the friction cone at the contact
point. This force and the resulting torque are resisted by frictional forces which are
determined by the velocities of the object. Thus, the range of forces and torques that can be
exerted on the object indirectly determine the range of velocities and therefore the range of
CORs.
Consider a circular axisymmetric object and presume, akin to the limiting cases of
impulsive manipulation, that this object can be pushed at any point on the boundary in
order to push the object in a single given direction. A force that produces no torque can
always be exerted on the object; this produces pure translation, corresponding to a COR at
infinity. As illustrated in Figure 8.8, the maximum torque that can be exerted for a given
force is determined by the coefficient of friction between the pusher and the object, r .
Thenetforceandtorque,F ( ! v ) and T ( ! ), are monotonic decreasing and increasing
v
functions of ! respectively. The maximum torque to force ratio then determines the maximum
value of ! (the minimum value of v ) and therefore the closest possible COR.
v
v !

8.4. EXAMPLES FOR THE LIMITING CASES 107
-1
tan µ
Figure 8.8: The maximum value of T for pushing a disk is achieved by producing the
F
greatest amount of torque for a given force. Friction cones are drawn here at the contact
points which have the maximum lever arm for a frictional force in the ^x direction.
The maximum torque that can be generated for a given force is due to a force on the
edge of the friction cone. This torque is:
T = j~r jF sin ( tan ,1 r
r )=RFp (8.23)
1+
2
r
let 1=c min be the value of ! v
that satisfies:
F ( ! v )
T ( ! v ) =
R
q1 + 1 2 r
(8.24)
The set of rotation centers given the velocity direction ^v is
fc^z ^v j c 2 [c min ; 1]g (8.25)
Note that this is the same condition for the minimum COR as in the limiting case of
continuous tapping. (Compare Equations 8.24 and 8.20). With the possible exception of
the rotation centers at c min^z ^v , pushing and the limiting case of continuous tapping can
maintain the same range of CORs.
8.4 Examples for the limiting cases
Table 8.1 shows the results of analyzing the possible motions of a uniform disk and a ring
under the two limiting cases. These results are graphically illustrated in Figure 8.9. Note
that the “alignment” for the limiting case of continuous tapping occurs below the stable
eigendirection (i.e. the value of ! (slope in state space) for the limiting case is less than that
v
for the eigendirection).
This table also includes the size of the impact cone (i.e the maximum value of !0
v 0
achievable by impact) which is given by
!
v
R
2q1 + 1
2 r
(8.26)

108 CHAPTER 8. VIBRATORY MANIPULATION
20
int.
int.
cont.
continuous LC
15
10
5
cont.
continuous LC
intermittent LC
cm
intermittent LC
Figure 8.9: Graphic illustration of the motion constraints on a disk and a ring due to tapping.
For each limiting case, the object can move about CORs from infinity (pure translation)
up to the closest COR, indicated by a dot; a radius is drawn to the path of maximum
curvature that the object can follow. The shaded regions represent the locations the objects
can reach; the set of locations for intermittent tapping is a subset of the locations for
continuous tapping for these examples.
and will is a function of the radius of the object. The smaller the radius, the smaller the
radius of gyration, and the wider the impact cone will be.
Note that for both examples, the range of CORs possible for the limiting cases is greater
than that of the impact cone, i.e. there are states at which the limiting case can be sustained
but which cannot be reached directly when starting the object from rest. Also note that
the range of possible CORs for both limiting cases in both examples lies below the eigendirection;
therefore the constraints on object motion for pushing and for the limiting case of
continuous tapping differ with respect to motion about the CORsatc min (^z ^v).
8.5 Discussion
8.5.1 Limiting case of intermittent tapping
The limiting case of intermittent tapping is not in itself useful because the average velocity
approaches zero in the limit, but approximations to the limiting case have several useful
properties.
First of all, an approximation to this limiting case would be the simplest and perhaps
most practical to implement since the object comes to rest in between taps. This makes it
easier to incorporate sensory feedback while executing a series of taps and allows for more
exact execution of planned sequence of taps. In contrast, the limiting case of continuous
tapping is more dynamic, so it would be difficult to repeatedly strike the object at exactly
the same state and “kick” it back precisely to the initial state.
Intermittent tapping also provides some independence between planning a path and
executing it because the exact same path that can be executed with a single tap can be
executed by arbitrarily many taps. In fact, since error tends to grow with increased energy,

8.5. DISCUSSION 109
ω
stable eigendirection
alignment
value
edge of impact cone
Figure 8.10: When the initial impact cone lies below the eigendirection, we can reach points
in the state space outside the this cone by repeated impacts. Since all points below the
eigendirection asymptotically approach the eigendirection, ! increases beyond the impact
v
cone. By repeatedly striking the object, we can maintain the energy of the object and thus
reach any point in the state space for which the limiting case of continuous tapping exists.
The solid lines show the object sliding; the dotted lines correspond to impacts.
v
smaller taps yield smaller error, so it may be advantageous to use a sequence of smaller
taps, possibly with feedback.
With the range of possible CORs a given object, a nonholonomic motion planner could
be adapted to plan complex motions which avoid obstacles, as has been done for pushing
by Lynch and Mason [41]. One caveat is that motion about a COR may need to be followed
arbitrarily closely in order to avoid obstacles.
8.5.2 Continuous tapping limiting case
In the examples presented in Section 8.4, the closest possible COR for the limiting case of
continuous tapping was closer than the COR corresponding to the edge of the impact cone,
i.e. the limiting case of continuous tapping can be sustained at states which cannot be
directly reached from rest.
When !0
v 0
is below the eigendirection, the trajectory will eventually leave the negative
impact cone. If the object is tapped again before it leaves the negative impact cone, the
value of !0
v 0
can be increased. Figure 8.10 illustrates this process repeated in order to reach
states outside of the impact cone at the origin. If the object is tapped again after it leaves
the negative impact cone, then the value of !0
v 0
will decrease, regardless of the tap.
8.5.3 Tracking versus following the object
One issue in the two dimensional limiting cases is whether to track or follow the object, i.e.
whether to always strike the object at the same contact point relative to the world frame
or relative to the object frame. The question is essentially whether to ignore rotation or to
track rotation. The distinction is clearest for objects with circular boundaries because the
boundary of a circle is same regardless of its orientation, so either choice may be taken. For

110 CHAPTER 8. VIBRATORY MANIPULATION
noncircular object boundaries, only tracking the object is possible. (Following an object
with a noncircular boundary would not generally lead to a stable periodic motion.) The
decision between following or tracking affects whether the COR of the motion remains fixed
within the world frame or relative to the body.

Chapter 9
Conclusions
9.1 Contributions
This thesis has demonstrated, both analytically and experimentally, how tapping can be
used for manipulation. The main results are:
Analysis of the mechanics of impact and friction to solve the planning problem for
tapping.
Experimental validation that the models can predict motion of real objects with reasonable
accuracy.
Development of planning methods (or feedback control strategies) to compensate for
experimental errors.
Experimental demonstration that tapping can be used to position an object to greater
precision than the manipulator can position a tapping device.
Development of devices to generate a single controlled repeatable impact and identification
of design goals and issues for these devices.
In the analysis portion, this thesis presents a solution to planning a single tap by solving
the inverse sliding problem and the impact problem. For the inverse sliding problem,
I have described methods to determine initial velocities for an object to slide a given displacement
(translation and rotation) and have proven the existence and uniqueness of the
solution for the axisymmetric case. For the impact problem, I have applied the mechanics
of impact to determine how to strike an object to produce desired initial velocities and have
also described a method to characterize the velocities that can be produced by an impact.
Not all displacements can be reached in a single tap, so a sequence of taps may be
required. For axisymmetric and nearly axisymmetric objects, I have described constructs
and methods for planning multiple-tap plans and have shown that any goal configuration
can be reached within two taps. These constructs are also used to show that tapping is
small time locally controllable for all axisymmetric and at least some nearly axisymmetric
objects. Thus, tapping can be used to make these object follow any path arbitrarily closely.
111

112 CHAPTER 9. CONCLUSIONS
Tapping can also be used in a manner which I refer to as vibratory manipulation;
here, an object interacts with a striker which follows some regular or servoed periodic
motion (generally a high frequency low amplitude motion). I have studied this problem
in one dimension, applying results from related work to determine conditions for stable
periodic object motion, and I have laid the foundation for two dimensions by formulating
the limiting cases of the mechanics for a single tap.
The experiments are a major part of the contributions of this thesis. My experimental
efforts began with an exploration of ways to produce a controlled impact. I have designed
and used five different tapping devices in the course of this work; this thesis describes
those devices and their performance and discusses the design issues I have found to be
most important.
The experimental setup did not fully conform to the idealized assumptions used in
modelling and analysis, primarily with respect to the uniformity of support distributions.
The key question in the first experiments was whether the impact and sliding models
would still be adequate to predict object motion. These experiments showed that the models
can be modified to predict object motion with acceptable accuracy; this modification is
the use of a scaling factor to increase the amount of torque acting upon the object. For the
three experiments described in this thesis, one required no scaling factor; the other two required
scaling factors of 2.17 and 3.64. The reason for this deviation is not completely clear.
It is likely due in part to the nonuniformity of the support distribution and to the three dimensional
extent of the object. However, it also seems plausible that frictional torque may
be caused by a mechanism that differs from classical mechanics since Coulomb friction is
an approximation to a complex nonlinear phenomena.
In order to use tapping for positioning tasks, planning methods must take into account
both errors due to modelling and errors due to variation in the impact and friction
phenomena. I incorporated different error models into three different planning methods
that essentially serve as feedback control laws. The modified mechanics models and these
planning methods were used to demonstrate a positioning task via tapping. Finally, it was
experimentally shown that tapping can be used to position an object to greater precision
than the tapping device itself is positioned by rounding positions and orientations of the
tapping device with respect to the global frame.
9.2 Future work
This thesis is a first step in the formal study and application of impulsive manipulation.
I have shown that tapping, one form of impulsive manipulation, is a viable and useful
method of manipulation; it possesses a unique combination of attributes, namely its speed,
nonprehensile nature, and limited interaction between the manipulator and the object. This
thesis covers a wide range of topics directed towards the analysis and experimental use of
tapping, but there are many areas that deserve more attention.
Friction modeling
It is unclear whether the necessity of a torque scaling factor is due to some fundamental
mechanism or whether it is simply due to deviations from the idealized models. This will

9.2. FUTURE WORK 113
require many careful experiments with precise measurements.
Stability analysis
A formal proof of the stability of tapping under some planning method should be possible,
building upon the sensitivity analysis of Chapter 7. This would result in a characterization
of the bounded input bounded output stability of the system in the presence of errors.
The planning methods described in this thesis have different distributions of the probability
of reaching the goal as a function of the number of taps. It would be interesting to
characterize the tradeoff between the average plan length and its variation.
Adaptive control and active perception
Although it is not possible in general to determine both material properties and tapping
device calibration at the same time, with knowledge of one we can determine the other. It
should be possible to formulate an adaptive controller to learn the properties of an object,
essentially implementing a form of active perception.
Vibratory manipulation
Experimental demonstration of one dimensional vibratory manipulation would validate
the analysis presented in Chapter 8 and would pave the way for a two dimensional implementation.
Further analysis is required to combine the limiting cases of impulsive manipulation
with a given striker behavior in order to characterize the motion of an object subject to such
excitation.
Kinematic constraint on objects
Tapping an object subject to some kinematic constraint (such as an object in contact with
other fixed objects) likely provides some additional stability or reduction in error for a
given tap. This sort of tapping would be useful for parts assembly.
Other forms of impulsive manipulation
Exploration of impulsive manipulation where the object becomes airborne in between taps
would involve some different mechanics but would be applicable to different types of vibratory
parts feeding systems.

114 CHAPTER 9. CONCLUSIONS

Appendix A
Impact Analysis
This appendix contains the details of the mechanics of impact which are used throughout
this thesis. The primary result of this appendix is that any impulse within a friction cone in
impulse space at the contact point can be generated by an impact. The details of why this
is so and how to determine the striker parameters for a given impulse are covered in this
appendix.
Since I will be making detailed references to the mechanics of two dimensional impact
with friction, as described in Wang and Mason [63], I first summarize their results. With
a few simplifying assumptions, I then show that any impulse within a friction cone in impulse
space can be generated and describe the procedure to calculate the required striker
parameters for a particular impulse. Finally, the impact parameters for some specific strikers
and striker/object geometries are presented.
A.1 Two dimensional rigid-body collisions with friction
I have used slightly different notation and terminology than Wang and Mason [63] to summarizing
their results here. What Wang and Mason call Body 1 and Body 2, I refer to as
the “object” and “striker” respectively. Quantities related to the striker have a subscript s,
e.g. M s , x s ,andy s ; quantities related to the object do not have a subscript, e.g. M, x, and
y. Whereas Wang and Mason express moments of inertia in terms of the mass and radius
and gyration, here I simply use the letter I. Finally, instead of using for the coefficient of
friction between the striker and the object, I have substituted r to differentiate it from the
coefficient of friction between the object and the support surface.
A.1.1
Equations of motion
We assume that two rigid bodies, the striker and the object, collide at a single contact point.
(See Figure A.1.) We attach a frame at the contact point with the y axis normal to the
boundary and pointing into the object. The COM of the object relative to this frame is (x; y),
the COM of the striker, (x s ;y s ). Their respective linear velocities at the time of impact are
(_x 0 ; _y 0 ) and (_x s0 ; _y s0 ). There may also be rotational velocities, given by _ 0 and _ s0 . The
115

116 APPENDIX A. IMPACT ANALYSIS
initial normal (compression) velocity is:
C 0 =(_y 0 , _ 0 x) , (_y s0 , _ s0 x s )
(A.1)
The initial tangential (sliding) velocity is:
S 0 =(_x 0 + _ 0 y) , (_x s0 + _ s0 y s )
(A.2)
The impact process undergoes a compression and a restitution phase during which we
assume that there is no motion of the object. We will, however, assume that the velocity of
the object and striker changes continuously in accordance with accumulated impulse. The
effect of this impulse on the object velocities _x, _y, and _ is:
M (_x , _x 0 )=P x
M (_y , _y 0 )=P y
(A.3)
(A.4)
I( _ , _ 0 )=P x y , P y x (A.5)
There is an analogous set of equations for the effect of the impulse on the striker velocities.
As the collision progresses, the accumulated impulse will change the sliding velocity
(tangential velocity between the object and the striker) and the compression velocity (relative
velocity along the contact normal). By combining the dynamics and the kinematics,
we can express these velocities in terms of the accumulated impulse:
where
S = S 0 + B 1 P x , B 3 P y
C = C 0 , B 3 P x + B 2 P y
B 1 = 1 M s
+ 1 M + y2
I + y2 s
I s
(A.6)
(A.7)
(A.8)
B 2 = 1 + 1 M s M + x2
I + x2 s
(A.9)
I s
B 3 = xy
I + x sy s
(A.10)
I s
where M s is the mass of the striker and I s its moment of inertia. (Note that the above expression
for B 3 is correct; it appears incorrectly in Equation 21 in (Wang and Mason [63]).)
A.1.2
Impulse space constructs
Routh’s method [54] involves constructing a number of lines in impulse space along which
the state of the collision travels or at which the state of the collision changes.
The first two lines are related to the relative normal and tangential velocity of the two
objects. By setting the equation A.7 to zero, we get the line of maximum compression (C):
C 0 , B 3 P x + B 2 P y =0
(A.11)

A.1. TWO DIMENSIONAL RIGID-BODY COLLISIONS WITH FRICTION 117
object
y
(x, y)
x
(x s,y s )
striker
Figure A.1: Impact geometry and notation.
By setting the equation A.6 to zero, we get the line of sticking (S):
S 0 + B 1 P x , B 3 P y =0
(A.12)
We are also interested in the line of termination, obtained by setting equation A.7 to ,eC 0
(1 + e)C 0 , B 3 P x + B 2 P y =0 (A.13)
This is the line at which the collision will end under Newton’s law of restitution (to be
discussed further in section A.1.4. The fourth line of interest is the line of limiting friction
(L), given by
P x = ,s r P y
(A.14)
where r is the coefficient of friction between the striker and the object and s is the sign of
the initial sliding velocity S 0 :
s = S 0
jS 0 j
if S 0 6=0 (A.15)
There is one more line of interest which will be discussed in the next subsection.
A.1.3
Impact process
As impact progresses through the compression and restitution phases, impulse along the
P y axis will accumulate, changing the normal velocity until the collision is over. Impulse
along the P x axis accumulates due to friction between the striker and the object in accordance
with Coulomb friction.
The collision follows a path in impulse space starting at the origin, and initially following
the line of limiting friction. The collision continues (in a sliding mode) along this
line until the collision is over or until it crosses the line of sticking. At that point, the object
either sticks (then following the line of sticking) or slides in the opposite direction. If the
line of sticking lies within the friction cone in impulse space at this intersection point, then

118 APPENDIX A. IMPACT ANALYSIS
the object will stick, otherwise, the object will slide, entering the regime of reversed sliding
in which its progress through impulse space along the line of reversed limiting friction (RF):
dP x = s r dP y
(A.16)
To summarize, the collision accumulates impulse along the line of limiting friction until it
intersects with the line of sticking, at which point, it will follow either the line of sticking
or the line of reversed friction, whichever is steeper, until the collision ends.
A.1.4
End of the collision
The end of the collision is determined by the restitution law. Wang and Mason [63] give results
for both Newton’s law and for Poisson’s hypothesis. Since I have adopted Poisson’s
law in this thesis, I only summarize those results here. Note that Newton’s law, Poisson’s
hypothesis, and Stronge’s internal dissipation hypothesis all give the same answer
for direct central impacts (when B 3 =0and S 0 =0) and when the impact remains in a
sliding mode. Only when sliding stops or reverses direction do they differ. See Section 3.1,
Stronge [57], or Wang and Mason [63] for further discussion.
Poisson’s law terminates the impact based on the normal impulse accumulated during
compression and restitution (P r and P c respectively). It defines the coefficient of restitution
as:
e = P r
(A.17)
P c
A.1.5
Impulse solutions
The path of the accumulated impulse through impulse space depends upon the relative
slopes and intersection points of the line of sticking, the line of maximum compression,
and the line of limiting friction.
We classify a collision by three criteria: whether the collision stopped sliding or not; if
so, whether it entered a sticking mode or a reversed sliding mode; and whether it changed
modes during the compression or restitution phase of the impact. We can classify the collision
using the coefficient of friction between the striker and the object r and the quantities:
P d = sS 0 (B 2 + sB 3 )
P q = ,C 0 (B 1 + sB 3 )
s = , B 3
B 1
Table A.1 shows the different collision modes in terms of these four quantities.
The resultant impulses are:
(A.18)
(A.19)
(A.20)
sliding:
P x = ,s r P y
C 0
P y = ,(1 + e)
B 2 + s r B 3
(A.21)
(A.22)

A.2. GENERATING IMPULSE WITHIN A FRICTION CONE 119
r > j s j r < j s j
P d > (1 + e)P q
sliding
(1 + e)P q >P d >P q R-sticking R-reversed sliding
P q >P d C-sticking C-reversed sliding
Table A.1: Sliding modes classified by the parameters P d , P q , s ,and r . In addition, the
reversed sliding modes require that S 0 B 3 > 0.
R-sticking:
C-sticking:
R-reversed sliding:
C-reversed sliding:
where
P x = B 3P y , S 0
B 1
C 0
P y = ,(1 + e)
B 2 + s r B 3
P x = B 3P y , S 0
B 1
P y = ,(1 + e) B 1C 0 + B 3 S 0
B 1 B 2 , B 2 3
2S 0
P x = s r P y ,
B 3 + s r B 1
C 0
P y = ,(1 + e)
B 2 + s r B 3
2S 0
P x = s r P y ,
B 3 + s r B 1
1+e
P y = ,
C 0 + 2s rB 3 S 0
B 2 , s r B 3 B 3 + s r B 1
s =
S
0
jS 0j
if S 0 6=0
1 if S 0 =0
(A.23)
(A.24)
(A.25)
(A.26)
(A.27)
(A.28)
(A.29)
(A.30)
(A.31)
A.2 Generating impulse within a friction cone
The ability to generate any impulse within a friction cone in impulse space likely holds in
general, but I will only demonstrate it here under two assumptions appropriate for the sort
of manipulation in this thesis.
The first assumption is that the striker has no angular velocity when it collides with
the object. Thus the striker is controlled by two parameters: v s0 , the linear velocity of the
striker, and , the angle of incidence of the striker (measured CCW from the normal). The
initial compression and sliding velocities are then:
S 0 = v s sin
(A.32)

120 APPENDIX A. IMPACT ANALYSIS
C 0 = ,v s cos
(A.33)
This allows the striker velocity to be factored out from all the equations for impulse delivered
by an impact (equations A.21 through A.24). Consequently, each mode of impact can
be characterized by the range of values of Px
P y
that it can produce.
The second assumptions is:
the striker has infinite mass or infinite angular inertia, or
the COM of the striker always lies on the contact normal.
This serve the purpose of making the B i constant with respect to the angle of incidence
. The slopes of the line of sticking and the line of maximum compression then remain
constant, only moving up and down with changes in .
By varying the angle of incidence of the striker, a continuous range of impulse ratios
from , r to r can be generated, thus covering all the rays in the friction cone in impulse
space. Any particular point along a ray can be achieved by using the appropriate striker
velocity.
Although being able to generate any impulse in the friction cone in impulse space is
not too difficult to see by graphical reasoning (by taking advantage of Routh’s method),
I present the results of a more analytical approach here; these results are useful for determining
the actual striker velocities for a desired impulse.
A.2.1
General case
Table A.2 gives a summary of the impulse ratio ranges and impact modes for ranges of
under the assumption that reversed sliding does not occur, i.e. r > j s j. The collision
mode will be either sliding, R-sticking, or C-sticking. These expressions are the result of
applying the conditions and solutions in Section A.1.5 and in Table A.1.
Table A.3 gives the same information under the assumption that reversed sliding can
occur, i.e. r < j, s j. The collision mode will be either sliding, R-reversed sliding, or
C-reversed sliding. The behavior now depends upon whether B 3 is positive or negative
because S 0 B 3 must be positive in order for reversed sliding to occur. Although the impulse
ratio ranges were calculated from the impulse solutions in Section A.1.5 (originally from
Wang and Mason [63]), the expressions for the ranges for this case are somewhat different
than what Table A.1 (also from [63]) suggests.
In each case, the impulse ratio ranges, taken together, cover the entire friction cone,
from Px
=
P y
, r to Px
=
P y
r . In addition, there is always some angle of incidence beyond
which the magnitude of the impulse ratio does not increase. Note that the direction of the
impulse will in general be different from the angle of incidence of the striker.
In order to determine the striker velocities to generate a given impulse at a given contact
point, first determine the impulse ratio of the desired impulse. If j Px
P y
j > r , then this
impulse cannot be generated by an impact at this point. Next, determine whether reversed
sliding can occur, and use Table A.2 or A.3 as appropriate to determine which impact mode
is required to generate the desired impulse ratio. Then solve the corresponding equations
in Section A.1.5 for the initial compression and sliding velocities C 0 and S 0 . The striker
velocity v s0 and the angle of incidence can be solved using Equations A.32 and A.33.

mode
A.3. SPECIAL CASES 121
range
sliding tan ,1h,(1 r B1,B3
+ e)
B 2, rB 3i
R-sticking
tan
tan ,1h,(1 r B1,B3
+ e)
B 2, rB 3i
r B1,B3
tan ,1h,
B 2, rB 3i
0
C-sticking
,1h
C-sticking 0 tan
R-sticking
sliding
,1 rB1+B3
tan
B 2+ rB 3
tan ,1h(1 + e) rB1+B3
B 2+ r B 3i
rB1,B3
,1h,
B 2, rB 3i
rB 1+B 3
B 2+ rB 3i
B
tan ,1h(1 r B1+B3
+ e)
B 2+ rB 3i
P
P x
P y
x
P y
r Px
P y
B
B 3
+ rB1,B3
B 1 (1+e)B 1
3
B 1
3
B 1
, rB1+B3
(1+e)B 1
range
Px
P y
Px
P y
Px
P y
P x
P y
= r
= , r
B3
+ rB1,B3
B 1 (1+e)B 1
B3
B 1
B3
B 1
, rB1+B3
(1+e)B 1
, r
Table A.2: Generating impact in a friction cone in impulse space, sticking or sliding case.
This table shows the impact modes, ranges, and resulting Px
P y
ranges required to generate
an impulse in any part of the friction cone in impulse space.
A.2.2
Generalized central impacts
When B 3 =0, the mechanics of impact become much simpler. This occurs when (in addition
to the two assumptions of this section), the COM of the object lies on the contact
normal. The line of maximum compression becomes:
P y = ,C 0
(A.34)
B 2
and the line of sticking becomes:
P x = ,S 0
(A.35)
B 1
Like Tables A.2 and A.3 for the more general case, Table A.4 shows the relationship between
ranges and impulse ratio ranges for the different sliding modes. However, this situation
is far simpler than might appear from the table. So long as the desired impulse is within
the friction cone in impulse space, that impulse can be achieved by the initial compression
and sliding velocities:
C 0 = , B 2
1+e P y (A.36)
S 0 = ,B 1 P x
(A.37)
The striker velocity v s0 and the angle of incidence can again be solved using Equations
A.32 and A.33.
A.3 Special Cases
This section briefly looks at a number of specific instances of striker and striker/object
situations relevant to the experiments performed in this thesis.

122 APPENDIX A. IMPACT ANALYSIS
mode
B 3 > 0
range
P x
P y
range
P
sliding < 0
,1h x
=
P y
r
C-reversed sliding 0 < tan rB 1+B 3
B 2+ r B 3i
r > Px
e,1
P y
r e+1
R-reversed sliding tan ,1h(1 + e) B3+rB1
B 2+ rB 3i
tan ,1h(1 B3+r B1
+ e)
B 2+ r B 3i
e,1
r Px
e+1 P y
, r
sliding tan ,1h(1 + e) B3+rB1
B 2+ rB 3i
P
x
P y
= , r
mode
B 3 < 0
range
sliding tan ,1h(1 + e) B3,rB1
B 2+ rB 3i
tan ,1h(1 + e) B3,rB1
B 2+ rB 3i ,1h B
tan 3, rB 1
B 2+ rB 3i
R-reversed sliding
,1h B
C-reversed sliding tan
3, rB 1
B 2+ rB 3i
sliding 0 <
P
P x
P y
x
P y
range
= r
r Px
1,e
P y
r 1+e
1,e
< 0 r Px
1+e P y
> , r
P x
P y
= , r
Table A.3: Generating impact in a friction cone in impulse space, reversed sliding or sliding
case. This table shows the impact modes, ranges, and resulting Px
P y
ranges required to
generate an impulse in any part of the friction cone in impulse space, both when B 3 > 0
and when B 3 < 0.

A.3. SPECIAL CASES 123
mode
range
sliding tan ,1h, r (1 + e) B1
B 2i
R-sticking tan ,1h, r (1 + e) B1
B 2i
tan ,1h, B 1 r B 2i
P
P x
P y
range
x
P y
= r
r Px
P y
r
1+e
C-sticking tan ,1h, r
B 1
B 2i
0
r
1+e
Px
P y
0
C-sticking 0 tan ,1h r
B 1
B 2i
R-sticking tan ,1h r
B 1
B 2i
sliding
tan ,1h r (1 + e) B1
B 2i
0 Px
P y
, r
1+e
tan ,1h r (1 + e) B1
B 2i
, r
Px
1+e P y
, r
P
x
=
P y
, r
Table A.4: Generalized central impacts. This table shows the relationship between impulse
ratios and ranges for various impact modes under the assumption that B 3 =0. Reversed
sliding never occurs under this condition.
A.3.1
Spherical striker
With a spherical striker (I s = 2 5 M sR 2 s), the B i are:
B 1 = 7
2M s
+ 1 M + y2
I +
(A.38)
B 2 =
1 + 1 M s M + x2
I
B 3 = xy
I
(A.39)
(A.40)
A.3.2
Spherical striker and circular object
With a circular object (I = 1 2 MR2 ), the B i are:
B 1 = 7
2M s
+ 3 M
B 2 = 1 M s
+ 1 M
B 3 =0
(A.41)
(A.42)
(A.43)
A.3.3
Long thin rod striker
With a long thin rod striker with infinite angular inertia, the B i are:
B 1 =
1 M s
+ 1 M + y2
I
(A.44)

124 APPENDIX A. IMPACT ANALYSIS
B 2 =
1 M s
+ 1 M + x2
I +
B 3 = xy
I
(A.45)
(A.46)
A.3.4
Long thin rod striker with circular object
With a circular object (I = 1 2 MR2 ), the B i are:
B 1 = 1 M s
+ 3 M
(A.47)
B 2 = 1 M s
+ 1 M
B 3 =0
(A.48)
(A.49)

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