Trends in Grasping - Eric Paulos

Trends in Grasping - Eric Paulos

Nguyen observed that two point contacts with frictionat points P and Q form a planar force-closuregrasp if and only if the segment P Q, or QP , pointsstrictly out of and into the two friction cones of P andQ respectively. This important observation has beenexploited throughout the grasping literature and researchup to the present day. Nguyen's result can alsobe directly applied to the three-dimensional case withsoft-nger contacts 7 .Like Brost, Nguyen approached the synthesis problemby insisting that grasps require as little accuracyas possible. He constructed two-dimensionalgrasping regions using two point contacts with frictionby casting the problem into one of tting a twosidedcone cutting the two contact edges into twosegments of largest minimum length. Using similartechniques, he went on to describe algorithms forthree-dimensional grasps using two soft-nger contacts,three-dimensional grasps using three hard-ngercontacts 8 , two-dimensional grasps using four frictionlesscontacts, and three-dimensional grasps usingseven frictionless contacts. The complexity resultswere O(n) time to analyze whether a grasp with ngiven contacts was force-closure and O(n c c2 c ) time tosynthesize the n independent contact regions of an objectrequiring at least c contacts. The robust natureof Nguyen's solution using independent contact regionsmeans that ngers can be placed independently at anyposition in the regions and still satisfy a force-closuregrasp.6.2 Other Grasp Synthesis StrategiesAs mentioned in section 3, Mishra, Schwartz, andSharir included a grasp synthesis section their paper[MSS87] for polygonal/polyhedral objects. Usingpoint contacts without friction, they present a techniquefor constructing a grasp with O(n) ngers inO(n) time when n is the number of faces of the objectand the dimension of the object is xed.The construction is a two step process. First theyshow how to synthesis a point contact grip using O(n)ngers in O(n) time. This result comes from the factthat any object with n faces can be held at equilibriumby n point contacts, one at each face. In he next step,they describe how to reduce this grip to another withthe minimum number of ngers necessary for the graspusing portions of Steinitz's Theorem [Ste13].The reduction step takes O(n) time and involves7 Soft-nger contacts are contacts that can not only exertforces in any direction inside the friction cone of that objectbut also torques about the normal to the surface normal.8 Hard nger contacts are simply point contacts with friction.That is, they can exert forces in any direction that it within thefriction cone for that contact.choosing the correct coecients to achieve a positivelinear combination of a subset of the original contacts.They conclude with extending their construction techniqueto three dimensions yielding the same time complexities(for xed d).There are several problems with their technique.First, it is not useful in certain applications that requirethat the ngers be placed only in certain permissibleregions of the object. Also, there are no limitationsplaced on the forces that the ngers can exert.This means that large forces may be required by theresult of their algorithm that are not obtainable orare so large that they deform and/or break the objectand/or gripper. Finally, it is not robust to the uncertaintiesinvolved with real world multingered hands.Markensco and Papadimitriou [MNP90] devotethe majority of their work to showing the same resultsfor minimum number of nger contacts for variousgrasps and contact types. However, their analysisdoes provide some new research in that it considersobjects that include curved surfaces. In addition theyshow that for point contacts with friction, under themost relaxed assumptions three ngers are necessaryand sucient for force-closure in two-dimensions andfour ngers in three-dimensions. They also give interestingconstruction techniques for generalized objects 9including curved objects.Their grasp analysis and construction centers uponthe idea of maximal inscribed circles. They use thesecircles by choosing nger contact points either on orclose to points of contact of a maximal inscribed circlewith the boundary of the object. The use of maximalinscribed circles is not a new idea, in fact itwas rst utilized in a proof by Baker, Fortune, andGross [BFG85] in the context of stable grasps (notform-closure) of polygons.Throughout their analysis, they are forced to handlenumerous special cases. The most common specialcase occurs when the maximal circle touches theboundary of the object at only two points. Their proofruns through many of these weird special cases and ishence not elegant.For the three-dimensional case 10 , maximal inscribedspheres are used for their constructive proofsjust as maximal inscribed circles were used for the twodimensionalcase. Again we note that curved surfacesare considered, making this approach quite general forany two- or three-dimensional object.9 Object here means a closed, bounded, non-degenerate twodimensional body with a boundary that is connected and piecewisesmooth. Smooth is dened here to mean continuous andhaving continuous derivatives of any order10 Three-dimensional objects are bounded, closed, nondegenerate,and with piecewise smooth, connected boundaries4

When friction is considered, they show that twodimensionalform closure of any object (includingthe circle) can be achieved by three ngers (as opposedto four required without friction) and in threedimensions,form-closure of any object (even with anaxis of symmetry) is possible with four ngers (as opposedto seven). Since their proofs are all constructivein nature, they claim that the only dicult step in reformulatingtheir work into a grasp synthesis solverwould be to compute the maximal inscribed sphere ofthe object which they note can be carried out in lineartime in the number of faces of the object usingMegiddo's algorithm for linear programming [Meg80].7 Early Optimal Grasping TechniquesMarkensco and Papadimitriou [MP89] formulateand solve the problem of nding the optimal formclosuregrasp of any given polygon. They optimizethe grip with respect to minimizing the forces neededto balance the object's weight through friction. Inessence they minimize the worst-case forces needed tobalance any unit force acting on the center of gravityof the object. As we saw before, by minimizing thenecessary forces on the object, we avoid requiring largeforces that cause unnecessary stress and deformationon both the object and the robot hand. Also, thereare often force limitations in our actual system due tothe actuation of the joints in the hand, making this anatural constraint.They rst present an analytical method that usesthe metric from above to calculate the quality of a gripof a polygonal object by three ngers. In the secondpart of their paper, they attack the more dicult problemof optimizing the form-closure of a polygon whichthey do by solving a sequence of reductions using linearprogramming duality and some intricate plane geometry.This leads to an algorithm, that can rapidlyanalyze any reasonably complex object and computethe optimum grip by examining a nite number ofcases (growing as the cube of the number of sides ofthe polygon).In their analysis they run into special cases similarto those in [MNP90]. For two of those cases, concavevertices and parallel sides, they give special proceduresfor computing the optimum grip of a polygon withthree ngers. Although their results are more generalthan Nguyen's [Ngu88], the necessary consideration ofspecial cases, make this a somewhat messy solution.8 Robust Grasp Planning8.1 2D Polygons with 3 FingersPonce and Faverjon obtained excellent results usinga new approach to constructing three-nger forceclosuregrasps of polygonal objects [PF91]. They considera hand equipped with three hard-nger contactsfrom which they prove a sucient condition for forceclosuregrasps leading to a system of linear constraintsin the position of the ngers along the polygonal edges.Like Nguyen [Ngu88] for two nger grasps, they determinemaximal segments of the object boundarywhere the ngers can be positioned independentlywhile maintaining force-closure.The regions satisfying these constraints are foundby a projection algorithm based on linear parameterelimination accelerated by simplex techniques. Theysolve this problem of projecting linear convexes in n-dimensions along one of several coordinate axis directionswith linear programming, namely the simplexalgorithm. The search for a feasible vertex in theoriginal convex has to be done only once for all hyperplanes,thus making the simplex algorithm ecient.Once the contact regions are found they use a reasonablecriterion to maximize the minimum of thelengths of the three contact regions. However, in generalthere is not a unique solution to this problem, asfor suciently long edges, the size of the contact regionsdepends only on the size of the friction cones.They choose an additional criterion that attempts toplace the center of mass of the object at the center ofthe friction cones. This enables them to decrease theeect of gravity and internal forces during the motionof the robot.As we hinted at previously, the central idea behindtheir synthesis algorithm is that they translatethe problem of nding maximal independent contactregions into a linear programming problem with tenvariables that can be solved using the simplex algorithm.The function to maximize becomes simply aweighted combination of the two criteria: (1) the minimumof the lengths of the three intervals and (2) thethe position of the center of gravity in the frictioncones (at the center is maximal, hence optimal). Atypical result from this technique is shown in Figure 2.In principle their approach can be extended to morecomplicated problems such as the synthesis of three- orfour-nger grasps of three-dimensional objects. However,linear sucient conditions are dicult to nd forthese cases, and the number of variables and equationsinvolved make solving these problems much more dif-cult.5

Figure 2: An object and its best grasp using the techniquesfrom Ponce and Faverjon in 1991. The maximalindependent contact regions are shown as heavy lineswith security margin circles at the ends of the regions.8.2 2D Curved Objects with 2 FingersFaverjon and Ponce returned to the issue of computingforce-closure grasps in [FP91], considering theproblem of grasps on curved objects in two-dimensionsusing two-ngers with friction. The objects that theyconsidered were modeled by parametric curves, andthe force-closure grasps characterized by a systemsof polynomial constraints in the parameters of thesecurves. All conguration space regions satisfying theseconstraints were found by a numerical cell decompositionalgorithm based on curve tracing and continuationtechniques.As before, maximal object segments where ngerscan be positioned independently were found by optimizationwithin those grasp regions. This was an extensionof Nguyen's work and like Nguyen, they characterizetwo-nger force-closure grasps by the fact thatthe line joining the contact points must lie within thefriction cones at those points.As we have seen before in [PF91] and [Ngu88],the linear form of the force-closure constraints in thepolygonal case allowed for the design of fast and simplegeometric algorithms for the constructing stablegrasps. However, for smooth (and piecewise-smooth)curved objects, these constraints become highly nonlinear,calling for an approach of more algebraic natureto grasp synthesis.They begin their analysis by modeling the boundaryof the planar object to be grasped by a piecewisesmoothcollection of parametric curves. Each of thetwo contact points, lie on a parametric curve givenby (x i (u i ); y i (u i )) for i = 1; 2 and x i and y i are polynomialsin u i . For a given pair of curve segments,a grasp is completely determined by a pair (u 1 ; u 2 ).This pair takes values in the grasp conguration spacedened by the intervals of u 1 and u 2 . They thenre-write the friction cone conditions for each contactpoint for force-closure in algebraic terms. They observethat in the grasp conguration space the forceclosuregrasp regions are bounded by sets of algebraiccurves. To characterize the topology of the regionsand their bounding curves they use a numerical celldecomposition algorithm, whose output is a descriptionof the stable grasp regions, their bounding curves,and their adjacency relationships.During this step of the construction the most expensivepart of the algorithm is the computation of theextremal points and intersection points of the graspconguration space curves. Finding those featuresamounts to solving square systems of polynomial equations.For this they use the continuation method, aglobal numerical technique that can nd all solutionsof systems of polynomial equations having up to a fewthousand roots using [Mor87].Figure 3: The maximal rectangles found inside everyvalid region in the grasp space from work by Faverjonand Ponce in 1991.They next obtain maximal grasp rectangles withsides parallel to the coordinate axes since it is theserectangles that realize the local maximum size of thegrasp regions as desired. For each rectangle, they decomposethe four variables (the edges of the rectangle)into two embedded optimization process in twovariables. The resulting maximal area rectangle will6

have the maximal segments on the objects boundarywhere the ngers can be positioned independentlywhile maintaining force-closure. This results in a morerobust grasp planner, making implementation for areal robot with uncertainties possible. Typical resultsfrom this algorithm are shown in Figures 3{4.Figure 4: The ve grasps corresponding to the rectanglesof the previous gure.This method by Faverjon and Ponce is not withoutproblems and areas of possible improvement. Oneimportant area that is incomplete is in determining abound on the number of maximal rectangles withineach stable grasp region. This in turn would providebetter methods for generating starting points forthe corresponding optimization process since it wouldminimize the number of initial guesses. In addition,although their technique works well in practice, it hasnot been proven to yield all local maximal rectangles,making it incomplete. However, their approachis quite elegant and general. In principle, it can beextended to more complicated problems such as thesynthesis of three (or more) nger grasps of possiblythree-dimensional objects. However, as before, thenumber and degree of the equations involved makesuch an implementation dicult.8.3 3D Polyhedra with 3 and 4 FingersPonce, Sullivan, Boissonnat, and Merlet [PSBM93]extend the approach of [Ngu88], [PF91], and [FP91]to the synthesis of three- and four-nger force-closuregrasps of three-dimensional polyhedral objects againusing independent maximal grasp regions. They showthat for polyhedral objects, the sucient conditionsfor force-closure are linear in the unknown parameterswhich reduces the problem of computing force-closuregrasps of polyhedral objects to the problem of projectinga polytope onto some linear subspace.Their research uses the fact that a necessary conditionfor three points to form a force-closure grasp isthat there exists a point in the intersection of the planeformed by the three points with the double-sided frictioncone at these points. For four nger grasps theyderive results that will allow a stronger linear sucientcondition for three-nger force-closure grasps in thecase where gravity acts as a fourth nger. In the fournger case, they utilize the fact that a sucient conditionfor four-nger contact to be force-closure is thatthere exists a point in the intersection of the four openinternal friction cones with the tetrahedron formed bythese points.The sucient conditions of force-closure can bewritten as a set of linear inequalities in the positionof a point x 0 and the position of three or four contactpoints x i . For polyhedral objects, each contact can beparameterized linearly by two variables (u i ; v i ) whichspecify its position in the plane of the correspondingface. To obtain all the solutions (u i ; v i ) yielding forceclosuregrasps, they show that they must eliminatethe point x 0 . This results in a projection problem.The problem is to construct the orthogonal projectionof the d-polytope onto a k-dimensional subspaceof the original d-dimensional Euclidean space. A simpleimplementation of this projection algorithm takesO(tn) time where t is the size of the projection ofthe d-polytope and n is the number of intersectinghalf-spaces used to dene the polytope. They showthat this time complexity can be improved by preprocessingthe hyperplanes using a recent result byMatousek and Schwarzkopf [MS92]. In general theyshow that the projection algorithm for a d-polytopeonto a k-dimensional subspace can be computed intime and space of O(kn + t + ) where is any positiveconstant, t is the size of the projection of theoriginal d-polytope, and (roughly) 2 3 3.4Again because of uncertainties in robotics systemsthey attempt to minimize the sensitivity of a grasp topositioning errors. They achieve this by nding triplesof independent contact regions as. These regions, aswe saw before, are such that for any tuple of contactpoints chosen in them, the corresponding grasp isforce-closure. From before, we know that nding maximalindependent contact regions reduces to solving alinear programming problem, making their overall solutionquite nice.7

9 Optimal GraspsFerrari and Canny [FC92] address the problem ofcomputing optimal force-closure grasps of polygonalobjects by computing a maximal ball included in theconvex hull of the contact wrenches. Their notion of anoptimal grasp, called quality criteria, actually focusesaround two criteria that consider the total nger forceand the maximum nger force. They p dene the magnitudeof the wrench as kwk = kF k 2 + k k 2 withthe choice for being somewhat arbitrary. The rstof the quality criteria is concerned with nding thegrasp congurations that maximize the wrench, givenindependent force limits (i.e. that minimize the worstcaseforce applied at any point contact). The secondcriteria minimizes the sum of all applied forces. Sincethe magnitude of the force is proportional to the totalcurrent in the motors and ampliers, this second criteriaresults in the minimization of the power neededto actuate the gripper.Their major insight is that in the wrench space,the convex hull of the nger wrenches on the objecthave important geometrical interpretations. Namely,that their quality measure is simply the distance of thenearest facet of the convex hull from the origin. Thismeans that optimal contacts can be synthesized in thewrench space by simply computing the convex hullof candidate contacts, determining the facet of minimumdistance from the origin in the wrench space,and choosing the maximum of these distances. Thishas to be repeated for each side of a polygon wherethe number of possible congurations grows linearlyas the number of edges of the polygons while the computationof the convex hull of the primitive wrenchesin the wrench space takes constant time. This resultsin an O(n) time algorithm with n the size of the polygon.Essentially the quality criteria of the grasp isgiven by the value of the radius of the largest closedball centered in the origin of the wrench space, containedin the set of all the possible wrenches that canbe resisted by applying at most unit forces at the contactpoints. Their solution can also be applied to anythree-dimensional object.The advantage of this solution is that these qualitycriteria can be easily calculated for a wide variety ofobjects/parts in hopes of feeding these quality valuesto a planner that will choose optimal tool/gripper selectionfor each object/part in an assembly task. Themotivation for using dierent grippers, each of whichis suitable for a small subset of operations, along withan automated tool-changer, is the philosophy of RISCRobotics.RISC Robotics (Reduced Intricacy in Sensing andControl) is a class of robotics that relies on simple controland sensory information. Sensory information in aRISC Robotics system (1) require little or no processingtime (typically single bit digital I/O signals are used),(2) is inexpensive (thus making redundancy a feasibleoption), and (3) is highly accurate (typically on the orderof 70 microns). RISC Robotics follows many of thesame paradigms of RISC for computer architecture byusing simple tools to build up and perform more complicatedtasks. In terms of grasping, RISC Roboticsmoves away from multingered hands because of theirintricate design making them expensive, unreliable,computationally heavyweight, and dicult to controland create planners for. In general, a RISC Roboticssystem maintains a small set of simple, inexpensive,and accurate machines that can be easily interchangedand/or combined to perform more complex tasks.10 Future and Current WorkExciting new work is ongoing into many of the nonholonomicconstraints in grasping. Recent work by Li,Murray, and Sastry [LMS93] examines rolling ngercontacts and object sliding constraints. Related workin a recent paper by Chen and Burdick [CB93] investigatesnger gates on planar objects. Essentially, mostof the past grasping research was for static grasps,whereas new work looks at various motions that canbe done once an object is being held and exactly howto plan and control such operations. Some of the motivationfor this work comes from the observation thatthe grasp used for picking up a pencil is entirely differentfrom the one we use for writing, even thoughthe geometry remains the same.There is also still exciting work ongoing into graspplanning, in particular planning optimal grasps. Mostof the work in this areas that was presented in this paperhave come out of research within the past year. Inaddition, work into formalizing a synthesis techniqueto accompany the 1992 paper [FC92] is in process byMirtich and Canny.11 ConclusionWe have also seen that manipulation is complex,typically involving combinations of open and closedloop kinematic chains, non-holonomic constraints, redundantdegrees of freedom, and singularities. In additiontheir are non-linearities in the contact conditionsbetween soft ngers and grasped objects as wellas the actuator dynamics. As a result, many assumptionshad to be made to make the problem tractable.Yet, humans can perform such tasks easily suggestingthat a simple approach exists.In addition, an interesting insight made by8

Nguyen [Ngu88] and others, observed that larger frictioncones were produced when nger contacts wereat edges or vertices of an object. This explains whypeople grasp objects at the edges and corners and whythe contacting surface of human ngers need to be softrather than hard like the ngernails. However, thisfact has yet to be properly addressed and exploited inthe grasping community.Finally, research pointed out by Ferrari andCanny [FC92] as well as others, moves grasping towardsa new direction. This new research centersaround the recent movement away from anthropomorphichands towards specialized grippers and toolchangers.Such specialized tools are simpler in termsof kinematics and controllability, and far more accuratethan a human like hands. After all, the hand hasevolved over millions of years as an organ used as muchfor sensation and communication as for manipulation.In fact, for many manufacturing tasks the human handis less than ideal. When a mechanic starts to work ona machine, the rst thing he reaches for is his/her toolbox,with pliers, wrenches, tweezers and work glovesto help him/her nish the job.References[Asa79][BFG85]H. Asada. Studies on Prehension and Handlingby Robot Hands with Elastic Fingers.PhD thesis, Kyoto University, April 1979.B. Baker, S. Fortune, and E. Grosse. Stableprehension with three ngers. In ProceedingsSymp. Theory of Computing, pages114{120, 1985.[Bro88] Randy C. Brost. Automatic grasp planningin the presence of uncertainty. InternationalJournal of Robotics Research,7(1):3{17, 1988.[CB93]I-M. Chen and J. W. Burdick. A qualitativetest for n-nger force-closure grasps onplanar objects with applications to manipulationand nger gaits. In IEEE InternationalConference on Robotics and Automation,pages 814{820, 1993.[Cut89] Mark R. Cutkosky. On grasp choice,grasp models, and design of hands formanufacturing tasks. IEEE InternationalTransations on Robotics and Automation,3(3):814{820, June 1989.[FC92][FP91][HA77][Lak78][Erd86] Michael Erdmann. Using backprojectionsfor ne motion planning with uncertainty.International Journal of RoboticsResearch, pages 19{45, 1986.C. Ferrari and J.F. Canny. Planning optimalgrasps. In IEEE International Conferenceon Robotics and Automation, pages2290{2295, 1992.B. Faverjon and J. Ponce. On computingtwo-nger force closure grasps of curved 2dobjects. In IEEE International Conferenceon Robotics and Automation, pages 424{429, 1991.H. Hanafua and H. Asada. Stable prehensionby a robot hand with elastic ngers.In International Symposium of IndustrialRobotics, pages 361{368, 1977. (Tokyo).K. Lakshminarayana. Mechanics of formclosure. ASME Paper, 78-DET-32, 1978.[LMS93] Zexiang Li, Richard M. Murray, andS. Shankar Sastry. Robotics: Manipulationand Planning. Pre-print, 1993.[LMT84][Meg80][MNP90][Mor87]T. Lozano-Perez, M.T. Mason, and R.H.Taylor. Automatic synthesis of ne-motionstrategies for robots. International Journalof Robotics Research, pages 3{24, 1984.M. Megiddo. Linear algorithms for linearprogramming in d dimensions. In ProceedingsFOCS Conference, October 1980.Xanthippi Markensco, Lugun Ni, andChristos H. Papadimitriou. The geometryof grasping. International Journal ofRobotics Research, 9(1):61{74, 1990.A.P. Morgan. Solving Polynomial SystemsUsing Continuation for Engineering andScientic Problems. Prentice Hall, 1987.[MP89] Xanthippi Markensco and Christos H.Papadimitriou. Optimum grip of a polygon.International Journal of Robotics Research,8(2):17{29, 1989.[MS92][MSS87]J. Matousek and O. Schwarzkopf. Linearoptimization queries. In ACM Symposiumon Computational Geometry, pages 16{25,1992.B. Mishra, J. T. Schwartz, and M. Sharir.On the existence and synthesis of multingeredpositive grips. Algorithmica, 2:541{558, 1987.9

[Nap56]J. Napier. The prehensive movements ofthe human hand. In Journal of BoneJoint Surgery, volume 38B-4, pages 902{913, November 1956.[Ngu88] Van-Duc Nguyen. Constructing forceclosuregrasps. International Journal ofRobotics Research, 7(3):3{16, 1988.[Pes90][PF91]Michael A. Peshkin. Programmed compliancefor error corrective assembly. In IEEETransations on Robotics and Automation,pages 473{482, 1990.J. Ponce and B. Faverjon. On computingthree-nger force closure grasps of polyhedralobjects. In International Conferenceon Advanced Robotics, pages 1018{1023,1991.[PSBM93] Jean Ponce, Steven Sullivan, Jean-DanielBoissonat, and Jean-Pierre Merlet. Oncharacterizing and computing three- andfour-nger force-closure grasps of polyhedralobjects. In IEEE International Conferenceon Robotics and Automation, pages821{827, 1993.[Reu75] F. Reuleaux. Kinematics of machinery.Dover, 1875.[SR83] J.K. Salisbury and B. Roth. Kinematicand force analysis of articulated mechanicalhands. In Journal of Mechanisms,Transmission, and Automation in Design,pages 105:35{41, 1983.[Ste13] E. Steinitz. Bedingt konvergente reihenund konvexe systeme i. Journal ReineAngew. Mathematics, 143:128{175, 1913.[Ste86] E. Steinitz. Analysis of multingeredhands. International Journal of RoboticsResearch, 4(4):3{17, 1986.[Whi82] D.E. Whitney. Quasi-static assembly ofcompliantly supported rigid parts. ASMEJournal of Dynamic Systems Measurementand Control, pages 65{77, 1982.10

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