Analysis of Cartesian Stiffness And Compliance With ... - helix

Analysis of Cartesian Stiffness
And Compliance With Applications
by
NAMIK CIBLAK
George W. Woodruff School of Mechanical Engineering
G e o r g i a I n s t i t u t e of T e c h n o l o g y
May 1998

What is Stiffness and Compliance ?
∆(configuration)
Stiffness
∆(load)
∆(load)
Compliance
∆(configuration)
1

Elastically Suspended Rigid Body Model
P´
loads
rigid
body
P´
loads
elastic system
elastic
connection
2

Robotic Examples
Cooperating robots, assembly robots
z
x
Insertion tasks
Grasp problems:
• stability
• shifting contacts
• object manipulation
3

Why is Stiffness Important ?
high
stiffness
→
greater
accuracy
low
stiffness
→
better interaction
with environment
4

Examples of High and Low Stiffness Devices
Actuators
Deformable
beams
RCC center
Base
Geometric center
A generic Stewart Platform device.
An example of high stiffness
parallel mechanism. Good
positional accuracy due to high
stiffness.
A Remote-Center-of-Compliance
(RCC) device. An example of a
compliant parallel mechanism. Low
stiffness facilitates insertion and
assembly tasks.
5

•
P
Screws: Twist
v P
: linear velocity : angular velocity
•
hω
Q
≡
ω PQ
v P
ω
•
P
ω
• Q
Chasle’s Theorem: The instantaneous velocity can be given as a
rotation about a certain axis plus a translation parallel to the axis.
T
^
P =
v P
ω
6

Screws: Wrench
f
m P
f
•
P
PQ
Q
≡
Poinsot’s Theorem: Any load on a body can be represented by a force along a
certain axis plus a moment parallel to the axis.
^
W P
=
h f
• Q
•
P
f
m P
7

Screw Axis and Pitch
ha
h: pitch
b P
a
a
•
P
•
screw axis
On the screw axis,
• displacements and velocities are similar to the kinematics of a simple screw.
• loads are similar to the force and torque relation for a simple screw.
8

n-Systems of Screws
All screws in an n-system can be given as linear
combinations of n basis elements.
A 2-system
9

Principal Screws & Center of 3-Systems
There exists a special basis for 3-systems whose elements
are called the principal screws (Ball).
A 3-system
10

Free-Vectors (Infinite Pitch Screws)
Infinite pitch screw
Pure translation
Infinite pitch screw
Pure moment
11

Line-Vectors (Zero Pitch Screws)
line of action
ω
ω
zero pitch screw
Pure rotation
f
f
zero pitch screw
Pure force
12

Line-Vector Subspaces
pencil
bundle
❂ Examples of bundles: Set of all rotations or forces through a point.
❂ 3-D case is not unique. Every point in space generates a bundle.
13

Spatial Cross Product
Screw space admits a vector multiplication called the
spatial cross product operation.
^
S 1
a
× 1
S ^ a ×
1
2 = × =
b 2
b 1
a 2
b 1
0
× a 1 ×
a 2
b 2
14

Definitions of Stiffness and Compliance
^
δq : infinitesimal spatial displacement (twist).
^
dW: infinitesimal spatial force (wrench).
stiffness
^ ^ ^
dW = K δq
6x1 6x6 6x1
compliance
^ ^ ^
dq = C δW
6x1 6x6 6x1
The 3x3 submatrices of stiffness and compliance
^
K =
A
B T
B
C
^
C =
D
E
E T
C
15

Eigenvalue (EV) Problems
❂ Complicated physical phenomena may be explained in terms that make sense
to human mind.
❂ Geometric and constitutive contents are separated.
^ a
K = λ
b
twist
wrench
a
b
twist
not meaningful
16

Free-Vector EV Problems (Lipkin and Patterson)
^ f
C =
τ
wrench
a f
f
0
parallel
translation
^ δ 0
K = k γ
γ γ
twist
parallel
couple
eigenwrench
^
W fi
^
T γi
eigentwist
parallel
translation
f i
γ i
parallel
couple
(Dimentberg’s problem)
(Correct dualization)
• There exist three eigentwists and three eigenwrenches.
• Eigenvalues are the stationary values.
17

Free-Vector Decompositions
K^ =
f
τ
k f
0
γ 0
0
k γ
f
τ
0
γ
T
a f
f δ
C^ =
0 γ 0
0
a γ
f δ
0 γ
T
geometric
constitutive
geometric
18

Center of Elasticity
The centers of the eigentwist and eigenwrench 3-systems
coincide (Lipkin and Patterson).
19

Coplanar Perpendicular Vectors
f 2
f 3
r f2
E
r f3
E
E
r f1
E
f 1
Lipkin and Patterson:
Σ r E
fi
=
Σ r E
γ i
=
0
20

Location of Center of Elasticity
f i
r fi
E
E
r fi
r E
O
r E
= 1 Σ 2 r ( fi r E
= 1 Σ 2 r ) γ i
21

Centers of Stiffness And Compliance
Loncaric:
B T = B
B T ≠ B
O
S
(Center of Stiffness)
E T ≠ E
C
E T = E
(Center of Compliance)
^
K =
A
B T
B
C
^
C =
D
E
E T
C
22

Constitutive Character of the Centers of
Stiffness and Compliance
Σ k r S fi fi = 0
Σk r C γ i γ i = 0
23

Line-Vector EV Problems
^ n
C =
m
wrench
a m
0
m
parallel
rotation
^ t
K =
w
twist
t
k t 0
parallel
force
co-eigenwrench at G
co-eigentwist at G
force
G
couple
at G
parallel
rotation
at G
parallel
force
at G
rotation
G
translation
at G
• Every point G in space generates a distinct second EV problem.
• There exist three co-eigentwists and three co-eigenwrenches for every G.
24

Line-Vector Decompositions
^
K G =
n
m
k m
t
0 0
0
k t
n
m
t
0
T
^
C G =
0
m
t a m 0 0
w 0 a t m
t
w
T
geometric
constitutive
geometric
Every generator G yields a distinct decomposition.
25

Co-Center of Elasticity
Symmetric A -1 B (E T F -1 ) ⇔ Center of Elasticity
Symmetric C -1 B T (ED -1 ) ⇔ Co-center of Elasticity
^
K =
A
B T
B
C
^
C =
D
E
E T
C
26

Co-Centers of Elasticity
C a
C b
G b
G d
C c
G a
G c
G e
C e
C d
C=G ⇒ E c
• There exists at least one co-center for every stiffness (compliance).
• There may be more than one co-centers.
27

Distribution of Co-centers
General case
E
Special cases
E
28

Compliant Axes
^
W fi
pure force
^
T γ i
pure rotation
h fi
=0 h γi
=0
f i
γ i
parallel
translation
parallel
couple
29

Relations Between Centers
^
W fi
^
( T γi
)
^
W fi
^
( T γi
)
E (E c )
⇔
S (C)
S
E
E c
C
a compliant axis
Two or three
compliant axes exist
⇒
E
E c
C
S
30

Generalization of Compliance Axes
G
t
f
generalized
force-translation
axis
⇔
G
γ
m
generalized
rotation-couple
axis
G
f
i
t
i
force-translation
axis
⇔
G
γ
i
m
i
rotation-couple
axis
31

A New Pair of Special Axes
pure force
(E*
cW/B )
h f = 0
f
force-rotation
axis
w
pure rotation
(E cT/A )
h w = 0
rotation-force
axis
32

Asymmetric Stiffness: Parallel Line-Springs
Model:
P
W
^
P ’
F’
B’ i B’ n
B’ 1
s i
l i
, k i
A 1
A i A n
O F
F’
F
Finite displacements are considered.
33

Stiffness Matrix of Line Springs
^ ^ ^
K = Σ
T ^
k i [ ρ i S i S i + (1-ρ i )M i ]
ρ i = l 0i
l i
For all springs are unextended (ρ i = 1):
^
^ ^
K = Σ k i S i S i
T
34

Skew-symmetric Property
^ ^ 1 ^
2
K = K sym - W ×
W^
net
external
load
35

Unloaded Equilibrium
The stiffness matrix of parallel line springs is symmetric
if and only if
the system is in an unloaded equilibrium
36

Model:
Asymmetric Stiffness: Parallel Torsional Springs
B’ 1
s i
F’ P’
B’ i B’ n
W
^
P’
cylindrical
joint
torsional
spring
A 1 l i , k i ,
A i
A n
F
θ i
Hook’s joint
O
• Stiffness matrix is determined in closed form.
• Skew-symmetric part is not a simple function of the applied wrench.
• Symmetry theorem does not apply.
• If all springs are unextended then
^
^ ^
K = Σ k i S i S i
T
37

Isotropic Vector Problem
Eigenvalue
problem
λ
Isotropic vector
problem
Au = w u T w = 0
38

Orthonormal Basis of Isotropic Vectors
U T AU =
0 x x
x 0 x
x x 0
...
...
...
...
...
...
A has a complete orthonormal set of isotropic vectors
if and only if
trace(A) = 0
3D Example:
c
a
a
b b
c
shear stress
σ 3
σ 2
σ 1
normal
stress
Pure shear state for deviatoric stress case
39

Synthesis of Stiffness by Springs
^
r = rank(K )
^
K = PP T
synthesis by
r springs
find an orthonormal set of r
isotropic vectors of P T ∆P
A pos. def. rank r stiffness can be realized by r springs
if and only if
the off-diagonals of the stiffness have zero trace
40

Results for the Synthesis Problem
❂ An algorithm is developed for synthesis by r springs.
❂ A method is developed for synthesis by n > r springs.
❂ Free-vector decomposition is applied. In general, 3 line and 3
torsional springs result. (free-vector synthesis)
❂ Line-vector decomposition is applied. In general, 6 line
springs result. (line-vector synthesis)
❂ Numerical results support the theory.
41

Remote Center of Compliance (RCC) Device
Deformable
beams
parallel
translation
parallel
rotation
RCC center
contact
force
contact
couple
Geometric center
Three compliant axis through RCC center. Combined centers case.
42

Design Equations for RCC Device
Earlier attempts at solving the RCC location and stiffness were not very
successful due to approximations (Whitney and co-workers).
Conical symmetry
Using conical symmetry and centers, the location of RCC and stiffnesses
are determined without approximations.
25
σ = 250
projection
ratio
20
15
10
5
σ = 50
0
0
5
10
15
20
25
30
cone angle
43

Rotational Symmetry Devices
E 3
S 3 S 2
connection
Serial or parallel
E 1
E 2
• The centers of elasticity, stiffness, compliance coalesce on the vertical axis.
• A co-center coincides with the combined center.
• Leads to generalized RCC definitions and classification.
44

All possible RCC Devices
→ ∞
Conical RCC (parallel)
Cylindrical RCC (parallel)
Planar RCC (parallel)
→ ∞
Conical RCC (serial)
Cylindrical RCC (serial)
Planar RCC (serial)
45

An RCC Construction Proposal
Generalization of Loncaric’s proposal:
RCC
center
A Theoretical Confirmation
generator
Exact stiffness is recovered by using
infinitely many beams
θ
k axial =
G r 4
4 R 3
46

Analysis of Spatial Mass of a Rigid Body
For impact problems:
^ ^ ^
W = M A
wrench
mass
matrix
spatial
acceleration
(twist)
^
M =
mI 0
0 J
6x6
at center of mass
• All the results of this study also applies to the mass matrix.
• Mass matrix is analogous to stiffness matrix.
• Center of mass is analogous to the combined centers case.
47

Eigenscrew Structure at Center of Mass
eigenwrench
a pure force
in any
direction
co-eigentwist
at
a pure
translation
in any
direction
eigentwist
a pure rotation
in a principal
direction
of inertia
co-eigenwrench
at
a pure couple
in a principal
direction of
inertia
48

Eigenscrew Structure away from Center of Mass
pure
rotation
pure
translation
pure
rotation
pure
rotation
pure
translation
pure
rotation
principal inertia directions
pure
force
pure
couple
pure
forces
pure
force
principal inertia directions
49

Center of Percussion
pure force
through CP
due to impact
pure translation
due to force
through CM
pure rotation
due to couple
at CM
resulting
pure rotation
through CG
center of grip
center
of
mass
center of
percussion
CP
Pure force through the CP results in a pure rotation: a force-rotation axis
50

Axes and Joints of Percussion
pure
force
a zero pitch
co-eigentwist
direction
pure
rotation
Similar procedure applies for the co-eigenwrench case
51

A Better Design Possibility for Golf Clubs
two revolute joints
minimum sting
for forces in the plane
plane of double principal inertia
52

Principal Contributions - Theory
❂ Closed form equations for the elastic center location are found.
❂ Physical and geometrical meanings of the centers of stiffness and
compliance are explained.
❂ Geometric relations between the centers are established using compliant
axes.
❂ A new set of eigenvalue problems for stiffness is proposed and solved.
❂ New decompositions of stiffness and compliance are determined.
❂ The free- and line-vector EV problems lead to generalized compliant axes
and a better classification of stiffnesses.
❂ Previously unknown special axes are predicted.
❂ Stiffness matrices of line- and torsional spring systems are found in closed
form. The asymmetry observed in other studies is explained.
❂ The skew-symmetric part of line-spring stiffness is equal to minus onehalf
the applied load in spatial cross-product form.
53

Principal Contributions - Applications
❂ The isotropic vector problem is proposed and solved. An algorithm is
developed to construct orthonormal sets of isotropic vectors.
❂ Using isotropic vectors, the stiffness synthesis by springs problem is fully
solved.
❂ Free-vector and line-vector decompositions are applied to the synthesis
problem leading to minimum syntheses.
❂ Design equations for RCC devices are determined.
❂ Rotational symmetry devices are proposed as generalized RCCs. Design
equations are found.
❂ The theory of stiffness is applied to the spatial mass matrix. The eigenand
co-eigenscrew structure is fully determined.
❂ The mass matrix results explains and generalizes the concept of
percussion center.
❂ Combining the stiffness and mass matrix results, the necessary and
sufficient condition for special free-vibration modes are found.
54