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Computer Science
A family of rigid body models: connections between quasistatic and
dynamic multibody systems
Jeff Trinkle
Computer Science Department
Rensselaer Polytechnic Institute
Troy, NY 12180
Jong-Shi Pang, Steve Berard, Guanfeng Liu
Computer Science
Motivation
Valid quasistatic plan exists
No quasistatic plan found, but dynamic plan exists
Dexterous Manipulation Planning
Part enters cg down
Part enters cg up
Parts Feeder Design
Parts feeder design goals:
1) Exit orientation independent of entering orientation
2) High throughput
Design geometry of feeder to guarantee 1) and maximize 2).
Feeder geometry has 12 design parameters
Evaluate feeder design via simulation
Computer Science
Simulation of Pawl Insertion
Computer Science
Past Work in Quasistatic Multibody Systems
Grasping and Walking Machines – late 1970s.Used quasistatic models with assumed contact states.
Whtney, “Quasistatic Assembly of Compliantly Supported Rigid Parts,” ASME DSMC, 1982
Caine, Quasistatic Assembly, 1982
Peshkin, Sanderson, Quasistatic Planar Sliding, 1986
Cutkosky, Kao, “Computing and Controlling Compliance in Robot Hands,” IEEE TRA, 1989
Kao, Cutkosky, “Quasistatic Manipulation with Compliance and Sliding,” IJRR, 1992
Peshkin, Schimmels, Force-Guided Assembly, 1992
Computer Science
Past Work in Quasistatic Multibody Systems
Mason, Quasistatic Pushing, 1982 - 1996Brost, Goldberg, Erdmann, Zumel, Lynch, Wang
Trinkle, Hunter, Ram , Farahat, Stiller, Ang, Pang, Lo, Yeap, Han, Berard, 1991 – present
Trinkle Zeng, “Prediction of Quasistatic Planar Motion of a Contacted Rigid Body,” IEEE TRA, 1995
Pang, Trinkle, Lo, “A Complementarity Approach to a Quasistatic Rigid Body Motion Problem,” COAP 1996
Computer Science
Hierarchical Family of Models
• Models range from pure geometric to dynamic with contact compliance
• Required model “resolution” is dependent on design or planning task
• Approach:– Plan with low resolution model first
– Use low resolution results to speed planning with high resolution model
– Repeat until plan/design with required accuracy is achieved
Model Space
Rigid Compliant
Dynamic
Quasistatic
Geometric
Kinematic
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Components of a Dynamic Model
Newton-Euler EquationDefines motion dynamics
Kinematic ConstraintsDescribe unilateral and bilateral constraints
Normal ComplementarityPrevents penetration and allows contact separation
Friction LawDefines friction force behavior:
Bounded magnitudeMaximum Dissipation
Leads to tangential complementarity
Maintains rolling or allows transition
from rolling to sliding
Quasistatic model: time-scale the Newton-Euler equation.
Computer Science
Let be an element of and
let be a given function in . Find such that:
Complementarity Problems
nz)(zw
w0 0zzn
bRzw w0 0z
Linear Complementarity Problem of size 1.
Given constants and , find such that:R b zw
z
nnzw :)(
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Newton-Euler EquationNon-contact
forces- configurationq
- generalized velocityv
- symmetric, positive definite inertia matrix
M
- non-contact generalized forces
f
))(),(,()())(( tvtqtftvtqM
)())(()( tvtqGtq
- Jacobian relating generalized velocity and time rate of change of configuration
G
dt
dxx where
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Kinematic Quantities at Contacts
Ni
tqt
tqt
tqt
io
it
in
,...,1
))(,(
))(,(
))(,(
Locally, C-space is represented as:
;0))(,( tqtin Ni ,,1
q
it̂ in̂
init
Normal and tangential displacement functions:
Computer Science
Normal Complementarity
Tioitini t ][)(
it
0)(),(0 tqt nn
whereT
inn ][ T
inn ][
Define the contact force
Normal Complementarity
in̂it̂
i
in
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Dry Friction
Friction
Slip
Coulomb
),( ioit
Assume a maximum dissipation law
)),,(),,(min(arg),( ioioititioit vqtvqt
Ni ,...,1);(),( iniioit where
is the contact slip rate
Slip
Friction
Linearized Coulomb
Slip
Friction
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Instantaneous-Time Dynamic Model
tTtot vqt ),,(min(arg),(
)),,( oTo vqt
Non-contact forces
oottnn WWWvqtfvM ),,(
0),(0 nn qt
Gvq
)(),( not
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Scale the Times of the Input Functions
)(),( not
)()()( tvqGtq
Scale the driving inputs. Replace with in the driving input functions.
))(),,()(),,(min(arg))(),(( tvqttvqttt oTot
Ttot
0)(),(0 tqt nn
)(),()(),()(),(),,()()( tqtWtqtWtqtWvqtftvqM oottnn
t t
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)~
()~
,~
( not
oottnn WWWvqfd
vdM
~~~
)~,~,(~
2
0~
)~,(0 nn q
)()(~ tqq Change variables
t
Time-Scaled Dynamic Model
)()(~
t )()(~ 1 tvv
Application of chain rule and algebra yields:
))~,~,(~
)~,~,(~
min(arg)~
,~
( vqd
dvq
d
d oTo
tTtot
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Approximate derivatives by:
where is the time step, , and is the
scaled time at which the state of the system was obtained.
Time Stepping Methods
hxxddx ll /)(/ 1 h )( l
l xx lthl
hvGqq lll 11 ~~~
)~
()~
,~
( 11n
lo
lt
112 ~)~,~,()~~( lll WvqfvvM
0~
)~(0 11
ln
nlTn
ln hvW
))~()~
()~()~
min((arg)~
,~
( 111111
olTo
Tlo
tlTt
Tlt
lo
lt vWvW
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LCP Time-Stepping Problem
0
/
//~
~~
00
00
000
00 2
1
1
1
12
1
1
1
n
nn
l
lf
ln
l
T
Tf
Tn
fn
l
lf
ln h
hfMvv
EU
EW
W
WWM
0~
~
01
1
1
1
1
1
l
lf
ln
l
lf
ln
hvGqq lll 11 ~~~
Constraint Stabilization Kinematic
Control
N NFB6
FNBSize 26
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Assume:
Particle is constrained from below
Non-contact force:
Fence is position-controlled
Wall is fixed in place
Expected motion:
Quasistatic: no motion when not in contact with fence.
Dynamic: if deceleration of paddle is large, then particle can continue sliding without fence contact
Example: Fence and Particle
Tmgf ]00[
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Time-Scaled Fence and Particle System
Dynamic
QuasistaticBoundary
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Time-Scaled Fence and Particle System
Dynamic
Quasistatic
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Introduce the friction work rate value function:
Cast Model as Convex Optimization Problem
)~
()~
,~
( not
)~()~
()~()~
()~
,~
,~( 1111111 lo
Tlo
lt
Tlt
lo
lt
l vbvbv
)~()~( 11
tlTt
lt vWvb
)~
,~
,~(min)~( 1111* lo
lt
ll vv
)~()~( 11
olTo
lo vWvb
Linear in1~ lv
Introduce the friction work rate minimum value function:
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Hypograph of is convex. Therefore is
concave and is convex.
KKT conditions are exactly the discrete-time model.
Equivalent Convex Optimization Problem
)~(~min 1*1 llT vvf 1~ lv
)~( 1* lv
)~(~.. 11 ln
lTn vbvWts
OPT :=
)~( 1* lv)~( 1* lv
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If solves the model with quadratic friction
cone, then is a globally optimal solutions of OPT corresponding
to . Conversely, if is a globally optimal solution to OPT for
a given and if is equal to an optimal KKT multiplier of the
constraint in OPT, then defining as below, the tuple
Theorem
)~
,~
,~
,~( 1111 lo
lt
ln
lv 1~ lv
1~ ln
)~
,~
( 11 lo
lt
1~ ln
1~ lv1~ l
n
)~
,~
,~
,~( 1111 lo
lt
ln
lv solves the model with quadratic friction cone.
22
11 ~~
ioit
itlini
lit
22
11 ~~
ioit
iolini
lio
Computer Science
where is a small change in
Corresponding to the solution of the
discrete-time model with quadratic friction cone, is the unique
solution of OPT, if and only if the following implication holds:
Proposition: Solution Uniqueness
1~ lv)
~,
~,
~,~( 1111 l
ol
tl
nlv
0~ 1 lvd
0|;0~ 1 in
lTin ivdW
0~ 1 lT vdfAdded motion does not decrease work
0,0|;0~ 1 inin
lTit ivdW
0,0|;0~ 1 inin
lTio ivdW
Added motion does not change friction work.
Added motion does not cause penetration
1~ lvd 1~ lv
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Example
SlipFriction
Solution is unique with non-zero quadratic friction on plane
Solution is not unique without friction
Solution is not unique with linearized friction on plane
Friction Slip
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Future Work
Convergence analysis
Experimental validation
Design applications
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Fini
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where the columns of are the vectors transformed into C-space.
is the vector of the components of relative velocity at the contact in the directions.
Maximum Work Inequalty: Unilateral Constraints
1lTi vD
Linearize the limit curve at contact
Friction Impulse
Relative Velocity
3id
1id
2id
4id
5id
6id
7id8id
Limit Curve0, 111 l
ilii
lif Dp
ijd
iD
00 111 li
Tlini
li ep
Boundary or Interior
00 111 li
lTi
li evD Maximum Work
Te ]11[ where
:i
ijd
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Tangential Complementarity: Example
0)(0 11
111 l
ilT
il vD
8,01 jlj
0)(0 18
118 l
ilT
il vD
118
lini
l p
Friction Impulse
Relative Velocity
3id
1id
2id
4id
5id
6id
7id8id
Limit Curve
811 )( lT
ili vD
Computer Science
0020
)()()(
0)()()(
0)()()(
tR
nR
Rn
tR
tR
nR
Rn
R
RRttRRtnRRtn
RRntRRnnRRnn
RRntRRnnRRnn
tR
tR
nR
Rn
b
b
b
a
s
c
c
IU
IAAA
AAA
AAA
s
a
a
a
Instantaneous Rigid Body Dynamics in the Plane
00
tR
tR
nR
Rn
tR
tR
nR
Rn
a
s
c
c
s
a
a
a
RU - diagonal matrix of friction coefficients at rolling contacts
RR NNSize 3
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Example: Sphere initially translating on horizontal plane.
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Simulation with Unilateral and Bilateral Constraints
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Time-Stepping with Unilateral Constraints
Solution always exists and Lemke’s algorithm can compute
one (Anitescu and Potra).
Admissible Configurations
lq
1lq
2lq
3lq
Without Constraint Stabilization
Admissible Configurations
lq
1lq
2lq
3lq
With Constraint Stabilization
Current implementation uses stabilization and the “path”
algorithm (Munson and Ferris).
Computer Science Solution Non-uniqueness:LCP Non-Convexity
bRca nn 00 nn ca
))sin()(cos(4
)cos(1 2
J
l
mR
m
glb ext )sin(
2
na
nc
Two Solutions
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Solution Non-Uniqueness:Contact Force Null Space
Both friction cones can “see” the other contact point.
Assume:Blue discs are fixed in space
Red disc is initially at rest
Solution 1 – disc remains at rest
Solution 2 – disc accelerates downward
External Load
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