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Computer Science
Computer Science
A family of rigid body models: connections between quasistatic and - - PowerPoint PPT Presentation
A family of rigid body models: connections between quasistatic and - - PowerPoint PPT Presentation
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
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Computer Science
LIGA Tribology Test “Vehicle”
LIGA – German acronym for process for making micro- scale parts from metals, ceramics, and plastics. Typical dimensions are on the
- rder of
Sandia wants to understand function, efficiency, robustness before building. Optimal design.
mm 0.001 . 1 ±
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Computer Science
Micro-Machine Assembly
Pawl (2.3 mm) and washer (1.0 mm) subassembly. Pins (0.169 mm) in holes (0.165 mm). Need fixture to hold and align washer and pawl. Fixture should guarantee unique positions and orientations of parts.
Tweezers
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Computer Science
Pawl in Fixture
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Simulation of Pawl Insertion
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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
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Past Work in Quasistatic Multibody Systems
Mason, Quasistatic Pushing, 1982 - 1996
Brost, 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
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Computer Science
Hierarchical Family of Models
- Models range from pure geometric to
dynamic with contact compliance
- Required model “resolution” is dependent
- n 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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Computer Science
Components of a Dynamic Model
Newton-Euler Equation
Defines motion dynamics
Kinematic Constraints
Describe unilateral and bilateral constraints
Normal Complementarity
Prevents penetration and allows contact separation
Friction Law
Defines friction force behavior: Bounded magnitude Maximum Dissipation Leads to tangential complementarity Maintains rolling or allows transition from rolling to sliding
Quasistatic model: time-scale the Newton-Euler equation.
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Computer Science
Let be an element of and let be a given function in . Find such that:
Complementarity Problems
n
ℜ z ) (z w ⊥ w ≤ ≥ z z
n
ℜ b Rz w + = ⊥ w ≤ ≥ z
Linear Complementarity Problem of size 1. Given constants and , find such that:
R b z w z
n n
z w ℜ → ℜ : ) (
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Computer Science
Newton-Euler Equation Non-contact forces
- configuration
q
- generalized velocity
v
- symmetric, positive definite
inertia matrix
M
- non-contact generalized
forces
f
)) ( ), ( , ( ) ( )) ( ( t v t q t f t v t q M = &
) ( )) ( ( ) ( t v t q G t q = &
- Jacobian relating generalized
velocity and time rate of change of configuration
G dt dx x = &
where
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Kinematic Quantities at Contacts
N i t q t t q t t q t
io it in
,..., 1 )) ( , ( )) ( , ( )) ( , ( = ψ ψ ψ
Locally, C-space is represented as:
; )) ( , ( ≥ t q t
in
ψ N i , , 1 L =
q
i
t ˆ
i
n ˆ
in
ψ
it
ψ
Normal and tangential displacement functions:
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Computer Science
Normal Complementarity
T io it in i t
] [ ) ( λ λ λ λ =
it
λ
) ( ) , ( ≥ ⊥ ≤ t q t
n n
λ ψ
where
T in n
] [ L L ψ ψ =
T in n
] [ L L λ λ =
Define the contact force Normal Complementarity
i
n ˆ
i
t ˆ
i
λ
in
λ
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Computer Science
Dry Friction
Friction Slip Coulomb
) , (
io it ψ
ψ & &
Assume a maximum dissipation law
) ) , , ( ) , , ( min( arg ) , (
io io it it io it
v q t v q t λ ψ λ ψ λ λ & & + =
N i ,..., 1 = ∀
); ( ) , (
in i io it
λ µ λ λ ℑ ∈
where is the contact slip rate Slip Friction Linearized Coulomb Slip Friction
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Computer Science
Instantaneous-Time Dynamic Model
t T t
- t
v q t λ ψ λ λ ) , , ( min( arg ) , ( & =
) ) , , (
- T
- v
q t λ ψ & +
Non-contact forces
- t
t n n
W W W v q t f v M λ λ λ + + + = ) , , ( &
) , ( ≥ ⊥ ≤
n n
q t λ ψ
Gv q = &
) ( ) , (
n
- t
µλ λ λ ℑ ∈
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Computer Science
Scale the Times of the Input Functions
) ( ) , (
n
- t
µλ λ λ ℑ ∈
) ( ) ( ) ( t v q G t q = &
Scale the driving inputs. Replace with in the driving input functions.
)) ( ) , , ( ) ( ) , , ( min( arg )) ( ), ( ( t v q t t v q t t t
- T
- t
T t
- t
λ ε ψ λ ε ψ λ λ & & + = ) ( ) , ( ≥ ⊥ ≤ t q t
n n
λ ε ψ
) ( ) , ( ) ( ) , ( ) ( ) , ( ) , , ( ) ( ) ( t q t W t q t W t q t W v q t f t v q M
- t
t n n
λ λ λ ε + + + = &
t t ε
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Computer Science
) ~ ( ) ~ , ~ (
n
- t
λ µ λ λ ℑ ∈
- t
t n n
W W W v q f d v d M λ λ λ ε τ τ ε ~ ~ ~ ) ~ , ~ , ( ~
2
+ + + = ~ ) ~ , ( ≥ ⊥ ≤
n n
q λ τ ψ
) ( ) ( ~ t q q = τ
Change variables
t ε τ =
Time-Scaled Dynamic Model
) ( ) ( ~ t λ τ λ =
) ( ) ( ~
1 t
v v
−
= ε τ
Application of chain rule and algebra yields:
)) ~ , ~ , ( ~ ) ~ , ~ , ( ~ min( arg ) ~ , ~ ( v q d d v q d d
- T
- t
T t
- t
ε τ τ ψ λ ε τ τ ψ λ λ λ + =
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Computer Science
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
h x x d dx
l l
/ ) ( /
1 −
≈
+
τ
h
) ( l
l
x x τ =
l
τ
th
l
h v G q q
l l l 1 1
~ ~ ~
+ +
+ =
) ~ ( ) ~ , ~ (
1 1 n l
- l
t
λ µ λ λ ℑ ∈
+ +
1 1 2
~ ) ~ , ~ , ( ) ~ ~ (
+ +
+ = −
l l l
W v q f v v M λ ε τ ε ~ ) ~ (
1 1
≥ ⊥ ∂ ∂ + + ≤
+ + l n n l T n l n
h v W λ τ ψ ψ
)) ~ ( ) ~ ( ) ~ ( ) ~ min(( arg ) ~ , ~ (
1 1 1 1 1 1
τ ψ λ τ ψ λ λ λ ∂ ∂ + + ∂ ∂ + =
+ + + + + +
- l
T
- T
l
- t
l T t T l t l
- l
t
v W v W
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Computer Science
LCP Time-Stepping Problem
∂ ∂ ∂ ∂ + − − + − − =
+ + + + + + +
/ / / ~ ~ ~
2 1 1 1 1 2 1 1 1
τ ψ τ ψ ψ ε σ λ λ ε ζ ρ ρ
n n n l l f l n l T T f T n f n l l f l n
h hf Mv v E U E W W W W M ~ ~
1 1 1 1 1 1
≥ ⊥ ≤
+ + + + + + l l f l n l l f l n
σ λ λ ζ ρ ρ
h v G q q
l l l 1 1
~ ~ ~
+ +
+ =
Constraint Stabilization Kinematic Control
N N F B 6
F N B Size + + = 2 6
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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
T
mg f ] [ − =
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Time-Scaled Fence and Particle System
Dynamic Quasistatic Boundary
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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
) ~ ( ) ~ , ~ (
n
- t
λ µ λ λ ℑ ∈
) ~ ( ) ~ ( ) ~ ( ) ~ ( ) ~ , ~ , ~ (
1 1 1 1 1 1 1 + + + + + + +
+ =
l
- T
l
- l
t T l t l
- l
t l
v b v b v λ λ λ λ θ
) ~ ( ) ~ (
1 1
τ ψ ∂ ∂ + =
+ + t l T t l t
v W v b
) ~ , ~ , ~ ( min ) ~ (
1 1 1 1 * + + + +
=
l
- l
t l l
v v λ λ θ θ
) ~ ( ) ~ (
1 1
τ ψ ∂ ∂ + =
+ +
- l
T
- l
- v
W v b
Linear in
1
~ +
l
v
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 + + −
−
l l T
v v f θ
1
~ +
l
v ) ~ (
1 * + l
v θ
) ~ ( ~ . .
1 1 + + + l n l T n
v b v W t s
OPT :=
) ~ (
1 * + l
v θ
) ~ (
1 * +
−
l
v θ
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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
) ~ , ~ , ~ , ~ (
1 1 1 1 + + + + l
- l
t l n l
v λ λ λ
1
~ +
l
v
1
~ +
l n
λ
) ~ , ~ (
1 1 + + l
- l
t
λ λ
1
~ +
l n
λ
1
~ +
l
v
1
~ +
l n
λ
) ~ , ~ , ~ , ~ (
1 1 1 1 + + + + l
- l
t l n l
v λ λ λ
solves the model with quadratic friction cone.
2 2 1 1
~ ~
io it it l in i l it
ψ ψ ψ λ µ λ Δ Δ Δ + − =
+ +
2 2 1 1
~ ~
io it io l in i l io
ψ ψ ψ λ µ λ Δ Δ Δ + − =
+ +
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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
~ +
l
v
) ~ , ~ , ~ , ~ (
1 1 1 1 + + + + l
- l
t l n l
v λ λ λ
~
1 =
⇒
+ l
v d
| ; ~
1
= ∀ ≥
+ in l T in
i v d W ψ
~
1 ≥ + l T v
d f
Added motion does not decrease work
, | ; ~
1
> = ∀ =
+ in in l T it
i v d W λ ψ , | ; ~
1
> = ∀ =
+ in in l T io
i v d W λ ψ
Added motion does not change friction work. Added motion does not cause penetration
1
~ +
l
v d
1
~ +
l
v
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Example
Slip Friction 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
1 + l T i v
D
Linearize the limit curve at contact
Friction Impulse Relative Velocity
3 i
d
1 i
d
2 i
d
4 i
d
5 i
d
6 i
d
7 i
d
8 i
d
Limit Curve
,
1 1 1
≥ =
+ + + l i l i i l if
D p β β
ij
d
i
D
1 1 1
≥ − ⊥ ≤
+ + + l i T l in i l i
e p β µ λ
Boundary or Interior
1 1 1
≥ + ⊥ ≤
+ + + l i l T i l i
e v D λ β
Maximum Work
T
e ] 1 1 [ L =
where : i
ij
d
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Computer Science
Tangential Complementarity: Example
) (
1 1 1 1 1
≥ + ⊥ ≤
+ + + l i l T i l
v D λ β M 8 ,
1
≠ ∀ =
+
j
l j
β ) (
1 8 1 1 8
≥ + ⊥ ≤
+ + + l i l T i l
v D λ β
1 1 8 + + = l in i l
p µ β
Friction Impulse Relative Velocity
3 i
d
1 i
d
2 i
d
4 i
d
5 i
d
6 i
d
7 i
d
8 i
d
Limit Curve
8 1 1
) (
+ +
− =
l T i l i
v D λ
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+ − =
+ − + − +
2 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (
tR nR R n tR tR nR R n R RR tt RR tn R R tn RR nt RR nn R R nn R R nt R R nn R R nn tR tR nR R n
b b b a s c c I U I A A A A A A A A A s a a a
Instantaneous Rigid Body Dynamics in the Plane
≥ ⊥ ≤
− + − + tR tR nR R n tR tR nR R n
a s c c s a a a
R
U
- diagonal matrix of friction coefficients at rolling contacts
R R
N N Size 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
- ne (Anitescu and Potra).
Admissible Configurations
l
q
1 + l
q
2 + l
q
3 + l
q
Without Constraint Stabilization
Admissible Configurations
l
q
1 + l
q
2 + l
q
3 + l
q
With Constraint Stabilization Current implementation uses stabilization and the “path” algorithm (Munson and Ferris).
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Solution Non-uniqueness:
LCP Non-Convexity
b Rc a
n n
+ = ≥ ⊥ ≤
n n
c a
)) sin( ) (cos( 4 ) cos( 1
2
θ µ θ θ − + = J l m R
m g l b
ext
− = ) sin( 2 θ θ &
n
a
n
c
Two Solutions
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