What's Wrong with Collision Detection in Multibody ...willij16/ICRA2013_presentation.pdf ·...

RensselaerPolytechnic

Institute

Daniel MontralloFlickinger,Jedediyah

Williams, Jeffrey CTrinkle

IntroductionBackground

The Problem

FormulationsGeometric Constraints

Stewart-Trinkle

Polyhedral Exact Geometry

ResultsSawtooth Benchmark

Hills Benchmark

Slender Rod Benchmark

What’s Wrong with Collision Detectionin Multibody Dynamics Simulation?

Daniel Montrallo Flickinger Jedediyah Williams Jeffrey C Trinkle

CS Robotics Laboratory


Institute




The Problem


Stewart-Trinkle



Hills Benchmark


Multibody Dynamics Simulation

Multibody System Dynamics involves the modeling and simulation of systems ofinterconnected bodies

Why should I care? So what is wrong with it?


Institute




The Problem


Stewart-Trinkle



Hills Benchmark


Why should I care?

Dynamic simulation is important ...

Ï for robotics and controls

Ï for designing complex machinery

Ï for virtual reality

Ï for the entertainment industry

Demand for high fidelity dynamic simulation is high as system complexityincreases under limited computational resources.

So what? I still don’t care Okay, that makes sense, but what is wrong with it?


Institute




The Problem


Stewart-Trinkle



Hills Benchmark


What’s wrong with collision detection in multibody dynamicssimulation

Ï Low fidelity

Ï Slow performance

Ï Tuning required (“magic numbers”)

Ï It’s slow

Why should I care? How do you solve it then?


Institute




The Problem


Stewart-Trinkle



Hills Benchmark


How we solve the problems with collision detection

P

1

23

4

ψ1n =ψ2n

Ï Formulate constraints that are geometrically accurate: the Polyhedral ExactGeometry formulation

Ï Devise efficient collision detection algorithms and complementarity systemsolvers

What are constraint formulations? Tell me more about PEG


Institute




The Problem


Stewart-Trinkle



Hills Benchmark


What are constraint formulations?

M

ψ

p1

p2

Ï Gap distance φ is calculated for each pair ofgeometric features

Ï Unilateral constraint equations constructed foreach gap distance

What are the complementarity conditions? What is the Stewart-Trinkle formulation?

What is the Polyhedral Exact Geometry formulation? I don’t care, just show me some pretty graphs


Institute




The Problem


Stewart-Trinkle



Hills Benchmark


Complementarity Constraints

M(q, t)ν̇=λvp(q, q̇, t)+λapp, (1)

where λvp is the sum of velocity dependent forces, and λapp are the applied forces.Complementarity constraint:

0 ≤λl+1n ⊥ GT

nνl+1 +Ψ

ln

h+ ∂Ψl

n

∂t≥ 0, (2)

where λn are forces normal to contact surfaces,Ψn are gap functions, or distancesbetween active bodies and contact surfaces, h is the step size, ν are the velocitiesof the active bodies, and Gn is a normal contact wrench.

What is the Stewart-Trinkle formulation? What is the Polyhedral Exact Geometry formulation?


Institute




The Problem


Stewart-Trinkle



Hills Benchmark


Stewart-Trinkle formulation

ψminimum

ε

t1

t2 t3

t4

Show me the Polyhedral Exact Geometry formulation So let’s see the equations


Institute




The Problem


Stewart-Trinkle



Hills Benchmark


Stewart-Trinkle formulation, mixed linear complementaritysystem

Equations of motion and complementarity equations (1) and (2) are formulatedas a mixed LCP: ∣∣∣∣ 0

ρl+1n

∣∣∣∣= ∣∣∣∣ M −Gn

GTn 0

∣∣∣∣ ∣∣∣∣νl+1

pl+1n

∣∣∣∣+ ∣∣∣∣−Mνl −plext

ψln/h

∣∣∣∣ , (3)

where M is the inertia matrix, ν is the generalized velocities, pn and pext arenormal and external impulsive forces, and h is the step size.Normal contact wrench:

Gnij =[

n̂ij

rij × n̂ij

], (4)

Show me the Polyhedral Exact Geometry formulation Jump right to the PEG equations


Institute




The Problem


Stewart-Trinkle



Hills Benchmark


Polyhedral Exact Geometry formulation

ε

t1

t2

t3

So let’s see the equations That’s cool, what are the results? What is the Stewart-Trinkle formulation again?


Institute




The Problem


Stewart-Trinkle



Hills Benchmark


Polyhedral Exact Geometry formulation, mixed linearcomplementarity system

The Stewart-Trinkle forumation in (3) is altered to introduce a heuristic:∣∣∣∣∣∣0

ρl+1n

ρl+1a

∣∣∣∣∣∣=∣∣∣∣∣∣

M −Gn 0GT

n 0 E1

GTa 0 E2

∣∣∣∣∣∣∣∣∣∣∣∣∣νl+1

pl+1n

cl+1a

∣∣∣∣∣∣∣+∣∣∣∣∣∣−Mνl −pl

extΨl

n/h∆Ψa/h

∣∣∣∣∣∣ , (5)

where M is the mass matrix, and Gn and Ga are the normal and auxiliary contactwrenches, respectively. Multiple adjacent contacts are grouped, and thecomponents of Gn from (4) are split into Gn and Ga in (5). Gn contains the contactwrenches of the contacts with the minimum value of ψn for each group ofcontacts.

Yes, but what is this auxiliary contact wrench? What is the Stewart-Trinkle formulation again? Show me an example

Whatever, just show me the results


Institute




The Problem


Stewart-Trinkle



Hills Benchmark


Polyhedral Exact Geometry, auxiliary contact wrench

The remaining contact wrenches in each manifold are put into Ga. The auxiliarygap functions are defined as

Ψa =

Ψa1

...Ψans

whereΨaj =

Ψ1 −Ψ2...

Ψ1 −Ψns

. (6)

Wait, what? Show me an example That’s cool, what are the results?


Institute




The Problem


Stewart-Trinkle



Hills Benchmark


Polyhedral Exact Geometry formulation, two constraintexample

M

ψ1ψ2

p1

p2

p3

max(ψ1,ψ2) ≥ 0 (7)

max(ψ1,ψ2) =ψ2 +max(0,ψ1 −ψ2). (8)

c = max(0,ψ1 −ψ2) (9)Complementarity constraints:

0 ≤ c− (ψ1 −ψ2) ⊥ c ≥ 0. (10)

0 ≤ max(ψ1,ψ2) ⊥λj ≥ 0, (11)

Which gives

0 ≤c+ψ2 ⊥λ1 ≥ 0 (12)

0 ≤c+ψ2 ⊥λ2 ≥ 0. (13)

Very interesting, do you have a general example? Forget I even asked, just tell me the results


Institute




The Problem


Stewart-Trinkle



Hills Benchmark


Polyhedral Exact Geometry formulation, general exampleExpanding the constraint in (10), a general system with m facets in contact is realized,

0 ≤ c2 −ψ2 +ψ1 ⊥ c2 ≥ 0 (14)

0 ≤ c3 −ψ3 + c2 +ψ1 ⊥ c3 ≥ 0

.

.

.

0 ≤ cm −ψm + cm +cm−1 + ...+ c2 +ψ1 ⊥ cm ≥ 0

0 ≤ d1 +ψ1 ⊥ d1 ≥ 0

0 ≤ d2 +ψ2 ⊥ d2 ≥ 0

.

.

.

0 ≤ dm +ψm ⊥ dm ≥ 0

0 ≤ d1 + (c2 + c3 + ...+ cm−1 + cm)+ψ1 ⊥λ1 ≥ 0

0 ≤ d2 + (c2 + c3 + ...+ cm−1 + cm)+ψ2 ⊥λ2 ≥ 0

.

.

.

0 ≤ dm + (c2 + c3 + ...+ cm−1 + cm)+ψm ⊥λm ≥ 0

0 ≤ (c2 +c3 + ...+ cm−1 +cm)+ψ1,

whereci = max(0,ψ1 −ψi), i = 2, . . . ,m, (15)

and d are slack variables. Note that ψi := [ψn

]i .

Okay, so what are the results?


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The Problem


Stewart-Trinkle



Hills Benchmark


Results

Trajectory error results are compiled for three uncomplicated benchmarksimulations

Ï sawtooth particle simulation

Ï hills particle simulation

Ï slender rod 3D simulation

1.99 2 2.01 2.02 2.03 2.04 2.05

0.995

1

1.005

1.01

1.015

1.02

1.025

X (meters)

Y(m

eter

s)

PEGS-TA-P


Institute




The Problem


Stewart-Trinkle



Hills Benchmark


Sawtooth benchmark simulation

0 2 4 6 8 12 140

0.51

1.5

X (meters)

Y(m

eter

s)PEG

SâT

4 4.2 4.4 4.6 4.8 5 5.2

0.6

0.8

1

1.2

1.4

1.6

1.8

X (meters)

Y(m

eter

s)

PEGSâT

Okay, how about another benchmark example? Particle simulations are junk, why don’t you show me a real system?


Institute




The Problem


Stewart-Trinkle



Hills Benchmark


Hills benchmark simulation

−1 −0.50 0.5 1 1.5 2 2.5 3

0

0.5

1

1.5

2

X (meters)

Y(m

eter

s) PEGSâT

0.5 1 1.5 2 2.50.5

1

1.5

X (meters)

Y(m

eter

s)

Alright, that’s nice. Now show me a 3D example Thanks, but I’m done


Institute




The Problem


Stewart-Trinkle



Hills Benchmark


Slender rod benchmark simulation

10−4 10−3 10−2 10−110−3

10−2

10−1

100

h (s)

med

ian

erro

r(m

)

PEGAPST

Cool beans


Institute




The Problem


Stewart-Trinkle



Hills Benchmark


...


Institute




The Problem


Stewart-Trinkle



Hills Benchmark


...

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