We present a formulation of nonpenetration constraint between pairs of polytopes which accounts for all possible combinations of active contact between geometric features. This is the first formulation that exactly models the body geometries near points of potential contact, preventing interpenetration while not overconstraining body motions. Unlike many popular methods, ours does not wait for penetrations to occur as a way to identify which contact constraints to enforce. Nor do we overconstrain by representing the free space between pairs of bodies as convex, when it is in fact nonconvex. Instead, each contact constraint incorporates all feasible potential contacts in a way that represents the true geometry of the bodies. This ensures penetration-free, physically correct configurations at the end of each time step while allowing bodies to accurately traverse the free space surrounding other bodies. The new formulation improves accuracy, dramatically reduces the need for ad hoc corrections of constraint violations, and avoids many of the inevitable instabilities consequent of other contact models. Although the dynamics problem at each time step is larger, the inherent stability of our method means that much larger time steps can be taken without loss of physical fidelity. As will be seen, the results obtained with our method demonstrate the effective elimination of interpenetration, and as a result, correction-induced instabilities, in multibody simulations.

References

References
1.
Ericson
,
C.
,
2004
,
Real-Time Collision Detection
(The Morgan Kaufmann Series in Interactive 3-D Technology),
Morgan Kaufmann Publishers
,
San Francisco, CA
.
2.
Eberly
,
D. H.
,
2010
,
Game Physics
(Interactive 3D Technology Series),
Elsevier Science
,
London
.
3.
Millington
,
I.
,
2010
,
Game Physics Engine Development: How to Build a Robust Commercial-Grade Physics Engine for Your Game
,
2nd ed.
,
Morgan Kaufmann Publishers
,
San Francisco, CA
.
4.
Terzopoulos
,
D.
,
Pltt
,
J.
,
Barr
,
A.
,
Zeltzer
,
D.
,
Witkin
,
A.
, and
Blinn
,
J.
,
1989
, “
Physically-Based Modeling: Past, Present, and Future
,”
SIGGRAPH Comput. Graphics
,
23
(
5
), pp.
191
209
.
5.
Baraff
,
D.
,
1994
, “
Fast Contact Force Computation for Nonpenetrating Rigid Bodies
,”
21st Annual Conference on Computer Graphics and Interactive Techniques
,
SIGGRAPH’94
, pp.
23
24
.
6.
Foster
,
N.
, and
Metaxas
,
D.
,
1996
, “
Realistic Animation of Liquids
,”
Graphical Models Image Process.
,
58
(
5
), pp.
471
483
.
7.
Witkin
,
A.
, and
Baraff
,
D.
,
1997
, “
Physically Based Modeling: Principles and Practice
,” SIGGRAPH ‘97 Course Notes, Course No. 19.
8.
Erleben
,
K.
,
Sporring
,
J.
,
Henriksen
,
K.
, and
Dohlmann
,
H.
,
2005
,
Physics-Based Animation
(Graphics Series),
Charles River Media
,
Rockland, MA
.
9.
Pfeiffer
,
F.
, and
Glocker
,
C.
,
1996
,
Multibody Dynamics With Unilateral Contacts
,
Wiley
,
New York
.
10.
Berard
,
S.
,
2009
, “
Using Simulation for Planning and Design of Robotic Systems With Intermittent Contact
,”
Ph.D. thesis
, Department of Computer Science, Rensselaer Polytechnic Institute, Troy, NY.
11.
Berard
,
S.
,
Nguyen
,
B.
,
Anderson
,
K.
, and
Trinkle
,
J.
,
2010
, “
Sources of Error in a Rigid Body Simulation of Rigid Parts on a Vibrating Rigid Plate
,”
ASME J. Comput. Nonlinear Dyn.
,
5
(
4
), p.
041003
.
12.
Macaluso
,
W.
,
2012
, “
Exploring the Domain of Applicability of Simulated 2D Rigid Body Dynamical Systems
,”
Master's thesis
, Department of Computer Science, Rensselaer Polytechnic Institute, Troy, NY.
13.
Haug
,
E.
,
Wu
,
S.
, and
Yang
,
S.
,
1986
, “
Dynamic Mechanical Systems With Coulomb Friction, Stiction, Impact and Constraint Addition–Deletion—I: Theory
,”
Mech. Mach. Theory
,
21
(
5
), pp.
407
416
.
14.
Leine
,
R. I.
, and
Nijmeijer
,
H.
,
2004
,
Dynamics and Bifurcations of Non-Smooth Mechanical Systems
(Lecture Notes in Applied and Computational Mechanics),
Springer
,
Berlin
.
15.
Moreau
,
J.
, and
Panagiotopoulos
,
P.
,
1988
, “
Unilateral Contact and Dry Friction in Finite Freedom Dynamics
,”
Nonsmooth Mechanics and Applications
,
Springer
,
New York
, pp.
1
82
.
16.
Stewart
,
D.
, and
Trinkle
,
J. C.
,
1996
, “
An Implicit Time-Stepping Scheme for Rigid Body Dynamics With Inelastic Collisions and Coulomb Friction
,”
Int. J. Numer. Methods Eng.
,
39
(
15
), pp.
2673
2691
.
17.
Stewart
,
D. E.
,
1998
, “
Convergence of a Time-Stepping Scheme for Rigid-Body Dynamics and Resolution of Painlev's Problem
,”
Arch. Ration. Mech. Anal.
,
145
(
3
), pp.
215
260
.
18.
Anitescu
,
M.
, and
Potra
,
F. A.
,
1997
, “
Formulating Dynamic Multi-Rigid-Body Contact Problems With Friction as Solvable Linear Complementarity Problems
,”
Nonlinear Dyn.
,
14
(
3
), pp.
231
247
.
19.
GitHub
,
2015
, “
DART Physics Engine
,”
GitHub
, Inc., San Francisco, CA.
20.
Mazhar
,
H.
,
Heyn
,
T.
,
Pazouki
,
A.
,
Melanz
,
D.
,
Seidl
,
A.
,
Bartholomew
,
A.
,
Tasora
,
A.
, and
Negrut
,
D.
,
2013
, “
CHRONO: A Parallel Multi-Physics Library for Rigid-Body, Flexible-Body, and Fluid Dynamics
,”
Mech. Sci.
,
4
(
1
), pp.
49
64
.
21.
Coumans
,
E.
,
2015
, “
Bullet Physics Library: An Open Source Collision Detection and Physics Library
,”
Real-Time Physics Simulation Forum
.
22.
Bender
,
J.
,
Erleben
,
K.
, and
Trinkle
,
J.
,
2014
, “
Interactive Simulation of Rigid Body Dynamics in Computer Graphics
,”
Comput. Graphics Forum
,
33
(
1
), pp.
246
270
.
23.
Williams
,
J.
,
2014
, “
A Contact Model for Geometrically Accurate Treatment of Polytopes in Simulation
,”
Ph.D. thesis
, Rensselaer Polytechnic Institute, Troy, NY.
24.
Cottle
,
R.
,
Pang
,
J.
, and
Stone
,
R.
,
1992
,
The Linear Complementarity Problem
(Classics in Applied Mathematics),
Society for Industrial and Applied Mathematics (SIAM)
,
Philadelphia, PA
.
25.
Billups
,
S. C.
, and
Murty
,
K. G.
,
2000
, “
Complementarity Problems
,”
J. Comput. Appl. Math.
,
124
, pp.
303
318
.
26.
Bender
,
J.
,
Erleben
,
K.
,
Trinkle
,
J. C.
, and
Coumans
,
E.
,
2012
, “
Interactive Simulation of Rigid Body Dynamics in Computer Graphics
,”
Conference of the European Association for Computer Graphics
, State of the Art Report (
STAR
), pp.
246
270
.
27.
Ferris
,
M.
, and
Munson
,
T.
,
2000
, “
Complementarity Problems in GAMS and the Path Solver
,”
J. Econ. Dyn. Control
,
24
(
2
), pp.
165
188
.
28.
Erleben
,
K.
,
2013
, “
Numerical Methods for Linear Complementarity Problems in Physics-Based Animation
,”
Special Interest Group on Computer Graphics and Interactive Techniques Conference
,
SIGGRAPH’13
,
ACM
,
New York
, pp.
8:1
8:42
.
29.
Pang
,
J.
, and
Trinkle
,
J.
,
1996
, “
Complementarity Formulations and Existence of Solutions of Dynamic Multi-Rigid-Body Contact Problems With Coulomb Friction
,”
Math. Program.
,
73
(
2
), pp.
199
226
.
30.
Todorov
,
E.
,
2011
, “
A Convex, Smooth and Invertible Contact Model for Trajectory Optimization
,”
IEEE International Conference on Robotics and Automation
(
ICRA
), May 0–13, pp.
1071
1076
.
31.
Signorini
,
A.
,
1959
, “
Questioni di elasticità non linearizzata e semilinearizzata
,”
Rend. Mat. Appl., V. Ser.
,
18
, pp.
95
139
.
32.
Nguyen
,
B.
,
2011
, “
Locally Non-Convex Contact Models and Solution Methods for Accurate Physical Simulation in Robotics
,”
Ph.D. thesis
, Rensselaer Polytechnic Institute, Troy, NY.
33.
Flickinger
,
D. M.
, and
Trinkle
,
J. C.
,
2013
, “
Evaluating the Performance of Constraint Formulations for Multibody Dynamics Simulation
,”
ASME
Paper No. DETC2013-12265.
34.
Williams
,
J.
,
2014
, “
A Complementarity Based Contact Model for Physically Accurate Treatment of Polytopes in Simulation
,”
Workshop on Computational Contact Mechanics: Advances and Frontiers in Modeling Contact
, Banff International Research Station, Banff, Canada, Talk No. 34.
35.
Williams
,
J.
,
Lu
,
Y.
, and
Trinkle
,
J.
,
2014
, “
A Complementarity Based Contact Model for Geometrically Accurate Treatment of Polytopes in Simulation
,”
ASME
Paper No. DETC2014-35231.
36.
Hu
,
J.
,
Mitchell
,
J. E.
,
Pang
,
J.
,
Bennett
,
K. P.
, and
Kunapuli
,
G.
,
2008
, “
On the Global Solution of Linear Programs With Linear Complementarity Constraints
,”
SIAM J. Optim.
,
19
(
1
), pp.
445
471
.
37.
Donald
,
B. R.
,
1984
, “
Local and Global Techniques for Motion Planning
,”
Ph.D. thesis
, Massachusetts Institute of Technology, Cambridge, MA.
38.
Latombe
,
J.-C.
,
1991
,
Robot Motion Planning
,
Kluwer Academic Publishers
,
Norwell, MA
.
39.
Lin
,
M. C.
,
1993
, “
Efficient Collision Detection for Animation and Robotics
,”
Ph.D. thesis
, University of California, Berkeley, CA.
40.
Jiménez
,
P.
,
Thomas
,
F.
, and
Torras
,
C.
,
1998
, “
Collision Detection Algorithms for Motion Planning
,”
Robot Motion Planning and Control
(Lecture Notes in Control and Information Sciences, Vol.
229
),
Springer
,
London
, pp.
305
343
.
41.
Mirtich
,
B.
,
1998
, “
V-Clip: Fast and Robust Polyhedral Collision Detection
,”
ACM Trans. Graphics
,
17
(
3
), pp.
177
208
.
42.
Zhang
,
L.
,
2013
, “
Physics-Empowered Perception for Robot Grasping and Dexterous Manipulation
,” Ph.D. thesis, Department of Computer Science, Rensselaer Polytechnic Institute, Troy, NY.
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