Stochastic Complementarity for Local Control of Discontinuous Dynamics

We present a method for smoothing discontinuous dynamics involving contact and friction, thereby facilitating the use of local optimization techniques for control. The method replaces the standard Linear Complementarity Problem with a Stochastic Linear Complementarity Problem. The resulting dynamics are continuously differentiable, and the resulting controllers are robust to disturbances. We demonstrate our method on a simulated 6-dimensional manipulation task, which involves a finger learning to spin an anchored object by repeated flicking.

[1]  L. S. Pontryagin,et al.  Mathematical Theory of Optimal Processes , 1962 .

[2]  Mihai Anitescu,et al.  A linearly implicit trapezoidal method for integrating stiff multibody dynamics with contact, joints, and friction , 2006 .

[3]  Xiaojun Chen,et al.  Expected Residual Minimization Method for Stochastic Linear Complementarity Problems , 2005, Math. Oper. Res..

[4]  Mihai Anitescu,et al.  Optimization-based simulation of nonsmooth rigid multibody dynamics , 2006, Math. Program..

[5]  Gui-Hua Lin,et al.  Stochastic Equilibrium Problems and Stochastic Mathematical Programs with Equilibrium Constraints: A Survey 1 , 2009 .

[6]  K. G. Murty,et al.  Complementarity problems , 2000 .

[7]  M. Anitescu,et al.  Formulating Dynamic Multi-Rigid-Body Contact Problems with Friction as Solvable Linear Complementarity Problems , 1997 .

[8]  Christopher G. Atkeson,et al.  Policies based on trajectory libraries , 2006, Proceedings 2006 IEEE International Conference on Robotics and Automation, 2006. ICRA 2006..

[9]  Emanuel Todorov,et al.  Implicit nonlinear complementarity: A new approach to contact dynamics , 2010, 2010 IEEE International Conference on Robotics and Automation.

[10]  Pieter Abbeel,et al.  An Application of Reinforcement Learning to Aerobatic Helicopter Flight , 2006, NIPS.

[11]  Richard W. Cottle,et al.  Linear Complementarity Problem. , 1992 .

[12]  Friedrich Pfeiffer,et al.  Multibody Dynamics with Unilateral Contacts , 1996 .

[13]  Paul Tseng,et al.  Non-Interior continuation methods for solving semidefinite complementarity problems , 2003, Math. Program..

[14]  W. Fleming,et al.  Controlled Markov processes and viscosity solutions , 1992 .

[15]  Olvi L. Mangasarian,et al.  A class of smoothing functions for nonlinear and mixed complementarity problems , 1996, Comput. Optim. Appl..

[16]  J. Moreau,et al.  Unilateral Contact and Dry Friction in Finite Freedom Dynamics , 1988 .

[17]  Marc Toussaint,et al.  Robot trajectory optimization using approximate inference , 2009, ICML '09.

[18]  Mihai Anitescu,et al.  Optimal control of systems with discontinuous differential equations , 2010, Numerische Mathematik.

[19]  E. Westervelt,et al.  Feedback Control of Dynamic Bipedal Robot Locomotion , 2007 .

[20]  M. L. Chambers The Mathematical Theory of Optimal Processes , 1965 .

[21]  William D. Smart,et al.  Receding Horizon Differential Dynamic Programming , 2007, NIPS.

[22]  Russ Tedrake,et al.  LQR-trees: Feedback motion planning on sparse randomized trees , 2009, Robotics: Science and Systems.

[23]  David E. Stewart,et al.  Rigid-Body Dynamics with Friction and Impact , 2000, SIAM Rev..