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High Accuracy Numerical Optimal Control for Rigid Bodies with Patch Contacts through Equivalent Contact Points -- Extended Version

Christian Dietz, Armin Nurkanović, Sebastian Albrecht, Moritz Diehl

Abstract

This paper extends the Finite Elements with Switch Detection and Jumps (FESD-J) [1] method to problems of rigid body dynamics involving patch contacts. The FESD-J method is a high accuracy discretization scheme suitable for use in direct optimal control of nonsmooth mechanical systems. It detects dynamic switches exactly in time and, thereby, maintains the integration order of the underlying Runge- Kutta (RK) method. This is in contrast to commonly used time-stepping methods which only achieve first-order accuracy. Considering rigid bodies with possible patch contacts results in nondifferentiable signed distance functions (SDF), which introduces additional nonsmoothness into the dynamical system. In this work, we utilize so-called equivalent contact points (ECP), which parameterize force and impulse distributions on contact patches by evaluation at single points. We embed a nondifferentiable SDF into a complementarity Lagrangian system (CLS) and show that the determined ECP are well-defined. We then extend the FESD-J discretization to the considered CLS such that its integration accuracy is maintained. The functionality of the method is illustrated for both a simulation and an optimal control example.

High Accuracy Numerical Optimal Control for Rigid Bodies with Patch Contacts through Equivalent Contact Points -- Extended Version

Abstract

This paper extends the Finite Elements with Switch Detection and Jumps (FESD-J) [1] method to problems of rigid body dynamics involving patch contacts. The FESD-J method is a high accuracy discretization scheme suitable for use in direct optimal control of nonsmooth mechanical systems. It detects dynamic switches exactly in time and, thereby, maintains the integration order of the underlying Runge- Kutta (RK) method. This is in contrast to commonly used time-stepping methods which only achieve first-order accuracy. Considering rigid bodies with possible patch contacts results in nondifferentiable signed distance functions (SDF), which introduces additional nonsmoothness into the dynamical system. In this work, we utilize so-called equivalent contact points (ECP), which parameterize force and impulse distributions on contact patches by evaluation at single points. We embed a nondifferentiable SDF into a complementarity Lagrangian system (CLS) and show that the determined ECP are well-defined. We then extend the FESD-J discretization to the considered CLS such that its integration accuracy is maintained. The functionality of the method is illustrated for both a simulation and an optimal control example.
Paper Structure (20 sections, 2 theorems, 68 equations, 5 figures)

This paper contains 20 sections, 2 theorems, 68 equations, 5 figures.

Table of Contents

  1. INTRODUCTION
  2. Contribution
  3. Notation
  4. RIGID BODY DYNAMICS WITH PATCH CONTACTS
  5. Signed Distance Function for Padded Polytopes
  6. Complementarity Lagrangian System
  7. Simultaneous Impact Law
  8. Well-Posedness of CLS and Uniqueness of ECP
  9. FINITE ELEMENTS WITH SWITCH DETECTION AND STATE JUMPS
  10. Runge-Kutta Integration
  11. Impact Equations
  12. Cross Complementarities
  13. Direct Optimal Control with Exact Switch Detection
  14. NUMERICAL EXAMPLES
  15. Simulation of a Falling Cuboid
  16. ...and 5 more sections

Key Result

Proposition 1

For any $q \in \mathbb{R}^{n_{q}}$, apart from the degenerate case $c_1 = c_2$, optimal dual variables for (distprob) are unique and depend continuously on $q$. Optimal primal variables are unique if there is no patch contact between the scaled padded polytopes.

Figures (5)

  • Figure 1: Visualization of two unscaled padded polytopes in bold color, their downsized interior polytopes (dotted) and exemplary circles used for padding (dashed). Further, scaled versions of the padded polytopes according to the scaling factor $\alpha$ determined by the distance problem (\ref{['distprob']}) are visualized in light color.
  • Figure 2: System configurations for the simulation example. For detailed velocity profiles, cf. Fig. \ref{['simplot']}. Orange and blue dots represent the approximation for $p(t^{-})$ and $p(t^{+})$, respectively. Red markers represent the ECP used for impulse resolution.
  • Figure 3: Evolution of differential and algebraic states for the simulation example. Active set changes of the force complementarity (\ref{['cls:c']}) and the SDF complementarity (\ref{['cls:e']}) are marked by purple and orange dotted lines, respectively.
  • Figure 4: System configurations for the optimal control example. For detailed velocity profiles, cf. Fig. \ref{['cntplot']}. Orange and blue dots represent the approximation for $p(t^{-})$ and $p(t^{+})$, respectively. Red markers represent the ECP used for impulse resolution.
  • Figure 5: Evolution of differential and algebraic states for the optimal control example. Active set changes of the force complementarity (\ref{['cls:c']}) and the SDF complementarity (\ref{['cls:e']}) are marked by purple and orange dotted lines, respectively.

Theorems & Definitions (2)

  • Proposition 1
  • Proposition 2