The Time-Freezing Method for Numerical Optimal Control of Dynamics with State Jumps

Armin Nurkanovic

Siemens Corporate Technology and Department of Microsystems Engineering (IMTEK), University Freiburg

Wednesday, October 07, 2020, 14:00 - 15:30

Room 01-012, Georges-Köhler-Allee 102, Freiburg 79110, Germany

In this talk we present a novel method for numerical simulation and optimal control of dynamic systems where the solution trajectories have state jumps. In such cases we cannot in general speak of ODEs and we must use tools such as Measure Differential Inclusions (MDIs), which are difficult to treat both numerally and theoretically. The Time-Freezing Method reformulates MDIs into ODEs with a discontinuous right-hand side. These equations are significantly simpler than MDIs, since the jump discontinuity is moved from the trajectory into its time derivative. Now we can use the much richer toolkit developed for discontinuous ODEs and treat systems with state jumps. 

The main idea in the time-freezing method is to introduce an auxiliary differential equation to mimic the state jump map. Thereby, a clock state is introduced which does not evolve during the runtime of the auxiliary system. The pieces of the trajectory that correspond to the parts when the clock state was evolving recover the solution of the original system with jumps.

In the first part of the talk we focus on the simpler case of (partially) elastic impacts and show how to satisfy the assumptions we made. The relevant tasks of locomotion, grasping and manipulation in robotics require however inelastic impacts. In the second part of the talk we focus on recent developments and extensions of the method for systems with inelastic impacts. After an inelastic impact, the dynamic system evolves on the boundary of the feasible set. In the case of mechanic impact problems, the difficulty lies in the requirements that the generalized velocity jumps to zero along the constraint normal and the trajectory must evolve on the manifold defined by this constraint. This requires switching between ODEs and Differential Algebraic Equations of index 3. 

We provide numerical examples demonstrating the ease of use of this reformulation in both simulation and optimal control. In the optimal control example, we solve a sequence of nonlinear programming problems (NLPs) in a homotopy penalization approach and recover a time-optimal trajectory with state jumps.


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