Koopman Based Trajectory Optimization with Mixed Boundaries

TL;DR

This paper tackles non-convex trajectory optimization with mixed boundary constraints that couple $x(0)$ and $x(T)$ and feature an unknown terminal time $T$. It employs a Koopman generator surrogate to lift nonlinear dynamics into a linear, high-dimensional space and formulates a bilevel problem $\hat{\mathcal P}$ with a convex lower-level and a low-dimensional upper-level that optimizes $(x_0, x_T, T)$. Three lifted-boundary formulations are analyzed ($\hat{\boldsymbol b}_0$, $\hat{\boldsymbol b}_T$, and $\hat{\boldsymbol b}_{soft}$), with hard lifting on a single boundary providing the best agreement with the original NLP while soft constraints trade off accuracy. The approach is validated on a mathematical pendulum and a compass-gait walker, showing close matches in state, input, and period, and moderate computational advantage. The work points to future improvements by learning the Koopman observables and generator with deep models and extending the framework to broader constraint classes and co-design problems.

Abstract

Trajectory optimization is a widely used tool in the design and control of dynamical systems. Typically, not only nonlinear dynamics, but also couplings of the initial and final condition through implicit boundary constraints render the optimization problem non-convex. This paper investigates how the Koopman operator framework can be utilized to solve trajectory optimization problems in a (partially) convex fashion. While the Koopman operator has already been successfully employed in model predictive control, the challenge of addressing mixed boundary constraints within the Koopman framework has remained an open question. We first address this issue by explaining why a complete convexification of the problem is not possible. Secondly, we propose a method where we transform the trajectory optimization problem into a bilevel problem in which we are then able to convexify the high-dimensional lower-level problem. This separation yields a low-dimensional upper-level problem, which could be exploited in global optimization algorithms. Lastly, we demonstrate the effectiveness of the method on two example systems: the mathematical pendulum and the compass-gait walker.

Figures (10)

• Figure 2: Illustrated are two mechanical systems: (a) a mathematical pendulum and (b) a compass-gait walker. The MBCs $\boldsymbol b$ include periodicity, operating, and anchor constraints. For (a), the operating constraint is the desired amplitude $a$, and for (b), it is the desired average forward velocity $v_\text{avg}$. The system states are $\boldsymbol x^\top = [\boldsymbol q^\top~\dot{\boldsymbol q}^\top]$, with all parameters normalized by gravity $g$, length $l_\circ$ and mass $m$.
• Figure : (a)
• Figure : (a)
• Figure : (a)
• Figure : (a)