Koopman Based Trajectory Optimization with Mixed Boundaries

TL;DR

This paper introduces a Koopman operator-based approach to trajectory optimization with mixed boundary conditions, transforming the problem into a bilevel formulation to enable convexification and improve solution tractability.

Contribution

It proposes a novel bilevel optimization framework leveraging Koopman operators to handle mixed boundary constraints in trajectory optimization.

Findings

Effective on mathematical pendulum system

Successfully applied to compass-gait walker

Enables convexification of high-dimensional lower-level problem

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.