Efficient Optimal Path Planning in Dynamic Environments Using Koopman MPC

This paper presents a data-driven model predictive control framework for mobile robots navigating in dynamic environments, leveraging Koopman operator theory. Unlike the conventional Koopman-based approaches that focus on the linearization of system dynamics only, our work focuses on finding a global linear representation for the optimal path planning problem that includes both the nonlinear robot dynamics and collision-avoidance constraints. We deploy extended dynamic mode decomposition to identify linear and bilinear Koopman realizations from input-state data. Our open-loop analysis demonstrates that only the bilinear Koopman model can accurately capture nonlinear state-input couplings and quadratic terms essential for collision avoidance, whereas linear realizations fail to do so. We formulate a quadratic program for the robot path planning in the presence of moving obstacles in the lifted space and determine the optimal robot action in an MPC framework. Our approach is capable of finding the safe optimal action 320 times faster than a nonlinear MPC counterpart that solves the path planning problem in the original state space. Our work highlights the potential of bilinear Koopman realizations for linearization of highly nonlinear optimal control problems subject to nonlinear state and input constraints to achieve computational efficiency similar to linear problems.