Linear Temporal Logic-based Mixed-Integer Linear Problem Planning with the Koopman Operator

We present a formulation for a linear temporal logic (LTL)-based task planning using the Koopman operator. The dynamics of nonlinear systems can be represented as linear systems by lifting them to a space of augmented states using the Koopman operator. On the other hand, the lifted linear system cannot capture the nonlinear effects of inputs, which appear in many robotic systems. Therefore, instead of a lifted linear system, we can consider representing control-affine bilinear systems. However, since the lifted bilinear systems are nonlinear, we need to solve nonlinear programming problems for trajectory optimization. This paper presents a methodology for the trajectory optimization problem of the lifted bilinear system. Using the mixed-integer convex approximation, we can solve the trajectory optimization problem of the lifted bilinear systems as a mixed-integer linear programming problem. This formulation allows us to solve LTL-based task planning problems for nonlinear systems. The effectiveness of the proposed method was confirmed by numerical simulations.