Reachability Analysis in Ground Vehicle System using Koopman Operator Theory

Recent advances in autonomous vehicles have encouraged researchers to provide accurate and efficient safety verification tools. Reachability analysis is one way to provide these safety and stability certificates for any dynamical systems. In this paper, we are proposing a model-based reachability approach for the performance analysis of vehicle dynamics. Reachable sets demonstrate all the possible solutions of a dynamical system that can be reached starting from a set of initial conditions. This analysis particularly has importance in both on-road and off-road vehicles, where the driver can lose control of the vehicle for some uncertain initial conditions, resulting in an unwanted path, leading to collision or missing the target. The computation of reachable sets becomes challenging for the vehicle due to the nonlinearity present in the model. In this paper, we are proposing a Koopman theory-based approach using principal eigenfunctions and the Koopman spectrum. The principal eigenfunction reduces the computational complexity and provides a formal approximation of reachable sets. We demonstrate the proposed approach's usefulness in computing the reachable sets for the nonlinear quarter-car dynamics. On comparing the results with the time-domain simulations, we found our approach to be quicker.