A Koopman Reachability Approach for Uncertainty Analysis in Ground Vehicle Systems

Recent progress in autonomous vehicle technology has led to the development of accurate and efficient tools for ensuring safety, which is crucial for verifying the reliability and security of vehicles. These vehicles operate under diverse conditions, necessitating the analysis of varying initial conditions and parameter values. Ensuring the safe operation of the vehicle under all these varying conditions is essential. Reachability analysis is an important tool to certify the safety and stability of the vehicle dynamics. We propose a reachability analysis approach for evaluating the response of the vehicle dynamics, specifically addressing uncertainties in the initial states and model parameters. Reachable sets illustrate all the possible states of a dynamical system that can be obtained from a given set of uncertain initial conditions. The analysis is crucial for understanding how variations in initial conditions or system parameters can lead to outcomes such as vehicle collisions or deviations from desired paths. By mapping out these reachable states, it is possible to design systems that maintain safety and reliability despite uncertainties. These insights help to ensure the stability and reliability of the vehicles, even in unpredictable conditions, by reducing accidents and optimizing performance. The nonlinearity of the model complicates the computation of reachable sets in vehicle dynamics. This paper proposes a Koopman theory-based approach that utilizes the Koopman principal eigenfunctions and the Koopman spectrum. By leveraging the Koopman principal eigenfunction, our method simplifies the computational process and offers a formal approximation for backward and forward reachable sets. First, our method effectively computes backward and forward reachable sets for a nonlinear quarter-car model with fixed parameter values. Furthermore, we applied our approach to analyze the uncertainty response for cases with uncertain parameters of the vehicle model. When compared to time-domain simulations, our proposed Koopman approach provided accurate results and also reduced the computational time by half in most cases. This demonstrates the efficiency and reliability of our proposed approach in dynamic systems uncertainty analysis using the reachable sets.