A Koopman operator approach for the vertical stabilization of an off-road vehicle

Abstract

Vehicles traversing off-road terrains experience modes of dynamics that are not usually encountered on-road. Deformable terrains with significant variations in height and the presence of bumps can lead to significant vertical dynamics in a vehicle. Besides rider discomfort and safety of the vehicle, in the case of unmanned vehicles, these vertical dynamics coupled with pitch motion can produce significant disturbances to sensors such as cameras. Reducing these vertical dynamics is therefore important. However any control design for this problem has to first contend with the modeling complexity of vehicle-tire-terrain interaction. Motivated by recent developments in dynamical systems, we propose the use of the Koopman operator to obtain a linear representation for both the vehicle and terrain interaction dynamics as well as the effect of the control input. As a test case for this framework we address the problem of stabilizing the vertical dynamics of a half-car model moving on a deformable terrain using a propulsive force on the car as the sole control input. The terramechanics are modeled using the Bekker-Wong equations. Using the Koopman operator framework, we obtain a lifted, linear representation of the nonlinear control system, which is then used to formulate the optimal control problem as a constrained linear quadratic model predictive control problem on the lifted system. The framework proposed in this paper can potentially be extended to design a combination of data-driven and physics-based control algorithms for intelligent suspensions, motion control and path-tracking for unmanned ground vehicles on unstructured terrains.