Steering control of nonholonomic systems with drift: The extended nonholonomic double integrator example

This paper presents two different kinds of steering control strategies for a system of extended nonholonomic double integrator, which is an example of nonholonomic control systems with drift. The first strategy presents piece-wise constant, states-dependent feedback control law. The method is based on the construction of a cost function V (not a Lyapunov function), which is the sum of two semi-positive definite functions V1 and V2. The semi-positive definite function V1 is dependent on first m state variables which can be steered along the given vector fields and V2 is dependent on the remaining n-m state variables which can be steered along the missing Lie brackets. The values of the functions V1 and V2 allow determination of a desired direction of system motion and permit to construct a sequence of controls such that the sum of these functions decreases in an average sense. The second strategy presents a time-varying feedback law based on the model reference approach, where the trajectory of the extended system is chosen as the model reference trajectory. The controllers are designed in such a way that after each time period T, the trajectory of the nonholonomic double integrator intersects the trajectory of the model reference, which can be made asymptotically stable. The proposed feedback law is as a composition of a standard stabilizing feedback control for a Lie bracket extension of the original system and a periodic continuation of a specific solution to an open loop control problem stated for an abstract equation on a Lie group. This approach does not rely on a specific choice of a Lyapunov function, and does not require transformations of the model to chained forms.