Tracking control of the orbitally flat kinematic car with a new time-scaling input

Abstract

The paper defines a time-scaling scheme such that the scaling function depends on the state variables of the system and on an external variable referred to as the time-scaling input. The system evolving according to the scaled time has thus one more input variable compared to the original system. This time- scaling scheme is applied to the kinematic car with one input, namely the steering angle. The longitudinal velocity of the car is a measurable external signal and cannot be influenced by the controller. Using an on-line time-scaling, which is driven by the longitudinal velocity of the car, by a time-scaling output of the tracking controller, and by their time derivatives up to the second order, one can achieve exponential tracking of any sufficiently smooth reference trajectory with non-vanishing velocity, similar to the two-input differentially flat model of the kinematic car. The time derivatives of the longitudinal velocity and the time-scaling output are no longer necessary if one designs the tracking controller for the linearized error dynamics around the reference path.