Nonholonomic Robot Parking by Feedback -- Part II: Nonmodular, Inverse Optimal, Adaptive, Prescribed/Fixed-Time and Safe Designs

Abstract: For the unicycle system, we provide constructive methods for the design of feedback laws that have one or more of the following properties: being nonmodular and globally exponentially stabilizing, inverse optimal, robust to arbitrary decrease or increase of input coefficients, adaptive, prescribed/fixed-time stabilizing, and safe (ensuring the satisfaction of state constraints). Our nonmodular backstepping feedbacks are implementable with either unidirectional or bidirectional velocity actuation. Thanks to constructing families of strict CLFs for the unicycle, we introduce a general design framework and families of feedback laws for the unicycle, which are inverse optimal with respect to meaningful costs. These inverse optimal feedback laws are endowed with robustness to actuator uncertainty and arbitrarily low input saturation due to the unicycle's driftlessness. Besides ensuring robustness to unknown input coefficients, we also develop adaptive laws for these unknown coefficients, enabling the achievement of good transient performance with lower initial control effort. Additionally, we develop controllers that achieve stabilization within a user-specified time horizon using two systematic methods: time-dilated prescribed-time design with smooth-in-time convergence to zero of both the states and the inputs and homogeneity-based fixed-time control that provides an explicit bound on the settling time. Finally, with a nonovershooting design we guarantee strictly forward motion without curb violation. This article, along with its Part I, lays a broad constructive design foundation for stabilization of the nonholonomic unicycle.