Stabilization of Perturbed Nonholonomic Systems in Chained Form
Many real-world systems exhibit velocity-dependent and/or acceleration-dependent constraints in their mathematical models. If these constraints are non-integrable then these systems are known as nonholonomic systems. Examples of such nonholonomic systems include hopping robots, unmanned aerial vehicles (UAVs), car-like robots, autonomous underwater vehicles (AUVs), surface vessels, vertical take-off and landing systems and many more. These systems are special as, in general, the stabilization problem of these systems cannot be solved by smooth (or continuous) static state-feedback and, thus, requires time-varying or discontinuous state-feedback control. In this research, we are considering first-, secondand higher-order nonholonomic systems that can be transformed into chained or power form which are canonical representations of these mechanical systems. The importance of stabilization problem of perturbed nonholonomic systems is further magnified by the variety of real-world day-to-day applications.
This research presents the solution to the stabilization problems for a selected class of perturbed first-, second- and higher-order nonholonomic mechanical systems. The methodologies are based on adaptive integral sliding mode control (AISMC). For the perturbed nonholonomic system, the original system is transformed into perturbed chained form. Then this perturbed chained form system is further transformed into a special structure containing nominal part and some unknown terms through input transformation. The unknown terms are computed adaptively. Later the transformed system is stabilized using integral sliding mode control (ISMC). The stabilizing controller for the transformed system is constructed which consists of the nominal control plus some compensator control. The compensator controller and the adaptive laws are derived in such a way that derivative of a Lyapunov function becomes strictly negative. A similar approach is applied to the third-order nonholonomic system with a jerk constraint. The validity of the proposed controllers is ascertained by simulating the perturbed first-, secondand higher-order nonholonomic systems in MATLAB / SIMULINK. The proposed control algorithms globally steer the whole system to the origin.