A Continuous Control for Stabilizing the Extended Nonholonomic Double Integrator: Theory and Simulation Results

The extended nonholonomic double integrator (ENDI) cannot be asymptotically stabilized by a continuous and time-invariant feedback controller since it violates the Brockett’s condition. In this paper, a new continuous control scheme is proposed to stabilize the ENDI system without drift. The dynamics of an ENDI system is enlarged to a higher dimensional space where a continuous but time-varying control law is designed for its stabilization. Besides the theoretical proofs, simulations conducted on the dynamics of a mobile robot are also presented to demonstrate the validity and performance of the proposed method. Introduction For an under-actuated vehicle or a vehicle suffering from nonholonomic constraints, its point stabilization control (PSC) will become particularly challenging due to the existence of either under-actuated dynamics or the second-order constraints. Since Brockett proved that a nonholonomic integrator system cannot be stabilized by continuously differentiable or time invariant state feedback control laws [1], a large number of approaches have been proposed to overcome the limitations imposed by Brockett’s result and stabilize it to the equilibrium points. Among the proposed solutions are smooth time-varying controllers [2], sinusoidal excitation [3], discontinuous or piecewise smooth control laws [5], and hybrid controllers [6], and also [2,4] including comprehensive survey of the relative field. Special attention is being paid to the continuous control schemes, which in some cases can overcome the complexity and performance degradation, such as low rates of convergence and oscillating trajectories, due to the discontinuous control algorithms. In this paper, a continuous control scheme is proposed to stabilize the so-called extended nonholonomic double integrator (ENDI) with drift. The dynamics of an ENDI system is enlarged to a higher dimensional space where a simple continuous but time-varying control law is designed to stabilize the system. Problem Statement The so called nonholonomic integrator can be described as,