Stabilization of non-admissible curves for a class of nonholonomic systems

Abstract

The problem of tracking an arbitrary curve in the state space is considered for underactuated driftless control-affine systems. This problem is formulated as the stabilization of a time-varying family of sets associated with a neighborhood of the reference curve. An explicit control design scheme is proposed for the class of controllable systems whose degree of nonholonomy is equal to 1. It is shown that the trajectories of the closed-loop system converge exponentially to any given neighborhood of the reference curve provided that the solutions are defined in the sense of sampling. This convergence property is also illustrated numerically by several examples of nonholonomic systems of degrees 1 and 2. where x = (x 1 , . . . , x n ) is the state and u = (u 1 , . . . , u m ) is the control. The stabilization of such systems has been the subject of numerous studies over the last few decades, and many important results have been obtained in this area. In particular, it follows from the famous result of R.W. Brockett [7] that the trivial equilibrium of (1) is not stabilizable by a regular time-invariant feedback law if the vectors f 1 (0), f 2 (0), ..., f m (0) are linearly independent. Despite the significant progress in the development of control algorithms to stabilize the solution x = 0 of system (1) (see, e.g., [4,6,8,17,21,23,25,26,30,31], and references therein), the stabilization of nonholonomic systems to a given curve remains a challenging problem. This issue can be formulated as the trajectory tracking problem. In many papers, this problem has been addressed under the assumption that the trajectory is admissible, i.e. satisfies the system equations with some control inputs [1-3, 10, 11, 20, 27-29]. Since the number of controls m may be significantly smaller than the dimension of the state space n, not every path in the state space is admissible for system (1). However, in many applied problems, it is important to stabilize system (1) along an arbitrary curve, which is not necessarily admissible. As it is mentioned in [22], although it is not possible to asymptotically stabilize nonholonomic systems to non-admissible curves because of the non-vanishing tracking error, the practical stabilization can be achieved. It has to be noted that such problem has been addressed only for particular classes of systems, e.g., for unicycle and car-like systems [13,22,24] This paper deals with rather general formulation of the stabilization problem with non-admissible reference curves. The main contribution of our paper is twofold. First, we introduce a class of control functions for the first degree nonholonomic systems, which allows stabilizing the system in a prescribed neighborhood of an arbitrary (not necessarily admissible) curve. We also show how the obtained results can be extended to higher degree nonholonomic systems. The proposed feedback design scheme is based on the approach introduced in [14,30,32] for the stabilization and motion planning of nonholonomic systems. However, it has to be noted that the results o...

Optimisation-based Quasi Algebraic Controller for trajectory-tracking of a Class of Affine Systems with Constrained Controls

“…Moreover, when dealing with the trajectory-tracking tasks, it is usual to constrain the trajectories to that feasible with the mobile platform. However, by exploiting the structure of driftless nonlinear control-affine systems, the trajectory-tracking of non-admissible curves can also be achieved [19]. In these scenarios, the problem is formulated as the stabilisation of a time-varying family of sets associated with a neighbourhood of the reference curve.…”

Partial stabilization of nonholonomic systems with application to multi-agent coordination

“…The analogy between the equations of motion of nonholonomic systems and underwater vehicles has been pointed out in [2], where driftless control-affine systems have been used to model the kinematics of an autonomous submarine. These equations have been analyzed in the paper [10] in the context of trajectory tracking problem with oscillating inputs. A survey of recent advances in the motion planning of autonomous underwater vehicles (AUV) is presented in [26].…”