This paper investigates the regulation problem for a class of nonholonomic systems that includes power form systems and an approximated system of the rolling sphere as special cases. The basic idea is first to introduce a virtual periodic moving trajectory that satisfies certain persistent excitation condition (PE) and has the zero value at some time instants. Based on LaSalle invariance principle, the associated tracking problem is then solved under a necessary condition for stabilization and particularly true for the power form systems and the rolling sphere. With the help of virtual trajectory, the achieved tracking result is applied to the regulation problem and used to guarantee practical stability. The proposed controllers have a simple and explicit form, and hence are easily implemented. Simultaneously, fast convergence is guaranteed, thanks to the K-exponential convergence. More interestingly, the used approach is adding sufficiently exciting signals to the systems by considering virtual tracking signals so that the attractivity of the origin can be guaranteed based on LaSalle invariance principle. Thus, it is possible to extend the proposed results to more general systems. To verify the effectiveness of the proposed scheme, interesting simulation results are presented.

In this paper, the basic function field of an asymptotically optimal tower of function fields by Garcia and Stichtenoth is studied, where bases of one-point linear systems can be explicitly constructed. A representation of finite GF(q2)-rational points on the associated plane curve is derived and exploited to develop a real-time hardware architecture for the calculation of syndromes needed in the decoding of AG codes built on this curve.

This paper presents a simple stabilizing control scheme for the rolling sphere. A tracking control problem with respect to a "virtual moving trajectory" is first solved. Then, it is used to achieve practical stability for the regulation problem. The derived controllers have a simple form and can guarantee fast convergence. To verify the effectiveness of the proposed results, an interesting simulation result is presented.

The paper investigates the global ρ-exponential stabilizability for nonholonomic Caplygin systems that are linear in certain state variables. A simple and easily verified controllability condition is proposed to guarantee the global ρ-exponential stabilizability. For an important subclass of Caplygin systems, the controllability condition is further reduced to some conditions relating to the degree and non-zero property of the lowest nonzero polynomials in the Taylor expansion of constraint function. A new feature in this paper is that all parameters can be explicitly determined from the constraint function. Moreover, a novel coordinate transformation between Caplygin systems can be used to enhance the proposed criterion so that it can be applied to various situations. Several interesting examples, including of the rolling wheel system and a set-point problem of hopping robot, are provided to validate the effectiveness of the proposed results.

This paper studies the general parking problem for unicycle-modeled mobile robots by combining the motion planning method with tracking controller design. We add a re-designed virtual trajectory to the original trajectory such that the parking problem can be transformed into a tracking problem. Consequently, a new global tracking controller is proposed to achieve the locally exponential convergence. Further improvement of the tracking performance is achieved using linearization and pole placement method. The proposed controllers are quite simple and can be easily implemented. The effectiveness of the proposed method is validated by several interesting experiments including the parallel parking and the back-into-garage parking.

In this paper, issues of practical stabilization problem for nonholonomic chained systems are discussed. A novel controller that can guarantee a fast convergence, like the exponential convergence, is proposed based on the switching controllers approach. The steady-state error can be specified in advance. Moreover, the convergent rate can be pre-assigned and the transient response can be improved using the pole-placement method. The proposed controller is shown to be robust with respect to some uncertainties in the model. Simulation results for a unicycle-modeled mobile robot that has a small bias in orientation, an unknown radius in the rear wheels and an unknown distance in between them are presented to validate the effectiveness of our approach.

A general tracking control problem with saturation constraint for nonholonomic mobile robots is proposed and solved using the backstepping technique. A global result is given in which some artificial assumptions about the linear and the angular velocities of mobile robots from recent literature are dropped. The proposed controller can simultaneously solve both the tracking problem and the regulation problem of mobile robots. With the proposed control laws, mobile robots can now globally follow any path such as a straight line, a circle and the path approaching to the origin using a single controller. Computer simulations are presented which confirm the effectiveness of the tracking control laws. Moreover, practical experimental results concerning the tracking control are reported with saturation constraint for mobile robots.