A new control design for nonholonomic systems based on controllability of linear systems

In this paper, we deal with an optimal control problem for nonholonomic systems. We would be confronted with the Hamilton-Jacobi-Bellman (HJB) equation, whenever we try to obtain the feedback solution. Moreover, it is well known that a viscosity solution is necessary for the HJB equations in the case of nonholonomic systems. Generally, it is very difficult to obtain a standard solution for the HJB equations and so it is much more difficult to obtain a viscosity solution. Therefore, in the design methodology for the nonholonomic systems, little research has been reported so far. We introduce an approximation technique for the nonholonomic problems and construct a feedback controller based on the standard solution of the HJB equation, instead of the viscosity solution. Therefore, the existing research results to obtain the standard solution of the HJB equations is usable in our approach. Our control design consists of two steps. First step is to construct a controller using this standard solution. However, it should be noted that the trajectories do not converge to the origin because of the approximation error. Second step is to design a control strategy without performance index in order to let the trajectories converge to the origin from the neighborhood using a characteristic of nonholonomic systems.