Regular approximations of the fastest motion of mobile robot under bounded state variables

The time minimization problem for a mobile controlled robot under state constraints is studied. A numerical solution method based on the Pontryagin maximum principle is proposed. It is known that the problem of controlling the mobile robot, as any motion complying with the unicyclic model, belongs to the class of essentially nonregular problems with respect to state constraints. The solution to this problem using the maximum principle is complicated by the fact that there is no formula for the Lagrange multiplier measure. It is not clear how this measure can be expressed in terms of other extremum values and, therefore, it is not clear how the conditions of the maximum principle can be reduced to the corresponding boundary value problem. To overcome this difficulty, a regularization method for the problem based on $\varepsilon$-perturbation is proposed. Results of a numerical experiment that demonstrate the continuity of the measure multiplier are presented.