A new approach to solve nonlinear path planning problem via measure theory

In this paper a new approach is proposed to find an approximate solution for nonlinear path planning problem. In this approach, first a new problem is defined in the calculus of variations which is equivalent to the original problem. The new problem can be expressed as an optimal control problem by introducing slack variable. Then a metamorphosis is performed in the space of problem by defining an injection from the set of admissible trajectory-control pairs of control problem into the space of positive Radon measures. Thereby, using properties of Radon measures, the problem is changed to a measure-theoretical optimization problem. This problem is an infinite dimensional linear programming (LP) and it is approximated by a finite dimensional LP. Finally, solution of finite dimensional LP is used to construct an approximate solution for the original problem. The proposed approach in comparison with other numerical methods works well; especially it is practical and accurate enough for systems with too complicated nonlinear terms. Moreover, accuracy can be improved as fine as desired. In addition, the obtained control function is piecewise constant and so it is suitable for switching systems. Finally, a numerical example is presented to show the effectiveness of the proposed approach to solve nonlinear path planning problems.