Spline-based Approach to Optimal Control of Trajectories under Inequality Type Constraints

The paper is devoted to an optimal trajectory planning problem considered as a problem of constrained optimal control for dynamical systems. It is one of the fundamental problems in robotics, biomechanics, aeronautics and many other areas of application of control theory. The simplest version of this problem supposes that there are given sequences of target points and prescribed times, and we are required to be at the given point at the prescribed time. However, in most of the applications, it is enough when the trajectory passes close to the assigned point at the prescribed time. So, the location conditions could be considered as the inequality type constraints. The aim of this research is to reduce such an optimal control problem to the problem of splines in convex sets, which could be analysed and solved by methods of the general theory of splines. Dynamical systems associated with the second order linear differential equation with initial conditions are investigated in the paper (the restriction on the order of equations is not essential). We consider this system as a curve generator. The goal is to find a control law by minimization of the quadratic cost function under inequality type constraints on location conditions. A spline-based numerical scheme for some cases of such optimal control problems is proposed in this paper. In particular, the method of adding-removing spline interpolation knots is applied to the construction of its solution. The suggested technique is illustrated by numerical examples.