Programming and control of robots by means of differential algebraic inequalities

The method of programmed constraints has recently been proposed as an executable specification language for robot programming. The mathematical structures behind such problems are viability problems for control systems described by ordinary differential equations (ODE) subject to user-defined inequality constraints. This paper describes a method for the numerical solution of such problems, improving and extending some of our previous results. The algorithm presented is composed of three parts: delay-free discretization, local control, and local planning. Delay-free discretizations are consistent discretizations of control systems described by ODEs with discontinuous inputs. The local control is based on the minimization of an artificial, logarithmic barrier potential function. Local planning is a computationally inexpensive way to increase the robustness of the solution procedure, making it a refinement to a strategy based on viability alone. Simulations of a mobile robot are used to demonstrate the proposed strategy. Some complementarity is shown between the programmed-constraints approach to robot programming and optimal control. Moreover, we demonstrate the relative efficiency of our algorithm compared to optimal control: Typically, our method is able to find a solution on the order of 100 times faster than an optimal-control solver.