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Solution methods for differential systems subject to algebraic inequality constraints

Abstract:

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 subject to user-defined inequality constraints. Although these types of problems are common in applications, practical algorithms and software are generally lacking. This thesis describes a new method for the numerical solution of such viability problems. The algorithm presented is composed of three parts: delay-free discretization, local planning, and local control. Delay-free discretizations are shown to be consistent discretizations of control systems described by ordinary differential equations with discontinuous inputs. The generalization of delay-free discretizations to higher order in the context of implicit-explicit Runge-Kutta methods represents a potentially powerful new class of time integrators for ordinary differential equations that contain terms requiring different discretizations. The local planning aspect is a computationally inexpensive way to increase the robustness in finding a solution to the viability problems of interest, making it a refinement to a strategy based on viability alone. The local control is based on the minimization of an artificial potential function in the form of a logarithmic barrier. Theoretical examples are given of situations where the choice of such a control can be interpreted to yield heavy solutions. Simulations of two robotic systems are then used to validate the particular strategy investigated. Some complementarity is shown between the programmedconstraint approach to robot programming and optimal control. Moreover, we demonstrate the relative efficiency of our algorithm compared to optimal control in the case of programming a mobile robot: our method is able to find a solution on the order of one hundred times faster than a typical optimal control solver. Some simulations on the control of a simple robot arm are also presented.

Author: Raymond J. Spiteri

Download: ubc_1997-251659

Link: Ph.D. Thesis