Optimal control of mechanical system motion with limited reactions of constraints

Abstract

The problem of determining the mechanical system limited control forces, with simultaneously imposed limitations to reactions of constraints is solved. Both external and internal constraints that can be holonomic or nonholonomic are analyzed. In this regard. differential equations of motion are formed with explicitly present Lagrange’s multipliers. Phase space dimensions and structure depend of the constraints that are of interest for the present problem. Dependence of multipliers of the constraints on corresponding reactions of constraints intensity is established. Control forces limitations and reactions of constraints intensity are given in the form of inequations. The problem of optimal control is solved by Pontryagin’s Maximum Principle. The proposed method has been applied to two examples of motion in a minimal time between two specified positions. The first example provides solution for the case of the motion of a material point along holonomic constraint of limited reaction. The second example refers to solution for the case of a rigid body with nonholonomic constraint — by the Chaplygin blade type, whose reaction of constraint is limited in intensity.