Min-max redundancy resolution for a mobile manipulator

We have considered the problem of determining the values of the joint variables of a mobile manipulator with many redundant degrees of freedom that will minimize an objective function when the position and orientation of the end of the manipulator are given. The objective function is the weighted sum of three components: distance, torque, and reach. Each of the three components is a max or min. We have converted the min-max optimization problem into a nonlinear programming problem and used the Kuhn-Tucker conditions to derive necessary conditions for the optimum solutions. The necessary conditions require that one or more of each of the three sets (distance, torque, and reach) of nonnegative Lagrange multipliers must be positive. If one of the Lagrange multipliers is positive, the corresponding slack variable must be zero. When two or more of the Lagrange multipliers from a single set are positive, the slack variables place constraints on the joint variables. Specification of the Cartesian position and orientation of the end of the arm also places constraints on the joint variables. If the mobile manipulator has N degrees of freedom and the total number of constraints is M, the constraints define a manifold of dimensions N - M. When N = M, the dimension of the manifold is zero (it consists of isolated points). When N > M, a search of the manifold may yield a submanifold that maximizes the Lagrangian function. We discuss examples where the number of slack variable constraints (M) is two or more