On catching an object by a robotic manipulator

The problem of a robotic manipulator catching a moving object (target) is viewed here as pursuer-evader problem. It is formulated as an optimization problem to determine the max-min = min-max (a saddle point) solution. After the feedback linearization, the models in the world coordinate system for the manipulator and the object are assumed to be linear, and the performance criterion is quadratic. The continuous-time models are combined to a single vector differential equation constraint. The optimum control theory determines the optimal inputs to the manipulator and the object in the feedback form. The gains of the feedback control laws contain the solution to the Riccati equation. It is shown that the positive definiteness of this solution yields a requirement for the capture: The difference of the reduced controllability matrix of the manipulator and that of the object (target) must be positive definite.