Control of symmetric mechanical systems with incomplete model information using reduction, dynamic feedback and flatness

The authors present a strategy for controlling symmetric systems. Given initial and final states of the full system, we find the corresponding outputs and derivatives of outputs for each state and then find a smooth curve y(t) that connects them while respecting the required differential inequalities. Then by restricting one of the relevant equations to the invariant coordinates of the reduced symmetric system, we find the desired trajectory of the reduced system which we can track because of the full actuation of that system. The final issue we address is the implementation of this strategy in such as way as to compensate for uncertainty in the details of the system. The part of this problem corresponding to the control of the reduced system is an extension of work done by the authors (1996) that show that for mechanical systems an arbitrary smooth and bounded trajectory can be asymptotically tracked to arbitrary accuracy using a dynamic feedback controller that does not require explicit knowledge of the inertial or potential properties of the system and also requires only measurement of the configuration of the system and not of velocities or momenta. This latter item is of substantial practical importance since in many applications, such as robotics, position sensors built into the apparatus provide accurate configuration information, but velocity measurements are typically of poor quality if available at all