An intrinsic PID controller for mechanical systems on Lie groups

This paper shows how “integral action” may be implemented intrinsically in the general setting of fully-actuated mechanical systems on Lie groups. When combined with an intrinsic formulation of PD control, integral action ensures that a suitably defined configuration tracking error will converge to zero in response to constant velocity commands and disturbance forces. Convergence is locally exponential and almost global, and is robust to bounded parametric uncertainty. For more general bounded commands and disturbances, the tracking error can be made arbitrarily small by choosing sufficiently large controller gains. A separation principle allows the addition of an intrinsic velocity observer to complete a geometric analog to the familiar output feedback PID controller for second-order linear systems. The controller extends naturally to accommodate tracking of equivariant output functions. The result is explicitly shown for the Lie group SO(3), and supported by simulations of a 6DOF quadrotor model.