Integral Control on Lie Groups

In this paper, we extend the popular integral control technique to systems evolving on Lie groups. More explicitly, we provide an alternative definition of “integral action” for proportional(derivative)-controlled systems whose configuration evolves on a nonlinear space, where configuration errors cannot be simply added up to compute a definite integral. We then prove that the proposed integral control allows to cancel the drift induced by a constant bias in both first order (velocity) and second order (torque) control inputs for fully actuated systems evolving on abstract Lie groups. We illustrate the approach by 3-dimensional motion control applications. Key words— PID control, Riemannian manifolds, Lie groups, bias rejection