State and configuration feedback for almost global tracking of simple mechanical systems on a general class of Lie groups

We present a general intrinsic tracking controller design for fully-actuated simple mechanical systems, when the configuration space is one of a general class of Lie groups. We show that if a suitable error function can he found, then a general smooth and hounded reference trajectory may be tracked asymptotically from almost every initial condition, with locally exponential convergence. Such functions may be shown to exist on any compact Lie group, or on any product of a compact Lie group and R/sup n/. In the case of compact Lie groups, we show that the full-state feedback law composed with an exponentially convergent velocity estimator, also converges globally for almost every initial tracking error. We explicitly compute these controllers on SO(3), and simulate their performance for the axisymmetric top problem.