Stabilization of Relative Equilibria II

In this paper, we obtain feedback laws to asymptotically stabilize relative equilibria of mechanical systems with symmetry. We use a notion of stability `modulo the group action' developed by Patrick [1992]. We deal with both internal instability and with instability of the rigid motion. The methodology is that of potential shaping , but the system is allowed to be internally underactuated, i.e., have fewer internal actuators than the dimension of the shape space. Contents 1 Introduction 2 2 Patrick's Stability Result 4 3 Description of Forces 5 4 G ¯e -Stability of the Feedback System 8 5 Showing convergence of z t to an invariant set M 10 6 Characterization of the invariant set M 12 7 Relaxing the positivity condition on d 2 (H \Gamma J ¸e )(z e ). 14 7.1 Stability Analysis at a Relative Equilibrium . . . . . . . . . . . . . . 14 7.2 The Case A ¯e ? 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 7.3 The Case A ¯e \Pi 0. . . . . . . . . . . . . . . . . . . . . . . ...