In contour following applications, the various degrees of freedom of a mechanical system have to be well coordinated, but very often, the speed at which the contour is followed is not critical. Moreover, in the context of machining, the system has to interact closely with its physical environment. When the contour following task is represented by a velocity field on the configuration manifold of the system, the coordination aspect of the problem is made explicit. The passive velocity field control (PVFC) scheme developed in the Part I companion paper  can them be applied to track the defined velocity field so that the desired contour is followed, and to ensure that the interaction of the closed-loop system with the physical environment is passive to enhance safety and stability. Unfortunately, for some contours, an encoding velocity field on the configuration manifold does not exist or is difficult to define and, as consequence, the PVFC cannot be directly applied. For systems whose configuration manifolds are compact Lie groups and the desired contour is represented by a parameterized trajectory, a general methodology is developed, using a suspension technique, to define a velocity field on a manifold related to the configuration manifold of the system for which PVFC can be applied. With this strategy, timing along the contour can be naturally varied on-line by a self-pacing scheme so that the contour tracking performance can be improved. The experimental results for a 2 degree of freedom robot following a Lissajous contour illustrates and verifies the convergence and robustness properties of the PVFC methodology.
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- Lie groups
- Velocity field