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We derive a reduced model, that is, a model in terms of independent generalized coordinates, for the equations of motion of closed-chain mechanisms. We highlight the fact that the model has two special characteristics which make it different from models of open-chain mechanisms. First, it is defined locally in the generalized coordinates. We therefore characterize the domain of validity of the model in which the mechanism satisfies the constraints and is not in a singular configuration. Second, it is an implicit model, that is, parts of the equations of motion are not expressed explicitly. Despite the implicit nature of the equations of motion, we show that closed-chain mechanisms still satisfy a skew symmetry property, and that proportional derivative (PD)-based control with so-called simple gravity compensation guarantees (local) asymptotic stability. We discuss the computational issues involved in the implementation of the proposed controller. The proposed modeling and PD control approach is illustrated experimentally using the Rice planar delta robot which was built to experiment with closed-chain mechanisms.
Ghorbel et al. (Sat,) studied this question.