A case study of inverse dynamics control of manipulators with passive joints

Manipulators with both active and passive joints are examples of underactuated systems, featured by less control inputs than degrees of freedom. Due to the underactuation, in the trajectory tracking (servo-constraint) problem, the feed-forward control obtained from an inverse model is influenced by internal dynamics of the system, leading to a more involved control design than in the fully actuated case. It is demonstrated that a convenient approach to the problem solution is to formulate the underactuated system dynamics in the input-output normal form, with the arising governing equations formulated either as ODEs (ordinary differential equations) or DAEs (differential-algebraic equations). The interrelationship between the inverse dynamics control and the associated internal dynamics is then studied and illustrated using a planar manipulator with two active and one passive joint. Some simulation results for the sample case study are reported.