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.
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