Flatness and realization of virtual holonomic constraints in Lagrangian systems

The virtual holonomic constraints technique used widely to decompose possibly complex control design tasks for the mechanical system into several simpler tasks will be analyzed here. Following Spong's terminology from the underactuated mechanical systems theory, the so-called collocated virtual holonomic constraints are defined. It is then shown that a certain type of collocated constraints for Lagrangian system are state and feedback equivalent to the so-called at virtual holonomic constraints for Lagrangian system. The at holonomic constraints have the simplest possible form and therefore can be straightforwardly imposed by the feedback. Moreover, the Lagrangian system constrained with at constraints give the constrained system that is the Lagrangian one as well. Brief illustrative example is included and further discussion of the possible applications of these results is provided as well.