On the collocated virtual holonomic constraints in Lagrangian systems

In this paper, the collocated virtual holonomic constraints are introduced in the coordinate free manner and their flat coordinates representation is derived. Flat coordinates are those where the virtual holonomic constraints are given by a subset of coordinates components put to be zero. The flat coordinates representation enables a clear description of both the constraint system and the feedback imposing it. Results are illustrated by swinging up the mechanical four link chain. Possible outlooks for walking design are discussed as well.