Virtual holonomic constraint approach for planar bipedal walking robots extended to double support

The concept of virtual holonomic constraints is extended to the case of double support, which is characterized by a closed kinematic chain and redundancy in the state variables. An appropriate coordinate transformation is used to present the equations of motion in the coordinates of the actual degrees of freedom of the system. Virtual constraints of proper dimension lead to reduced dynamics of dimension two. The reduced dynamics is derived in a generalized way which includes previous results for the case of a single point contact. Control of the reduced dynamics is discussed in particular for the double support case. The derived concept is illustrated for a biped robot walking gait with a double support phase and an underactuated single support phase.