  In this paper we present a new approach for dynamic motion planning for
legged robots. We formulate a trajectory optimization problem based on a
compact form of the robot dynamics. Such a form is obtained by projecting the
rigid body dynamics onto the null space of the Constraint Jacobian. As
consequence of the projection, contact forces are removed from the model but
their effects are still taken into account. This approach permits to solve the
optimal control problem of a floating base constrained multibody system while
avoiding the use of an explicit contact model. We use direct transcription to
numerically solve the optimization. As the contact forces are not part of the
decision variables the size of the resultant discrete mathematical program is
reduced and therefore solutions can be obtained in a tractable time. Using a
predefined sequence of contact configurations (phases), our approach solves
motions where contact switches occur. Transitions between phases are
automatically resolved without using a model for switching dynamics. We present
results on a hydraulic quadruped robot (HyQ), including single phase (standing,
crouching) as well as multiple phase (rearing, diagonal leg balancing and
stepping) dynamic motions.
