  We propose a novel and efficient lifting approach for the optimal control of
rigid-body systems with contacts to improve the convergence properties of
Newton-type methods. To relax the high nonlinearity, we consider the state,
acceleration, contact forces, and control input torques, as optimization
variables and the inverse dynamics and acceleration constraints on the contact
frames as equality constraints. We eliminate the update of the acceleration,
contact forces, and their dual variables from the linear equation to be solved
in each Newton-type iteration in an efficient manner. As a result, the
computational cost per Newton-type iteration is almost identical to that of the
conventional non-lifted Newton-type iteration that embeds contact dynamics in
the state equation. We conducted numerical experiments on the whole-body
optimal control of various quadrupedal gaits subject to the friction cone
constraints considered in interior-point methods and demonstrated that the
proposed method can significantly increase the convergence speed to more than
twice that of the conventional non-lifted approach.
