  Linear models for control and motion generation of humanoid robots have
received significant attention in the past years, not only due to their well
known theoretical guarantees, but also because of practical computational
advantages. However, to tackle more challenging tasks and scenarios such as
locomotion on uneven terrain, a more expressive model is required. In this
paper, we are interested in contact interaction-centered motion optimization
based on the momentum dynamics model. This model is non-linear and non-convex;
however, we find a relaxation of the problem that allows us to formulate it as
a single convex quadratically-constrained quadratic program (QCQP) that can be
very efficiently optimized. Furthermore, experimental results suggest that this
relaxation is tight and therefore useful for multi-contact planning. This
convex model is then coupled to the optimization of end-effector contacts
location using a mixed integer program, which can be solved in realtime. This
becomes relevant e.g. to recover from external pushes, where a predefined
stepping plan is likely to fail and an online adaptation of the contact
location is needed. The performance of our algorithm is demonstrated in several
multi-contact scenarios for humanoid robot.
