  This paper presents a contact-implicit model predictive control (MPC)
framework for the real-time discovery of multi-contact motions, without
predefined contact mode sequences or foothold positions. This approach utilizes
the contact-implicit differential dynamic programming (DDP) framework, merging
the hard contact model with a linear complementarity constraint. We propose the
analytical gradient of the contact impulse based on relaxed complementarity
constraints to further the exploration of a variety of contact modes. By
leveraging a hard contact model-based simulation and computation of search
direction through a smooth gradient, our methodology identifies dynamically
feasible state trajectories, control inputs, and contact forces while
simultaneously unveiling new contact mode sequences. However, the broadened
scope of contact modes does not always ensure real-world applicability.
Recognizing this, we implemented differentiable cost terms to guide foot
trajectories and make gait patterns. Furthermore, to address the challenge of
unstable initial roll-outs in an MPC setting, we employ the multiple shooting
variant of DDP. The efficacy of the proposed framework is validated through
simulations and real-world demonstrations using a 45 kg HOUND quadruped robot,
performing various tasks in simulation and showcasing actual experiments
involving a forward trot and a front-leg rearing motion.
