Contact-Implicit Model Predictive Control: Controlling Diverse Quadruped Motions Without Pre-Planned Contact Modes or Trajectories

TL;DR

A contact-implicit model predictive control framework for the real-time discovery of multi-contact motions, without predefined contact mode sequences or foothold positions, and proposes the analytical gradient of the contact impulse based on relaxed complementarity constraints to further the exploration of a variety of contact modes.

Abstract

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.

Figures (23)

• Figure 1: Experimental demonstration of front-leg rearing motion with the HOUND quadruped robot using the proposed framework. Only a desired body pitch angle was set, without any predefined contact modes.
• Figure 2: The overall framework of the contact-implicit model predictive control (MPC): The optimal control problem is tackled using a variant of the differential dynamic programming (DDP) algorithm, with the integration of a hard contact model within the system dynamics. The strict complementarity constraint, arising from the Signorini condition, is exclusively employed during the roll-out process, while the relaxed complementarity constraint is employed when computing the gradient component.
• Figure 3: Illustration of a contact impulse and contact velocity in the cases of (a) Separating, (b) Clamping, and (c) Sliding
• Figure 4: Illustration of (a) a Complementarity condition and (b) relaxed Complementarity condition with the parameter $\rho$
• Figure 5: The effect of foot slip and clearance cost is depicted. Two scenarios are considered: "With Foot Cost", incorporating both regulating cost and foot slip and clearance cost, and "Without Foot Cost", using only the regulating cost. For both cases, the target position is set 2.0 m ahead of the initial position. The resultant swing leg trajectory is depicted in (a), and the planned sliding contact is illustrated in (b).