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.

翻译：本文提出了一种接触隐式模型预测控制（MPC）框架，用于实时发现多接触运动，而无需预定义接触模式序列或足端落点位置。该框架采用接触隐式微分动态规划（DDP）方法，将硬接触模型与线性互补约束相融合。我们提出了基于松弛互补约束的接触冲量解析梯度，以进一步探索多种接触模式。通过采用基于硬接触模型的仿真及基于平滑梯度的搜索方向计算，我们的方法能够同时识别动态可行的状态轨迹、控制输入和接触力，并揭示新的接触模式序列。然而，扩展后的接触模式范围并不总能保证实际应用的可行性。为此，我们引入了可微成本项来引导足端轨迹并生成步态模式。此外，为应对MPC框架中初始滚动不稳定性的挑战，我们采用了DDP的多重打靶变体。通过使用45千克HOUND四足机器人进行的仿真与实物验证，所提框架的有效性得到证实——仿真中完成了多种任务，实际实验则展示了前向小跑和前腿直立动作。