State-of-the-art approaches to footstep planning assume reduced-order dynamics when solving the combinatorial problem of selecting contact surfaces in real time. However, in exchange for computational efficiency, these approaches ignore joint torque limits and limb dynamics. In this work, we address these limitations by presenting a topology-based approach that enables~\gls{mpc} to simultaneously plan full-body motions, torque commands, footstep placements, and contact surfaces in real time. To determine if a robot's foot is inside a contact surface, we borrow the winding number concept from topology. We then use this winding number and potential field to create a contact-surface penalty function. By using this penalty function,~\gls{mpc} can select a contact surface from all candidate surfaces in the vicinity and determine footstep placements within it. We demonstrate the benefits of our approach by showing the impact of considering full-body dynamics, which includes joint torque limits and limb dynamics, on the selection of footstep placements and contact surfaces. Furthermore, we validate the feasibility of deploying our topology-based approach in an~\gls{mpc} scheme and explore its potential capabilities through a series of experimental and simulation trials.

翻译：当前先进的落脚点规划方法在实时解决接触面组合选择问题时，通常采用降阶动力学模型。然而，这种以计算效率为代价的方法忽略了关节力矩约束与肢体动力学。本研究提出一种基于拓扑的方法，使模型预测控制（MPC）能够实时同步规划全身运动、力矩指令、落脚点位置与接触面选择。为判定机器人足部是否位于接触面内部，我们借鉴拓扑学中的卷绕数概念，并利用该卷绕数及势场构建接触面惩罚函数。通过引入该惩罚函数，MPC可从邻近所有候选接触面中选取最优接触面，并确定其内部的落脚点位置。通过展示考虑全身动力学（包括关节力矩约束与肢体动力学）对落脚点及接触面选择的影响，我们验证了本方法的优势。此外，通过系列实验与仿真试验，我们验证了将基于拓扑的方法部署于MPC框架的可行性，并探索了其潜在应用能力。