四足机器人支撑相模型预测控制策略研究

Research on Support Phase Model Predictive Control Strategy for Quadruped Robots

摘要: 针对四足机器人支撑相非线性动力学特性导致非线性模型预测控制(MPC)计算复杂度高、难以满足嵌入式系统实时控制需求的问题，本文提出一种适用于四足机器人支撑相的MPC控制策略。首先阐述了MPC的基本原理，基于腿部质量可忽略的核心假设建立四足机器人单刚体动力学模型；针对SO(3)流形上旋转矩阵带来的旋转动力学非线性问题，采用基于变分法的线性化方案完成系统线性化处理，通过克罗内克积实现矩阵变量的向量化转换，构建了包含轨迹跟踪目标与控制量平滑约束的二次代价函数，将摩擦锥约束线性化后结合执行器输出限制约束，最终把支撑相MPC优化问题转化为可高效求解的标准二次规划问题。通过MATLAB/Simulink仿真与自主研发的液压四足机器人原型机实物实验对控制策略进行验证，结果表明：在模拟崎岖路面的地面晃动扰动下，该策略可将机身横滚角、俯仰角最大误差分别控制在0.09 rad、0.08 rad以内，足端接触力变化平稳且严格满足摩擦锥与执行器出力约束；算法在ARM Cortex-A9嵌入式平台上单次优化求解平均耗时5.3 ms，最大耗时7.8 ms，完全满足10 ms采样周期的实时控制要求。该控制策略具备优异的姿态跟踪能力、抗扰动性能与工程实时性，能够有效实现四足机器人支撑相的稳定控制。

Abstract: To address the problem that the nonlinear dynamic characteristics of the support phase of quadruped robots result in high computational complexity of nonlinear model predictive control (MPC) and make it difficult to meet the real-time control requirements of embedded systems, this paper proposes an MPC strategy for the support phase of quadruped robots. Firstly, the basic principle of MPC is elaborated. Based on the core assumption that the mass of the legs is negligible, a single rigid body dynamic model of the quadruped robot is established. Aiming at the nonlinear problem of rotational dynamics introduced by rotation matrices on the SO(3) manifold, a linearization scheme based on the variational method is adopted to complete the linearization of the system. The matrix variables are vectorized via the Kronecker product, and a quadratic cost function containing the trajectory tracking objectives and smoothness constraints on the control variables is constructed. After linearizing the friction cone constraints and combining the output limit constraints of the actuators, the optimization problem of the support phase MPC is finally transformed into a standard quadratic programming problem that can be solved efficiently. The proposed control strategy is verified through MATLAB/Simulink simulations and physical experiments on a self-developed hydraulic quadruped robot prototype. The results show that underground shaking disturbances simulate rough terrain conditions, this strategy can limit the maximum errors of the robot body’s roll and pitch angles to within 0.09 rad and 0.08 rad, respectively. Meanwhile, the foot-end contact forces change smoothly and strictly satisfy the friction cone constraints and actuator output constraints. The algorithm achieves an average computation time of 5.3 ms for a single optimization solution on the ARM Cortex-A9 embedded platform, with a maximum time of 7.8 ms, which fully meets the real-time control requirements for a 10 ms sampling period. This control strategy has excellent attitude tracking capabilities, superior disturbance rejection performance, and good real-time performance for engineering applications, and can effectively realize stable control of the support phase of quadruped robots.