We provide an algorithm for adaptive legged locomotion via online learning and model predictive control. The algorithm is composed of two interacting modules: model predictive control (MPC) and online learning of residual dynamics. The residual dynamics can represent modeling errors and external disturbances. We are motivated by the future of autonomy where quadrupeds will autonomously perform complex tasks despite real-world unknown uncertainty, such as unknown payload and uneven terrains. The algorithm uses random Fourier features to approximate the residual dynamics in reproducing kernel Hilbert spaces. Then, it employs MPC based on the current learned model of the residual dynamics. The model is updated online in a self-supervised manner using least squares based on the data collected while controlling the quadruped. The algorithm enjoys sublinear \textit{dynamic regret}, defined as the suboptimality against an optimal clairvoyant controller that knows how the residual dynamics. We validate our algorithm in Gazebo and MuJoCo simulations, where the quadruped aims to track reference trajectories. The Gazebo simulations include constant unknown external forces up to $12\boldsymbol{g}$, where $\boldsymbol{g}$ is the gravity vector, in flat terrain, slope terrain with $20\degree$ inclination, and rough terrain with $0.25m$ height variation. The MuJoCo simulations include time-varying unknown disturbances with payload up to $8~kg$ and time-varying ground friction coefficients in flat terrain.

翻译：我们提出了一种通过在线学习和模型预测控制实现自适应腿式运动的算法。该算法由两个交互模块组成：模型预测控制（MPC）与残差动力学的在线学习。残差动力学可表征建模误差与外部扰动。我们的研究动机源于自主化的未来愿景——四足机器人需在未知载荷、不平坦地形等现实世界不确定因素下自主执行复杂任务。该算法采用随机傅里叶特征在再生核希尔伯特空间中逼近残差动力学，并基于当前学习到的残差动力学模型执行MPC。模型通过最小二乘法以自监督方式在线更新，所用数据来自四足机器人控制过程中的实时采集。本算法具有次线性动态遗憾度，其定义为相对于知晓残差动力学变化的最优预见控制器的性能次优性。我们在Gazebo与MuJoCo仿真环境中验证了算法有效性，其中四足机器人需跟踪参考轨迹。Gazebo仿真包含平坦地形、20°倾斜坡道地形及0.25m高度变化的崎岖地形中高达$12\boldsymbol{g}$（$\boldsymbol{g}$为重力矢量）的恒定未知外力。MuJoCo仿真则包含平坦地形中高达$8~kg$的时变未知载荷扰动与时变地面摩擦系数。