Task-Space Riccati Feedback based Whole Body Control for Underactuated Legged Locomotion

TL;DR

This work addresses the challenge of tuning floating-base feedback in whole-body control for underactuated legged robots by deriving a Riccati-based task-space feedback from a linearized unactuated model and incorporating friction-cone constraints via a log-barrier approximation. The proposed method replaces heuristic user-tuned gains with a time-varying Riccati gain $F^{\text{lqr}}_k$ obtained from a constrained LQR problem around a feasible trajectory, integrated into the QP-based inverse dynamics framework. Validation on MuJoCo simulations with a point biped and hardware experiments on a quadruped demonstrate consistent improvements in linear and angular velocity tracking and a reduction in oscillations, while substantially lowering parameter tuning effort. The approach leverages model information to capture underactuation and contact constraints, offering practical gains for real-time WBC, though its linear model is best within trajectories close to the reference and benefits from replanning when deviations grow.

Abstract

This manuscript primarily aims to enhance the performance of whole-body controllers(WBC) for underactuated legged locomotion. We introduce a systematic parameter design mechanism for the floating-base feedback control within the WBC. The proposed approach involves utilizing the linearized model of unactuated dynamics to formulate a Linear Quadratic Regulator(LQR) and solving a Riccati gain while accounting for potential physical constraints through a second-order approximation of the log-barrier function. And then the user-tuned feedback gain for the floating base task is replaced by a new one constructed from the solved Riccati gain. Extensive simulations conducted in MuJoCo with a point bipedal robot, as well as real-world experiments performed on a quadruped robot, demonstrate the effectiveness of the proposed method. In the different bipedal locomotion tasks, compared with the user-tuned method, the proposed approach is at least 12% better and up to 50% better at linear velocity tracking, and at least 7% better and up to 47% better at angular velocity tracking. In the quadruped experiment, linear velocity tracking is improved by at least 3% and angular velocity tracking is improved by at least 23% using the proposed method.

Figures (5)

• Figure 1: Illustration of the proposed approach for stabilizing the floating-base trajectory $\bm{y}_b(t)$ along its reference $\bm{y}_b^*(t)$. For floating-base task, the user-tuned PD feedback is repalced by a Riccati-based feedback. The Riccati gain is derived by considering the impact of friction-cone constraints and the column space of the contact Jacobian within the formualted LQR.
• Figure 2: (a) Bipedal robot P1 used for simulation tests (b) Quadruped robot A1 for experiments.
• Figure 3: Framework for robot control encompassing state estimation, reference generation(planning), and low-level controller. The control module integrates convex MPC, LQR in eq. \ref{['opt: constrained_LQR_approx']} and whole-body controller.
• Figure 4: Linear velocity in quadruped hardware Test 7 (Blue: measured, Red: reference).
• Figure 5: Angular velocity in quadruped hardware Test 9 (Blue: measured, Red: reference).