Control of Humanoid Robots with Parallel Mechanisms using Differential Actuation Models

TL;DR

The study addresses control and learning for humanoid robots with parallel actuation by introducing a compact, differentiable actuation model that exactly captures the non-linear transmissions of knee and ankle mechanisms. This Actuated Serial model enables efficient first- and second-order derivatives for trajectory optimization and provides an analytical impedance transfer to motor space for reinforcement learning. By integrating the model into full-body trajectory optimization and RL, and validating on hardware, the approach yields higher accuracy and robustness than constant-ratio approximations and supports transferring serial-trained policies to hardware with actuator-space gains. The results demonstrate practical benefits for WB-MPC and RL in parallel-actuated humanoids, reducing computational overhead while expanding feasible motion envelopes and improving real-world performance.

Abstract

Several recently released humanoid robots, inspired by the mechanical design of Cassie, employ actuator configurations in which the motors are displaced from the joints to reduce leg inertia. While studies accounting for the full kinematic complexity have demonstrated the benefits of these designs, the associated loop-closure constraints greatly increase computational cost and limit their use in control and learning. As a result, the non-linear transmission is often approximated by a constant reduction ratio, preventing exploitation of the mechanism's full capabilities. This paper introduces a compact analytical formulation for the two standard knee and ankle mechanisms that captures the exact non-linear transmission while remaining computationally efficient. The model is fully differentiable up to second order with a minimal formulation, enabling low-cost evaluation of dynamic derivatives for trajectory optimization and of the apparent transmission impedance for reinforcement learning. We integrate this formulation into trajectory optimization and locomotion policy learning, and compare it against simplified constant-ratio approaches. Hardware experiments demonstrate improved accuracy and robustness, showing that the proposed method provides a practical means to incorporate parallel actuation into modern control algorithms.

Figures (12)

• Figure 1: Recent humanoids using parallel architectures. From top-left to bottom-right: Adam, A2, Atlas, T1, Digit V2, GR1, G1, H1, Walker S1 (Non exhaustive list)
• Figure 2: The Bipetto robot includes serial-parallel architecture, with parallel actuation for the knees and ankles.
• Figure 3: Planar 4-bar mechanism, with the serial link rotating around O, of angle $q_s$, motor rotating around M of angle $q_m$, B the attachment of the linkage on the lower limb and A the joint of the closed-loop linkage. A concrete example is given with the knee of the Bipetto robot.
• Figure 4: Sub projected planar four-bar from the ankle in purple. A concrete example is given on the right (ankle of the Bipetto robot)
• Figure 5: The constant serial gains generate an affine (with slopes $K_{Ps}$ and $K_{Ds}$) torque control in the joint-space (left). Computing the corresponding motor torques using the Actuation Model gives a non-linear control law (middle) that is approximated with a tangent plane (i.e. with correct $K_{Pm}$ and $K_{Dm}$) at the desired point $\tau_m^* = J_A^{-1}\tau_s^*$. Using the reference torque $\tau_m^*$ without feedback gains would lead to a horizontal plane which is a very wrong approximation of the curved manifold (in the middle).