MS-PPO: Morphological-Symmetry-Equivariant Policy for Legged Robot Locomotion

Abstract

Reinforcement learning has recently enabled impressive locomotion performance on quadrupeds and other articulated robots, yet most policy architectures remain morphology- and symmetry-agnostic, leading to inefficient training and weak generalization. This work introduces MS-PPO, a morphology-symmetry-equivariant policy learning framework that encodes robot kinematic structure and morphological symmetries directly into the policy network. We construct a graph-based neural architecture that is provably equivariant to morphological group actions, ensuring consistent responses under sagittal-plane reflections while maintaining invariance in value estimation. This design eliminates the need for reward shaping or data augmentation, which are typically required to enforce symmetry. We evaluate MS-PPO in simulation on Unitree Go2 and Xiaomi Cyberdog2 robots across multiple locomotion tasks, including trotting, pronking, slope walking, and bipedal turning, and further deploy policies on hardware. Extensive experiments show that MS-PPO achieves superior training stability, command generalization ability, and sample efficiency in challenging tasks, compared to state-of-the-art baselines. These findings demonstrate that embedding both kinematic structure and morphological symmetry into policy learning provides a powerful inductive bias for legged robot locomotion control.

---

Method

Overall framework of MS-PPO: Common and privileged observations are first converted into a graph data structure where the graph is constructed from the robot kinematic connectivity. These are then fed into the MS-GNN-Inv critic network to predict the state value, while common observations are input to the MS-GNN-Equ actor network to output the mean target joint position at each joint. By design, MS-PPO imposes hard symmetry constraints on the policy learning.

Walk-to-One-Side with Go2 Robot

In this task, we evaluate the task-level symmetry performance of the learned policies on Go2 robot. The locomotion policy is trained using commands restricted to one side, i.e., $c_x \in [-1,0, 1.0] (m/s)$, $c_y \in [0.0, 0.6] (m/s)$, $c_{\text{yaw}}\in[0.0, 1.0](rad/s)$, and evaluated under commands in both directions to validate symmetry generalization.

Bipedal Locomotion with Cyberdog2 Robot

Walk-on-Slope: This task requires the Cyberdog2 robot to stand up and walk on an inclined flat surface using two rear feet.

Stand-and-Turn: This task requires the Cyberdog2 robot to stand up on two feet and follow the input heading commands.

Real-World Experiments with Go2 Robot

We deployed the trot gaits of the four policies on a physical Unitree Go2 robot to test sim-to-real adaptation. Each policy was tested with $c_\omega = \pm 1$ $rad/s$ and $c_x=c_y=0$ $m/s$ via a remote controller for 15 seconds. Thus, the desired behavior is in-place rotation with no drift. MS-PPO achieved nearly in-place turning for both commands, PPO-EMLP exhibited moderate drift, consistent with its behavior in the training set, where the robot is turning left. PPO-MLP and MI-PPO, however, showed significant drift. Furthermore, MI-PPO failed to maintain a stable gait or accurately track the command velocities. These results are consistent with our observation in the simulation test and demonstrate MS-PPO's superior capability for out-of-distribution generalization on physical hardware.