Mind the Phase: Effective Rank and Representation Health in Legged Locomotion

Felipe Tommaselli; Thiago H. Segreto; Juliano D. Negri; Ricardo V. Godoy; Marcelo Becker

cs.ROSep 7, 2026

Mind the Phase: Effective Rank and Representation Health in Legged Locomotion

Authors: Felipe Tommaselli, Thiago H. Segreto, Juliano D. Negri, Ricardo V. Godoy, Marcelo Becker

Organizations: Mobile Robotics Group, University of S˜ao Paulo

Abstract

Reinforcement learning has become the leading paradigm in legged locomotion, enabling complex behaviors from backflips to parkour through massively parallel simulation. Under PPO's non-stationarity, shallow networks remain the de facto architecture, supported by carefully staged curricula and environments, yet the representations these policies learn stay poorly understood, leaving no training-time signal of how they will behave on hardware. In this work, we empirically study locomotion policies through the effective rank of the policy Jacobian and show that conditioning rank on the gait phase exposes architectural structure that global rank averages away. In particular, we find that standard architectural choices, namely layer normalization and residual connections, allocate roughly two more dimensions of effective rank to swing than to stance, which is fully absent in vanilla MLPs. Building on this, we propose a simple recipe that turns these representational signatures into smoother, more reliable sim-to-real transfer. In practice, this results in roughly 3x lower joint jitter that holds from simulation onto a physical Spot, suggesting that representation health is an effective training-time lens to track sim-to-real smoothness.

Figures & tables

Figure 1: Background: locomotion rollouts for the three simulated embodiments studied here, two quadrupeds and a humanoid. Center: real deployment on a Boston Dynamics Spot, walking on flat terrain. Website: https://felipe-tommaselli.github.io/MindThePhase/

Figure 2: Overview of the rank objects studied here, namely the actor weight matrices $W_{\ell}$ , the penultimate feature representation $F_{\theta}(X)$ , and the policy Jacobian $J_{\theta}(x_{i})$ measuring local input-output sensitivity.

Figure 3: Phase-conditioned Jacobian rank separates the architecture families. (a) Phase gap $\Delta_{\phi}=\mathrm{erank}(J_{\mathrm{swing}})-\mathrm{erank}(J_{\mathrm{stance}})$ on flat terrain. Every SimBa configuration is swing-dominant, and every MLP is not. (b) Pooled swing, stance, and global (dashed) Jacobian erank per family. The global values are nearly equal ( $9.3$ vs $9.2$ ), masking the split that phase conditioning exposes.

Figure 4: Jacobian matrix for the Simba policy. (a) Untrained matrix exhibiting near-random structure; (b) Trained matrix showing emergent primitive relationships.

Figure 5: Global feature effective rank for MLP versus SimBa across all three robots, with SimBa holding a consistent $\Delta\approx 20$ gap. Bars show mean $\pm$ spread over runs.

Figure 6: Sim-to-Real Transfer and Cross-Robot Generalization. (a) Spot joint control jitter % and (b) center-of-mass linear velocity tracking RMSE (m/s) in simulation (faded, ghost bars) and physical hardware deployment (solid bars) across MLP and SimBa architectures. (c) Simulation-only cross-robot control jitter across Boston Dynamics Spot (12-DoF), Anymal-D (12-DoF), and H1 (19-DoF biped). In physical hardware deployment, classical MLPs suffer severe velocity-tracking degradation and chattering, while SimBa maintains robust, low-jitter real-world transfer and generalizes consistently across quadrupeds and bipeds.

Figure 7: Representation collapse under over-training (healthy baseline desaturated, over-trained saturated). (a) Inflation: feature space erank (left axis) and functional Jacobian erank (right axis) inflate under over-training. SimBa feature rank explodes (+66%) while functional rank saturates. (b) Erasure: the specialized phase split $\Delta\phi$ collapses directly to the zero line for SimBa, while MLP remains flat near zero. (c) Blind reward: standard gait reward is insensitive to this representation collapse, with the Big SimBa config even showing a deceptive reward increase.

Figure 8: Gram matrix visualization under different training regimes. (a) Healthy training produces well-defined geometric structure, indicative of organized representation learning. (b) Collapsed training yields smooth, non-linear patterns, reflecting a loss of structure.

Appendix figures & tables6 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 9: Bayesian optimization search for learning rate on Simba Architecture.

Figure 10: Overall performance across task depths. We compare the classical MLP baseline and SimBa architecture. Markers denote model capacity (parameters, $M$ ) in logarithmic scale. Both in their actuation range show similar performance, which is the necessary condition for our analysis.

Figure 11: Boston Dynamics Spot used for the real-world experiments.

Category	Parameter	Value
Rollout	Steps per environment	$24$
	Parallel environments	$4{,}096$
	Buffer size	${\approx}\,49{,}152$
PPO	Clipping parameter $\epsilon$	$0.2$
	Value loss coefficient	$1.0$
	Entropy coefficient	$0.005$

Appendix

Table 1: Network architecture and PPO hyperparameters.

Term	Equation	Weight	Description
Base linear velocity	Eq. ( 5 )	$+5.0$	XY velocity tracking
Base angular velocity	Eq. ( 6 )	$+5.0$	Yaw rate tracking
Base angular vel. (fine)	Eq. ( 6 )	$+5.0$	Yaw rate tracking (high precision)
Gait	Eq. ( 7 )	$+10.0$	Enforces trotting via foot sync
Foot clearance	Eq. ( 11 )	$+0.5$	Encourages 0.1 m swing height
Action smoothness	Eq. ( 12 )	$-1.0$	Penalizes sudden action changes

Appendix

Table 2: Reward and penalty terms of the locomotion training objective.

Event	Stage	Range / Parameter
Ground Friction	Startup	Friction $\in[0.3,1.0]$
Base Mass Perturbation	Startup	$\Delta m\in[-2.5,2.5]$ kg
Arm Mass Perturbation	Startup	$\Delta m\in[-0.5,0.5]$ kg
Leg Mass Scaling	Startup	Scale $\in[0.8\times,1.5\times]$
Impulsive Wrench	Reset	$F=10$ N, $\tau=2$ Nm
Base Pose Reset	Reset	Yaw $\in[-\pi,\pi]$ , Pos $\in[-0.5,0.5]$ m

Appendix

Table 3: Domain randomization events applied during training.

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Learning-based approaches to locomotion have risen in popularity in recent years, showing the capability for complex legged locomotion and whole-body control. Reinforcement learning (RL), the primary learning-based approach for locomotion, often utilizes a high-performance simulation tool, providing a controlled and efficient training and development environment. However, policies that perform well in simulation frequently encounter unexpected challenges when deployed on a physical system, known as the sim-to-real gap. This work presents a robust RL locomotion framework capable of whole-body control. The proposed RL framework utilizes Nvidia's new set of simulation tools, Isaac Sim, and its companion RL framework, Isaac Lab, for training, achieving a zero-shot sim-to-real policy. The performance of our policy is validated on physical hardware using the Unitree Go1, with experimental results showing similar velocity tracking performance to the quadruped's integrated controller, with a greater ability to recover from large disturbances, and achieve linear velocities of 2.0 m/s and angular velocities of 1.8 rad/s.

Abstract

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