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arxiv: 2604.02744 · v1 · submitted 2026-04-03 · 💻 cs.RO

Recognition: 2 theorem links

· Lean Theorem

Learning Locomotion on Complex Terrain for Quadrupedal Robots with Foot Position Maps and Stability Rewards

Matthew Hwang , Yubin Liu , Ryo Hakoda , Takeshi Oishi
Authors on Pith no claims yet

Pith reviewed 2026-05-13 20:30 UTC · model grok-4.3

classification 💻 cs.RO
keywords quadrupedal locomotionreinforcement learningfoot position mapstability rewardcomplex terraingeneralizationout-of-domainattention-based policy
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The pith

Adding foot position maps to heightmaps and stability rewards to policies lets quadrupedal robots walk more precisely and stably on complex terrains.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper tries to establish that embedding an explicit foot position map into the terrain heightmap and adding a dynamic locomotion-stability reward to an attention-based reinforcement learning policy produces more precise and stable quadrupedal movement. Standard RL methods infer foot placements only implicitly from joint angles, which limits accuracy on rough ground. A sympathetic reader would care because reliable navigation over unpredictable surfaces is essential for practical robot deployment. The work validates the approach through extensive tests on both training terrains and out-of-domain ones, reporting higher success rates.

Core claim

By integrating a foot position map into the heightmap and employing a locomotion-stability reward in an attention-based policy, the method achieves precise and stable quadrupedal locomotion on complex terrain, with demonstrated improvements in success rates for both in-domain and out-of-domain cases.

What carries the argument

The foot position map integrated into the heightmap observation together with a dynamic locomotion-stability reward inside an attention-based reinforcement learning framework, supplying explicit placement data and stability signals to the policy.

If this is right

  • Locomotion success rates increase on terrains seen during training.
  • Performance improves on out-of-domain terrains not encountered in training.
  • Foot placement becomes more precise than in policies that infer positions only from joint angles.
  • Movement stability rises during traversal of complex surfaces.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The explicit map could narrow the difference between learning-based and classical optimization methods for foot placement.
  • Similar observation and reward structures might transfer to other legged platforms with minimal redesign.
  • Further tests with altered terrain generation parameters in simulation would help isolate whether gains are truly distribution-robust.

Load-bearing premise

That the foot position map and stability reward produce genuine generalization to new terrains rather than overfitting to the specific simulation distributions and reward shaping.

What would settle it

Transferring the trained policy to a physical quadruped and measuring locomotion success rates on real-world complex terrains that differ from simulation; a large drop relative to simulation results would falsify the generalization claim.

Figures

Figures reproduced from arXiv: 2604.02744 by Matthew Hwang, Ryo Hakoda, Takeshi Oishi, Yubin Liu.

Figure 1
Figure 1. Figure 1: Overview of our method. With the proposed foot posi [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: We propose a foot position map which is concatenated with the heightmap providing information of the foot positions [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: CoP-based stability rewards. We calculate the mini [PITH_FULL_IMAGE:figures/full_fig_p002_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The training environment consists of smooth, rough, [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The success rate results. (a) Our proposed method produces higher success rates across the board for all terrains. (b) [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Qualitative locomotion results of our proposed method on different terrains. (a) stones + stairs up, (b) stairs up, (c) [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Qualitative locomotion results of our method on [PITH_FULL_IMAGE:figures/full_fig_p006_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: We implement stability rewards based on the CoM [PITH_FULL_IMAGE:figures/full_fig_p006_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: When using global velocity tracking the issue of [PITH_FULL_IMAGE:figures/full_fig_p007_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Locomotion results on gaps (top) of 20cm and stairs [PITH_FULL_IMAGE:figures/full_fig_p007_10.png] view at source ↗
read the original abstract

Quadrupedal locomotion over complex terrain has been a long-standing research topic in robotics. While recent reinforcement learning-based locomotion methods improve generalizability and foot-placement precision, they rely on implicit inference of foot positions from joint angles, lacking the explicit precision and stability guarantees of optimization-based approaches. To address this, we introduce a foot position map integrated into the heightmap, and a dynamic locomotion-stability reward within an attention-based framework to achieve locomotion on complex terrain. We validate our method extensively on terrains seen during training as well as out-of-domain (OOD) terrains. Our results demonstrate that the proposed method enables precise and stable movement, resulting in improved locomotion success rates on both in-domain and OOD terrains.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript presents a reinforcement learning approach for quadrupedal robot locomotion on complex terrain. It integrates a foot position map into the heightmap input and incorporates a dynamic locomotion-stability reward within an attention-based policy network. The method is evaluated on terrains encountered during training as well as out-of-domain terrains, with the central claim being that this combination enables more precise and stable locomotion, yielding higher success rates compared to prior methods.

Significance. Should the quantitative improvements be substantiated, the work would represent a meaningful step toward combining the adaptability of learning-based methods with the precision of optimization-based foot placement in robotic locomotion. The use of explicit foot position mapping and stability rewards could help address common failure modes in RL policies on uneven terrain. The emphasis on OOD generalization is particularly relevant for real-world deployment, though the current presentation leaves the magnitude of the advance unclear.

major comments (3)
  1. Abstract: The assertion of improved locomotion success rates on in-domain and OOD terrains supplies no quantitative results, ablation studies, error bars, or formulation of the stability reward. This is load-bearing for the central claim and prevents verification of the reported gains.
  2. Experiments: The OOD terrains are described as procedurally generated heightmaps with similar roughness and slope ranges to the training distribution, but no explicit distribution-shift metric (e.g., Wasserstein distance on local curvature or frequency content) is provided. This weakens the generalization claim and leaves open the possibility that gains arise from shared statistics rather than the foot-position map or stability reward.
  3. Method/Experiments: No ablation is reported that removes only the foot position map and stability reward while holding the attention-based backbone and training budget fixed. Without this isolation, it is impossible to attribute any success-rate lift specifically to the proposed mechanisms.
minor comments (2)
  1. Abstract: The phrase 'dynamic locomotion-stability reward' is introduced without a mathematical definition or reference to its components (e.g., which stability criteria are encoded).
  2. Method: Clarify the precise integration of the foot position map into the heightmap observation (dimensionality, encoding, and update frequency).

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the thoughtful and constructive feedback. We address each major comment below and commit to revisions that strengthen the manuscript's clarity and rigor without altering its core contributions.

read point-by-point responses

  1. Referee: Abstract: The assertion of improved locomotion success rates on in-domain and OOD terrains supplies no quantitative results, ablation studies, error bars, or formulation of the stability reward. This is load-bearing for the central claim and prevents verification of the reported gains.

    Authors: We agree that the abstract should better substantiate the central claims. In the revised version we will expand the abstract to report specific success-rate improvements (including standard deviations across runs) on both in-domain and OOD terrains, along with a concise statement of the stability-reward formulation. The detailed reward definition already appears in Section 3.2 and the full ablation results in Section 4.3; these will be referenced explicitly in the updated abstract. revision: yes

  2. Referee: Experiments: The OOD terrains are described as procedurally generated heightmaps with similar roughness and slope ranges to the training distribution, but no explicit distribution-shift metric (e.g., Wasserstein distance on local curvature or frequency content) is provided. This weakens the generalization claim and leaves open the possibility that gains arise from shared statistics rather than the foot-position map or stability reward.

    Authors: We acknowledge that an explicit shift metric would strengthen the OOD claims. We will add a new paragraph in the Experiments section that quantifies the distributional difference using mean/variance of local slopes, roughness (standard deviation of height gradients), and frequency content via Fourier analysis between the training and OOD heightmap sets. While we did not originally compute Wasserstein distances on curvature, the added statistics will allow readers to assess the degree of shift. revision: yes

  3. Referee: Method/Experiments: No ablation is reported that removes only the foot position map and stability reward while holding the attention-based backbone and training budget fixed. Without this isolation, it is impossible to attribute any success-rate lift specifically to the proposed mechanisms.

    Authors: We agree that a controlled ablation isolating these two components is necessary. In the revision we will add a dedicated ablation table (new Table X) that evaluates four configurations—full method, without foot-position map, without stability reward, and without both—while keeping the attention-based policy architecture, observation space, and training budget identical. The new results will be reported with the same success-rate metric and error bars used in the main experiments. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in derivation or claims

full rationale

The paper introduces a foot position map integrated into the heightmap and a dynamic stability reward inside an attention-based RL policy for quadrupedal locomotion. No equations, derivations, or parameter-fitting steps are described that reduce the reported success rates or generalization claims to the inputs by construction. The validation on in-domain and OOD terrains is presented as empirical measurement of policy performance, not as a quantity defined from the same reward weights or fitted parameters used in training. No self-citations are invoked as load-bearing uniqueness theorems, and no ansatz or renaming of known results is used to support the central claims. The method is self-contained against external benchmarks with no circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The central claim rests on the unstated assumptions that the simulation environment accurately models real terrain contact dynamics and that the attention mechanism can effectively use the added foot-position channel without additional training instabilities.

pith-pipeline@v0.9.0 · 5423 in / 1124 out tokens · 25581 ms · 2026-05-13T20:30:34.458365+00:00 · methodology

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Reference graph

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