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arxiv: 2606.19504 · v1 · pith:YUJYJKV5 · submitted 2026-06-17 · cs.RO · cs.SY· eess.SY

Simulating Robotic Locomotion in Sand: Resistive Force Theory in an Open-Source Physics Engine

Ryan Walker Brown , Laura K. Treers , Kathryn A. Daltorio This is my paper

Reviewed by Pith T0 review T1 audit T2 compute T3 formal T4 reserved 2026-06-26 20:43 UTCgrok-4.3pith:YUJYJKV5record.jsonopen to challenge →

Figure 1
Figure 1. Figure 1: Walking of legged robots in granular media is modeled using MuJoCo. As a physical robot walks through sand (top), the legs and feet penetrate the ground at different angles throughout the gait cycle, especially when taking advantage of crab-like pointed end effectors (dactyls). To simulat… reproduced from arXiv: 2606.19504
classification cs.RO cs.SYeess.SY
keywords resistive force theoryrobotic locomotiongranular mediaphysics simulationMuJoCohexapod robotsand
0
0 comments X

The pith

Implementation of 3D resistive force theory in MuJoCo predicts hexapod robot walking distance and foot sinkage in sand within 20 percent of experiments.

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

The paper implements 3D Granular Resistive Force Theory in the MuJoCo physics engine to approximate ground reaction forces during locomotion in sand. This approach avoids the high computational cost of modeling individual grains while integrating with standard dynamics. The authors verify that the simulations preserve trends in end effector shape, speed, and loading. They show that the model predicts walking distance and foot sinkage for a 12-degree-of-freedom hexapod robot within 20 percent of real experiments. This matters because it offers an efficient tool for testing and improving robot designs that must traverse granular media.

Core claim

We implement 3D Granular Resistive Force Theory in MuJoCo. The resulting simulations preserve key trends due to end effector shape, speed, and loading. Our implementation predicts walking distance and foot sinkage of a 12-Degree of Freedom hexapod robot within 20% of experiments in sand.

What carries the argument

3D Granular Resistive Force Theory (3D RFT) integrated with MuJoCo's standard dynamics calculations, which approximates ground reaction forces to simulate a stable substrate for walking robots.

If this is right

  • The simulations preserve trends due to end effector shape, speed, and loading.
  • The open-source tool can help develop new and improved robot designs to traverse granular media substrates.
  • Resistive force approximations provide a stable substrate for a freely walking robot when integrated with standard dynamics.
  • Key performance metrics like walking distance and foot sinkage can be predicted accurately enough for practical use.

Where Pith is reading between the lines

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

  • This approach may reduce the need for expensive physical prototyping when designing robots for sandy or granular environments.
  • Similar integrations could be tested in other physics engines to increase accessibility for researchers.
  • Extending the model to other types of granular media or more complex robot gaits could be a next step.
  • The 20% accuracy level might suffice for initial design iterations but may require refinement for precise control tasks.

Load-bearing premise

Resistive force approximations are sufficient, when integrated with standard dynamics calculations, to provide a stable substrate for a freely walking robot.

What would settle it

Running the implemented simulation on the 12-DoF hexapod and comparing the predicted walking distance and foot sinkage to physical experiments in sand; a discrepancy larger than 20% would falsify the prediction accuracy claim.

Figures

Figures reproduced from arXiv: 2606.19504 by Kathryn A. Daltorio, Laura K. Treers, Ryan Walker Brown.

Figure 2
Figure 2. Figure 2: Resistive Force Theory, RFT, is a fundamentally different approach for predicting ground contact forces, which estimates intrusion forces on rigid bodies passing through sandy ground. (Left) MuJoCo’s fast default handling of many contacts can be attributed to key simplifications of Coulomb friction. (A) First, the geometry is simplified and contact points where rigid convex hulls intersect are identified. … view at source ↗
Figure 3
Figure 3. Figure 3: When rotating an irregular object at constant angular velocity, RFT simulation correctly predicts increases in required actuation torque when the object is in sand. (A) We verified the RFT implementation by rotating an irregular object 180◦ about a fixed point such that the object enters and exits a flat bed of sand. (B) During rotation, we measured the torque required for the Dynamixel actuator to maintai… view at source ↗
Figure 4
Figure 4. Figure 4: When simple legs of different shape are rotated through the sand, RFT predicts motion along a fixed-height rail, with total displacements agreeing within 15%. (A) Carriage displacement vs. intruder rotation in both simulation and experiment is shown. Experimental data represents the mean and standard deviation over 5 trials. When the leg is a flat rectangular intruder, the carriage begins to move as the in… view at source ↗
Figure 5
Figure 5. Figure 5: The simulated articulated leg pulls the supporting carriage along the guide rail by using an end effector stepping into and out of sand. (A) Time evolution of the guide rail carriage in simulation and experimental trials are shown. (B) Final carriage displacement after three steps of the articulated leg are shown at different simulated ζ. The third baseline test compared the carriage motion when propelled … view at source ↗
Figure 6
Figure 6. Figure 6: We used our implementation to predict the motion of a walking hexapod robot. (A) The legs and dactyl surfaces are discretized into plate elements, each with RFT “sites” (gray dots), with the deepest points generating the most force (graded in red). (B) Translation of the robot without payload is tracked over six steps. (C) Mean sinkage vs. resistive coefficient for three different payloads is shown. The da… view at source ↗
Figure 7
Figure 7. Figure 7: RFT (ζ = 3.75) captures the local maxima in walking speed better than existing options in MuJoCo. We plot average walking speed vs. payload for five different simulation conditions, and compare experimental and simulation results. Errors represent one standard deviation. (A) We simulated additional payload increments (gray bars) with ζ = 3.75 to show refinement of trend in previous figure (colored bars). T… view at source ↗
read the original abstract

Recent advancements in Resistive Force Theory (RFT) enable approximation of ground reaction forces for locomotion in sand without the computational expense of modeling interactions with individual grains. However, these tools have been absent in 3D physics engines commonly used for robot simulation. We explore if resistive force approximations are sufficient, when integrated with standard dynamics calculations, to provide a stable substrate for a freely walking robot. To determine this, we implement 3D Granular Resistive Force Theory (3D RFT) in a physics simulation engine, MuJoCo. We verify simulations in multiple scenarios to demonstrate that key trends due to end effector shape, speed, and loading are preserved. Our implementation predicts walking distance and foot sinkage of a 12-Degree of Freedom hexapod robot within 20\% of experiments in sand. While RFT has inherent approximations, the open source tool described here has potential to help develop new and improved robot designs to traverse granular media substrates.

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 paper implements 3D Granular Resistive Force Theory (3D RFT) inside the MuJoCo physics engine to approximate ground-reaction forces during locomotion in sand. It reports that the implementation preserves qualitative trends with end-effector shape, speed and loading, and that a 12-DOF hexapod simulation reproduces experimental walking distance and foot sinkage to within 20%. The central claim is that resistive-force approximations, once inserted into a standard dynamics solver, supply a sufficiently stable substrate for free walking.

Significance. An open-source RFT module in a widely used engine would lower the barrier to simulating granular locomotion and could accelerate design iteration for robots that traverse sand or regolith. The work supplies reproducible code, which is a concrete strength.

major comments (3)
  1. [Hexapod locomotion results] Hexapod results paragraph: the statement that distance and sinkage lie 'within 20% of experiments' supplies neither error bars, the number of trials, nor any description of how RFT parameters were chosen or cross-validated; without these the quantitative claim cannot be assessed.
  2. [Verification scenarios] Verification section: preservation of trends with shape, speed and load is asserted but no quantitative metric (e.g., relative error, statistical test) or comparison against a null model is given, leaving the claim that RFT integration yields a 'stable substrate' unsupported by the reported data.
  3. [Discussion and limitations] Discussion: the manuscript does not examine whether resistive-force vectors remain accurate under stance-phase acceleration, lateral slip, or multi-foot loading sequences that differ from the validation set; such mismatches could accumulate into instability while still keeping scalar distance/sinkage within 20%.
minor comments (2)
  1. [Abstract] Abstract: the phrase 'key trends due to end effector shape, speed, and loading' is never defined; a short enumeration of the trends would improve clarity.
  2. [Figures] Figure captions: several plots lack axis units or legend entries for the experimental versus simulated curves.

Simulated Author's Rebuttal

3 responses · 0 unresolved

Thank you for the opportunity to respond to the referee's report on our manuscript. We address each major comment point by point below.

read point-by-point responses

  1. Referee: [Hexapod locomotion results] Hexapod results paragraph: the statement that distance and sinkage lie 'within 20% of experiments' supplies neither error bars, the number of trials, nor any description of how RFT parameters were chosen or cross-validated; without these the quantitative claim cannot be assessed.

    Authors: We agree with the referee that these details are necessary for assessing the claim. The revised manuscript will include error bars from repeated trials (n=5), specify the number of trials performed, and describe the RFT parameter selection and cross-validation process based on single-leg experiments. This addresses the concern directly. revision: yes

  2. Referee: [Verification scenarios] Verification section: preservation of trends with shape, speed and load is asserted but no quantitative metric (e.g., relative error, statistical test) or comparison against a null model is given, leaving the claim that RFT integration yields a 'stable substrate' unsupported by the reported data.

    Authors: The verification section shows preservation of trends via comparative plots. To better support the stability claim, we will add quantitative relative error metrics for the trends and include a comparison to the default contact model in MuJoCo as a null baseline in the revised manuscript. revision: yes

  3. Referee: [Discussion and limitations] Discussion: the manuscript does not examine whether resistive-force vectors remain accurate under stance-phase acceleration, lateral slip, or multi-foot loading sequences that differ from the validation set; such mismatches could accumulate into instability while still keeping scalar distance/sinkage within 20%.

    Authors: We acknowledge this limitation in the current validation. The discussion will be expanded to explicitly discuss these conditions as potential sources of error accumulation, while maintaining that the 20% agreement holds for the tested steady locomotion. This will clarify the scope without overclaiming. revision: yes

Circularity Check

0 steps flagged

No significant circularity; validation uses independent physical experiments

full rationale

The paper implements 3D RFT inside MuJoCo and reports that simulated hexapod walking distance and foot sinkage match separate physical experiments within 20%. The abstract and reader's summary indicate direct comparison to external data rather than fitting any parameter to the same trajectories later presented as predictions. No equations, self-citations, or ansatzes are shown that would make the reported accuracy tautological by construction. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no equations or parameter tables are supplied, so the ledger cannot list specific fitted values or axioms. The central claim rests on the unstated assumption that the RFT force model remains valid when coupled to rigid-body dynamics inside MuJoCo.

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