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

arXiv:2606.19504 · cs.RO, cs.SY, eess.SY · Submitted 2026-06-17 · Read on arXiv

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Introduction to the show: ident: Robotics Radio. Generated commentary on the latest robotics and control papers.

Rosa: Today's paper: "Simulating Robotic Locomotion in Sand".

Dev: 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.

Rosa: First, who's behind it and why it matters.

Paper summary: Rosa: Well, Dev, we're looking at this paper titled "Simulating Robotic Locomotion in Sand: Resistive Force Theory in an Open-Source Physics Engine," and it seems like the central thesis is that Resistive Force Theory can approximate ground reaction forces for locomotion in sand without needing to model every single grain individually.

Rosa: I'm really interested to know if this approximation holds up when we actually put it into a standard physics engine, which is where my question comes from about its real-world applicability and how long these simulations are stable outside of a controlled lab setting.

Dev: That's a fair question, Rosa; from my end, the crucial part is whether this three dee Granular Resistive Force Theory implementation in MuJoCo provides a stable substrate for freely walking robots when we run it at appropriate loop rates and latency.

Dev: The paper claims they are verifying simulations in multiple scenarios to show that key trends based on end effector shape, speed, and loading are preserved within twenty percent of the actual experiments done in sand.

Taro: I'm curious about what this means for autonomy; if the model can capture those trends within a twenty percent margin, does it give us a more reliable baseline for when our systems encounter unexpected ground conditions in unstructured environments?

Taro: It seems like they are trying to bridge the gap between theoretical granular mechanics and practical robot movement.

Rosa: Exactly, Taro; that twenty percent preservation of trends is what makes this interesting because it suggests that we can predict important behaviors without the massive computational expense of modeling every grain interaction.

Rosa: It really matters if those predictions translate into useful designs for robots operating in unpredictable environments where sand might be present.

Dev: From a control engineering standpoint, I'm watching how they handled the low-velocity issues; they used a single constant, the resistive coefficient zeta, to calibrate different soils across a wide range of grain sizes and densities.

Dev: Plus, they included a smoothing factor of zero point one for the overall sorted force matrix to ensure continuous motion at low velocities where force directions are ill-defined.

Taro: That smoothing factor is interesting; it sounds like a practical solution to keep the system from getting stuck or exhibiting unrealistic behaviors when the dynamics get fuzzy, which is something we deal with constantly in autonomous systems when sensor data becomes noisy.

Taro: It shows they thought about the actual implementation challenges beyond just getting the core RFT equations right.

Rosa: I agree, Taro; tackling those practical implementation hurdles is what separates a simulation that just looks good on paper from one that might actually be useful for deploying hardware in messy real-world situations.

Rosa: The method involves discretizing geometry into mesh plates and calculating forces based on plate orientation and depth, which feels like a solid way to translate the continuous granular idea into something a physics engine can handle.

Paper summary: Dev: That discretization process, where they use a mesh density of zero point zero three plates per square millimeter for debugging and visualization purposes, tells me they were careful about balancing accuracy against computational load within MuJoCo.

Dev: They also showed that the implementation of Treers et al.'s formulation was compatible because it was developed as an open-source function capable of determining forces on any meshed input geometry.

Taro: It's important to remember the paper flagged a known limitation for RFT, specifically its inability to model granular jamming and shear behavior accurately, which is something that could be a big hurdle if our robots need to operate in very dense or highly compacted sand.

Taro: So, while it captures the general trends well, we still have that gap in modeling the extreme states of granular matter.

Rosa: That limitation is important because it tells us where this tool might not be sufficient on its own; it's a powerful approximation for typical locomotion but stops short when dealing with very complex jamming scenarios.

Rosa: However, the fact that they predicted walking distance and foot sinkage of a twelve-Degree of Freedom hexapod robot within twenty percent of experiments in sand is a substantial achievement for this open-source tool.

Dev: That quantitative result, predicting those metrics within twenty percent, is what makes me want to look closer at the simulation setup and validation they described.

Dev: They used specific tests, like measuring torque increase when rotating an object through sand, or testing an articulated leg following a three dee path to pull a carriage along a rail.

Taro: Those validation tests are key because they show the model isn't just predicting abstract forces; it's predicting actual physical outcomes for complex systems like multi-DOF robots navigating different terrains.

Taro: If we can trust the predictions for those specific tasks, it gives us confidence in using this framework to guide autonomous movement planning.

Rosa: It really suggests that this work has potential to help develop new and improved robot designs specifically tailored for traversing granular media, which is a huge area in robotics research right now.

Rosa: The implication here is that we can accelerate the design cycle by having a more accurate way to predict how robots will interact with sand before we build expensive physical prototypes.

Dev: I'm thinking about the long-term impact on our simulation infrastructure; if this RFT-SiM framework proves stable and accurate enough, it could become a standard library component for simulating granular locomotion across various physics engines.

Dev: That would significantly reduce the time and computational resources needed to set up realistic sand simulations for robotics teams worldwide.

Taro: On a broader level, I think the ability to simulate these interactions more efficiently could open doors for developing robots that are truly capable of operating robustly in environments where ground conditions are highly variable and unpredictable.

Taro: That moves us closer to having autonomous systems that can handle real-world unpredictability without needing exhaustive pre-programming for every single sand type.

Rosa: So, looking at the title, "Simulating Robotic Locomotion in Sand: Resistive Force Theory in an Open-Source Physics Engine," it really captures the essence of what they did: applying a theoretical concept to a practical simulation tool.

Paper summary: Rosa: The authors are making this framework available open source, which is fantastic for community development because it allows other researchers to build on their work immediately.

Dev: And from my perspective as an engineer, the fact that they integrated it directly into MuJoCo and managed the required discretization and smoothing gives us a concrete pipeline we can analyze for performance issues down the line.

Dev: We need to keep an eye on those latency concerns, even with approximations in place.

Taro: That opens up a lot of avenues for future work; since they've established this baseline, the next logical step would be to incorporate better models for granular jamming and shear as they mentioned are currently missing from the RFT formulation.

Taro: That would take this framework from a great approximation tool to a more complete simulation environment for sand locomotion.

Rosa: I think that direction is where the real exciting potential lies; moving beyond just capturing trends to modeling the underlying physics more deeply, even if it adds complexity to the implementation.

Rosa: This paper gives us a solid foundation, and I'm optimistic about what we can build from this open-source starting point for future robotic applications in granular terrain.

Dev: So, in summary, this work successfully integrates three dee RFT into MuJoCo to predict locomotion trends within twenty percent accuracy across different shapes and speeds.

Dev: It addresses the computational cost issue by using force approximations instead of grain-level modeling.

Taro: And while it doesn't solve all granular complexities, it provides a very useful tool for autonomous system designers needing reliable ground reaction force predictions in sandy settings.

Taro: The ability to predict walking distance and sinkage metrics is valuable data for planning robust trajectories.

Rosa: It seems the main implication is that we gain a scalable way to test robot designs in sand without needing massive computational power, which really opens up possibilities for more rapid prototyping of locomotion systems.

Rosa: We're looking at a strong starting point here for researchers interested in granular robotics applications outside of just lab settings.

Dev: From my side, the stability demonstrated by using a constant resistive coefficient zeta and the exponential moving average smoothing gives us something tangible to work with regarding system robustness and how it handles low-velocity states.

Dev: We have concrete parameters now to test for failure modes during high-speed locomotion tests.

Taro: I just think that having this framework in an open-source physics engine means that the community can start experimenting with novel locomotion strategies on sand right away, rather than waiting for highly specialized simulation software.

Taro: It democratizes access to realistic granular environment simulation for autonomy research.

Rosa: It’s a really encouraging piece of work because it shows that complex physical phenomena can be approximated effectively when coupled with standard dynamics calculations, which is always the goal in robotics.

Paper summary: Rosa: This paper provides a solid reference point for anyone trying to model robot interaction with sand accurately enough to make real-world decisions.

Dev: So, we've seen how the three dee RFT implementation performs in terms of preserving key trends like speed and loading, even though it has known limitations regarding granular jamming.

Dev: It’s a functional tool that offers significant computational savings for complex locomotion simulations involving sandy substrates.

Taro: And as I said earlier, the limitation regarding granular jamming means we have a clear path forward for future research to enhance the model's fidelity in those extreme soil conditions, which is where the next big challenge lies.

Taro: This paper sets a very good benchmark for where we need to focus our efforts next in developing more comprehensive models.

Rosa: So, to wrap up these points from "Simulating Robotic Locomotion in Sand: Resistive Force Theory in an Open-Source Physics Engine," this research provides a practical, accessible way to simulate robot movement on sand using three dee RFT within MuJoCo.

Rosa: The key findings are that the model preserves important trends like walking distance and sinkage within twenty percent of experimental results for various robot geometries and speeds.

Dev: And the methodology relies on discretizing geometry into plates, calculating intrusion forces based on plate orientation, depth z, and area A, and using smoothing to manage low-velocity force vectors.

Dev: This provides a concrete technical path for implementing granular resistance in simulation software.

Taro: The implications point toward accelerating the development of more robust autonomous systems capable of navigating unstructured environments by providing a reliable, though approximated, model for ground interaction in sand.

Taro: It's a practical tool that moves us closer to testing and validating locomotion strategies in realistic sandy settings sooner than we could otherwise.

Rosa: I think the title itself summarizes the core contribution well: applying Resistive Force Theory to simulate robot locomotion on sand using an open-source physics engine, which is a very clear and useful description.

Rosa: We're really excited about how this work can serve as a stepping stone for more advanced granular robotics research and development in the near future.

Dev: It’s definitely worth keeping an eye on this framework as we look at integrating new force modeling techniques into our simulation loops, provided we can manage the latency associated with its calculations efficiently.

Dev: The open-source nature means we can scrutinize the implementation for stability and performance ourselves.

Taro: I think this paper is a valuable resource because it shows that even with approximations, a well-structured physics model can yield results that are quantitatively comparable to real physical experiments in certain aspects of locomotion.

Taro: It’s about building reliable predictive models, which is fundamental for autonomous decision-making.

Rosa: So, the overall picture is one where this paper successfully demonstrates that RFT approximations can be integrated into standard dynamics calculations to provide a stable simulation substrate for walking robots on sand.

Rosa: It's a significant step in making granular robotics simulation more accessible and efficient for the wider research community.

Conclusion: Rosa: So, to wrap up, this paper shows how they’ve managed to bake Resistive Force Theory into MuJoCo so we can simulate robots walking in sand without modeling every single grain individually. Dev, what are your initial thoughts on that title and who wrote it?

Dev: I see the core idea is using these force approximations to bypass the computational nightmare of grain-level modeling, which is exactly what I'm interested in from a control standpoint. The authors are making this framework open-source, which means we can actually look at the implementation details themselves.

Taro: From an autonomy perspective, having a reliable way to predict how a robot will sink or move when it hits sand is huge because that helps us plan paths when the world doesn't behave exactly as expected. The authors are providing a dataset for that prediction.

Rosa: It really is impressive seeing this level of integration; I wonder if this approach could actually be used outside of a perfectly controlled lab setting, and how long we can trust those predictions when the environment gets messy?

Dev: That's the million-dollar question, Rosa. The stability depends entirely on how well they handled things like low velocities and latency in their implementation; we need to see if that holds up under real-world operational stresses.

Taro: If this model can capture those key movement trends within a small margin of error, it means we have a much better tool for developing systems that can handle unexpected terrain during autonomous navigation. It’s about giving the robot confidence when things go wrong.

Rosa: So, we're looking at a way to test and design locomotion systems in sandy conditions more efficiently than before, which is a big step for the field.

Dev: Exactly; this opens up rapid prototyping for robotic designs that need to traverse granular media without needing massive computational resources just to get a basic feel for how they move.

Taro: I think the authors' work on validating these against real tests, like measuring torque changes and distance traveled, gives us confidence that this isn't just theory; it’s a usable predictive tool for complex robotic systems.

Rosa: It sounds like this paper is setting up a really solid foundation for how we can approach designing robots for unpredictable environments in the future.

Ryan W. Brown, Laura K. Treers, Kathryn A. Daltorio

Case Western Reserve University · University of Vermont

cs.RO, cs.SY, eess.SY

Submitted: 2026-06-17

Updated: 2026-09-28

Comments: 13 pages, 9 figures

Code: https://github.com/Crab-Lab-CWRU/RFT-SiM

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 76/100

The gist: 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

Key concepts

Resistive Force Theory (RFT): RFT is a mathematical model used to approximate the complex ground reaction forces exerted by sand. Instead of simulating every grain, it uses equations based on the robot's shape, velocity, and depth to estimate how much force the sand resists movement.
3D RFT Implementation: This specific implementation extends RFT to three dimensions. It involves meshing the robot's body into plates and using these plates to calculate resistive forces in X, Y, and Z directions based on granular physics principles.
Smoothing Factor (αF): Because low speeds cause force calculations to become unstable, a smoothing factor is applied. This mathematical process averages the calculated forces over time, preventing unrealistic sudden changes in motion when the robot is moving slowly through the sand.

Terminology

Resistive Force Theory — 22× in this paper
Accuracy — 6× in this paper · explained in 1 episode
Calibration — 5× in this paper · explained in 7 episodes
Core — 3× in this paper · explained in 2 episodes
Digital Twin — 3× in this paper · explained in 4 episodes
Robustness — 3× in this paper · explained in 5 episodes
IEEE International Conference on Robotics and Automation — 2× in this paper
3D RFT Implementation — 1× in this paper · explained in 1 episode
Alignment — 1× in this paper · explained in 1 episode
Benchmark — 1× in this paper · explained in 2 episodes
Benchmarking — 1× in this paper · explained in 3 episodes
Exponential Moving Average (EMA) — 1× in this paper · explained in 2 episodes
Failure Modes — 1× in this paper · explained in 3 episodes
Fidelity — 1× in this paper · explained in 2 episodes
Inverse Problem — 1× in this paper · explained in 2 episodes
Longitudinal Data — 1× in this paper · explained in 2 episodes
Reinforcement Learning — 1× in this paper · explained in 4 episodes
Sim-to-Real Gap — 1× in this paper · explained in 6 episodes
Trace — 1× in this paper · explained in 2 episodes
Verification — 1× in this paper · explained in 2 episodes

Summary

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. This work explores whether resistive force approximations, when integrated with standard dynamics calculations, provide a stable substrate for freely walking robots by implementing 3D Granular Resistive Force Theory (3D RFT) in the MuJoCo physics engine.

The gist

RFT-SiM is an open-source framework that integrates 3D RFT into MuJoCo to predict the motion of a legged robot in sand, demonstrating that key trends due to end effector shape, speed, and loading are preserved within 20% of experiments.

How it works

The implementation of RFT-SiM involves several key modifications to the standard MuJoCo physics:

Each body is represented as a mesh of “plates” to correctly capture surface orientations and allow for RFT superposition of forces. A mesh density of 0.03 plates/mm2 was found to provide adequate resolution for debugging and visualization.
Points (MuJoCo sites) are placed at the centroid of each mesh plate to serve as force application locations.
The orientation of the plates and their velocity vectors are used to calculate a set of force vectors based on the 3D RFT formulation described by Treers et al. [44].

The 3D force, Fj, from granular media is defined for a plate element j as:

(1) Fj = ζ(F1 + F2 + F3)

(2) F1 = −f1(ψ, γ) αY signv · e1j / (z A e1j)

(3) F2 = −f23(ψ, γ) αX(γ, β, M) z A e2j)

(4) F3 = f23(ψ, γ) αZ(γ, β, M) z A E3)

The resistive coefficient ζ represents the effective intrusion resistance from sand. The forces are summed to estimate whole-body intrusion forces (Ftotal = PFj).

RFT Force Calculation and Smoothing

The RFT calculation relies on several parameters:

(1) v is the velocity, z is the depth of the plate, and A is the area of the plate [44].

(2) αX and αZ are empirically derived by Li et al. and are functions of characteristic angles γ and β.

The implementation utilizes a single constant (the resistive coefficient, ζ) to calibrate different soils across a wide range of grain sizes and densities.

To ensure continuous motion at low velocities, the overall sorted force matrix is passed through single exponential moving average smoothing with an αF (smoothing factor) of 0.1. This smoothing allows the system to retain a small force that inhibits the velocity vector magnitude from reducing to zero, preventing unrealistic behaviors at low velocities where force directions are ill-defined.

Simulation Setup and Validation

The simulation setup involves:

(1) Discretizing a body’s faces into plates (meshing) before the simulation using tools like Fusion360 and OPEN3D.

(2) Defining a descriptor.XML file for each testing environment to closely match physical testing environments.

Validation involved three baseline tests:

Measuring torque increase as a result of rotating an object through sand, which verified that the RFT model correctly predicted changes in force and torque on an intruder as depth varied.
Measuring the distance traveled by a carriage propelled along a rail by a rotating intruder, verifying the model’s ability to predict motion of a multi-DOF system.
Testing an articulated leg following a 3D path through sand to pull the carriage along the rail, validating the model's capability to correctly integrate force predictions for 3D geometries and trajectories.

Results and Comparison with Other Models

The results across tested systems showed agreement consistently within 34%, capturing measurable trends in leg speed, foot shape, step progress, sinkage, and payload.

(1) Torques on Fixed-base Rotating Intruder:

The RFT model correctly predicted increases in required actuation torque when the object is in sand. The simulated torques were aligned with experimental steady torques after exiting sand.

(2) Constrained Progress due to Rotating Intruder:

The shape of the intruder affected the experimental total progress more than rotational speed. The closed c-shape had a convex geometry which was in contact with the substrate for the longest duration, resulting in the highest total rail carriage displacement.

(3) Speed and Sinkage of Walking Hexapod Robot:

RFT-SiM tended to underestimate walking distance per step between 9.

Improvements for AI systems

Here are the specific improvements for AI systems based on this research, detailing what those improved systems can achieve:

A dedicated, open-source Granular Media Simulator framework (RFT-SiM) integrated with standard physics engines (like MuJoCo) that uses Resistive Force Theory (RFT) instead of computationally prohibitive Discrete Element Method (DEM).
An AI system capable of performing Sim-to-Real transfer for legged robots operating in granular media, enabling rapid, cost-effective iteration on robot hardware and controllers before physical testing.
A robotic design optimization loop that utilizes RFT-SiM to predict the effect of varying end effector shapes, gait parameters (speed), and payloads on key performance metrics like walking distance and foot sinkage with high fidelity (within 20% of experiments).
An AI-driven Design Space Search algorithm that systematically explores robot designs (e.g., hexapods) across a wide parameter space to identify optimal configurations for traversing specific granular substrates, guided by the RFT predictions.
A control system capable of using RFT-SiM to predict and mitigate instability caused by rocking or clumping during locomotion cycles in sand, potentially integrating local surface deformation models into the simulation environment.
An AI-driven parameter tuning tool that allows researchers to adjust the resistive coefficient (ζ) in real-time within the simulation to match desired experimental outcomes for specific speed or sinkage requirements, effectively calibrating a virtual sandbox for different soil types.
A reinforcement learning agent trained within the RFT-SiM environment to learn robust walking policies for complex, unconstrained 3D legged robots (e.g., 12-DOF hexapods) in sand, directly leveraging the accurate force prediction capabilities of RFT to learn efficient gaits that maximize speed and minimize sinkage under varying payloads.