MILD: Tractable Terrain Modeling for Learning Improved Bipedal Locomotion on Deformable Surfaces

AI summary

Overview

Research area: Robotics — bipedal locomotion control, contact mechanics for deformable terrain, and reinforcement learning (RL) for sim-to-real transfer.

Technical level: Advanced. The paper combines a physics-grounded granular-media contact solver derived from first principles with an RL training pipeline, and it assumes familiarity with rigid-body dynamics, Jacobians, friction cones, variational autoencoders, and PPO-style policy optimization.

Scope (one sentence): The paper introduces MILD, a tractable discrete-element foot–terrain contact model plus a terrain-aware RL controller, and validates it on the EngineAI SA01 bipedal robot walking across rubber tiles, polyurethane foams of several densities, grass, and sand.

Metadata: arXiv:2608.19955v1 [cs.RO], 20 Aug 2026. Authors are affiliated with the Adaptive Robotic Controls Lab (ArcLab), Department of Mechanical Engineering, The University of Hong Kong; the School of Mechanical and Electrical Engineering, Beijing Institute of Technology; and the Graduate School of Information Science and Technology, The University of Tokyo. Manuscript received April 28, 2025; revised August 30, 2025; accepted December 4, 2025; recommended for publication by Editor Aleksandra Faust. Supported by General Research Fund Grant No. 17204222.

What This Paper Is About

Bipedal robots are trained mostly in simulators that treat the ground as rigid, so they cannot reproduce how a yielding surface (sand, foam, grass, soil) deforms differently under different parts of a foot at different moments. Because bipedal robots have large footplates, the common assumption that contact force is spread uniformly across the contact area is wrong for them. The paper's goal is to build a contact solver that is accurate enough to generate realistic training data for soft surfaces yet cheap enough to run inside a massively parallel RL training loop, then to train a controller that adapts to terrain compliance online.

Key Contributions

1. A tractable foot–terrain interaction model for large contact areas. The foot is discretized into independent elements, each forming its own developing "jamming cone" in the granular medium, so forces vary spatially and temporally rather than being assumed uniform. The model explicitly handles eccentric insertion and heterogeneous penetration depths, and generalizes the underlying cone formulation to scenarios with multiple interacting target areas that the original model overlooked.
2. A kinematic-aware whole-body contact solver with continuity and drift constraints. Element-wise impulses are computed in isolation and then reconciled: pairwise kinematic continuity constraints (Eq. 11–12) penalize inconsistency between neighboring elements, a friction-cone projection (Eq. 13) keeps tangential impulses physical, and a drift-resistive force (Eq. 14) opposes horizontal motion as a function of accumulated path length.
3. An RL training framework with a terrain estimator and a modulation network. A multi-head variational autoencoder (VAE) estimates base velocity, feet velocities, and feet heights from proprioception and outputs a latent variable encoding robot–environment information; a modulation network then applies an affine transformation to that latent using estimated feet velocity, producing a dynamically adjusted latent for the policy.
4. Simulation and hardware validation, including online terrain identification. Comparisons against five contact-model baselines and two state-of-the-art learning-based controllers, plus demonstrations of automatic adaptation during abrupt rubber-to-foam-to-rubber transitions that were never seen in training.

Main Findings

• Discretization resolution: A resolution of 154 elements (0.0195 m per unit) was chosen as the best balance between contact richness and computational efficiency; beyond this the paper reports the marginal gain in contact richness is outweighed by a prohibitive increase in computation time.
• Penetration depth: MILD achieves deeper penetration, reported as +38% versus Ecc-Spring, along with longer slip distances, which the authors interpret as broader contact interaction characteristics.
• Baseline ranking in simulation: The two concentric single-point-contact models (Con-Spring, Con-Cone) show the most limited performance of the surface-expanded methods, after the rigid-surface baseline, because they fail to capture edge interactions and severely underestimate contact area.
• Versus a passivity-based controller: Against a whole-body passivity-based controller designed for compliant terrain, MILD produces significantly smaller Center of Mass oscillations and greater stability during dynamic walking, without requiring the baseline's computationally expensive online optimization.
• Terrain adaptability on hardware: Across rubber tiles, 60d, 45d, and 28d polyurethane (PU) foam, grass, and sand, the controller automatically increases foot clearance on softer surfaces to compensate for ground deformation, and achieves the highest success rate and energy efficiency among the compared approaches. Smaller foam density values indicate softer, more malleable surfaces.
• Reliability: The robot completes 10 forward/backward walking cycles at 1.2 m/s on each of seven terrain types (including a rigid plane) without a single failure.
• Latent space structure: t-SNE visualization of the modulated latent state, built from 15-second walking trials on each terrain type, shows distinct clusters along the stiffness–deformability spectrum, with softer terrains occupying cluster interiors and stiffer terrains distributed peripherally.
• Energy efficiency versus learning baselines: Cost of Transport (COT) on hardware at commanded velocities of 0.3, 0.6, 0.9, and 1.2 m/s — MILD: 0.82, 0.75, 0.78, 0.83; HT-2: 1.05, 0.96, 1.20, 1.08; Clock: 1.38, 1.30, 1.35, 1.42. COT is computed as Power/(Weight × Velocity) with Power the sum over actuators of |τ·q̇|.
• Peak joint torque on hardware (Nm): MILD: 58.2, 63.2, 73.3, 88.4; HT-2: 62.9, 75.8, 96.0, 91.6; Clock: 67.6, 82.5, 102.8, 115.2.
• Velocity tracking in simulation (measured m/s for commands of 0.3, 0.6, 0.9, 1.2): MILD: 0.28, 0.57, 0.84, 1.15; HT-2: 0.28, 0.61, 0.98, 1.28; Clock: 0.23, 0.54, 0.81, 1.09. MILD is the closest to the commanded values at the two highest speeds.
• Baseline failure mode: Controllers trained with baseline contact models and deployed on 45d foam adopt overly conservative gaits with insufficient foot clearance during swing, leading to joint-limit violations or stability loss.
• Online identification: During rubber → foam → rubber transitions, the robot automatically increases stride length when moving from high-stiffness to low-stiffness ground, and distinct latent-space trajectories appear through the transition cycle. The authors report the pipeline adapts even to abrupt changes never encountered in training, which they read as the latent representation learning ground-stiffness properties rather than memorizing training conditions.
• Training cost: 4096 agents trained in parallel on Isaac Gym for 20,000 episodes, taking approximately 42.4 hours on a desktop PC with an NVIDIA RTX 4080 GPU. Linear velocity commands span [0.0, 1.2] m/s and angular velocity commands span [−1.5, 1.5] rad/s.
• Robot and control stack: The EngineAI SA01 bipedal robot has 12 degrees of freedom, 6 per leg. A PD controller tracks desired joint angles at 100 Hz with Kp = [50, 50, 70, 70, 20, 20] and Kd = [5.0, 5.0, 7.0, 7.0, 0.2, 0.2] per leg.

Methodology in Plain English

The researchers start from an existing physics model that describes how a small object pushed into a granular medium builds up a compacted "cone" of grains — a cone whose accumulated mass exerts extra force. Existing versions of this model assume a single circular or rectangular contact penetrating symmetrically. That assumption breaks for a bipedal robot's large footplate, where one edge may dig in deeper than another.

Their fix is to slice the foot's envelope surface into discrete elements and let each element penetrate independently, producing its own cone and therefore its own force. Deeper elements produce larger added-mass effects, so the total force distribution becomes naturally non-uniform. Forces are split into a quasistatic stiffness term, a cone stress term, and added-mass inertial terms, and the integrals involved have closed-form solutions in terms of penetration depth, which keeps computation fast.

Because elements are solved separately, the foot could appear to "tear" or partially slip. The solver therefore iterates: it predicts each contact point's velocity, measures how inconsistent the predicted velocities are between pairs of elements, and projects the tangential impulses back into a friction cone that includes corrective terms for both desired zero velocity and kinematic coherence. A drift-resistive force then opposes horizontal sliding, scaled by penetration depth and by the accumulated path length from the initial intrusion point — the authors specify path length rather than straight-line Euclidean distance, which they found works better. The resulting per-element forces are summed to get the resultant force and torque, from which the zero moment point is computed.

This solver is plugged into Isaac Gym. For control, the researchers use an asymmetric actor–critic setup: the actor sees a history of proprioception (base angular velocity, commanded velocity, projected gravity, joint angles, joint velocities, and the previous action), while the critic additionally sees ground-truth simulation states and true terrain parameters. The actor also receives estimated states and a latent vector. A multi-head VAE trained jointly with the policy loss (weighted by ε = 0.3) reconstructs observations and regresses base velocity, feet velocities, and feet heights, while its latent is passed through a modulation network that scales and shifts it based on estimated feet velocity. The action is a 12-dimensional deviation of desired joint angles from a nominal pose. Training uses a periodic gait schedule as a reference, with reward terms and weights listed in Table I (velocity tracking, foot clearance, gait alignment, foot-slip penalty, base acceleration, and action smoothness).

Terrain diversity is produced by randomizing physical parameters — shear band angle, compaction rate, surrounding mass scale, inertial drag scale, volume fraction, friction coefficient, penetration stiffness, and conical stress — alongside training perturbations such as added base mass, center-of-mass offset, motor latency, motor offset, IMU latency, and command latency (ranges are in Table II).

Why This Matters

Impact on research. The paper attacks a specific, named modeling assumption in the literature: that reaction force is uniformly distributed across the contact area. It shows that removing this assumption changes the quality of training data enough to change learned behavior, and it argues that the benefit for RL comes from the agent witnessing a wider range of contact scenarios during randomization. It also supplies a tractable, closed-form-integrable extension of the added-mass cone model to multi-area eccentric contact, which is reusable by other legged-robot researchers.

Real-world applications (the paper names disaster response and planetary exploration; the tested surfaces point to these as well):

• Disaster response, where robots must traverse rubble, debris, sand, and soil that yield under load.
• Planetary exploration, where rovers and legged systems operate on regolith-like granular surfaces.
• Outdoor inspection and field robotics over grass, sand, and soft soil — surfaces the paper explicitly tests.
• Indoor and service robotics crossing rubber tiling, carpet padding, and foam, where compliance changes abruptly at doorways and surface boundaries.

Industry relevance. The reported cost-of-transport and peak-torque advantages over HT-2 and Clock translate directly into battery life and actuator sizing margins for commercially deployed legged platforms. The demonstrated online adaptation to abrupt stiffness changes without prior knowledge matters for robots that must keep walking when the floor underneath them changes, and the trained policy runs as a pre-optimized network rather than requiring the online optimization that the passivity-based baseline needs.

Future Directions

• Visual terrain deformation perception. The authors explicitly propose incorporating visual sensing of terrain deformation to improve adaptability.
• Multi-gait generalization. The paper names jumping and running maneuvers as future work, along with implementing the model on model-based control architectures rather than learned policies.
• Broader terrain coverage and quantification. The evaluation uses six terrains plus a rigid plane; whether the learned latent identifies stiffness outside this range, and how it behaves on the natural grass and sand surfaces beyond the reported trials, is left open. Absolute stiffness values for the foam grades are not reported numerically beyond their density labels.
• Scaling and portability. The training run used 4096 parallel agents on one RTX 4080 GPU for approximately 42.4 hours; how the discrete-element solver's cost scales with finer foot discretization or with higher-degree-of-freedom platforms is not reported.

Target Audience

This paper is most valuable to graduate students and researchers working on legged-robot locomotion, contact dynamics simulation, and sim-to-real reinforcement learning, particularly those dealing with compliant or granular ground. It also suits robotics engineers who need to decide whether a high-fidelity contact model is worth the training-time cost compared with domain randomization over rigid terrain, and control engineers interested in how latent-variable estimation and modulation can substitute for explicit terrain classification online. Readers without a background in rigid-body dynamics, friction cones, and policy-gradient RL will find the method sections demanding.

Authors’ abstract

Enabling robots to walk on yielding terrain is vital for applications ranging from disaster response to planetary exploration. While bipedal robots hold immense potential, their locomotion on deformable surfaces remains limited as current simulators fail to capture the spatiotemporal heterogeneity of such yielding substrates. We present MILD, featuring a physics-grounded discrete-element contact solver that accurately simulates spatially varying foot-terrain interactions. Complementing this model, we train a terrain-aware locomotion controller via deep reinforcement learning with latent modulation and proprioceptive estimation. Quantitative comparisons against state-of-the-art methods show our approach generates more diverse and realistic contact scenarios during training, resulting in controllers that exhibit natural adaptation on real deformable surfaces. Through hardware experiments, we demonstrate the system's capability for online terrain identification and adaptation across a wide range of surface stiffness.