Introduction
These limitations motivate NeSAM (Neuro-Symbolic Soil Adaptive Mobility), which combines learned terrain representations, differentiable Bekker–Wong terramechanics, and a Transformer-based residual model. The terramechanics component provides deformable-soil structure and adaptable physical parameters, while the learned residual accounts for systematic discrepancies between the analytical prediction and observed vehicle motion. The main contributions of this work are summarized as follows:
- We present a neuro-symbolic kinodynamics framework that integrates differentiable Bekker–Wong terramechanics, learned terrain representations, and Transformer-based residual kinodynamics for long-horizon, six-degree-of-freedom vehicle motion prediction over deformable terrain.
- We develop an EKF-based online soil adaptation method that updates physically interpretable terramechanics parameters using differences between predicted and measured vehicle motion.
- We evaluate long-horizon autoregressive prediction in Chrono-based simulation and on real-world vehicle trajectories, with comparisons against terrain-aware and terrain-free learning-based models.
- We validate online soil adaptation through closed-loop navigation in both simulation and on a physical Verti-4-Wheeler platform.
Related Work
This section reviews prior work on terramechanics and soil identification, learning-based off-road mobility modeling, and online adaptation of vehicle kinodynamics.
Learning-Based Off-Road Kinodynamics
Learning-based off-road mobility has progressed from inverse models and end-to-end control policies to forward kinodynamics models for predictive planning. Inverse kinodynamic models learn control corrections from vehicle motion data, while imitation learning directly maps onboard observations to steering and throttle commands. In contrast, learned forward models predict future vehicle states and can therefore be incorporated into model-predictive controllers.
These methods differ in their state representations, terrain modalities, and prediction structures, but they all treat terrain as rigid surfaces, and terrain-dependent effects are primarily encoded through features learned from scratch. They do not explicitly consider the load-dependent soil variables governing sinkage and shear on deformable terrain or provide corresponding physical parameters for online soil adaptation.
Online Adaptation and Hybrid Kinodynamics
Differentiable simulators provide another route for integrating physics with learning and have been applied to system identification, control, and residual modeling in rigid- and soft-body systems. Existing work, however, has not jointly addressed terrain-conditioned six-degree-of-freedom prediction, differentiable deformable-soil mechanics, and adaptation of interpretable soil parameters within an off-road planning framework.
Method
NeSAM predicts vehicle motion by coupling learned terrain-conditioned interaction modeling with differentiable terramechanics. A Transformer encodes the recent vehicle-terrain interaction history. A sinkage head predicts the wheel sinkages required by the terramechanics model, while a state-residual head predicts a single chassis-level correction to the physics-based state increment.
The predicted sinkages, together with the wheel angular velocities available to the model and the current soil estimate, determine the wheel-terrain interaction forces through differentiable terramechanics. These forces are propagated through Newton–Euler kinodynamics, and the learned state residual corrects the resulting vehicle-state prediction.
NeSAM Learning and Adaptation
The learned components of NeSAM are optimized through multi-step autoregressive prediction. Starting from the initial vehicle state and available input history, the rollout is generated for multiple future time steps by recursively applying the NeSAM state-transition model.
Here, the terrain observation aligned with the predicted vehicle state is the terrain observation used by the model, and the corresponding semantic soil prior is the associated soil information. Thus, each predicted state is used to construct the subsequent model input without access to the future ground-truth state.
The model is trained using a multi-step rollout objective, where the weighted vehicle-state prediction error measures the prediction error and the rollout-step weighting factor controls the contribution of each rollout step. Gradients are propagated through the soil predictor, Transformer prediction heads, differentiable terramechanics, and Newton–Euler kinodynamics model, while the pretrained terrain encoders remain fixed.
Experiments
NeSAM is evaluated from two complementary aspects. First, long-horizon kinodynamic modeling accuracy is evaluated against learning-based state-of-the-art baselines in both simulation and physical experiments. Second, the effect of NeSAM’s online soil adaptation is evaluated in closed-loop trajectory tracking by comparing the same NeSAM model with and without EKF-based soil adaptation. The latter serves as an adaptation ablation rather than a comparison against other kinodynamic baselines.
Experiments are conducted in the Verti-Bench simulator and on the physical off-road testbed Verti-Arena.
NeSAM is compared against a pure end-to-end Transformer predictor and TAL using elevation and semantic observations jointly, referred to as TAL Elevation and Semantics, or either modality alone. Two NeSAM ablations are also evaluated:
- NeSAM without state residual: Removes the learned state-residual branch.
- NeSAM without sinkage: Suppresses the predicted wheel sinkage while retaining the remaining neuro-symbolic pipeline.
Conclusion
NeSAM is a neuro-symbolic kinodynamic modeling framework for off-road mobility over deformable terrain. By combining learned terrain representations, prediction of physically meaningful wheel-terrain interaction variables, differentiable Bekker–Wong terramechanics, and a learned state residual, NeSAM improves long-horizon vehicle-motion prediction in both simulation and physical experiments.
When integrated with MPPI, the EKF-based online soil adaptation further improves closed-loop trajectory tracking by refining the soil parameters from observed vehicle motion as terrain conditions change.
A current limitation is that NeSAM assumes wheel-terrain interaction can be adequately represented by the underlying Bekker–Wong terramechanics formulation. Although online adaptation can refine the associated soil parameters, it cannot modify the constitutive pressure–sinkage and shear relationships themselves. Consequently, terrain behaviors that deviate substantially from these assumptions may remain difficult to model even after parameter adaptation.
Future work will investigate richer differentiable terrain models and learned physics corrections that can relax these structural assumptions while retaining the interpretability and adaptability of the neuro-symbolic framework.
Frequently Asked Questions
What is NeSAM? NeSAM is a neuro-symbolic kinodynamic framework for predicting off-road vehicle motion over deformable terrain.
How does NeSAM model soil interaction? It combines predicted wheel sinkages, soil estimates, wheel angular velocities, and differentiable Bekker–Wong terramechanics to determine wheel-terrain interaction forces.
What does online soil adaptation do? An EKF updates physically interpretable terramechanics parameters using differences between predicted and measured vehicle motion.
Where was NeSAM evaluated? NeSAM was evaluated in the Verti-Bench simulator, on real-world vehicle trajectories, and on the physical Verti-Arena off-road testbed.