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ProTerrain: Probabilistic Physics-Informed Rough Terrain World Modeling

Golnaz Raja, Ruslan Agishev, Miloš Prágr, Joni Pajarinen, Karel Zimmermann, Arun Kumar Singh, Reza Ghabcheloo

TL;DR

ProTerrain tackles unreliable off-road trajectory forecasting under perceptual uncertainty by modeling spatially correlated aleatoric terrain uncertainty in a probabilistic world model and propagating it through a differentiable physics engine. It introduces a structured covariance formulation $\Sigma=L D L^T$ to enable efficient, high-resolution uncertainty estimation via matrix-free conjugate gradient methods, coupled with a Lift-Splat-Shoot based backbone and per-cell mean/variance heads. The approach yields improved trajectory accuracy and uncertainty calibration over baselines, demonstrated on real ROUGH data with superior probabilistic samples and risk-aware predictions. The work advances practical, end-to-end uncertainty-aware planning for autonomous navigation in unstructured terrain, with potential extensions to epistemic uncertainty and richer terrain semantics.

Abstract

Uncertainty-aware robot motion prediction is crucial for downstream traversability estimation and safe autonomous navigation in unstructured, off-road environments, where terrain is heterogeneous and perceptual uncertainty is high. Most existing methods assume deterministic or spatially independent terrain uncertainties, ignoring the inherent local correlations of 3D spatial data and often producing unreliable predictions. In this work, we introduce an efficient probabilistic framework that explicitly models spatially correlated aleatoric uncertainty over terrain parameters as a probabilistic world model and propagates this uncertainty through a differentiable physics engine for probabilistic trajectory forecasting. By leveraging structured convolutional operators, our approach provides high-resolution multivariate predictions at manageable computational cost. Experimental evaluation on a publicly available dataset shows significantly improved uncertainty estimation and trajectory prediction accuracy over aleatoric uncertainty estimation baselines.

ProTerrain: Probabilistic Physics-Informed Rough Terrain World Modeling

TL;DR

ProTerrain tackles unreliable off-road trajectory forecasting under perceptual uncertainty by modeling spatially correlated aleatoric terrain uncertainty in a probabilistic world model and propagating it through a differentiable physics engine. It introduces a structured covariance formulation to enable efficient, high-resolution uncertainty estimation via matrix-free conjugate gradient methods, coupled with a Lift-Splat-Shoot based backbone and per-cell mean/variance heads. The approach yields improved trajectory accuracy and uncertainty calibration over baselines, demonstrated on real ROUGH data with superior probabilistic samples and risk-aware predictions. The work advances practical, end-to-end uncertainty-aware planning for autonomous navigation in unstructured terrain, with potential extensions to epistemic uncertainty and richer terrain semantics.

Abstract

Uncertainty-aware robot motion prediction is crucial for downstream traversability estimation and safe autonomous navigation in unstructured, off-road environments, where terrain is heterogeneous and perceptual uncertainty is high. Most existing methods assume deterministic or spatially independent terrain uncertainties, ignoring the inherent local correlations of 3D spatial data and often producing unreliable predictions. In this work, we introduce an efficient probabilistic framework that explicitly models spatially correlated aleatoric uncertainty over terrain parameters as a probabilistic world model and propagates this uncertainty through a differentiable physics engine for probabilistic trajectory forecasting. By leveraging structured convolutional operators, our approach provides high-resolution multivariate predictions at manageable computational cost. Experimental evaluation on a publicly available dataset shows significantly improved uncertainty estimation and trajectory prediction accuracy over aleatoric uncertainty estimation baselines.
Paper Structure (30 sections, 1 theorem, 21 equations, 2 figures, 1 table)

This paper contains 30 sections, 1 theorem, 21 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Related Work
  3. Geometric Terrain Mapping
  4. Probabilistic Terrain Mapping
  5. Physics-Informed and Dynamics-Aware Approaches
  6. Summary and Positioning
  7. Problem Formulation
  8. Probabilistic World Model Definition
  9. Robot State and Controls
  10. Probabilistic Trajectory Prediction
  11. Preliminaries and Background
  12. MonoForce: Physics-informed World Modeling
  13. Terrain Encoder
  14. Differentiable Physics Engine
  15. Conjugate Gradient (CG)
  16. ...and 15 more sections

Key Result

Theorem 1

Let $\Sigma = L D L^\top$ be a structured covariance matrix, where $L \in \mathbb{R}^{n \times n}$ is a fixed matrix representing convolution with a kernel $g$, $D = \operatorname{diag}(\sigma^2)$ is a diagonal matrix of per-cell predicted variances, and $n = H' \times W'$ is the total number of spa where $\mathbf{b} = D^{1/2} L^\top \mathbf{a}$ and $\mathbf{a} = \Sigma^{-1} \mathbf{r}$. Evaluatio

Figures (2)

  • Figure 1: Architecture overview: Uncertainty‐aware terrain parameter estimation chained with differentiable physics.
  • Figure 2: Qualitative comparison of our proposed method versus the standard per-cell independent uncertainty formulation. (a) Ground-truth geometric heightmap (from LiDAR) within the camera field of view, with corresponding RGB input images. (b) Predicted mean geometric heightmap $\mu_{\phi_{\mathrm{geom}}}$, example probabilistic samples, and the residual standard deviation over 50 samples (visualizing overall uncertainty). (c) Predicted trajectory samples from both methods, compared with ground truth, visualized in the $xy$-plane and as height over time.

Theorems & Definitions (3)

  • Definition 1: 2D Convolution as Toeplitz Matrix Multiplication
  • Theorem 1: NLL for Structured Covariance
  • proof