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ProTrav: Probabilistic Traversability with Elevation Uncertainty Propagation for Risk-aware Off-road Navigation

Chuanlin Zuo, Miloš Prágr, Arun Kumar Singh, Mohammad Loni, Reza Ghabcheloo

arXiv:2610.30839v1cs.RO

TL;DR

Off-road traversability estimates can miss spatially correlated elevation uncertainty as it passes through nonlinear feature mappings. ProTrav samples terrain functions to propagate that uncertainty into cost distributions and incorporates a UCB risk term into MPPI; it is evaluated in simulation and on a real-world robot.

  • Problem

    Deterministic elevation-based traversability overlooks spatial correlations and does not account for elevation uncertainty passing through nonlinear traversability mappings.

    Source: §I · p. 1, A. Function-Space Elevation Map Sampling · p. 4

  • Method

    ProTrav uses SVGP-based terrain realizations and function-space sampling to propagate elevation uncertainty into traversability costs, then adds a moment-based UCB risk term to MPPI.

    Source: §I · pp. 1–2, C. Risk-Aware MPPI Navigation · p. 4

  • Results

    ProTrav is evaluated in simulation and real-world deployment, and its uncertainty representation helps the planner better account for uncertain traversability than a conventional deterministic baseline.

    Source: §I · p. 2, §VI · p. 8

  • Takeaways & Limitations

    MPPI rollout evaluation can jointly account for expected terrain difficulty and uncertainty in cost estimation.

    Source: C. Risk-Aware MPPI Navigation · p. 5

Abstract

from arXiv · show

Reliable traversability estimation is essential for autonomous navigation on unstructured terrain. State-of-the-art approaches typically evaluate traversability from deterministic terrain elevation, thereby overlooking spatial correlations in the terrain and failing to account for how elevation uncertainty propagates through nonlinear traversability mappings. We propose ProTrav, a probabilistic terrain representation that uses function-space sampling to generate spatially correlated terrain realizations and propagate elevation uncertainty to traversability cost. First, a Sparse Variational Gaussian Process (SVGP) reconstructs a probabilistic elevation field from observations. Then, posterior function-space sampling generates spatially correlated terrain realizations, preserving spatial dependencies between neighboring locations while propagating elevation uncertainty through geometric features into traversability cost. Consequently, a distribution of traversability cost at arbitrary location is established, where the mean and standard deviation quantify the expected traversability cost and associated uncertainty, respectively. A risk term is defined by moment-based Upper Confidence Bound (UCB) and incorporated into a Model Predictive Path Integral (MPPI) controller for risk-aware navigation. Simulation experiments demonstrate navigation success rates exceeding 90\% on different terrains, compared with below 40\% for a baseline, while reducing attitude variations. Furthermore, the proposed approach is validated on a robot traversing real-world off-road terrain.

I. INTRODUCTION

Off-road terrain perception is uncertain because LiDAR observations are sparse, irregular, noisy, and occluded. ProTrav propagates elevation uncertainty through nonlinear geometric features into traversability costs, then uses a moment-based UCB risk term in MPPI.

I. INTRODUCTION

  • Mean-first terrain descriptors can underestimate risk because nonlinear feature mappings and spatially correlated elevation uncertainty are not captured by a single expected elevation.A partially observed negative obstacle can appear flat in the mean elevation prediction despite elevated uncertainty.Source: §I · p. 1, Figure 1 · p. 1
  • ProTrav models terrain as a continuous stochastic elevation function with an SVGP and draws spatially correlated posterior realizations for feature extraction.Slope, roughness, and step height are computed for each realization and combined into a traversability cost.Source: §I · p. 2
  • The resulting traversability-cost distribution uses its mean and standard deviation to represent expected cost and associated uncertainty, with moment-based UCB risk incorporated into MPPI.The controller evaluates rollouts using the risk term.Source: §I · p. 2
  • The proposed risk-aware MPPI framework is validated across diverse terrains in simulation and real-world experiments.The contribution statement reports validation but gives no success-rate figures here.Source: §I · p. 2

II. RELATED WORK

Prior traversability work spans handcrafted geometric descriptors, learned representations, and probabilistic terrain or risk models. ProTrav addresses the stated gap by propagating elevation uncertainty through nonlinear geometric mappings to obtain probabilistic traversability costs.

A. Terrain Traversability Estimation

  • Traditional methods use handcrafted geometric descriptors, while learning-based approaches infer traversability from terrain observations and increasingly combine multimodal or long-range inputs.Examples include height-map CNNs, voxelized point-cloud maps, multi-camera and LiDAR models, and adaptation to unseen environments.Source: A. Terrain Traversability Estimation · p. 2
  • These learning-based representations generally produce deterministic traversability despite their increasingly expressive terrain representations.Source: A. Terrain Traversability Estimation · p. 2

B. Probabilistic Traversability

  • Probabilistic terrain work models uncertainty from sensing, localization, or terrain structure, while other methods represent uncertainty directly in traversability or risk.Examples include predictive terrain uncertainty, learned wheel-slip distributions, and CVaR-based navigation.Source: B. Probabilistic Traversability · p. 2
  • Closest geometric approaches estimate terrain features or use elevation uncertainty to adapt confidence or mask poorly observed regions, but do not explicitly propagate it to traversability cost.ProTrav instead uses function-space sampling to propagate elevation uncertainty through nonlinear mappings and construct a risk term.Source: B. Probabilistic Traversability · p. 2

III. PRELIMINARIES

The preliminaries model terrain as a probabilistic elevation field inferred from spatial observations. A GP posterior supplies elevation predictions, while an SVGP with inducing points makes inference suitable for online terrain mapping.

A. Gaussian Process Terrain Modeling

  • A terrain surface is represented as a continuous elevation field inferred from spatial observations, with measured elevations modeled using measurement noise.Source: A. Gaussian Process Terrain Modeling · p. 3
  • The GP uses a prior mean and covariance kernel, with an RBF kernel specified by output scale and length scale, to induce a posterior elevation distribution.The posterior at a query location depends on observed elevations and their covariance with that query location.Source: A. Gaussian Process Terrain Modeling · p. 3
  • Exact GP inference incurs cubic complexity in the number of observations, so an SVGP uses a small set of inducing points to reduce inference cost for real-time mapping.Source: A. Gaussian Process Terrain Modeling · p. 3

B. Terrain Geometric Features Extraction

The section defines slope, roughness, and step height as geometric descriptors of terrain, each capturing a different aspect relevant to robot mobility.

IV. METHODOLOGY

ProTrav propagates elevation uncertainty through geometric features and traversability costs, then uses the resulting cost distribution for risk-aware MPPI navigation.

  • A Sparse Variational Gaussian Process builds a probabilistic terrain elevation model from terrain observations.Source: §IV · p. 4
  • Posterior function-space elevation maps are used to independently compute geometric descriptors and traversability costs, forming a traversability-cost distribution map.Source: §IV · p. 4
  • The cost map supports risk-aware navigation using an MPPI controller with UCB-based risk.Source: §IV · p. 4

A. Function-Space Elevation Map Sampling

Function-space sampling addresses the mismatch between cost at mean elevation and expected cost, while geometry-based mapping propagates elevation uncertainty into a traversability-cost distribution.

A. Function-Space Elevation Map Sampling

  • Nonlinear traversability mapping means cost evaluated at mean elevation can differ from expected cost, making mean-only estimation an inaccurate approximation.The discrepancy follows from Jensen’s inequality and can miss terrain variations induced by elevation uncertainty.Source: A. Function-Space Elevation Map Sampling · p. 4
  • The method jointly samples the GP posterior over the local map using its full covariance, preserving spatial correlations within each plausible elevation realization.The local map is discretized into vehicle-relative grid locations.Source: A. Function-Space Elevation Map Sampling · p. 4

B. Geometry-Based Traversability Mapping

  • Geometric features are normalized by robot-capability-based critical thresholds, combined with relative weights, and transformed by a Hill-type function.The transformation bounds and smoothens the geometric score; its sharpness parameter suppresses small variations in benign terrain and limits disproportionate influence from extreme feature values.Source: B. Geometry-Based Traversability Mapping · p. 4
  • Across sampled terrain realizations, nonlinear feature extraction and cost computation produce an empirical traversability-cost distribution represented using Dirac delta masses.Source: B. Geometry-Based Traversability Mapping · p. 4

C. Risk-Aware MPPI Navigation

The navigation framework uses a moment-based UCB risk term in MPPI trajectory evaluation to account for terrain difficulty and uncertainty.

  • The moment-based UCB risk term uses the first two moments of the generally non-Gaussian traversability-cost distribution.The stated confidence level is 95%, giving κδ = 1.645.Source: C. Risk-Aware MPPI Navigation · p. 4
  • The Gazebo simulation environments show start-goal pairs and expected paths on baseline mountainous terrain and custom-built cliff terrain.Source: Figure 3 · p. 5
  • MPPI trajectory cost balances goal-reaching performance against terrain risk, and evaluating rollouts with cUCB accounts for expected terrain difficulty and cost-estimation uncertainty.Source: C. Risk-Aware MPPI Navigation · p. 5

V. EXPERIMENT AND RESULTS

Experiments compare probabilistic and deterministic traversability on a Husky robot in Gazebo, then report deployments on a small differential-drive robot.

  • The evaluation combines quantitative and qualitative Gazebo comparisons with a deterministic baseline, uncertainty-sensitivity ablations, and onboard deployments on a small differential-drive robot.Source: §V · p. 5

A. Simulation Experiments

In simulation, Prob-MPPI combines probabilistic traversability with MPPI and is tested against mean-first configurations across mountain and cliff terrain. Its advantages are clearest on cliffs with partially unobserved obstacles, while mountain trials show similar attitude responses across uncertainty settings.

A. Simulation Experiments

  • The comparison uses Base-RRT* with mean-first cost and RRT*, Base-MPPI with the same cost and MPPI, and Prob-MPPI with probabilistic traversability and MPPI.Source: A. Simulation Experiments · p. 5
  • The confidence study varies κδ: 1.645 corresponds to a one-sided 95% confidence level, 0.842 to 80%, and 0 removes the uncertainty penalty while retaining function-space sampling.Source: A. Simulation Experiments · p. 5
  • 100% success across tested mountain start-goal pairs for all MPPI configurations; Prob-MPPI also has lower mean and maximum roll and pitch deviations than Base-MPPI, even at κδ = 0.Table I reports mean ± standard deviation over 10 trials and success rate.Source: A. Simulation Experiments · p. 5, Table I · p. 6
  • The three uncertainty settings produce similar mountain attitude responses, where dense observations and limited local-horizon uncertainty keep the uncertainty term small relative to expected traversability cost.Source: A. Simulation Experiments · p. 6
  • 80–90% success for Prob-MPPI in the first cliff scenario, versus 10% for Base-RRT* and 20% for Base-MPPI; in the second, success ranged from 20% at κδ = 0.842 to 90% at κδ = 1.645.The cliff scenarios include partially unobserved negative obstacles; Table II reports falls and success over 10 trials.Source: A. Simulation Experiments · p. 6, Table II · p. 6
  • Prob-MPPI maintains high reliability across substantially different mountain and cliff terrain types using the same terrain-cost weight without specific retuning; baselines vary more across terrains.The authors attribute this pattern to uncertainty-aware traversability improving robustness to terrain-structure changes without parameter adjustment.Source: A. Simulation Experiments · p. 6

B. Deployment on a Mobile Robot

Onboard deployments test ProTrav on a mound, a downhill ramp, and a waypoint sequence. The reported demonstrations include mound avoidance and successful navigation despite a supervisor-stopped trial on loose soil.

B. Deployment on a Mobile Robot

  • The deployment uses a four-wheeled differential-drive robot with Livox Mid-360 LiDAR and a fused RTK-GNSS-IMU-camera positioning unit; mapping runs at 1 Hz and MPPI at 5 Hz.Source: B. Deployment on a Mobile Robot · p. 6
  • The Sequence scenario follows waypoints for approximately 125 m, including a downhill traversal over the Ramp.Source: B. Deployment on a Mobile Robot · p. 7
  • In the Mound scenario, the robot avoided the mound by driving over flat ground and reached the goal in three repeated trials.A separate trial initialized too close to the mound was stopped by a supervisor because loose soil offered limited friction.Source: B. Deployment on a Mobile Robot · p. 7
  • The Ramp scenario requires traversing a slope to reach the downhill goal; the system reports relatively low costs on the locally flat slope.Source: B. Deployment on a Mobile Robot · p. 7
  • The geometric approach cannot distinguish contextual visual properties such as loose soil, and extending uncertainty propagation to visual descriptors is left for future work.Source: B. Deployment on a Mobile Robot · p. 8

VI. CONCLUSION

ProTrav propagates elevation uncertainty into traversability cost and integrates the probabilistic representation into MPPI for risk-aware navigation. The conclusion identifies fine geometric structures, visual sensing, and temporal consistency as future-work areas.

VI. CONCLUSION

  • ProTrav samples spatially correlated elevation fields, computes geometric features and traversability cost across realizations, and integrates the resulting probabilistic representation into MPPI.The approach is evaluated in simulation and real-world deployment.Source: §VI · p. 8
  • Explicitly representing traversability uncertainty helps the planner account for uncertain terrain geometry, improves attitude stability on uneven terrain, and enhances safety around negative obstacles.Source: §VI · p. 8
  • Future work includes capturing fine structures such as small pits that sparse observations may miss, extending to visual sensing, and improving temporal consistency amid observation changes.Source: §VI · p. 8
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