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When Obstacles Bend: Modeling Vegetation Deformation in the context of Field Robotics

Muhammad Hsaeeb Zaar Khizar, Tom Montagnon, Roland Lenain, Romuald Aufrère, Johann Laconte

arXiv:2608.26050v1cs.RO

TL;DR

Robot-specific traversability measures do not capture vegetation’s intrinsic mechanics, limiting transfer across platforms and obscuring interaction-relevant properties. The paper combines force and deformation measurements with rod theory to estimate distributed and lumped stiffness, finding that the spatial profile transfers across contact heights while lumped stiffness remains geometry-dependent.

  • Problem

    Robot-specific traversability measures entangle environmental characterization with platform dynamics instead of describing vegetation’s intrinsic mechanical properties.

  • Method

    The paper uses rod theory with contact-force and deformation measurements to estimate vegetation’s spatial flexural-rigidity profile and lumped rotational stiffness.

  • Results

    The spatial profile EI(s) remains consistent across contact heights, whereas lumped stiffness kθ depends on interaction geometry and applies within the small-deformation regime.

  • Takeaways & Limitations

    Intrinsic mechanical descriptors can support vegetation-aware robotic interaction and traversability assessment beyond platform-dependent metrics.

Abstract

from arXiv · show

Autonomous robots operating in natural environments must often interact with vegetation rather than simply avoid it. In this context, traversability is typically defined from the robot's perspective, by measuring how a specific platform responds when moving through the environment. While practical, this viewpoint entangles the assessment of the environment with the robot's own dynamics, making the resulting characterization difficult to transfer across different platforms. More importantly, it does not directly reflect the properties of the vegetation itself, which are the true source of interaction and potential damage in applications such as agriculture and environmental monitoring. To address this limitation, we propose to characterize vegetation through its intrinsic mechanical properties, independently of any specific robot. By combining deformation measurements with contact force data, we estimate the underlying mechanical parameters and reconstruct the vegetation's response to interaction. This enables vegetation-aware navigation based on intrinsic environmental properties rather than platform-dependent metrics.

I. INTRODUCTION

The paper argues that robot-specific traversability metrics do not describe vegetation independently, and proposes recovering intrinsic mechanical properties from ordinary robot–vegetation interaction. A rod-theoretic framework combines force and shape sensing to support both distributed and lumped stiffness descriptions.

  • Vegetation resistance depends on mechanical behavior, so appearance and geometry alone cannot determine whether a robot should push through, deflect, or circumvent it.
  • Robot-specific traversal costs depend on the robot, sensor mounting, and control policy rather than describing vegetation’s own mechanical properties.
  • A physically grounded description requires a mechanical model connecting observed deformation and contact interaction to the plant’s underlying structure.
  • The paper uses rod theory to model both smooth bending along a stem and near-rigid rotation about a compliant base.
  • The framework recovers a distributed stiffness profile for detailed characterization and a single lumped stiffness for real-time traversability decisions.

III. ROD-THEORETIC MODELING OF VEGETATION

The modeling framework represents vegetation as a planar Kirchhoff rod and derives mechanical stiffness from contact force and observed stem shape. Its assumptions reduce interaction to quasi-static, planar, inextensible bending with localized contact and defined boundary conditions.

  • A force sensor measures the contact force Fc as the robot advances, and the model can use any sensor providing this force or torque output.
  • The RGB-D pipeline observes the stem side-on and supplies centerline shape and curvature for spatial stiffness estimation.
  • The framework models a stem as a planar Kirchhoff rod whose centerline and tangent angle describe its configuration along arc length.
  • The model assumes planar bending, quasi-static loading, inextensibility, unshearability, linear elasticity, and single-point contact.
  • The base is clamped and the tip is free, while contact is approximated as a concentrated transverse force along an equivalent single rod.

B. Rod Kinematics, Equilibrium, and Flexural-Rigidity Estimation

The paper models vegetation as a planar Kirchhoff rod and combines measured contact force with observed shape to estimate flexural rigidity along the stem.

  • Rod kinematics: The planar, unshearable Kirchhoff rod represents stem configuration through centerline r(s) and tangent angle θ(s).It specializes the general Cosserat rod using planarity, inextensibility, and no-twist assumptions.
  • Equivalent representation: The equivalent spring-rod abstraction replaces distributed bending below contact point P with one deflection angle θ and base torque τ.The Kirchhoff rod instead preserves continuously varying tangent angle and contact position along the stem.
  • Constitutive law: Curvature is the tangent-angle rotation rate κ(s) = θ′(s), and bending moment follows M(s) = EI(s)κ(s).The constitutive relation connects observed curvature to the stem’s flexural rigidity.
  • Equilibrium and contact: A horizontal point force Fc at arc length sc, with a free tip, determines the internal force and moment through Kirchhoff equilibrium.The contact geometry and boundary conditions make the moment computable from the measured force and stem shape.
  • Rigidity estimation: Using z(sc)=h, the contact moment is fixed by Fc and observed shape independently of EI(s), after which curvature recovers EI(s).The resulting frame-by-frame estimator combines shape-derived curvature with the contact-force-based moment.

2) Direct Estimation of EI(s):

Direct estimation recovers flexural rigidity pointwise without prescribing its spatial variation, but division by near-zero curvature makes the estimator ill-conditioned.

  • Direct estimator: The direct estimator is model-free because it makes no assumption about how EI(s) varies along the stem.It recovers rigidity from the constitutive relation using measured moment and curvature.
  • Conditioning limitation: Near-zero curvature causes unbounded relative error because measurement noise is divided by a vanishing curvature estimate.The problem is strongest near the base or early in a push, where curvature remains small.
  • Cross-sectional model: For circular cross-sections, I(s) = πD(s)^4/64, so diameter taper determines the geometric contribution to flexural rigidity.The second moment of area is independent of loading direction under the adopted circular-section assumption.

3) Flexural-Rigidity Parametrization and Fitting:

The parametric model reduces the spatial rigidity profile to base rigidity EI0 and taper exponent p, fitted from pooled curvature evidence and centerline positions.

  • Parametrization: EI(s) is characterized by two parameters: base flexural rigidity EI0 and taper exponent p.Estimating the full profile therefore becomes estimation of these two intrinsic quantities.
  • Identifiability: A single push identifies p poorly because it engages only a short stem span, allowing stiffness and taper combinations to produce similar responses.The limited engaged region begins near s ≈ h and extends only modestly as the push proceeds.
  • Taper fitting: The taper exponent is fit as p = b/4 from a weighted log-linear fit pooled across contact heights and binned by arc length.Weights use inverse squared relative standard error, reducing the influence of noisy near-zero-curvature bins.
  • Base-rigidity fitting: With p fixed, EI0 is recovered by minimizing residuals between predicted and observed contact-point centerline positions across engaged frames.This avoids the curvature-based ratio’s poor reliability at the base, where EI0 is defined.
  • Estimator relationship: Intrinsic profile estimation combines direct pointwise rigidity estimates with model-based shape prediction, whereas the direct estimator alone requires only equilibrium.Both parametric stages depend on the contact moment and stiffness model.

C. Shape Extraction from RGB-D Observations

The RGB-D pipeline extracts a metric stem centerline from segmented frames and smooths its tangent field before deriving curvature for rigidity estimation.

  • Pipeline outputs: Both direct and model-based rigidity estimators use observations from the same vision pipeline.The direct estimator requires curvature, while the model-based estimator requires the full position centerline.
  • Centerline extraction: Each RGB-D frame is segmented, converted into a base-to-tip centerline, and fit with a smoothing spline at equal arc-length increments.Depth converts pixel coordinates into metric arc length and provides the centerline coordinates (x(s), z(s)).
  • Curvature recovery: Gaussian-process smoothing produces a differentiable tangent angle field θ(s), enabling finite curvature κ(s) = θ′(s) without excessive shape smoothing.This addresses the sharp degradation caused by differentiating raw per-frame noise at higher sampling density.

D. Lumped-Parameter Reduction

The lumped model reduces a stem to a single rotational or translational stiffness, relating measured contact force to rotation or displacement. These small-deflection models can be replaced by a secant stiffness when larger rotations make linearization inaccurate.

  • D. Lumped-Parameter Reduction: The rotational formulation relates contact force Fc to rotation θ, while the translational formulation relates Fc to horizontal displacement ∆x at contact height h.The rotational stiffness enters torque balance, whereas translational stiffness enters an independent force balance.
  • D. Lumped-Parameter Reduction: The spring-rod model represents the stem as a rigid link on a rotational spring capturing distributed compliance below contact.Its stored energy is U = 1/2 kθθ^2, and the restoring torque is kθθ.
  • D. Lumped-Parameter Reduction: For constant EI, the stiffnesses reduce to kθ = 2EI/h and kx = 3EI/h^3, matching standard cantilever results.These expressions correspond to end-point-load rotation and tip-deflection stiffnesses.
  • D. Lumped-Parameter Reduction: The choice between kθ and kx depends on whether the downstream task requires restoring torque and rotation or restoring force and displacement.Both are small-deflection results derived from the rod model.
  • D. Lumped-Parameter Reduction: For larger rotations, the nonlinear elastica can provide the equivalent secant stiffness Fch/θ(h) instead of the small-deflection linearization.This alternative applies when the stem shape is observed.

2) Camera-Free Angle Recovery from the Wire Sensor:

The wire sensor provides lateral displacement from its measured elongation, while robot pose and mounting geometry recover contact location and deflection angle. These quantities, together with force and contact height, specify lumped stiffness without a camera.

  • 2) Camera-Free Angle Recovery from the Wire Sensor:: The wire sensor measures elongation as a stem displaces its midpoint laterally, stretching both wire halves from Lw/2 to (Lw/2)^2 + ∆x^2.Inverting the elongation model yields the lateral displacement ∆x required by the lumped force model.
  • 2) Camera-Free Angle Recovery from the Wire Sensor:: Robot pose and lateral displacement determine the contact coordinates (xc(t), zc(t)) in the base-fixed frame.xc is the horizontal distance advanced since first contact, and zc is the contact height above the base.
  • 2) Camera-Free Angle Recovery from the Wire Sensor:: The known wire height h and contact coordinates determine the deflection angle θ through a single trigonometric equation with a closed-form solution.The angle is recovered from geometry rather than camera-based shape observation.
  • 2) Camera-Free Angle Recovery from the Wire Sensor:: With θ and ∆x obtained from elongation and pose, and τ = Fch from force and height, the lumped stiffnesses are fully specified without vision.This camera-free route supplies the inputs to the rotational and translational stiffness models.
  • 2) Camera-Free Angle Recovery from the Wire Sensor:: The estimated stiffnesses describe material or geometric stem properties rather than properties of the sensor or robot.The distributed EI(s) is independent of contact height, whereas kθ depends on height and must be refit for each new height.

IV. EXPERIMENTS AND RESULTS

The experiments evaluate the rod-theoretic framework using push-through data from an artificial grass bush and a woody twig. The setup combines a wire force sensor on a UR10 arm with RGB-D vision for stem-shape extraction.

  • IV. EXPERIMENTS AND RESULTS: The evaluation uses physical push-through data from two vegetation specimens.The tested specimens are an artificial grass bush and a woody twig.
  • IV. EXPERIMENTS AND RESULTS: The artificial grass bush mimics a real grass tuft, while the woody twig was collected from the wild.A living rooted grass plant was excluded because repeated manipulation permanently changes its mechanical properties.
  • IV. EXPERIMENTS AND RESULTS: Both specimens were pushed from first contact to full traversal using a wire force sensor mounted on a UR10 arm at constant height.Different contact heights were tested to collect measurements across each specimen’s length.
  • IV. EXPERIMENTS AND RESULTS: RGB-D vision extracted each stem’s centerline by segmenting frames, skeletonizing the mask, and selecting a shortest path between base and tip.Depth converted pixel coordinates to metric arc length before spline fitting and tangent-angle estimation.

B. Flexural-Rigidity Profile EI(s)

The paper evaluates distributed flexural rigidity and lumped rotational stiffness as complementary models of vegetation deformation. The distributed profile supports predictions across contact heights, while the lumped model is simpler but limited by height dependence and yielding.

  • Distributed flexural rigidity: The direct EI(s) estimator becomes unreliable near the base because small curvature worsens its signal-to-noise ratio.A taper-law parametrization is fitted to provide a usable rigidity profile in this region.
  • Distributed flexural rigidity: Figure 5 compares pooled direct rigidity estimates with fitted taper laws and nearby taper-exponent curves for twig and grass specimens.The figure plots EI(s) against arc length s, with separate color-coded fitted curves for the two specimens.
  • Distributed flexural rigidity: The fitted rigidity matches observations outside the noisy base region, while the twig is stiffer and tapers more gradually than the grass.Without the parametrization, forward simulations of energy expenditure, stress, or deformation would be inaccurate.
  • Lumped rotational stiffness: Contact force versus deflection angle is fitted using torque balance, with the slope representing the lumped rotational stiffness kθ.The fit uses the linear response region and force measurements at three contact heights per specimen.
  • Lumped rotational stiffness: 0.31 N m/rad to 0.14 N m/rad for grass and 1.60 N m/rad to 0.98 N m/rad for twig: fitted kθ decreases with contact height.These values show that the lumped stiffness is not constant across interaction heights.
  • Lumped rotational stiffness: The lumped model misses the measured force plateau because its constant kθ assumption keeps predicted force rising with deflection angle.The plateau occurs after vegetation bends sufficiently below the interaction point.
  • Lumped rotational stiffness: The lumped model is insufficient for complete vegetation characterization because yield angle depends on contact point and robot geometry.It remains applicable for fixed-height interactions within a limited, small-deformation elastic regime.

D. Generality of the Models

The paper tests whether the two stiffness models generalize across contact heights and links model choice to interaction geometry. The distributed EI(s) profile is stable across heights, whereas kθ varies with the configuration but remains useful for fixed-height, limited-deflection tasks.

  • Cross-height generality: The generalization study compares kθ and EI(s) across several contact heights to determine when each model supports traversability assessment or crop handling.The models are evaluated beyond the setup at which their parameters were fitted.
  • Cross-height generality: The pooled flexural-rigidity profile EI(s) remains stable across contact heights, unlike the lumped stiffness kθ.Because rigidity varies along the stem, a single stiffness parameter cannot capture the same spatial complexity.
  • Model interpretation: EI(s) represents stem mechanical properties independently of the specific interaction used for estimation, whereas kθ describes a local response tied to contact height.This distinction explains why kθ varies with interaction configuration.
  • Model selection: The lumped-stiffness model is sufficient for fixed, known contact heights, while the rod model supports varying contact heights and loads.The rod model can reuse EI(s) in forward prediction for tasks such as harvesting or crop-lodging assessment.
  • Model selection: The framework combines a vision-and-force-based EI(s) profile with a force-only kθ estimate, trading transferability for additional visual sensing.EI(s) can be reused across robots, sensing configurations, and contact heights without additional mechanical characterization.
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