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An End-to-End Framework for Modelling Pneumatic Soft Robots Based on Differentiable Finite Element Methods

Shaohong Zhong, Yao Yao, Perla Maiolino, Ingmar Posner

arXiv:2610.04612v1cs.ROcs.SYeess.SY

TL;DR

Soft-robot modelling and system identification are challenging because of nonlinear materials, complex dynamics, intricate geometry, and demanding calibration. This paper presents a differentiable FEM framework using voxelised geometry and point-cloud observations to model pneumatic soft robots and optimise motion. On a complex bellow-shaped actuator, the model’s maximum positional error is less than 3 mm, and the framework is demonstrated in real-world motion-planning tasks.

  • Problem

    Soft robots’ nonlinear materials, complex dynamics, and intricate geometry make accurate modelling challenging, while system identification can require laborious robot-specific setups and substantial calibration data.

    Source: §I · p. 1

  • Method

    An end-to-end differentiable FEM framework automates voxelised mesh creation, identifies parameters from point clouds, and supports downstream motion planning for pneumatic soft robots.

    Source: §I · pp. 1–2

  • Results

    Maximum positional error was less than 3 mm, and the model supported real-world path following with an average target-waypoint error of 1.2 mm.

    Source: C. Reaching · p. 6, D. Path Following · p. 7

  • Takeaways & Limitations

    The framework models a complex bellow-shaped pneumatic actuator and applies the generated model to downstream robotics tasks.

    Source: §VI · p. 8

    Errors generally increase at higher pressures, and one chamber’s position behind the others obstructed a camera view and affected optimisation results.

    Source: C. Reaching · p. 6

Abstract

from arXiv · show

Soft robots present significant modelling challenges due to their non-linearity, complex dynamics and potentially intricate geometries. These difficulties in accurate system identification and dynamics modelling limit their applications in precise robotics tasks. Prior modelling approaches typically suffer from trade-offs in accuracy, computational efficiency, or speed. The differentiable finite element method (FEM) offers a promising balance between these desiderata for soft robot modelling, and enables the use of gradients for efficient calibration and trajectory optimisation. In this paper, we propose an end-to-end differentiable FEM-based framework designed to streamline modelling and system identification for pneumatic soft robots, enabling the generation of reliable models for downstream tasks. The framework automates the conversion of computer-aided designs into voxelised tetrahedral viscoelastic FEM meshes that accurately represent the robot's geometry and actuation mechanism. By integrating easily acquired point cloud data with differentiable FEM, the framework achieves precise identification of material and dynamic actuation parameters using a minimal experimental setup. Additionally, the differentiable nature of the model facilitates trajectory optimisation by leveraging gradients from robot dynamics and contact interactions. We validate the framework by modelling a complex bellow-shaped pneumatic soft robot and demonstrate its efficacy in real-world motion planning tasks. Experimental results indicate high modelling accuracy, with a maximum positional error of less than 3 mm, and successful application in tasks such as path following and grasping.

I. INTRODUCTION

Precise soft-robot tasks require models that capture nonlinear materials, complex dynamics, intricate geometry, and contact, while system identification can demand substantial manual effort and data. The paper proposes an end-to-end differentiable FEM framework for pneumatic robots, combining voxelised mesh generation, point-cloud-based parameter identification, and gradient-based motion planning.

I. INTRODUCTION

  • Nonlinear materials, complex dynamics, intricate geometry, and contact make soft-robot behaviours difficult to predict or simulate without simplifying assumptions.These challenges constrain the accuracy and computational cost of models for precise robotics tasks.Source: §I · p. 1
  • System identification can require laborious, robot-specific experimental setups and substantial calibration data.These demands add to the challenges of accurate dynamics modelling.Source: §I · p. 1
  • The proposed framework uses differentiable FEM to automate model creation and system identification for pneumatic soft robots, demonstrated on a complex bellow-shaped actuator.The framework overview proceeds from CAD to a voxelised finite element model, point-cloud-based parameter identification, and gradient-based motion planning.Source: §I · p. 1, Figure 1 · p. 1
  • CAD models are converted to volumetric meshes by voxelisation, while point clouds support gradient-based optimisation of material and dynamic actuation parameters.The framework also demonstrates downstream real-world motion-planning tasks.Source: §I · p. 2

II. LITERATURE REVIEW

The review contrasts soft-robot modelling approaches and prior differentiable FEM work, then introduces the paper’s FEM simulation choices. The paper positions its framework as addressing limitations in geometry, actuation, setup demands, and downstream task demonstrations.

II. LITERATURE REVIEW

  • The literature review surveys prior soft-robot modelling and differentiable FEM approaches before presenting the methodology.The paper’s methodology covers geometry generation, parameter calibration, and trajectory optimisation.Source: §II · p. 2, §III · p. 2

A. Soft Robot Modelling

  • Analytical models simplify geometry and deformation, data-driven models require substantial robot-specific data and computation, and FEM offers detailed representation at higher computational cost.The review frames these as trade-offs in accuracy, generalisation, and integration into downstream tasks.Source: A. Soft Robot Modelling · p. 2

B. Differentiable FEM for Soft Robot Modelling

  • Prior differentiable FEM studies used simplified geometries or actuation models, while other work addressed more accurate meshes for relatively simple robots or simulated simple trajectories.The paper also notes dedicated motion-capture requirements and limited downstream task exploration in much prior work.Source: B. Differentiable FEM for Soft Robot… · p. 2
  • The framework uses differentiable FEM with point-cloud observations for complex pneumatic-robot identification, a minimal setup, and no residual model compared with.The authors state that removing the residual model increases generalisability and utility for downstream tasks.Source: B. Differentiable FEM for Soft Robot… · p. 2

1) Finite Element Method (FEM):

  • Tetrahedral discretisation computes elastic forces and deformations using a Neo-Hookean model, semi-implicit Euler integration, penalty-based contact, and internal damping.The model uses Lamé parameters λ and µ and adds a speed-dependent damping factor to mimic real-world behaviour.Source: 1) Finite Element Method (FEM) · p. 2

2) Voxel-based Geometry Model:

The framework converts CAD-derived geometry into a tetrahedral viscoelastic mesh while preserving pneumatic chamber interiors, then models actuation through chamber pressure. It uses mesh vertices as point-cloud observations for system identification.

2) Voxel-based Geometry Model:

  • Voxelisation preserves empty pneumatic-chamber interiors and represents walls as filled voxels, which are converted into tetrahedra treated as viscoelastic FEM elements.Each voxel is split into five tetrahedra using alternating configurations; rigid and soft regions receive separate material parameters, with only soft-region parameters estimated.Source: 2) Voxel-based Geometry Model · pp. 2–3

3) Physics-based Pneumatic Actuation:

  • The actuation model applies uniform pressure directly to chamber interior surfaces and maps real-world pressures to simulated pressures through parameters K.The simulated pressure vector has one component per chamber.Source: 3) Physics-based Pneumatic Actuation · p. 3

B. System Identification

  • The high-fidelity geometry enables direct comparison of simulated and real robot behaviour using point clouds.Fig. 2 places voxelised mesh generation, point-cloud parameter identification, and gradient-based motion planning in sequence.Source: B. System Identification · p. 3, Figure 2 · p. 3

1) Point Cloud Generation:

  • Because ground-truth mesh states are unavailable, the framework uses point clouds captured with RGB-D cameras as observations without assuming a kinematic model.The model’s mesh vertices form an evenly distributed point cloud that retains gradient connections to the viscoelastic mesh and its parameters.Source: 1) Point Cloud Generation · p. 3

2) Parameter Identification:

Parameter identification aligns simulated and real point clouds using a unidirectional Chamfer-distance loss, optimising material properties before actuation parameters. Differentiable simulation also supports gradient-based trajectory optimisation for path following and grasping.

2) Parameter Identification:

  • Material and dynamic actuation parameters are estimated to match simulated robot behaviour to the real system.The material parameters are Lamé parameters {λ, µ}; K denotes the actuation parameters.Source: 2) Parameter Identification · p. 3
  • The unidirectional Chamfer distance compares cropped, aligned, projected real and simulated point clouds, mitigating some sensor-noise effects.Losses use steady-state observations; optimisation estimates {λ, µ} with the robot un-actuated, then estimates K at predefined pressure points.Source: 2) Parameter Identification · p. 3

C. Motion Planning

  • Differentiable simulation supplies gradients for task-space trajectory optimisation, including path following and contact-based grasping.The framework supports intuitive task-space metrics for downstream trajectory optimisation.Source: C. Motion Planning · pp. 3–4

IV. EXPERIMENTS

The experiments evaluate model generation, system identification, and downstream motion-planning utility using a custom bellow-shaped pneumatic actuator. Point clouds capture the unactuated robot and individually actuated chambers, while marker tracking provides ground-truth position.

IV. EXPERIMENTS

  • The experiments assess the framework’s modelling accuracy and utility for motion planning across robotics tasks.Source: §IV · p. 4

A. Implementation Details

  • The CAD model is voxelised at 0.8 mm, below the actuator’s 1.5 mm wall thickness, and the simulation is run under a quasi-static assumption.The simulation model is uniformly scaled up by 1000 times to avoid numerical instability from small voxel sizes.Source: A. Implementation Details · p. 4

B. Experimental Setup

  • The actuator has three bellow-shaped chambers and a central spine; its soft walls and spine are printed in Agilus30, with rigid VeroCyan ends.Source: B. Experimental Setup · p. 4
  • Material identification uses an unactuated robot, while dynamic actuation identification captures each chamber actuated individually; OptiTrack markers track free-end position for evaluation.The setup yields 19 identification data points: one unactuated and 18 actuated.Source: B. Experimental Setup · p. 4, Figure 3 · p. 4

C. System Identification

System identification estimates material and dynamic actuation parameters by comparing steady-state simulated and real point clouds. Material fitting uses an unactuated observation; dynamic fitting optimises simulated pressure against individually pressurised chambers.

C. System Identification

1) Material Parameter:

  • Material parameters {λ, µ} are fitted by gradient-based optimisation against one gravity-deformed point cloud of the unactuated robot.Source: 1) Material Parameter · p. 4

2) Dynamic Actuation Parameter:

  • Dynamic actuation parameters K map real-world to simulated pressure, fitted by comparing steady-state point clouds from two orthogonal views.Each chamber is pressurised individually; only that chamber’s simulated pressure is updated while the other two remain at zero.Source: 2) Dynamic Actuation Parameter · pp. 4–5

D. Evaluation

The evaluation uses gradient-based trajectory optimisation and transfers the resulting simulated pressures to the real actuator using identified actuation parameters. It describes steady-state position evaluation and a path-following task with waypoint and speed-based losses.

D. Evaluation

  • The evaluation optimises task-specific trajectories in a differentiable simulator and transfers simulated pressure sequences to the real robot using identified actuation parameters.Source: D. Evaluation · p. 5

1) Reaching:

  • Reaching accuracy is evaluated by comparing simulated and OptiTrack ground-truth free-end positions across chambers and pressure steps, after aligning their initial positions.Source: 1) Reaching · p. 5

2) Path following:

  • Path following optimises simulated pressures to reach waypoint positions while a speed-based loss encourages steady state before advancing to the next waypoint.The pressure sequence is transferred to the real actuator, and the waypoint interval τ is fixed at 20 seconds.Source: 2) Path following · p. 5

3) Grasping:

The grasping task uses differentiable surface-contact gradients to optimise pressure so the soft robot grasps a rigid rod. Its loss combines distance and contact terms, with contact detection implemented differentiably.

3) Grasping:

  • The grasping task maximises contact between the simulated robot surface and a rigid rod, using gradients from surface contacts.Source: 3) Grasping · p. 5
  • The grasping loss combines distance and contact terms; a thresholded differentiable selection marks points within distance ε of the rod as contact.Source: 3) Grasping · p. 5
  • Pressure is optimised through gradients computed from the differentiable contact-based loss.Source: 3) Grasping · p. 5

V. RESULTS AND ANALYSIS

Results show that voxelisation preserves the actuator’s internal chambers and supports fast GPU simulation, while point-cloud fitting estimates material and pressure-mapping parameters.

A. Soft Robot Modelling

  • ∼161,000 filled voxels yield ∼805,000 tetrahedra, retaining circular chambers and supporting GPU rates above 3,000 FPS forward and 160 FPS backward.CPU simulation runs below 15 FPS forward and 2 FPS backward.Source: A. Soft Robot Modelling · p. 5, Figure 4 · p. 6

B. System Identification

  • λ = 1.4 × 108 Pa and µ = 7.27 × 107 Pa were estimated from gravity-deformation point clouds and fall within the expected range for the printed soft material.Source: B. System Identification · p. 6
  • Six {preal, psim} pairs per chamber showed a linear real-to-simulated pressure relationship sufficient to reproduce similar deformation, validating point-cloud identification.The mapping is parameterized by K = {κ, ρ}.Source: B. System Identification · p. 6

C. Reaching

The model reproduces real-world reaching with under 3 mm maximum positional error and supports simulated and real-world path following. Real-world paths transfer from simulation even when multiple chambers are actuated together.

C. Reaching

  • Maximum reaching position error is less than 3 mm, or 3% of the actuator’s length, across pressures.Errors generally increase at higher pressures, where deformations are larger and more nonlinear.Source: C. Reaching · p. 6, Figure 6 · p. 6

D. Path Following

  • In simulation, path following reaches waypoints with 0.1 mm average error and has less than 0.5 mm average path deviation.The square, triangle, and figure-eight use 5, 4, and 8 waypoints, respectively; additional waypoints may improve tracking precision.Source: D. Path Following · p. 6, Figure 7 · p. 7
  • The framework supports gradient-based trajectory optimisation for arbitrary paths, with successful transfer of optimised pressure sequences to the physical robot.The gap in the real-world square trajectory is likely due to occlusions.Source: D. Path Following · p. 7, Figure 8 · p. 7
  • Real-world path following reaches waypoints with 1.2 mm average error and stays within 3.2 mm maximum path deviation.The transfer uses chamber-specific identified parameters even when multiple chambers are pressurised together, indicating the model handles their nonlinear interactions.Source: D. Path Following · p. 7, Figure 8 · p. 7

E. Grasping

The calibrated differentiable model optimises pressure to grasp a rod at different poses, using contact with the robot’s bellow-shaped surface. The framework also supports real-world robotics tasks without robot- or task-specific modifications, while further work is needed on computational requirements and broader actuation and material types.

E. Grasping

  • Optimising pressure to maximise contact enables successful rod grasping at different object poses.Accurate geometry modelling supports precise contact detection on the bellow-shaped surface.Source: E. Grasping · p. 7, Figure 9 · p. 7
  • The CAD-to-model framework is calibrated with point clouds and supports real-world tasks without robot- or task-specific modifications.The authors suggest potential generalisation to other tasks and pneumatic robots.Source: E. Grasping · pp. 7–8

VI. CONCLUSIONS

  • The framework is demonstrated on a complex bellow-shaped pneumatic actuator; future work includes other actuation mechanisms and materials, computational requirements, and alternative differentiable physics models to reduce potential numerical instability.Source: §VI · p. 8
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