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Soft Robots Modeling: a Structured Overview

Costanza Armanini, Frédéric Boyer, Anup Teejo Mathew, Christian Duriez, Federico Renda

arXiv:2112.03645v2cs.RO

TL;DR

Soft-robot modeling research has grown complex, and its theoretical foundations are often taken for granted, while the literature has rarely received a complete review. This paper classifies modeling approaches by their mathematical techniques and provides a critical analysis of their uses and applicability.

  • Problem

    Soft-robot modeling literature has rarely received a complete review, despite the growing diversity of approaches and theoretical foundations.

    Source: §I · p. 1

  • Method

    The paper reviews soft- and continuous-robot modeling techniques, classifying them by their mathematical techniques rather than their uses or designs.

    Source: §I · p. 1

  • Results

    The classification highlights structural similarities and differences among modeling approaches and supports analysis of their theoretical roots.

    Source: §I · p. 1

  • Takeaways & Limitations

    The overview provides a critical analysis of modeling approaches’ uses and applicability.

    Source: §I · p. 1

Abstract

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The robotics community has seen an exponential growth in the level of complexity of the theoretical tools presented for the modeling of soft robotics devices. Different solutions have been presented to overcome the difficulties related to the modeling of soft robots, often leveraging on other scientific disciplines, such as continuum mechanics, computational mechanics and computer graphics. These theoretical and computational foundations are often taken for granted and this leads to an intricate literature that, consequently, has rarely been the subject of a complete review. For the first time, we present here a structured overview of all the approaches proposed so far to model soft robots. The chosen classification, which is based on their theoretical and numerical grounds, allows us to provide a critical analysis about their uses and applicability. This will enable robotics researchers to learn the basics of these modeling techniques and their associated numerical methods, but also to have a critical perspective on their uses.

I. INTRODUCTION · II. BASIC CONCEPTS AND CLASSIFICATION

The paper reviews soft-robot modeling approaches using a classification based on their mathematical foundations, aiming to clarify their structure and support critical comparison. It frames the central modeling challenge as making nonlinear, infinite-dimensional continuum dynamics accurate yet numerically usable.

I. INTRODUCTION

  • Soft robots are infinite-dimensional systems governed by highly nonlinear partial differential equations, making accuracy balanced with numerical usability a central modeling challenge.The review covers continuum mechanics, numerical resolution, geometrical and discrete-body approaches, data-driven models, software, and applicability.Source: §I · pp. 1–2
  • The review classifies soft-robot models by their mathematical techniques rather than their applications or designs, to expose structural similarities and differences and support critical analysis.It also aims to explain the approaches’ theoretical origins, assumptions, and associated numerical methods.Source: §I · p. 1
  • The paper’s roadmap proceeds from continuum-mechanics formulations and numerical resolution to geometrical, discretized-body, and data-driven approaches, then software, applicability, and conclusions.The paper presents its general concepts and classification before these modeling approaches.Source: §I · pp. 1–2

II. BASIC CONCEPTS AND CLASSIFICATION

  • A mechanical model is described through configurations, generalized coordinates, a kinematic map, and a dynamic principle that determines equations of motion,.For soft robots, unlike rigid robots represented by discrete frames, the body is a continuum of particles.Source: §II · p. 2
  • Continuum-body models use material-coordinate fields, objective strain rates, dynamics, geometric boundary conditions, and constitutive laws, yielding generally nonlinear equations in an infinite-dimensional configuration space.The review distinguishes discretization for numerical solution from reduction, which selects a basis to describe the body’s kinematic fields.Source: §II · p. 2
  • Continuum-mechanics models derive from physical deformation theories and may use 3D, shell, membrane, beam, or rod formulations, while geometrical models assume particular deformed shapes.Geometrical models include functional space-curve models, usually with absolute coordinates, and piecewise-constant-curvature models with circular-arc sections and relative coordinates.Source: §II · p. 2
  • Discrete material models divide the body into finitely many components with an a priori finite-dimensional configuration space, while surrogate models infer configurations from data and learning.Discrete models use absolute and/or relative coordinates, with relative coordinates usually preferred; most surrogate approaches use neural networks and machine-learning algorithms.Source: §II · pp. 2–3
  • The classification groups models by their parametrization paths to clarify explanation and comparison, but the authors note that categories can overlap and that this is not the only possible classification.Potential overlaps are revisited in the paper’s applicability analysis.Source: §I · p. 2, §II · p. 3

III. CONTINUUM MECHANICS MODELS · A. Classical 3D Models

Classical three-dimensional continuum mechanics provides a general framework for modeling soft bodies, with robotics applications seeking computational efficiency while retaining realism. The section develops its balance laws and constitutive models, then notes that many soft-robotics models still assume linear elasticity despite the prevalence of hyperelastic materials.

III. CONTINUUM MECHANICS MODELS

  • Continuum mechanics is a major theoretical foundation for soft-robot modeling, and classical elasticity offers a general framework whose robotics use aims to balance computational efficiency with realism.For slender bodies, the overview next turns to Cosserat-based director approaches, encompassing Kirchhoff and nonlinear Euler–Bernoulli theories.Source: §III · p. 3

A. Classical 3D Models

  • A three-dimensional theory treats the soft body as material particles Ω labeled by coordinates X and aims to predict the time evolution of its configuration r.Source: A. Classical 3D Models · p. 3
  • Balance laws account for volume forces and boundary tractions on any material subpart, with Cauchy’s theorem relating traction to stress as t = σn.Applying the divergence theorem gives the Cauchy equilibrium equations for the continuum.Source: A. Classical 3D Models · p. 3
  • Closing the balance equations requires an objective strain measure, such as Green–Lagrange strain, and a constitutive relation linking strain to stress.Source: A. Classical 3D Models · p. 3
  • Hooke’s law gives the linear-elastic 1D case, while Green hyperelasticity derives stress from stored energy over principal stretches; rubber-like materials are often treated as hyperelastic under static deformation.Incompressible materials satisfy λ1λ2λ3 = 1, and rubber may also exhibit hysteresis, viscoelasticity, or stress softening.Source: A. Classical 3D Models · p. 3
  • Ogden’s general hyperelastic model yields Neo-Hookean for n = 1, α = 2 and incompressible Mooney–Rivlin for n = 2, α1 = 2, α2 = −2, with pressure enforcing incompressibility.Cauchy principal stresses relate to engineering stresses by σe,i = σiλi^-1.Source: A. Classical 3D Models · pp. 3–4
  • Although most materials used in soft robotics can be considered hyperelastic, many models assume linear elasticity because they focus on large deformations rather than large strains.Source: A. Classical 3D Models · p. 4

B. Directors Approaches

Director-based reductions model slender soft bodies with a centerline or surface and fields describing their cross-sections. For robotic beams, the Cosserat rod framework supports formulations of rod dynamics, tendon actuation, and simpler beam models.

B. Directors Approaches

  • Most soft-robot director models are beam models, representing a rod by a centerline and cross-sectional directors; octopus-inspired robots are an exception.These reduced models apply when one or two material dimensions dominate, with small dimensions incorporated analytically into the thickness.Source: B. Directors Approaches · p. 4
  • A Cosserat rod’s configuration comprises its centerline position and cross-section orientation along one material coordinate, with balance laws closed by strain measures and a stress-strain law.Newton-Euler balances are derived over rod subintervals; the formulation assumes finite deformations and small strains for the stated strain measures.Source: B. Directors Approaches · p. 4
  • The Lie-group formulation represents the rod configuration as a curve on SE(3), with space-twists describing variation along the rod and first-order PDEs governing velocity evolution.A kinematic model reconstructs configuration over time by integration; the formulation also expresses equilibrium in cross-sectional frames.Source: B. Directors Approaches · pp. 4–5
  • For tendon-driven robots, actuation is modeled either as external wrench densities or as internal stress added through an active constitutive law.Both approaches idealize tendons as inelastic, frictionless force transmitters; one derives rod-tendon forces by Newton’s laws and action-reaction, while the other uses virtual work.Source: B. Directors Approaches · p. 5
  • The Reissner beam theory contains Kirchhoff rods as submodels, while planar static Kirchhoff rods reduce to the elastica, an elastic line of minimal curvature energy.The Kirchhoff constraints prevent stretching and shearing; the elastica description uses planar curvature, bending stiffness, and external bending moments.Source: B. Directors Approaches · p. 5

IV. NUMERICAL RESOLUTION OF 3D CONTINUUM FORMULATIONS

Numerical resolution of 3D continuum soft-manipulator models requires solving a closed formulation for each material sub-domain. Time integration typically uses finite differences, while spatial discretization may use finite differences, boundary elements, or finite elements.

IV. NUMERICAL RESOLUTION OF 3D CONTINUUM FORMULATIONS

  • Finite elements preserve the formulation’s variational structure and, through the Ritz method, yield a robust nonlinear optimization problem.Source: §IV · p. 5
  • Spatial discretization can use finite differences, boundary elements, or finite elements; finite differences are difficult for complex shapes, and boundary elements are restricted to linear problems.Source: §IV · p. 5
  • Time is usually discretized with finite differences using explicit or implicit integration schemes.Source: §IV · p. 5

A. Finite Element Method (FEM) for 3D formulations

FEM models soft robots by discretizing a continuum into finite elements and solving the resulting equations for static or dynamic behavior. The section describes its nonlinear formulations, software and actuation methods, and a differentiable MLS-FEM variant for design optimization.

A. Finite Element Method (FEM) for 3D formulations

  • FEM meshes the domain and approximates continuous fields with polynomial interpolation at nodes, converting PDEs into algebraic equations for steady states or ODEs for transients.The nodal coordinates are the generalized coordinates used in the discrete equations.Source: A. Finite Element Method (FEM) for 3D… · pp. 5–6
  • Element-wise integration and assembly yield discrete forces and mass; geometric and material nonlinearities can be handled with corotational methods or ANCF-FEM.Geometric nonlinearities arise from large element displacements, especially rotations, while material nonlinearities arise from constitutive laws.Source: A. Finite Element Method (FEM) for 3D… · p. 6
  • General-purpose FEM software supports multiphysics studies and soft-robot models,,, but its breadth can increase computational cost and slow dynamic convergence.Examples include Abaqus, ANSYS, and Comsol, with applications spanning structural dynamics, fluid-structure interactions, contact, and thermodynamics.Source: A. Finite Element Method (FEM) for 3D… · p. 6
  • SOFA models deformable continua with dynamic or quasi-static simulation nodes and includes a SoftRobots plugin for design, modeling and control,, and self-collision.SOFA was introduced as an open-source computational environment for medical simulations,,.Source: A. Finite Element Method (FEM) for 3D… · p. 6
  • SOFA represents actuation with constraints, linearizes internal forces, solves a free configuration, projects constraint violations via Lagrange multipliers, then corrects the configuration.For fluidic actuation, the multiplier and violation represent cavity pressure and volume; for cables, they represent tension and length. In inverse modeling, the multiplier can be solved through a QP-type problem; cable-driven direct models can impose nonnegative tensions as unilateral complementarity constraints.Source: A. Finite Element Method (FEM) for 3D… · p. 6
  • MLS-FEM makes the deformation gradient twice differentiable, enabling gradient-descent design for automated actuation routing and sensor design.The formulation addresses the nondifferentiability of standard finite elements at element boundaries.Source: A. Finite Element Method (FEM) for 3D… · p. 6

V. NUMERICAL RESOLUTION OF DIRECTORS-BASED FORMULATIONS

Directors-based soft-robot models use Cosserat-rod kinematics, stress balance, constitutive laws, and strain definitions, with either absolute or relative configuration fields. Numerical approaches divide into non-energetic spatial BVP discretizations and energetic Ritz methods, further classified as nodal or modal.

V. NUMERICAL RESOLUTION OF DIRECTORS-BASED FORMULATIONS

  • A closed Cosserat-rod model combines kinematics, stress balance with connection-dependent boundary conditions, a constitutive law, and a strain definition.Configurations may be expressed using absolute fields g or relative fields ξ or ϵ, with equation (11) reconstructing g along the rod.Source: §V · p. 6
  • Time-differentiating the relative space-reconstruction equation yields models whose spatial integration recovers absolute velocity and acceleration from strain velocities and accelerations.This parameterization is termed relative, in contrast to the classical absolute formulation based on g.Source: §V · p. 6
  • Non-energetic methods discretize the original spatial boundary-value problem directly, while energetic Ritz methods reduce configuration fields on a truncated basis and derive Lagrange ODEs.Ritz bases are nodal when formed from finite-element interpolation polynomials, and modal when defined over the full rod domain.Source: §V · pp. 6–7

A. Non-energetic approaches … 1) Absolute nodal-Ritz reduction: GE-FEM

The overview distinguishes non-energetic methods, which recast closed model formulations into standard numerical-analysis forms, from energetic methods, which reduce continuous models to finite Lagrangian ODEs while retaining their variational structure.

A. Non-energetic approaches; 1) Finite Differences

  • Finite differences convert Cosserat-rod boundary-value models into time ODEs at grid nodes, while non-energetic approaches recast closed formulations into standard numerical-analysis forms.For a cable-driven multibending arm, forward differences approximate the kinematic-field derivatives and backward differences the strain-field derivatives, yielding a finite-dimensional system integrated with explicit Runge-Kutta.Source: 1) Finite Differences · p. 7, A. Non-energetic approaches · p. 7

2) Shooting Methods

  • Shooting solves Cosserat-rod boundary-value problems by iteratively correcting predicted proximal conditions until the distal boundary conditions are satisfied.Applications extend from tendon-driven static and dynamic models to multisegment, parallel, and concentric-tube robots, with boundary conditions or segment domains handled in cascade as appropriate.Source: 2) Shooting Methods · p. 7

3) Continuous Newton-Euler based Method

  • The continuous Newton-Euler formulation decouples inverse-dynamics boundary-value problems into two passes, avoiding shooting for those problems.When pose-dependent external forces are neglected, forward statics has the same decoupling property; this supports quasi-static tendon-robot simulation.Source: 3) Continuous Newton-Euler based Method · pp. 7–8

4) Collocation Methods

  • Collocation replaces an unknown strain field with a polynomial and enforces zero residual at selected points to determine its parameters.For Cosserat-rod kinematics, combines orthogonal collocation with forward integration using Magnus Expansion on SE(3), representing ξ(X) with three Chebyshev polynomials.Source: 4) Collocation Methods · p. 8

B. Energetic approaches; 1) Absolute nodal-Ritz reduction: GE-FEM

  • Energetic approaches reduce continuous models to finite Lagrangian ODEs, retaining their variational structure.GE-FEM generalizes interpolation and spatial and temporal integration from positional fields to SO(3), with nodal positions, rotations, velocities, and angular velocities in its dynamic equations.Source: B. Energetic approaches · p. 8, 1) Absolute nodal-Ritz reduction: GE-FEM · p. 8
  • GE-FEM models positions and rotations together on SO(3), applying finite-element operations without simplifying finite rotations beyond spatial and temporal discretization.Its implicit geometric time integrator preserves the Lie-group structure. Although used as a reference and commercialized in SAMCEF/MECANO, full-Cosserat dynamics become ill-conditioned for very slender rods; soft-robotics adaptations include an inflatable-robot model with an added radial-inflation strain.Source: 1) Absolute nodal-Ritz reduction: GE-FEM · p. 8

2) Absolute modal-Ritz reduction · 3) Relative nodal-Ritz reduction · 4) Relative modal-Ritz reduction

The sections compare absolute and relative Ritz reductions, including strain-based approaches that reduce soft-robot models to generalized coordinates. They describe the methods’ accuracy, computational tradeoffs, and extensions to other robot models.

2) Absolute modal-Ritz reduction

  • Absolute modal Ritz approximates parameterized rod fields on truncated Ritz-function bases that satisfy the boundary conditions.The fields can include backbone position and cross-section orientation, represented by time-dependent modal coordinates.Source: 2) Absolute modal-Ritz reduction · p. 8
  • In, Ritz-Galerkin reduction yields algebraic statics equations solved by root finding and configuration-dependent dynamic ODEs integrated explicitly.The formulation uses an inertial base frame, Kirchhoff assumption, and Bishop frames, with torsion represented by an additional field.Source: 2) Absolute modal-Ritz reduction · p. 8

3) Relative nodal-Ritz reduction

  • Strain-interpolated GE-FEM is more accurate but more computationally complex than conventional GE-FEM; uses an exponential-map helical shape function based on base pose and strain.This strain-parametrized approach was presented for soft robotics after related work in.Source: 3) Relative nodal-Ritz reduction · p. 9

4) Relative modal-Ritz reduction

  • In GVS, Cosserat-rod strain fields are expanded in Ritz modes, whose coefficients act as generalized coordinates for homogeneous transformations and yield highly reduced ODEs.The coordinates are comparable to rigid-robot joint angles.Source: 4) Relative modal-Ritz reduction · p. 9
  • GVS was validated against standard statics and dynamics benchmarks, providing good accuracy with few generalized coordinates.The comparison used GE-FEM as the reference.Source: 4) Relative modal-Ritz reduction · p. 9
  • Unlike absolute Ritz methods, GVS strain functions need not satisfy boundary conditions, but the formulation involves difficult double space integrals.Recent work addresses the integral difficulty in deriving the reduced model.Source: 4) Relative modal-Ritz reduction · p. 9
  • An analytical GVS reduction uses Magnus expansions and a kinematic Jacobian to project pose-space virtual work into strain coordinates and derive Lagrangian equations.The Jacobian relates twist and generalized-coordinate velocities; inertial and external forces are transformed through it, while elastic and actuation terms are projected onto strain modes.Source: 4) Relative modal-Ritz reduction · p. 9
  • GVS also has a numerical Newton–Euler alternative and extensions to concentric-tube, constant-strain, and planar robot models.The Newton–Euler approach performs the double integrals through two passes of the continuous computed-torque algorithm [22]. Extensions include tube insertion in quasi-steady and unsteady conditions, SoRoSim, PCS, and closed chains, constant-curvature models with hyperelastic and viscoelastic materials, and planar PC, and flexure-joint models.Source: 4) Relative modal-Ritz reduction · p. 9

C. Analytical-based resolutions · VI. GEOMETRICAL MODELS · A. Functional approaches

Analytical approaches can yield elliptic-function solutions under specific loading conditions, while geometrical models represent soft robots through chosen shape coordinates. Functional approaches use mathematical curves to describe robot geometry, with different representations and modeling techniques across applications.

C. Analytical-based resolutions

  • Elliptic-function solutions can describe Euler–Bernoulli rod deformation under specific loads, particularly concentrated rather than distributed loads.For a simply supported beam under a tip compressive force, the base angle is given implicitly as a function of load; integrating the resulting rotation yields the displacement field.Source: C. Analytical-based resolutions · p. 9
  • Euler–Bernoulli theory supports soft-arm and actuator models for snap-back, post-buckling, backbone deflection, disk equilibrium, and pressure-driven bending.The fiber-reinforced actuator model balances stresses and bending moment against internal air pressure; an extended approach includes pressure on the chamber’s lateral surface. A longitudinal pneumatic arm model discretizes the arm and numerically solves static balance.Source: C. Analytical-based resolutions · pp. 9–10

VI. GEOMETRICAL MODELS

  • Geometrical models represent soft bodies using shape coordinates mapped to a Cosserat-rod configuration, then derive equilibrium through Euler–Lagrange equations or equivalent dynamics principles.These approaches approximate the Cosserat model and divide into functional and piecewise constant-curvature models; depending on material consistency, they can be equivalent to Ritz-based approaches.Source: §VI · p. 10

A. Functional approaches

  • Functional approaches describe robot geometry with chosen mathematical curves, from serpenoid snake kinematics to backbone-based modal models for hyper-redundant robots.The modal backbone is a curve formed by integrating a direction field scaled by a length factor, with orthonormal frames and roll distribution completing the geometry.Source: A. Functional approaches · p. 10
  • Modal models reduce robot shape to generalized coordinates representing length scaling, tangent angles, and roll, with modal functions chosen to satisfy task constraints.The inverse-kinematics problem finds shape functions satisfying constraints such as end-effector position; variational methods optimize curves under joint and task constraints, and the model was extended to coiling-robot dynamics. Constant curvature makes the approach equivalent to the following section’s transformation.Source: A. Functional approaches · p. 10
  • Other functional representations include Pythagorean-hodograph curves optimized through potential energy and neural networks that predict load effects, and experimentally fitted cubic Hermite splines for pneumatic actuators.The spline uses two control points and two control orientations; its orientation along the curve is obtained by minimizing backbone torsion.Source: A. Functional approaches · pp. 10–11
  • Euler arc splines represent tendon-driven robot backbones as directed arcs with curvature varying in arithmetic progression, combining functional representation with a constant-curvature assumption [99].Elliptic Fourier descriptors instead fit closed curves to experimentally extracted contours, describing two-dimensional robot morphologies with four Fourier coefficients.Source: A. Functional approaches · p. 11

B. Constant Curvature Models · 1) Kinematics-based models

Constant-curvature models represent soft bodies as finite sets of constant-curvature arcs, simplifying kinematic modeling and control. Kinematics-based approaches map actuator or arc parameters to robot poses, with alternative parameterizations addressing a singularity as curvature vanishes.

B. Constant Curvature Models

  • Constant-curvature models represent a soft body as circular arcs with spatially constant curvature, a shape motivated by parallel-path actuators without external forces.The overview distinguishes kinematics-based models, which relate actuators to arc parameters, from mechanics-based models.Source: B. Constant Curvature Models · p. 11

1) Kinematics-based models

  • Arc triplets κ, φ, and l are a popular parameterization, and proposed kinematic maps yield identical transformations from the arc base to any point along it.The triplet comprises curvature κ, the plane angle φ, and arc length l.Source: 1) Kinematics-based models · p. 11
  • Modified Denavit–Hartenberg parameters fit the continuous backbone to a virtual rigid-link manipulator, while a later approach maps three parallel tendon lengths to arc parameters,.The modified parameters account for coupling imposed by curvature in a continuum system.Source: 1) Kinematics-based models · p. 11
  • As curvature approaches zero, common PCC parametrizations and maps can become numerically singular, with an infinite or undefined radius of curvature.Taylor-series modal approximations of transformation entries in actuator lengths at zero were used to address this limitation; a Lagrangian spatial-dynamics model was developed for single-section arms and extended to multi-section arms –.Source: 1) Kinematics-based models · p. 11
  • An alternative derives the circular-arc transformation using Lie-group exponential coordinates, composing a rotation with an in-plane transformation –.The approach was applied to concentric tubes assuming equal tube stiffness and torsional rigidity; torque-mediated curvature was modeled with Euler–Bernoulli bending and force balance analogous to parallel springs.Source: 1) Kinematics-based models · pp. 11–12
  • An axis-rotation parameterization describes an arc by (ρ, l) or (ω, l), obtains forward kinematics through exponential maps, and maps configuration to tendon lengths by geometry.Its screw-motion axis ω is perpendicular to ρ, whose norm is the arc-curvature radius.Source: 1) Kinematics-based models · p. 12
  • A nonsingular alternative represents each section as linear combinations of four arcs and maps conventional parameters (κ, φ, l) to (Δx, Δy, δl).Here Δx and Δy are differences in length between opposite arcs, and δl is the section-length change from its resting position.Source: 1) Kinematics-based models · p. 12

2) Mechanics-based models

Mechanics-based models apply static or dynamic mechanics to soft-robot configurations, including PCC representations. They also address PCC limitations involving cross-sectional deformation and torsion, particularly for continuum and concentric-tube robots.

2) Mechanics-based models

  • [25] derives forward and inverse kinematics for tendon-driven manipulators using static force and moment balances, relating arm configuration to tendon tensions and displacements.The planar model uses curvature and axial stretch as generalized coordinates and represents equilibrium as Kq = BT u, with tendon displacements y = Cmu.Source: 2) Mechanics-based models · p. 12
  • For braided pneumatic manipulators, finds that PCC fails to capture cross-sectional deformation, while releasing PCC constraints gives better results.The study models cross-sectional effects using a strain-energy function based on Cauchy-Green stretch.Source: 2) Mechanics-based models · p. 12
  • To model torsion absent from bending-focused PCC, and use virtual power for TDCR dynamics, adding a lumped torsional angle to subsegment curvatures.Their robot has an elastic backbone, rigid disks, and three tendon cables; the disks and cable subsegments are modeled as rigid and circular arcs, respectively.Source: 2) Mechanics-based models · p. 12
  • For CTRs, gives PCC validity conditions and a general model using arc-length-dependent tube twist, moment equilibrium, centerline compatibility, and Cosserat-rod torsional strain.The resulting boundary-value problem is solved over the input-output set; and use the model for stability analysis.Source: 2) Mechanics-based models · p. 12
  • A 3D static model is also developed in.Source: 2) Mechanics-based models · p. 12

VII. DISCRETE MODELS … C. Discrete rods

Discrete models begin with a discrete system representation rather than discretizing a continuous model; the overview groups them as lumped-mass, pseudo-rigid, and discrete-rod approaches. Their foundations and tradeoffs span simple mass-spring-damper systems, rigid-link chains, and discrete elastic rod formulations.

VII. DISCRETE MODELS

  • Discrete models start with a discrete system representation and comprise lumped-mass, pseudo-rigid, and discrete-rod approaches.Unlike approaches that discretize a continuous model or assume a backbone geometry, these systems are discrete from the beginning.Source: §VII · p. 12

A. Lumped-mass Models

  • Lumped-mass models represent continuous bodies with discrete masses, dampers, and springs, deriving equations through energy methods or Newton’s laws.Forces propagate between particles in configuration space.Source: A. Lumped-mass Models · pp. 12–13
  • Lumped-mass models are simple and capture nonlinear friction, material hysteresis, and inertia, but matching continuum or FEM fidelity requires many DOFs and data-expensive identification.Applications include continuum surfaces and tendon-driven catheters; Titan uses parallel computing for massive-scale simulations.Source: A. Lumped-mass Models · p. 13

B. Pseudo-rigid Models

  • Pseudo-rigid models represent soft bodies as rigid-link series joined by revolute, universal, or spherical joints, leveraging established rigid-robotics theories.They can model hyper-redundant or snake-like robots, but continuous elastic structures have low spatial approximation accuracy and expensive identification; examples include catheter models, and a 6-DOF continuum manipulator model.Source: B. Pseudo-rigid Models · p. 13
  • Pseudo-rigid applications include compliant beams, hyper-flexible manipulators, steering catheters, and an octopus-inspired swimmer.These models use, respectively, a torsion spring, a serial rigid chain, D-H forward kinematics, and arm chains with actuated and passive joints.Source: B. Pseudo-rigid Models · p. 13

C. Discrete rods

  • Discrete elastic rods use inextensible Kirchhoff assumptions, a centerline, and Bishop frames with parallel transport to describe cross-sectional poses.The cross-sectional orientation is given by the angle θ(X) between the Bishop and cross-sectional frames.Source: C. Discrete rods · p. 13
  • DER discretizes rods into vertices and straight edges, defining curvature from the turning angle and expressing bending and twist energies in generalized coordinates.Each vertex carries averaged edge mass, and edge mass moments of inertia come from volume integration.Source: C. Discrete rods · pp. 13–14
  • DER extensions model slender-rod dynamics,, planar folding architectures, contact and friction, and compliant mechanisms with a pseudo-rigid hybrid.The 3D filament model includes extension, shear, and axial deformations and uses a symplectic second-order scheme; the hybrid supports computational design optimization.Source: C. Discrete rods · p. 14

VIII. SURROGATE MODELS · A. Neural Networks

Surrogate models use external data rather than requiring a physical model, but depend on large amounts of representative data that can be difficult to collect. Neural networks are used in most soft-robotics data-driven modeling approaches.

VIII. SURROGATE MODELS

  • Surrogate models avoid requiring a physical model but rely on large amounts of representative data, which can be difficult to collect.The overview identifies neural networks as the approach used in most soft-robotics data-driven modeling.Source: §VIII · p. 14

A. Neural Networks

  • Neural networks map actuation variables u to shape parameters q through weighted nonlinear layers; back-propagation adjusts weights, with feed-forward networks typical for kinematics and statics and recurrent networks for dynamics.Source: A. Neural Networks · p. 14
  • In, a feed-forward neural-network controller compensates continuum-robot dynamic uncertainty; models a bionic trunk’s forward kinematics from experimentally measured tip positions across actuation pressures.Source: A. Neural Networks · p. 14
  • The trunk is also modeled with a modified Elman network versus a four-prismatic-joint pseudo-rigid model, while feed-forward networks learn cable-manipulator inverse kinetics from tip position to cable tension.Source: A. Neural Networks · p. 15
  • Continuous motor babbling supplies samples for a single-hidden-layer network to learn global inverse kinematics, mapping end-effector pose and robot configuration to the next configuration.Source: A. Neural Networks · p. 15
  • For an underwater cable-driven robot, PCS dynamics support task-space forward dynamics and a learned state-to-control mapping, enabling trajectory-optimized open-loop predictive control; a closed-loop controller is also implemented.Source: A. Neural Networks · p. 15
  • For a bionic trunk, inverse kinematics is trained over target vertices in a desired Cartesian volume, minimizing the distance between each target and its actual position.Source: A. Neural Networks · p. 15

B. Data-driven Order Reduction … XI. CONCLUSIONS

The overview compares soft-robot modeling approaches by their accuracy, computational cost, and suitability for design and control, while outlining software tools and data-driven reduction methods. It emphasizes the field’s interdisciplinary foundations and the growing focus on control and optimization.

B. Data-driven Order Reduction

  • Koopman theory lifts nonlinear dynamics into a linear operator on an infinite-dimensional observable space, approximated from experimentally measured snapshot pairs.The passage defines observables as real-valued functions on the system state domain.Source: B. Data-driven Order Reduction · p. 15
  • Finite-element reduction projects equations onto lower-dimensional subspaces; snapshot POD builds constraint-dependent bases, extended to low-order control and observation, contact-force reduction, and cable-actuated hyperelastic robots.Reduced models lower simulation size and computational time, enabling FEM methods for control.Source: B. Data-driven Order Reduction · p. 15

IX. SOFTWARE IMPLEMENTATION

  • Software spans physics-based and reduced-order tools: Sofa’s SoftRobots plugin supports multimodal interaction, PyElastica implements Elastica, SoRoSim uses PVS, TMTDyn uses lumped reduced models, and SoMo couples pseudo-rigid models with a Python physics engine.Sofa represents mechanical, thermal, and visual aspects of objects and supports interactions among rigid bodies, soft bodies, and fluids.Source: §IX · p. 15
  • Graphics-oriented ChainQueen uses moving-least-squares material point simulation, IPC-Sim handles nonlinear volumetric elastodynamics, and DiffAqua targets underwater-robot optimization.ChainQueen combines particles and grids in a hybrid Eulerian/Lagrangian FEM approach.Source: §IX · p. 15

X. CRITICAL ANALYSIS AND VIEW POINTS

  • Modeling choices balance realism against computational efficiency: surrogates enable fast, design-specific control but need large measurement databases, while commercial FEM handles complex 3D multiphysics at high computational cost.Continuum-mechanics models and discrete rods are identified as physically consistent across geometry, kinematics, and dynamics; commercial FEM is poorly adapted to robotics-specific control.Source: §X · p. 16
  • GE-FEM with Cosserat beams offers accuracy and computational efficiency, while director-based alternatives suit beam-like robots and can support real-time simulation or model-based control.GE-FEM remains difficult to implement and lacks dedicated commercial robotics software; modal Ritz approaches such as GVS yield reduced explicit Lagrangian models.Source: §X · pp. 16–17
  • Geometric models are efficient but simplify kinematics: PCC is consistent only for specific actuators with negligible gravity, while reference-curve methods may not capture torsion consistently.The discrete rod method is mechanically consistent and offers a promising route to fast interactive simulation.Source: §X · p. 17

XI. CONCLUSIONS

  • Soft-robot modeling draws on continuum and solid mechanics, computational mechanics, machine learning, and computer graphics, as research increasingly extends simulation methods toward control and optimization, [161],,.The overview aims to clarify the distinct and shared features of techniques across this broad interdisciplinary field.Source: §XI · p. 17

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