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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 has developed a complex, interdisciplinary literature that lacked a comprehensive structured review. The paper classifies modeling approaches by theoretical and numerical grounds, critically analyzes their uses and applicability, and provides researchers with foundations for evaluating them.

  • Problem

    Soft-robot modeling literature is extensive and intricate because approaches draw on diverse theoretical and computational disciplines, while their foundations are often taken for granted.

  • Method

    The paper presents a comprehensive review classified by mathematical techniques and their associated numerical methods rather than by application or robot design.

  • Results

    The review provides a structured overview of proposed soft-robot modeling approaches and critically analyzes their uses and applicability.

  • Takeaways & Limitations

    Researchers can use the classification to learn modeling foundations and assess the assumptions underlying different approaches.

Abstract

from arXiv · show

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

The paper reviews soft-robot modeling approaches through a classification based on their mathematical and numerical foundations. This structure connects theoretical origins, numerical methods, uses, and applicability across a diverse literature.

  • The review comprehensively covers techniques proposed to model soft and continuous robots.
  • Its classification groups approaches by mathematical techniques rather than by uses such as design, simulation, or control.This highlights structural similarities and differences among modeling approaches.
  • The overview traces theoretical roots across continuum mechanics, computational mechanics, computer graphics, and other disciplines.
  • The paper organizes models into continuum-mechanics, geometrical-curve, a priori discretization, data-driven, and software-implementation approaches.
  • Soft-robot modeling requires distinguishing discretization for numerical solution from reduction of the model’s infinite-dimensional equations.

A. Classical 3D Models

Classical three-dimensional continuum models represent a body through material-coordinate-dependent positional fields, balance equations, strain definitions, and constitutive laws. These formulations are general but require assumptions about material behavior to close the equations.

  • Three-dimensional theories predict a body’s configuration as a time-evolving positional field over material particles labeled by coordinates X.
  • The formulation starts from balance equations for volume and surface forces, then derives local Cauchy equilibrium equations.
  • Strain definitions and constitutive relations supplement equilibrium equations by relating deformation measures to material stress response.
  • Hyperelastic materials derive stress–strain behavior from a strain-energy density function, with principal stretches constrained by λ1λ2λ3 = 1 for incompressible materials.
  • Many soft-robot models still assume linear elasticity while focusing on large deformations rather than large strains, although many materials are hyperelastic.

B. Directors Approaches

Director approaches reduce continuum bodies to reference lines or surfaces augmented by director fields, with beam models dominating soft-robot applications. Cosserat rods provide geometrically nonlinear kinematics and dynamics, while specialized submodels impose further constraints.

  • Reduced beam and shell models replace the full three-dimensional position field with a reference line or surface plus director fields.
  • Most soft-robot applications use beam models, derived from continuum theories such as Kirchhoff and Reissner formulations.
  • Cosserat rod configurations are represented by a curve of cross-sectional frames, with spatial variation described by twist fields.
  • Cosserat dynamics combine first-order equilibrium PDEs with a kinematic model that reconstructs configuration by integration.
  • Actuation is incorporated case by case, including tendon effects represented through external wrench terms or active constitutive laws.
  • Kirchhoff rods arise by preventing axial stretching and transverse shearing, while planar static assumptions permit further integration of the equilibrium equations.

IV. NUMERICAL RESOLUTION OF 3D CONTINUUM FORMULATIONS

Numerical resolution of three-dimensional continuum formulations requires solving a closed PDE system containing equilibrium equations, strain and constitutive relations, and boundary conditions. Common methods discretize time and space to obtain approximate solutions.

  • A continuum-manics simulation requires solving equilibrium PDEs together with strain definitions, constitutive laws, and boundary conditions.
  • Finite differences commonly discretize the time axis using explicit or implicit integration schemes.

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

FEM approximates soft-robot continuum mechanics by subdividing the domain into finite elements and solving the resulting equilibrium equations in generalized coordinates. Actuation constraints are incorporated through Lagrange multipliers and projected corrections.

  • FEM subdivides the problem domain into smaller finite elements to obtain approximate solutions to partial differential equations.
  • Static equilibrium uses generalized coordinates q, with internal and external generalized forces represented by Qint(q) and Qext(q).
  • Element-by-element integration computes the generalized terms, while mesh assembly positions each element and concentrates nonlinearities in Qint(q, v).
  • Actuation constraints enter static equilibrium through Lagrange multipliers, with B as the Jacobian of the actuator constraint equations.
  • The solver finds a free configuration, projects constraint violations into constraint space, and corrects the configuration using the resulting constraint response.

V. NUMERICAL RESOLUTION OF DIRECTORS-BASED FORMULATIONS

Directors-based formulations provide closed soft-manipulator models that can be reshaped into standard numerical-analysis forms. Their configurations may be absolute, relative, or mixed, with Ritz methods reducing fields to finite generalized coordinates.

  • V. NUMERICAL RESOLUTION OF DIRECTORS-BASED FORMULATIONS: A closed Cosserat-rod formulation combines kinematics, stress balance, constitutive laws, and strain definitions for each rod.
  • V. NUMERICAL RESOLUTION OF DIRECTORS-BASED FORMULATIONS: Boundary conditions in the stress balance can depend on how multiple rods are connected.
  • V. NUMERICAL RESOLUTION OF DIRECTORS-BASED FORMULATIONS: Relative configuration parameterization uses strain fields whose spatial integration reconstructs absolute velocity and acceleration fields along the rod.
  • V. NUMERICAL RESOLUTION OF DIRECTORS-BASED FORMULATIONS: Ritz-Galerkin methods reduce absolute or relative vector fields onto truncated spatial bases, whose components become generalized coordinates governed by Lagrange ODEs.
  • V. NUMERICAL RESOLUTION OF DIRECTORS-BASED FORMULATIONS: Ritz methods are classified as nodal or modal according to whether their basis uses finite-element interpolation polynomials or other spatial basis functions.
  • A. Non-energetic approaches: These approaches extract a closed absolute, relative, or mixed formulation and reshape it into a standard numerical-analysis form.

1) Finite Differences

The reviewed numerical approaches solve soft-robot boundary-value and dynamic problems using finite differences, shooting, collocation, and Newton-Euler formulations. Their applicability spans Cosserat cables, soft arms, and continuum robots, with method-specific numerical trade-offs.

  • Finite differences solve partial differential equations by a simple and established discretization approach, including dynamic Cosserat-beam models of towed submarine cables.
  • Shooting solves one-dimensional boundary-value problems by integrating a sequence of initial-value problems while determining unknown proximal conditions.
  • The forward static Cosserat-rod problem is written as z′ = f(z) after removing velocities and accelerations and inverting the constitutive law.
  • Newton-Euler algorithms originally developed for rigid multibody dynamics have been extended to inverse dynamics of Cosserat-rod hyperredundant robots.
  • Newton-Euler methods have been applied to soft and continuum robotics, including quasi-static tendon-driven simulation through a two-pass approach.
  • When pose-dependent external forces are neglected, forward static and inverse boundary-value problems share a decoupling property, and Newton-Euler and finite-difference methods can yield the same numerical solutions from different viewpoints.
  • Collocation replaces the unknown strain field with a polynomial and enforces vanishing residuals at finitely many collocation points.
  • Energetic approaches reduce continuous models to finite Lagrangian ordinary differential equations while retaining their variational structure.

2) Absolute modal-Ritz reduction

Modal-Ritz methods reduce soft-robot pose or strain fields using spatial modes and time-dependent generalized coordinates. Strain-based reductions offer compact coordinates and accuracy, while requiring more complex spatial integrations and specialized analytical or numerical treatments.

  • The modal Ritz method parameterizes rod configurations with vector fields such as backbone position and cross-section orientation before applying separation of variables.
  • Spatial shape functions are stacked in matrices Φ(X) and Ψ(X), while q contains the time-dependent generalized modal coordinates.
  • Ritz-Galerkin reduction has been applied to Cosserat-rod statics and dynamics using inertial-frame kinematics and approximated backbone position fields.
  • Strain-interpolation geometrically-exact FEM is more accurate but computationally more complex than conventional geometrically-exact FEM.
  • In the GVS approach, strain fields are represented as ϵ(X, t) = Φ(X)q(t), making strain coefficients the generalized coordinates of homogeneous transformations.
  • The reduced dynamics use the Lagrangian form M(q)¨q + C(q, ˙q) ˙q + Qint(q, ˙q) = Qext(q, ˙q).
  • GVS was validated against GE-FEM and produced very good accuracy with few generalized coordinates, while strain functions avoid boundary-condition requirements.
  • Strain-based reduction requires double space integrals, addressed through analytical Magnus-expansion methods or numerical Newton-Euler two-pass computation.

C. Analytical-based resolutions

Analytical-based resolutions solve selected soft-robot models directly, often using elliptic-function solutions or prescribed geometric curves. Their applicability depends on loading conditions, geometry assumptions, and how the curve representation is chosen.

  • Analytical solutions: Elliptic-function solutions can be obtained analytically for Euler–Bernoulli rod problems under specific concentrated-loading conditions.For a simply supported beam with a concentrated compressive tip force, the solution gives an implicit relation between base angle and applied load.
  • Analytical solutions: The resulting base angle determines the rotational field through integration, which then yields the rod’s displacement field.The displacement field is expressed using additional elliptic functions.
  • Model relationships: Curve-based approaches can approximate Cosserat models, with equivalence depending on material consistency and the chosen formulation.The paper relates these approximations particularly to Ritz-based approaches.
  • Functional approaches: Functional approaches describe the robot’s desired geometry with a chosen mathematical space curve, such as a serpenoid curve or backbone curve.The modal approach represents the backbone using modal functions and participation factors.
  • Functional approaches: In the modal approach, modal factors serve as generalized coordinates that fully describe the robot shape.The modal functions are selected by the robot programmer to satisfy task constraints.

B. Constant Curvature Models

Constant-curvature models simplify continuum-robot kinematics by representing bodies as finite sequences of curved segments. Their practical use is broad, but common parameterizations can become singular as curvature vanishes and may omit mechanically relevant effects.

  • Motivation: Constant curvature is attractive because it simplifies kinematic modeling, real-time control, and related computations.The assumption is motivated by cylindrical manipulators with actuators parallel to the mid-line when external forces are absent.
  • Kinematic parameterization: A constant-curvature segment is commonly parameterized by curvature κ, arc-plane angle φ, and arc length l.These arc parameters define the segment’s task-space mapping.
  • Kinematic mapping: Denavit–Hartenberg parameters can map a constant-curvature backbone to a virtual rigid-link manipulator and produce a homogeneous transformation matrix.The modified parameters account for curvature-induced coupling in the continuum system.
  • Limitations: PCC parameterizations can exhibit a numerical singularity as κ →0 because the radius of curvature becomes infinite or undefined.Taylor-series approximations and alternative parameterizations are among the proposed remedies.

2) Mechanics-based models

Mechanics-based models supplement constant-curvature kinematics with force and moment balances. They establish relationships between actuator configurations and beam shapes, while extensions address cross-sectional deformation and torsion.

  • Mechanics-based formulation: Force and moment balances can derive forward and inverse kinematics for tendon-driven manipulators represented with piecewise constant curvature.The tendon is modeled as experiencing constant tension along its length.
  • Mechanics-based formulation: Equation (30) provides a mechanics-based relationship between beam configuration and tendon configuration.The relationship is analogous to the kinematic transformations used for constant-curvature models.
  • Mechanical consistency: Cross-sectional deformation can make PCC parameterization inaccurate for braided pneumatic continuum manipulators.A strain-energy formulation based on Cauchy–Green stretch is used to study this effect.
  • Mechanical consistency: Extensions incorporate torsion through an additional lumped torsional angle alongside two constant orthogonal curvatures.This formulation is applied to dynamics of tendon-driven continuum robots with elastic backbones and rigid disks.

VII. DISCRETE MODELS

Discrete models represent soft bodies as discrete systems from the outset, including lumped-mass, pseudo-rigid, and discrete-rod approaches. They trade structural simplicity and adaptability for higher degrees of freedom, costly identification, or lower spatial accuracy.

  • Overview: Discrete modeling approaches include lumped-mass, pseudo-rigid, and discrete-rod models.The system is discrete from the beginning rather than discretized only for numerical resolution or through geometric assumptions.
  • Lumped-mass models: Lumped-mass models represent a continuous body with discrete masses, dampers, and springs.Their governing equations can be derived using energy methods or Newtonian force propagation.
  • Lumped-mass models: Lumped-parameter models are simple and adaptable to nonlinear friction, material hysteresis, and inertial dynamics.Mass–spring–damper arrays can model large deformations while allowing bending and shear.
  • Pseudo-rigid models: Pseudo-rigid models represent soft bodies as rigid-link chains connected by revolute, universal, or spherical joints.This formulation exploits established rigid-robotics theories.
  • Pseudo-rigid models: For continuous elastic structures, pseudo-rigid models have low spatial approximation accuracy and require expensive identification procedures.They can nevertheless provide satisfactory results for hyper-redundant or snake-like robots.

C. Discrete rods

Discrete-rod approaches model soft robots by discretizing rod geometry into nodes and segments, enabling discrete curvature, mass, bending, and twist representations. The review also places neural-network models among data-driven alternatives that learn mappings between actuation and shape parameters.

  • Discrete elastic rods: Discrete elastic rod (DER) models discretize an inextensible Kirchhoff rod into nodes connected by straight segments.Each node has a position vector, and each segment has an edge vector and tangent.
  • Discrete elastic rods: DER assigns discrete curvature to each vertex from the turning angle between consecutive edges.The curvature is expressed as κ_i = 2 tan φ_i, where φ_i is the turning angle.
  • Discrete elastic rods: Each vertex carries a mass, while edge integration supplies mass moments of inertia and supports discrete bending and twist energies.The generalized coordinates include node positions, twist angles, and turning angles.
  • Data-driven models: Data-driven soft-robot models use external data rather than requiring a physical model, but they rely on large amounts of data.The review identifies surrogate models as a distinct alternative to the modeling approaches described earlier.
  • Neural networks: Neural networks are common data-driven models whose inputs and outputs typically represent actuation variables u and shape parameters q.Training adjusts network weights through back-propagation after forward and backward passes.
  • Neural networks: A modified Elman neural network has been used to model a handling-assistant trunk, while feedforward networks address inverse kinematics of cable-driven soft manipulators.The trunk study compares data-driven modeling with a pseudo-rigid model.

B. Data-driven Order Reduction

Data-driven order-reduction methods approximate soft-robot dynamics using learned or lifted representations, including Koopman operators and reduced physical models. The review emphasizes that modeling choices balance realism against computational efficiency and application needs.

  • Koopman operator methods: Koopman operator theory represents nonlinear dynamics linearly in an infinite-dimensional space of observables.The operator lifts system dynamics from state space to observable-function space.
  • Koopman operator methods: Approximate Koopman operators can be obtained from experimentally measured state snapshot pairs.These data provide the measurements needed to estimate the discrete operator.
  • Reduced physical models: Data-driven order reduction is also applied to finite-element models to reduce their computational burden.The review identifies model-reduction methods based on FEM as another route within this family.
  • Accuracy–efficiency trade-off: Soft-robot modeling selects a compromise between realism and computational efficiency according to the user’s needs.Design may prioritize realism, optimization may prioritize repeated fast simulations, and closed-loop control may accept lower accuracy for online inverse algorithms.
  • Finite-element methods: Commercial FEM handles arbitrary 3D geometries and multiphysics but is often computationally expensive and poorly adapted to robotics-specific control.Large numbers of nodal degrees of freedom remain a barrier despite efforts toward faster simulation.
  • Classification: The review organizes modeling families by their parametrization paths and acknowledges overlap between approaches.The classification is intended to clarify structural relationships among methods rather than exhaust all possible classifications.
  • Director-based methods: Director-based methods can provide comparable accuracy to geometrically exact FEM for beam-like robots while supporting robotics-specific real-time simulation and control.The review lists concentric tubes, continuous parallel robots, soft arms, and tendon-driven continuum robots as suitable examples.
  • Review motivation: The paper motivates this synthesis by the difficulty of grasping a widespread field and uses it to identify commonalities and distinctions among techniques.The stated goal is to untangle the breadth of soft-robotics modeling research.
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