Soft Robots Modeling: a Structured Overview

In plain words

Soft robots bend, stretch and squeeze instead of moving at rigid joints. That makes them hard to predict: the shape of the robot is a continuous object, and the equations that describe it are difficult to solve. Engineers have therefore developed many different ways to model these robots, each with its own trade-off between how faithful the model is and how fast it runs.

This paper is a map of those methods. It sorts them into families by the mathematics they use: full physical models of the material, simplified models that assume the robot bends into a known shape such as a circle, models that replace the robot with a chain of rigid pieces or small masses, and models learned from data. For each family it explains the core idea, what it is good for (simulation, design or control), and where it breaks down. It does not introduce a new method and it contains no experiments of its own.

TL;DR

• A review, not a method: it classifies soft-robot models into four families and explains the equations behind each §I §II.
• Physics-based continuum and Cosserat-rod models are the most consistent but costly; constant-curvature and discrete models are fast but make simplifications §X.
• Data-driven models are fast at run time but need large datasets that must be rebuilt for each robot design §VIII §X.
• The paper states no numerical results of its own; there is nothing to reproduce from it.

Problem

Soft robots are continuous bodies, so their state is described by fields over the body, not by a finite set of joint angles §II. Their governing equations are highly nonlinear partial differential equations and are generally not solvable in closed form §I. The review's stated problem is to find a trade-off between accuracy and numerical usefulness for control and optimisation §I. It also notes that the assumptions behind many models are often not stated explicitly §II.

Method

The paper does not propose a method. Its contribution is a classification and a critical comparison. The classification uses four families §II:

1. Continuum mechanics models §III. The body is treated as a continuum, either in full 3D §III.A or as a reduced beam or shell with directors §III.B. Solved numerically by finite elements §IV, by finite differences, shooting, Newton–Euler or collocation methods, or by Ritz reductions §V.
2. Geometrical models §VI. The deformed body is assumed to follow a chosen curve. Functional models use a fixed mathematical curve §VI.A. Piecewise-constant-curvature (PCC) models split the body into circular arcs §VI.B.
3. Discrete models §VII. The body is discretised from the start: lumped masses and springs §VII.A, rigid links with joints (pseudo-rigid) §VII.B, or discrete elastic rods §VII.C.
4. Surrogate models §VIII. A network or other learned map is fitted to measured data, typically feed-forward networks for statics and recurrent networks for dynamics §VIII.A. Data-driven order reduction is treated separately §VIII.B.

The paper also lists software packages for these families §IX.

The paper gives no algorithm. Its own trade-off advice §X can be read as the following selection sketch. This is our paraphrase, not a procedure stated in the paper.

# Family selection sketch, paraphrasing [§X] and [§XI]
# Input:  need ("design", "simulation" or "control"), body_shape
# Output: list of model families to consider

def candidate_families(need, body_shape):
    # Design of new actuators: physics-consistent full models  [§X]
    if need == "design":
        return ["continuum (FEM)", "directors (GE-FEM)"]
    # Slender, beam-like bodies: directors with Ritz or shooting  [§X]
    if body_shape == "beam":
        return ["directors (modal Ritz, GVS)", "discrete rods"]
    # Arc-like bodies and kinematic control: geometric models  [§X]
    if need == "control" and body_shape == "arc":
        return ["piecewise constant curvature", "functional"]
    # Otherwise, a surrogate model if enough measured data exists  [§VIII]
    return ["surrogate (neural network)"]

Continuum mechanics

Balance of forces in the body gives the Cauchy equilibrium equations §III.A:

\nabla \cdot \bm{\sigma} + \bm{b} = \rho \dot{\bm{v}}

The paper closes this with the Green–Lagrange strain and a constitutive law. Its Green–Lagrange definition is §III.A:

\bm{E} = \frac{1}{2}\left(\nabla\bm{r}^{T}\nabla\bm{r} - \nabla\bm{r}_o^{T}\nabla\bm{r}_o\right)

For hyperelastic (Green) materials, the paper gives the Ogden strain energy in principal stretches §III.A:

U(\lambda_1, \lambda_2, \lambda_3) = \sum_{j=1}^{n} \frac{\mu_j}{\alpha_j}\left(\lambda_1^{\alpha_j} + \lambda_2^{\alpha_j} + \lambda_3^{\alpha_j} - 3\right)

It states that the neo-Hookean model is the case n = 1, α = 2, and the Mooney–Rivlin model is the case n = 2, α₁ = 2, α₂ = −2 §III.A. The paper also states the Cauchy stress relation with a pressure term p §III.A.

For slender bodies the paper uses the Cosserat rod. A rod is a centreline r(X) with a rotating cross-section R(X), with X the rest-length coordinate §III.B. Its static and dynamic balance of forces and moments is §III.B:

\bm{n}' + \bar{\bm{f}} = \rho A \dot{\bm{v}}, \qquad \bm{m}' + \bm{r}' \times \bm{n} + \bar{\bm{l}} = \partial_t(\rho \bm{I}\cdot\bm{\omega})

with the linear Hookean law n = R H_ℓ ε_ℓ and m = R H_a ε_a, where H_ℓ = diag(EA, GA, GA) and H_a = diag(GI₁, EI₂, EI₃) §III.B.

The same model can be written on SE(3) using the homogeneous pose g(X) and the twist ξ, with g' = g ξ̂ §III.B. The paper gives the matching balance equation in twist form §III.B:

\bm{\mathcal{M}}\dot{\bm{\eta}} - \mathrm{ad}_{\bm{\eta}}^{T}\bm{\mathcal{M}}\bm{\eta} = \bm{\Lambda}' - \mathrm{ad}_{\bm{\xi}}^{T}\bm{\Lambda} + \bar{\bm{F}}

Tendon actuation enters either as external wrenches §III.B or as an active constitutive law §III.B. The paper gives the external-wrench form as a sum over tendons i with tension u_i and routing vector d_i §III.B, and the active-law form as the passive law plus the same tendon terms §III.B.

Kirchhoff and non-linear Euler–Bernoulli beams are presented as special cases of the Cosserat model §III.B. For a planar elastica the paper gives EI(X) θ' = M_ext §III.B.

Finite elements

For 3D bodies the paper describes the finite element method and writes the static equilibrium as Q_int(q) = Q_ext(q), with a dynamic ODE system for transient problems §IV.A. Nonlinearities are either geometric or material. Corotational and ANCF formulations are mentioned as ways to capture finite rotations §IV.A. Sofa's actuation handling is written as a constraint linearisation with Lagrange multipliers λ and a three-step projection procedure §IV.A.

Numerical resolution of directors-based models

Non-energetic methods discretise the boundary-value problem directly: finite differences §V.A, shooting §V.A, a continuous Newton–Euler scheme §V.A and collocation §V.A. Energetic (Ritz) methods reduce the configuration to Lagrange ODEs §V.B. The paper's main examples are the geometrically-exact finite element (GE-FEM) §V.B, absolute and relative modal Ritz reductions §V.B, and the geometric variable-strain (GVS) approach, which parameterises the rod by strain modes and whose reduced dynamics take the form §V.B:

\bm{M}(\bm{q})\ddot{\bm{q}} + \bm{C}(\bm{q},\dot{\bm{q}})\dot{\bm{q}} + \bm{Q}_{int}(\bm{q},\dot{\bm{q}}) = \bm{Q}_{ext}(\bm{q},\dot{\bm{q}})

The analytical Euler–Bernoulli case for a tip-loaded beam is given in terms of an elliptic integral §V.C:

F = \frac{4EI}{l^2}\left[\mathcal{K}\sin\frac{\theta(0)}{2}\right]^2

Geometrical models

Functional models describe the backbone with a chosen curve: a serpenoid curve, a modal backbone integral, Pythagorean-hodograph curves, cubic Hermite splines, Euler arc splines, or elliptic Fourier descriptors §VI.A.

PCC models describe each segment as a circular arc. The paper gives the arc transform in the form §VI.B:

\bm{T}(\kappa, \phi, s) = \begin{bmatrix} \cos\phi\cos\kappa s & -\sin\phi & \cos\phi\sin\kappa s & \tfrac{1}{k}\cos\phi(1-\cos\kappa s) \\ \sin\phi\cos\kappa s & \cos\phi & \sin\phi\sin\kappa s & \tfrac{1}{k}\sin\phi(1-\cos\kappa s) \\ -\sin\kappa s & 0 & \cos\kappa s & \tfrac{1}{\kappa}\sin\kappa s \\ 0 & 0 & 0 & 1 \end{bmatrix}

The paper attributes this transform to Webster and Jones §VI.B and gives the conversion from three tendon lengths l₁, l₂, l₃ and routing distance d to the arc parameters as §VI.B:

l = \frac{l_1+l_2+l_3}{3}, \qquad \phi = \tan^{-1}\!\left(\frac{\sqrt{3}}{3}\, \frac{l_2+l_3-2l_1}{l_2-l_3}\right), \qquad \kappa = \frac{2}{d}\, \frac{\sqrt{l_1^2+l_2^2+l_3^2-l_1l_2-l_1l_3-l_2l_3}}{l_1+l_2+l_3}

Mechanics-based PCC models add static balances to the arc model; the paper writes the Camarillo-type tendon response as y = A† q §VI.B.

Discrete models

A lumped-mass model is a set of masses connected by springs and dampers §VII.A, fig. 4. Pseudo-rigid models are chains of rigid links with joints §VII.B. Discrete elastic rods (DER) use vertices and edges, with the discrete curvature at vertex i given by κ_i = 2 sin φ_i / (1 + cos φ_i) = 2 tan(φ_i/2) §VII.C, fig. 5.

Surrogate models

A neural network is written as a nested composition of layers with weights and activations §VIII.A. The paper states that input and output layers usually represent actuation and shape parameters §VIII.A. It gives no architecture, layer sizes or training settings for any specific robot.

Hyperparameters

Not applicable. The paper is a review and fixes no numerical settings. Its equations use symbols (for example E, G, A, I, μ_j, α_j, ρ, d, n_a) and do not assign values to them §III.A §III.B §VI.B. Where the paper gives a model parameter count, it does so only in prose, for example that Sofa's FE models have "a huge number of dofs" tab. 1.

Results

Not applicable. The paper reports no experiments and no benchmark numbers. The comparative statements are qualitative judgements, summarised in tab. 1 and discussed in §X. These judgements are the authors' opinion, not measured results. The validation of the GVS approach against GE-FEM is cited to Boyer et al. §V.B, not reproduced here.

Paper vs code

No official code. The paper links no repository for its own content. The paper describes third-party software (Sofa, Elastica, SoRoSim, TMTDyn, SoMo, ChainQueen, IPC-Sim, DiffAqua) in §IX. These belong to other groups and are not part of this card. Searched: the paper and its bibliography for code links (none found), and the GitHub repository search for the exact title with the first author and for "soft robots modeling review" (no repository by an author of this paper).

Pitfalls

• Nothing to implement. This is a survey. An implementer needs the cited primary papers, not this card, for any specific model.
• Strain-invariant wording. The paper says the principal stretches are "the eigenvalues of" the Green–Lagrange tensor E §III.A. In standard continuum mechanics, stretches are obtained from the right Cauchy–Green tensor (C = 2E + I), not from E itself. Our reading; check the primary source before using the Ogden expression.
• Inconsistent notation in the PCC matrix. The entries use 1/k in two places and 1/κ in another §VI.B. Read them as 1/κ, the curvature, unless a primary source says otherwise. Our reading.
• Tendon-to-arc formula without derivation. The equation for κ from the tendon lengths §VI.B is given without derivation, citing Jones et al. We did not verify it against the cited paper.
• Euler–Bernoulli case under-specified. §V.C calls the example a simply supported beam under a tip force, but does not define the elliptic-integral modulus or the full boundary conditions. Treat the formula as "not stated" in those details.
• Cross-references disagree. The conclusions refer to "Section IV" for the directors-based numerical methods, which are §V in the paper's own numbering §XI. Some text refers to "Sec. X" for the critical analysis, which matches the numbering. Table 1 contains a typo ("ANCS-FEM" for ANCF-FEM).
• Equation numbers not given. The fetched source does not include the compiled PDF, so equation numbers cannot be checked. Citations here point to sections instead.
• Opinion versus measurement. Statements such as "the optimal trade-off" tab. 1 and "should be advantageously replaced" §X are the authors' judgement, not experiments.
• Figures. Six figures are described by caption and cited as fig. 1 to fig. 6. None is reproduced on this card.

Lineage

• Builds on: Cosserat rod theory §III.B, Simo's geometrically-exact FEM §V.B, and Webster and Jones' constant-curvature kinematics §VI.B.
• Compared against: earlier reviews on soft-robot modelling, cited in §I.
• Superseded by: not stated. The paper itself recommends the GVS approach in place of the PCC models it covers, as a more consistent alternative §X.