Pith. sign in

REVIEW 4 major objections 4 minor 99 references

SoRoMoX: Fast, Differentiable, and Parallelizable Soft Robot Models

T0 review · 4 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read SoRoMoX claims to be the first rod/strain-based soft-robot model framework that runs directly on GPUs and is end-to-end differentiable, enabling identification, control, safety, and learning workflows that were impractical before.

desk verdict A genuinely useful JAX-native soft-robot modeling framework whose headline speedup claim needs to be scoped to SoRoSim and backed by accuracy-matched benchmarks. read the letter →

arxiv 2608.06650 v1 pith:I6JQ5DAG submitted 2026-08-06 cs.RO cs.AI

Maximilian Stölzle , Solange Gribonval , Daniel Feliu-Talegon , Vito Daniele Perfetta , Michele Martini , Chuhan Zhang , Kiwan Wong , Mohammed Tarnini
show 4 more authors
Anup Teejo Mathew Federico Renda Daniela Rus Cosimo Della Santina
This is my paper · ORCID
classification cs.ROcs.AI
keywords softrobotmodelingCosseratroddifferentiablesimulationJAXGPUparallelizationmodel-basedcontrolsystemidentificationbarrierfunctions
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper's thesis is that the bottleneck in soft-robot control has moved from model derivation to model deployment: Cosserat-rod modeling theory is mature, but existing implementations do not support the differentiable, GPU-parallel, control-oriented workflows that rigid-robot engines provide. SoRoMoX claims to close this gap as the first rod/strain-based soft-robot modeling framework that runs directly on GPUs and is end-to-end differentiable with respect to states, inputs, and parameters, exposing inertia matrices, Coriolis and potential forces, Jacobians, and actuation maps through one common interface. The supporting benchmarks report up to 18.1x faster sequential CPU rollouts than the MATLAB toolbox SoRoSim and up to 234.6x higher throughput through GPU batching. If the claims hold, gradient-based system identification, model-based tracking, safety-constrained control, and massively parallel policy training for soft robots become as routine as they are for rigid robots.

What carries the argument

The central object is the control-oriented equations of motion $M(q)\ddot{q} + C(q,\dot{q})\dot{q} + G(q) + K(q) + D\dot{q} = A(q)u$, exposed as differentiable JAX primitives through a common SoftRobot interface so that controllers, optimizers, and simulators read the same model terms. The mechanism that carries the argument is a bundle of implementation practices: a static, fixed-shape runtime layout that converts heterogeneous GVS segment descriptions into padded arrays while restricting the state to active coordinates; structure-aware evaluation that propagates kinematics serially with lax.scan and batches independent quadrature points with vmap; caching of state-independent quantities; a fused forward-dynamics path that avoids materializing the Coriolis matrix; and protected evaluation of removable singularities such as $\sin\vartheta/\vartheta$, $(1-\cos\vartheta)/\vartheta^2$, and $(\vartheta-\sin\vartheta)/\vartheta^3$ via true conditionals (lax.cond) with Taylor limiting forms, so that values and gradients stay well defined at straight and zero-arclength configurations.

What would settle it

Re-run the CPU and GPU benchmarks with element counts chosen so that all three libraries agree on tip position to a prespecified tolerance (for example, equal tip-position RMSE within 0.1% of beam length) before timing rollouts; if the speedup ratios change substantially, the headline factors are discretization artifacts. Separately, instrument the HOCBF obstacle scenario with a physical force sensor and check whether the measured peak contact force, rather than the penetration proxy of Eq. 45, stays at or below the 5 N bound.

Watch

Extended reading notes

Core claim

The paper's central claim is that the reduced-order Cosserat-rod models the community already trusts — articulated soft chains, Piecewise Constant Strain (PCS), and Geometric Variable Strain (GVS) — can be implemented as fully numerical, JIT-compilable programs that expose the complete control-oriented equations of motion $M(q)\ddot{q} + C(q,\dot{q})\dot{q} + G(q) + K(q) + D\dot{q} = A(q)u$, together with forward kinematics, Jacobians, their time derivatives, and energy terms, all differentiable with respect to states, inputs, and parameters. The authors further claim that this makes SoRoMoX the first rod/strain-based soft-robot framework that runs directly on GPUs and is end-to-end differentiable, enabling workflows that were previously impractical or impossible: static-equilibrium identification of material parameters (marker RMSE reduced by 66%), residual-force learning (a further 64% reduction), computed-torque tracking with RMSE reduced by roughly a factor of 500 relative to model-free PD, control-gain optimization with up to 62% lower loss, safety-constrained control holding peak contact force within a 5 N bound, and reinforcement learning trained up to 7x faster than a CPU discrete-rod baseline.

Load-bearing premise

The quantitative claims rest on the premise that the benchmark configurations — two constant-strain elements for SoRoMoX and SoRoSim against 10 to 100 elements for PyElastica — deliver 'comparable simulation accuracy', which the paper asserts without reporting a concrete accuracy threshold; if the discretizations are not matched to the same physical accuracy, the 18.1x and 234.6x ratios may be artifacts of element count, and the 5 N safety bound likewise constrains a linear-elastic penetration proxy rather than a measured contact force.

Editorial extensions

If this is right

  • Gradient-based pipelines that are standard for rigid robots — system identification, trajectory optimization, and gain tuning — become directly applicable to strain-coordinate soft-robot models, because every model term and full rollout is differentiable end to end.
  • Massively parallel policy training for soft robots becomes practical on a single GPU, with the paper reporting roughly 4x and 7x wall-clock training speedups at 256 and 512 parallel environments relative to a two-core CPU PyElastica baseline.
  • Contact-force safety can be enforced in closed loop through high-order control barrier functions on differentiable soft-robot dynamics; in the paper's scenario the peak contact force stays within the prescribed 5 N bound, versus about 33.5 N without the safety filter.
  • Because articulated, PCS, and GVS models share one control-ready interface, controllers, coordinate transformations, renderers, and benchmarks can be written once and reused across model families, a step toward a standardized description format for continuum soft robots.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the speedups survive an accuracy-matched comparison — equalizing discretization error across libraries before timing — soft-robot modeling would gain the same computational maturity that rigid-body engines built over the past decade, making pre-built model libraries, batched design optimization, and large-scale sim-to-real pipelines routine engineering rather than research demonstrations.
  • The same implementation techniques (static runtime layout, early active-coordinate reduction, singularity-safe conditional branches) should extend to neighboring reduced-order formulations such as polynomial-curvature or finite-element strain parameterizations; this is a natural testable transfer the paper does not itself carry out.
  • The safety result is demonstrated on a linear-elastic penetration proxy for contact force rather than on a measured physical force, so a hardware experiment with a force sensor is the clear next test of whether the 5 N guarantee holds on a real soft robot.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The manuscript introduces SoRoMoX, a JAX/Python framework that implements articulated, Piecewise Constant Strain (PCS), and Geometric Variable Strain (GVS) soft-robot models through a unified, control-oriented interface. The framework exposes kinematics, Jacobians, dynamical matrices, actuation maps, coordinate transformations, packaged controllers, and differentiable closed-loop rollouts, with JIT compilation, automatic differentiation, and GPU/TPU batching. The paper reports CPU rollouts versus SoRoSim and PyElastica, GPU batch-scaling throughput, and six application case studies: static-equilibrium system identification, residual-force learning, model-based control, control-gain optimization, HOCLF/HOCBF safety-constrained control, and parallel reinforcement learning. The central advertised results are that SoRoMoX is the first rod/strain-based soft-robot framework running natively on GPUs, that sequential CPU rollouts are up to 18.1x faster than state-of-the-art alternatives, that GPU-parallel rollouts increase throughput by up to 234.6x, and that the framework enables several orders-of-magnitude improvements in downstream tasks.

Significance. If the technical claims are properly scoped, SoRoMoX is a significant open-source contribution to soft-robot simulation and control. The code is MIT-licensed, builds on JAX/Equinox/Diffrax, and provides a rare combination of differentiable dynamics, batched GPU execution, and a control-oriented API. The manuscript also includes strong software-engineering practices: a CI-based correctness suite that cross-checks analytical Jacobians against automatic differentiation, consistency across model families, and singularity-safe evaluation near zero-strain configurations. These are documented in the paper and are genuinely valuable to the community. The six case studies, while compact, demonstrate plausible end-to-end workflows. However, the headline performance claims need revision: Table IV shows that PyElastica is faster than SoRoMoX in every single-environment CPU case, and the 18.1x speedup is only versus SoRoSim, not versus all state-of-the-art alternatives. The equal-accuracy discretization underlying the CPU benchmark is also not demonstrated. The paper should be accepted only after these claims are corrected and the benchmarks are made more transparent.

major comments (4)
  1. [Abstract and Section IV-A, Table IV] The abstract's claim of 'up to 18.1x faster than state-of-the-art alternatives' is not supported as written. Table IV reports that PyElastica is the fastest library in every benchmark case (e.g., planar PCS: 2.25 s for PyElastica versus 4.18 s for SoRoMoX; spatial PCS: 2.45 s versus 13.26 s). The 18.1x figure is exclusively a speedup over SoRoSim. Please scope the CPU speedup claim to SoRoSim, or reframe it as 'faster than SoRoSim and competitive with PyElastica on single-CPU rollouts' with the precise qualifications.
  2. [Appendix B-A and Table V] The equal-accuracy premise of the CPU benchmark is not established. Appendix B-A fixes SoRoMoX and SoRoSim at two piecewise elements and PyElastica at 10-100 elements, stating that these counts were 'selected to achieve comparable simulation accuracy', but no accuracy threshold, convergence study, or head-to-head error metric against PyElastica is reported. Table V provides RMSE values only between SoRoSim and SoRoMoX (0.14% to 1.20% of beam length), which does not show that either reduced-order model matches the 10-100 element PyElastica discretization. Without matched physical accuracy, the speed ratios may be artifacts of discretization rather than of the software design. Please report a matched-accuracy comparison, or explicitly state that the speedups are implementation- and discretization-specific.
  3. [Section V-E, Eq. (45)] The safety-constrained control result is stated as a guarantee on contact force, but Eq. (45) defines b(y)=F_max + k_c min_{i,j} d_{ij}(q), with k_c=1000 N/m in Appendix D-E2. This is a linear-penetration proxy for contact force, not a measured or integrated physical contact force. The text repeatedly refers to the 'maximum pairwise normal force' being kept within the 5 N limit and reports a 33.5 N violation without it, which conflates the proxy with the physical quantity. Please revise the wording throughout Section V-E to state that the HOCBF constrains the linear-elastic proxy, and discuss the validity and limitations of this proxy for the safety claim.
  4. [Section V-F and Appendix D-F] The RL training speedup is an end-to-end comparison that mixes different model formulations, solvers, hardware, and batching strategies: one-segment PCS in SoRoMoX on a single GPU versus a DCM in PyElastica on two CPU cores. The appendix does acknowledge this, but the abstract and Section V-F present '7x faster' without the same qualification. Please scope the RL speedup specifically as a full-stack comparison and avoid implying that SoRoMoX is 7x faster at single-environment rollouts.
minor comments (4)
  1. [Appendix B-A] The solver name 'node45' appears to be a typo for 'ode45' (the ODE45 solver used in SoRoSim and SoRoMoX).
  2. [Footnote 2 and Abstract] The abstract claims that SoRoMoX is 'the first rod/strain-based soft-robot modeling framework that runs directly on GPUs', but footnote 2 mentions a recent experimental JAX implementation of PyElastica that enables GPU and TPU parallelization. The 'to our knowledge' qualifier may make the claim defensible, but the authors should clarify why that implementation is not a comparable framework.
  3. [Section I, acronym definition] The acronym expansion 'Soft Robot Models in jaX' is inconsistent with the stylized 'JAX'; consider using 'Soft Robot Models in JAX' throughout.
  4. [Appendix D-F] The RL reward function in Eq. (55) uses a term max(dot(epsilon_x), 0) where epsilon_x is the derivative of the error norm; the notation is understandable but the denominator epsilon_x + epsilon_RL should be defined explicitly in the main text.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: central claims rest on external benchmarks and published model equations, not on fitted or self-cited inputs.

full rationale

SoRoMoX's central claims are implementation and performance claims. The model equations are taken from published PCS/GVS formulations ([9]-[11], [34]) and the framework is benchmarked against two external open-source libraries (SoRoSim, PyElastica). No parameter is fitted to the benchmark outputs and then reported as a prediction; the system-identification and residual-learning case studies explicitly label their quantities as optimized fits, and the reported RMSE reductions are in-sample fitting improvements rather than derived predictions. The safety case study defines a force-like barrier b(y)=Fmax+kc min dij (Eq. 45) and the HOCBF controller enforces b(y)>=0; the reported 5 N bound is a constraint on this disclosed proxy, so the result is a controller-feasibility demonstration rather than an independent physical force prediction. Self-citations to [5], [17], [18], [74] are present, but they support controller formulations and prior method development, not the framework's core claims, which remain independently testable through the open-source code and external benchmarks. The unverified equal-accuracy discretization choice in Appendix B-A and the abstract's '18.1x faster than state-of-the-art alternatives' wording (inconsistent with Table IV, where PyElastica is fastest) are correctness and interpretation concerns, not circular reasoning.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The central contributions are software and empirical measurements, not new physical entities or new dynamical laws. The main pulls from prior work are the PCS/GVS strain parametrizations, the standard Euler-Lagrange structure, and a set of hand-chosen benchmark and safety-proxy parameters that load-bear on the reported quantitative claims.

free parameters (4)
  • Identified material parameters (E, nu, rho) = 0.315 MPa, 0.45, 1320 kg/m3
    Fitted to four-marker static equilibrium data in Section V-A and Appendix D-A; the reported 66% marker RMSE improvement depends on these optimized values.
  • Optimized control gains (Kp, Ki, Kd diagonals) = not reported
    Gradient-optimized over 100 iterations in batched runs in Section V-D; the reported 62% and 57% loss reductions depend on these fitted gains and on the batch initialization distribution.
  • HOCLF/HOCBF gains and contact stiffness = kc = 1000 N/m; alpha and gamma coefficients not reported
    Chosen by hand in Section V-E and Appendix D-E; the claimed 5 N force bound is enforced through this proxy model, so the safety result depends on these settings.
  • Benchmark discretization counts = 2 segments for SoRoMoX/SoRoSim, 10-100 elements for PyElastica
    Selected to achieve 'comparable simulation accuracy' in Appendix B-A, but no explicit accuracy threshold is given; the speedup ratios directly depend on this choice.
assumptions (4)
  • domain assumption Cosserat rod strain parametrizations (PCS, GVS) capture dominant soft robot deformations.
    The framework inherits this from cited modeling literature [8]-[11], [34]; the paper does not re-derive or validate the mechanical accuracy of these strain coordinates.
  • standard math Euler-Lagrange equations with M, C, G, K, D and actuation map A(q) describe the dynamics in Eq. 1.
    Used as the mathematical template for all model families; standard classical mechanics.
  • ad hoc to paper Five Gauss points and a zero-order GVS basis are sufficient for accurate dynamics in the GPU benchmark.
    Chosen in Appendix B-B without a reported convergence study; the real-time thresholds and GPU scaling boundaries depend on this.
  • ad hoc to paper A linear penetration stiffness kc maps signed clearance to a contact force proxy.
    Eq. 45 defines b(y) = Fmax + kc min dij; the 5 N safety limit is enforced on this proxy, not on a directly measured contact force.

how reviews work

0 comments
Cite this review

Pith. "Pith review of SoRoMoX: Fast, Differentiable, and Parallelizable Soft Robot Models." pith.science (2026). https://pith.science/paper/I6JQ5DAG

@misc{pith2026260806650,
  author       = {Pith},
  title        = {Pith review of: SoRoMoX: Fast, Differentiable, and Parallelizable Soft Robot Models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/I6JQ5DAG}},
  note         = {Machine review of arXiv:2608.06650}
}
read the original abstract

Reduced-order models based on Cosserat-rod theory are now well established, and modeling theory is no longer the primary bottleneck in soft-robot control. Their implementations, however, do not support the differentiable, GPU-parallel, and control-oriented workflows that underpin advanced rigid-robotics applications. Here, we fill this gap with SoRoMoX (Soft Robot Models in JAX), a fully numerical, JIT-compilable Python/JAX framework. SoRoMoX implements articulated, Piecewise Constant Strain, and Variable Strain models through a unified, control-ready interface that provides inertia matrices, gravitational and elastic forces, Jacobians, and their derivatives. To our knowledge, it is the first rod/strain-based soft-robot modeling framework that runs directly on GPUs and is end-to-end differentiable with respect to states, inputs, and parameters. Sequential CPU rollouts are up to 18.1x faster than state-of-the-art alternatives, while GPU-parallel rollouts increase throughput by up to 234.6x. This performance enables workflows that were previously impractical or impossible: static-equilibrium system identification with 66% lower marker RMSE; residual-force learning with a further 64% reduction; computed-torque tracking with RMSE reduced by a factor of approximately 500 relative to model-free PD; control-gain optimization with up to 62% lower loss than untuned gains; safety-constrained control using high-order control barrier functions to keep the peak contact force within a prescribed 5 N bound, compared with 33.5 N without the safety constraint; and reinforcement-learning policy training up to 7x faster than a CPU PyElastica discrete-rod baseline through massively parallel rollouts.

Figures

Figures reproduced from arXiv: 2608.06650 by the authors.

Figure 1
Figure 1. Overview of SoRoMoX. Implemented in Python/JAX, the framework [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. SoRoMoX user workflow. High-level path from installation and robot definition to querying model terms, constructing model-based controllers, running open- and closed-loop simulations, and rendering resulting trajectories and shape evolutions. strain components, and quadrature resolution, while fixed￾shape runtime arrays keep the implementation compatible with JAX compilation, differentiation, and batching. GVS syste… view at source ↗
Figure 3
Figure 3. Installing SoRoMoX. The PyPI command installs the core package, while the uv sync workflow installs the source checkout together with optional example dependencies. intended. The renderers share a common abstraction: they sam￾ple forward kinematics along the robot backbone and expose common methods for rendering a single configuration, a full trajectory, or an interactive scene. This keeps visualization independent … view at source ↗
Figures from the paper (10 more)
Figure 5
Figure 5. Figure 5: Constructing a GVS robot. In contrast to PCS, GVS is assembled from segment specifications that describe links, joints, basis functions, and integration settings. With no environment arguments, the robot points along the positive world z axis under gravity directed alo…
Figure 4
Figure 4. Figure 4: Minimal spatial PCS workflow. The named hanging constructor mounts the backbone along world −z under the default world-frame gravity. The typed parameter object defines the robot, the system class exposes kinematics and dynamics, rollout_to integrates the dynamics usin…
Figure 8
Figure 8. Figure 8: A shows the tip response for a representative [PITH_FULL_IMAGE:figures/full_fig_p007_8.png]
Figure 7
Figure 7. Figure 7: JAX transformations and parameter optimization. SoRoMoX methods can be compiled, vectorized, and differentiated, enabling gradient￾based updates of physical parameters while keeping the model structure fixed. PCS and spatial PCS show decreasing efficiency with in￾creas…
Figure 8
Figure 8. Figure 8: Benchmarking Comparison and Static-Equilibrium Identification/Residual-Learning Case Study. (A) CPU rollout benchmark of SoRoMoX against SoRoSim and PyElastica, showing the simulated tip response of a cylindrical soft beam initially aligned with the global z axis and d…
Figure 9
Figure 9. Figure 9: Benchmark of batched GPU simulation scaling for varying numbers [PITH_FULL_IMAGE:figures/full_fig_p009_9.png]
Figure 10
Figure 10. Figure 10: Application case studies for SoRoMoX: physical parameter iden [PITH_FULL_IMAGE:figures/full_fig_p010_10.png]
Figure 11
Figure 11. Figure 11: Model-based control with fully actuated spatial PCS robots. ( [PITH_FULL_IMAGE:figures/full_fig_p011_11.png]
Figure 12
Figure 12. Figure 12: Results of the control gain optimization demo. ( [PITH_FULL_IMAGE:figures/full_fig_p012_12.png]
Figure 13
Figure 13. Figure 13: Safety-constrained regulation under the HOCLF– HOCBF controller and parallel RL for goal tracking. (A) Rendered shape evolution of a safety-unaware HOCLF controller (top) and a safety-constrained HOCLF+HOCBF controller (bottom) on the task end-effector regulation for …

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

99 extracted references · 57 canonical work pages

  1. [1]

    Physical control: A new avenue to achieve intelligence in soft robotics,

    E. Milana, C. D. Santina, B. Gorissen, and P. Rothemund, “Physical control: A new avenue to achieve intelligence in soft robotics,”Science Robotics, vol. 10, no. 102, p. eadw7660, 2025. IEEE ROBOTICS AND AUTOMATION MAGAZINE 16

    2025

  2. [2]

    Design and kinematic modeling of constant curvature continuum robots: A review,

    R. J. Webster III and B. A. Jones, “Design and kinematic modeling of constant curvature continuum robots: A review,”The International Journal of Robotics Research, vol. 29, no. 13, pp. 1661–1683, 2010

    2010

  3. [3]

    Control oriented modeling of soft robots: The polynomial curvature case,

    C. Della Santina and D. Rus, “Control oriented modeling of soft robots: The polynomial curvature case,”IEEE Robotics and Automation Letters, vol. 5, no. 2, pp. 290–298, 2019

    2019

  4. [4]

    A geometrically exact model for soft continuum robots: The finite element deformation space formulation,

    S. Grazioso, G. Di Gironimo, and B. Siciliano, “A geometrically exact model for soft continuum robots: The finite element deformation space formulation,”Soft robotics, vol. 6, no. 6, pp. 790–811, 2019

    2019

  5. [5]

    Model-based control of soft robots: A survey of the state of the art and open challenges,

    C. Della Santina, C. Duriez, and D. Rus, “Model-based control of soft robots: A survey of the state of the art and open challenges,”IEEE Control Systems Magazine, vol. 43, no. 3, pp. 30–65, 2023

    2023

  6. [6]

    Control-oriented models for hyperelastic soft robots through differential geometry of curves,

    B. Caasenbrood, A. Pogromsky, and H. Nijmeijer, “Control-oriented models for hyperelastic soft robots through differential geometry of curves,”Soft Robotics, vol. 10, no. 1, pp. 129–148, 2023

    2023

  7. [7]

    Rod models in continuum and soft robot control: a review,

    C. Alessi, C. Agabiti, D. Caradonna, C. Laschi, F. Renda, and E. Falotico, “Rod models in continuum and soft robot control: a review,” arXiv preprint arXiv:2407.05886, 2024

    arXiv 2024

  8. [8]

    Reduced order modeling of hybrid soft-rigid robots using global, local, and state-dependent strain parameterization,

    A. T. Mathew, D. Feliu-Talegon, A. Y . Alkayas, F. Boyer, and F. Renda, “Reduced order modeling of hybrid soft-rigid robots using global, local, and state-dependent strain parameterization,”The International Journal of Robotics Research, vol. 44, no. 1, pp. 129–154, 2025

    2025

Show all 99 references
  1. [9]

    Discrete cosserat ap- proach for multisection soft manipulator dynamics,

    F. Renda, F. Boyer, J. Dias, and L. Seneviratne, “Discrete cosserat ap- proach for multisection soft manipulator dynamics,”IEEE Transactions on Robotics, vol. 34, no. 6, pp. 1518–1533, 2018

    2018

  2. [10]

    A geometric variable-strain approach for static modeling of soft manipula- tors with tendon and fluidic actuation,

    F. Renda, C. Armanini, V . Lebastard, F. Candelier, and F. Boyer, “A geometric variable-strain approach for static modeling of soft manipula- tors with tendon and fluidic actuation,”IEEE Robotics and Automation Letters, vol. 5, no. 3, pp. 4006–4013, 2020

    2020

  3. [11]

    Dynamics of continuum and soft robots: A strain parameterization based approach,

    F. Boyer, V . Lebastard, F. Candelier, and F. Renda, “Dynamics of continuum and soft robots: A strain parameterization based approach,” IEEE Transactions on Robotics, vol. 37, no. 3, pp. 847–863, 2020

    2020

  4. [12]

    Brax - a differentiable physics engine for large scale rigid body simulation,

    C. D. Freeman, E. Frey, A. Raichuk, S. Girgin, I. Mordatch, and O. Bachem, “Brax - a differentiable physics engine for large scale rigid body simulation,” 2021. [Online]. Available: http://github.com/google/brax

    2021

  5. [13]

    Isaac gym: High performance gpu based physics simulation for robot learning,

    V . Makoviychuk, L. Wawrzyniak, Y . Guo, M. Lu, K. Storey, M. Macklin, D. Hoeller, N. Rudin, A. Allshire, A. Handaet al., “Isaac gym: High performance gpu based physics simulation for robot learning,” inThirty- fifth Conference on Neural Information Processing Systems Datasets...

  6. [14]

    Analytical derivatives of strain-based dynamic model for hybrid soft-rigid robots,

    A. T. Mathew, F. Boyer, V . Lebastard, and F. Renda, “Analytical derivatives of strain-based dynamic model for hybrid soft-rigid robots,” The International Journal of Robotics Research, p. 02783649251346209, 2024

    2024

  7. [15]

    Sorosim: A matlab toolbox for hybrid rigid–soft robots based on the geometric variable-strain approach,

    A. T. Mathew, I. B. Hmida, C. Armanini, F. Boyer, and F. Renda, “Sorosim: A matlab toolbox for hybrid rigid–soft robots based on the geometric variable-strain approach,”IEEE Robotics & Automation Magazine, vol. 30, no. 3, pp. 106–122, 2022

    2022

  8. [16]

    JAX: composable transformations of Python+NumPy programs,

    J. Bradbury, R. Frostig, P. Hawkins, M. J. Johnson, C. Leary, D. Maclaurin, G. Necula, A. Paszke, J. VanderPlas, S. Wanderman-Milne, and Q. Zhang, “JAX: composable transformations of Python+NumPy programs,” 2018. [Online]. Available: http://github.com/jax-ml/jax

    2018

  9. [17]

    Safe yet precise soft robots: Incorporating physics into learned models for control,

    M. St ¨olzle, “Safe yet precise soft robots: Incorporating physics into learned models for control,” Dissertation (TU Delft), Mechanical Engi- neering, Delft University of Technology, 09 2025

    2025

  10. [18]

    Contact-aware safety in soft robots using high-order control barrier and lyapunov functions,

    K. Wong, M. St ¨olzle, W. Xiao, C. D. Santina, D. Rus, and G. Zardini, “Contact-aware safety in soft robots using high-order control barrier and lyapunov functions,”IEEE Robotics and Automation Letters, vol. 10, no. 12, pp. 12 485–12 492, 2025

    2025

  11. [19]

    Mujoco: A physics engine for model- based control,

    E. Todorov, T. Erez, and Y . Tassa, “Mujoco: A physics engine for model- based control,” in2012 IEEE/RSJ international conference on intelligent robots and systems. IEEE, 2012, pp. 5026–5033

    2012

  12. [20]

    MuJoCo XLA (MJX),

    Google DeepMind, “MuJoCo XLA (MJX),” https://mujoco.readthedocs. io/en/stable/mjx.html, 2023, accessed: 2026-06-19

    2023

  13. [21]

    Contact-aware morphology optimization via physi- cally consistent differentiable simulation,

    F. L. Ferretti, D. Ferigo, A. Croci, C. Sartore, O. G. Younis, S. Traver- saro, and D. Pucci, “Contact-aware morphology optimization via physi- cally consistent differentiable simulation,”IEEE Robotics and Automa- tion Letters, vol. 11, no. 5, pp. 6456–6463, 2026

    2026

  14. [22]

    Giuseppe L’Erario, Silvio Traversaro, Carlotta Sartore, Riccardo Grieco, Stefano Dafarra, Giulio Romualdi, Filippo Luca Ferretti, and Daniele Pucci, “adam,” Jan. 2026. [Online]. Available: https: //github.com/gbionics/adam

    2026

  15. [23]

    Newton: GPU-accelerated physics simulation for robotics, and simulation research

    Newton Contributors, “Newton: GPU-accelerated physics simulation for robotics, and simulation research.” Newton a Series of LF Projects, LLC,

  16. [24]

    Advanced soft robot modeling in chainqueen,

    A. Spielberg, T. Du, Y . Hu, D. Rus, and W. Matusik, “Advanced soft robot modeling in chainqueen,”Robotica, vol. 41, no. 1, pp. 74–104, 2023

    2023

  17. [25]

    Softzoo: A soft robot co-design benchmark for locomotion in diverse environments,

    T.-H. Wang, P. Ma, A. E. Spielberg, Z. Xian, H. Zhang, J. B. Tenenbaum, D. Rus, and C. Gan, “Softzoo: A soft robot co-design benchmark for locomotion in diverse environments,” inThe Eleventh International Conference on Learning Representations, 2023

    2023

  18. [26]

    Elastica: A compliant mechanics environment for soft robotic control,

    N. Naughton, J. Sun, A. Tekinalp, T. Parthasarathy, G. Chowdhary, and M. Gazzola, “Elastica: A compliant mechanics environment for soft robotic control,”IEEE Robotics and Automation Letters, vol. 6, no. 2, pp. 3389–3396, 2021

    2021

  19. [27]

    Somogym: A toolkit for developing and evaluating controllers and reinforcement learning algorithms for soft robots,

    M. A. Graule, T. P. McCarthy, C. B. Teeple, J. Werfel, and R. J. Wood, “Somogym: A toolkit for developing and evaluating controllers and reinforcement learning algorithms for soft robots,”IEEE Robotics and Automation Letters, vol. 7, no. 2, pp. 4071–4078, 2022

    2022

  20. [28]

    Softmanisim: A fast simulation framework for multi-segment continuum manipulators tailored for robot learning,

    M. Kasaei, H. Kasaei, and M. Khadem, “Softmanisim: A fast simulation framework for multi-segment continuum manipulators tailored for robot learning,” in8th Annual Conference on Robot Learning, 2025

    2025

  21. [29]

    Tmtdyn: A matlab package for modeling and control of hy- brid rigid–continuum robots based on discretized lumped systems and reduced-order models,

    S. H. Sadati, S. E. Naghibi, A. Shiva, B. Michael, L. Renson, M. Howard, C. D. Rucker, K. Althoefer, T. Nanayakkara, S. Zschaler et al., “Tmtdyn: A matlab package for modeling and control of hy- brid rigid–continuum robots based on discretized lumped systems and reduced-order ...

    2021

  22. [30]

    Dismech: A discrete differential geometry-based physical simulator for soft robots and structures,

    A. Choi, R. Jing, A. P. Sabelhaus, and M. K. Jawed, “Dismech: A discrete differential geometry-based physical simulator for soft robots and structures,”IEEE Robotics and Automation Letters, vol. 9, no. 4, pp. 3483–3490, 2024

    2024

  23. [31]

    Sorotoki: a matlab toolkit for design, modeling, and control of soft robots,

    B. J. Caasenbrood, A. Y . Pogromsky, and H. Nijmeijer, “Sorotoki: a matlab toolkit for design, modeling, and control of soft robots,”IEEE Access, vol. 12, pp. 17 604–17 638, 2024

    2024

  24. [32]

    Dynamic simulation of articulated soft robots,

    W. Huang, X. Huang, C. Majidi, and M. K. Jawed, “Dynamic simulation of articulated soft robots,”Nature communications, vol. 11, no. 1, p. 2233, 2020

    2020

  25. [33]

    Umarm: Untethered, modular, portable, soft pneumatic arm,

    R. Zuo, D. H. Han, R. Li, S. Jamal, and D. Bruder, “Umarm: Untethered, modular, portable, soft pneumatic arm,”arXiv preprint arXiv:2505.11476, 2025

    2025

  26. [34]

    Discrete cosserat approach for soft robot dynamics: A new piece-wise constant strain model with torsion and shears,

    F. Renda, V . Cacucciolo, J. Dias, and L. Seneviratne, “Discrete cosserat approach for soft robot dynamics: A new piece-wise constant strain model with torsion and shears,” in2016 IEEE/RSJ International Con- ference on Intelligent Robots and Systems (IROS). IEEE, 2016, pp. 5495–5502

    2016

  27. [35]

    I-support soft arm for assistance tasks: a new manufacturing approach based on 3D printing and characterization,

    L. Arleo, G. Stano, G. Percoco, and M. Cianchetti, “I-support soft arm for assistance tasks: a new manufacturing approach based on 3D printing and characterization,”Progress in Additive Manufacturing, vol. 6, no. 2, pp. 243–256, 2021. [Online]. Available: https://link.springer...

    2021 doi

  28. [36]

    An experimental study of model-based control for planar handed shearing auxetics robots,

    M. St ¨olzle, D. Rus, and C. Della Santina, “An experimental study of model-based control for planar handed shearing auxetics robots,” in Experimental Robotics. Cham: Springer Nature Switzerland, 2024, pp. 153–167

    2024

  29. [37]

    Mastering contact-rich tasks by com- bining soft and rigid robotics with imitation learning,

    M. Ram ´ırez Montero, E. Shahabi, G. Franzese, J. Kober, B. Maz- zolai, and C. Della Santina, “Mastering contact-rich tasks by com- bining soft and rigid robotics with imitation learning,”arXiv preprint arXiv:2410.07787, 2024

  30. [38]

    Soft robots,

    C. Della Santina, M. G. Catalano, and A. Bicchi, “Soft robots,” in Encyclopedia of Robotics. Springer, 2020, pp. 1–14

    2020

  31. [39]

    R. M. Murray, Z. Li, and S. S. Sastry,A mathematical introduction to robotic manipulation. CRC press, 2017

    2017

  32. [40]

    A recursive lagrangian formulation of manipulator dynamics and a comparative study of dynamics formulation complexity,

    J. M. Hollerbach, “A recursive lagrangian formulation of manipulator dynamics and a comparative study of dynamics formulation complexity,” IEEE Transactions on Systems, Man, and Cybernetics, vol. SMC-10, no. 11, pp. 730–736, 1980

    1980

  33. [41]

    Efficient dynamic computer simulation of robotic mechanisms,

    M. W. Walker and D. E. Orin, “Efficient dynamic computer simulation of robotic mechanisms,”Journal of Dynamic Systems, Measurement, and Control, vol. 104, no. 3, pp. 205–211, 1982

    1982

  34. [42]

    The calculation of robot dynamics using articulated- body inertias,

    R. Featherstone, “The calculation of robot dynamics using articulated- body inertias,”The International Journal of Robotics Research, vol. 2, no. 1, pp. 13–30, 1983

    1983

  35. [43]

    Soft robots modeling: A structured overview,

    C. Armanini, F. Boyer, A. T. Mathew, C. Duriez, and F. Renda, “Soft robots modeling: A structured overview,”IEEE Transactions on Robotics, vol. 39, no. 3, pp. 1728–1748, 2023

    2023

  36. [44]

    Modelling handed shearing auxetics: Selective piecewise constant strain kinematics and dynamic simulation,

    M. St ¨olzle, L. Chin, R. L. Truby, D. Rus, and C. Della Santina, “Modelling handed shearing auxetics: Selective piecewise constant strain kinematics and dynamic simulation,” in2023 IEEE International Con- ference on Soft Robotics (RoboSoft). IEEE, 2023, pp. 1–8. IEEE ROBOTICS...

    2023

  37. [45]

    A unified approach for motion and force control of robot manipulators: The operational space formulation,

    O. Khatib, “A unified approach for motion and force control of robot manipulators: The operational space formulation,”IEEE Journal on Robotics and Automation, vol. 3, no. 1, pp. 43–53, 2003

    2003

  38. [46]

    Input decoupling of lagrangian systems via coordinate transformation: General characterization and its application to soft robotics,

    P. Pustina, C. Della Santina, F. Boyer, A. De Luca, and F. Renda, “Input decoupling of lagrangian systems via coordinate transformation: General characterization and its application to soft robotics,”IEEE transactions on robotics, vol. 40, pp. 2098–2110, 2024

    2024

  39. [47]

    On Neural Differential Equations,

    P. Kidger, “On Neural Differential Equations,” Ph.D. dissertation, Uni- versity of Oxford, 2021

    2021

  40. [48]

    On an improved state parametrization for soft robots with piecewise constant curvature and its use in model based control,

    C. Della Santina, A. Bicchi, and D. Rus, “On an improved state parametrization for soft robots with piecewise constant curvature and its use in model based control,”IEEE Robotics and Automation Letters, vol. 5, no. 2, pp. 1001–1008, 2020

    2020

  41. [49]

    Model- based dynamic feedback control of a planar soft robot: trajectory track- ing and interaction with the environment,

    C. Della Santina, R. K. Katzschmann, A. Bicchi, and D. Rus, “Model- based dynamic feedback control of a planar soft robot: trajectory track- ing and interaction with the environment,”The International Journal of Robotics Research, vol. 39, no. 4, pp. 490–513, 2020

    2020

  42. [50]

    Analysis and control of the underactuation in contin- uum soft robots: a kinematic independent approach,

    P. Pustinaet al., “Analysis and control of the underactuation in contin- uum soft robots: a kinematic independent approach,” Ph.D. dissertation, Universit`a degli Studi di Roma” La Sapienza”, 2025

    2025

  43. [51]

    Guiding soft robots with motor-imagery brain signals and impedance control,

    M. St ¨olzle, S. Baberwal, D. Rus, S. Coyle, and C. Della Santina, “Guiding soft robots with motor-imagery brain signals and impedance control,” in2024 IEEE 7th International Conference on Soft Robotics (RoboSoft). IEEE, 2024, pp. 1–8, received the Best Paper Award

    2024

  44. [52]

    Exact task execution in highly under-actuated soft limbs: An operational space based approach,

    C. D. Santina, L. Pallottino, D. Rus, and A. Bicchi, “Exact task execution in highly under-actuated soft limbs: An operational space based approach,”IEEE Robotics and Automation Letters, vol. 4, no. 3, pp. 2508–2515, 2019

    2019

  45. [53]

    Open3d: A modern library for 3d data processing,

    Q.-Y . Zhou, J. Park, and V . Koltun, “Open3d: A modern library for 3d data processing,”arXiv preprint arXiv:1801.09847, 2018

    2018 arXiv

  46. [54]

    Viser: Imperative, web- based 3d visualization in python,

    B. Yi, C. M. Kim, J. Kerr, G. Wu, R. Feng, A. Zhang, J. Kulhanek, H. Choi, Y . Ma, M. Tancik, and A. Kanazawa, “Viser: Imperative, web- based 3d visualization in python,”arXiv preprint arXiv:2507.22885, 2025

    2025 arXiv

  47. [55]

    Runge–kutta pairs of order 5 (4) satisfying only the first column simplifying assumption,

    C. Tsitouras, “Runge–kutta pairs of order 5 (4) satisfying only the first column simplifying assumption,”Computers & Mathematics with Applications, vol. 62, no. 2, pp. 770–775, 2011

    2011

  48. [56]

    Forward and inverse problems in the mechanics of soft filaments,

    M. Gazzola, L. H. Dudte, A. G. McCormick, and L. Mahadevan, “Forward and inverse problems in the mechanics of soft filaments,” Royal Society open science, vol. 5, no. 6, p. 171628, 2018

    2018

  49. [57]

    Learning to walk in minutes using massively parallel deep reinforcement learning,

    N. Rudin, D. Hoeller, P. Reist, and M. Hutter, “Learning to walk in minutes using massively parallel deep reinforcement learning,” in Conference on robot learning. PMLR, 2022, pp. 91–100

    2022

  50. [58]

    Underwater soft robot modeling and control with differentiable simulation,

    T. Du, J. Hughes, S. Wah, W. Matusik, and D. Rus, “Underwater soft robot modeling and control with differentiable simulation,”IEEE Robotics and Automation Letters, vol. 6, no. 3, pp. 4994–5001, 2021

    2021

  51. [59]

    Diffusebot: Breeding soft robots with physics-augmented generative diffusion models,

    T.-H. J. Wang, J. Zheng, P. Ma, Y . Du, B. Kim, A. Spielberg, J. Tenen- baum, C. Gan, and D. Rus, “Diffusebot: Breeding soft robots with physics-augmented generative diffusion models,”Advances in Neural Information Processing Systems, vol. 36, pp. 44 398–44 423, 2023

    2023

  52. [60]

    Aggressive driving with model predictive path integral control,

    G. Williams, P. Drews, B. Goldfain, J. M. Rehg, and E. A. Theodorou, “Aggressive driving with model predictive path integral control,” in2016 IEEE International Conference on Robotics and Automation (ICRA), 2016, pp. 1433–1440

    2016

  53. [61]

    Sampling-based model predictive control leveraging parallelizable physics simulations,

    C. Pezzato, C. Salmi, E. Trevisan, M. Spahn, J. Alonso-Mora, and C. H. Corbato, “Sampling-based model predictive control leveraging parallelizable physics simulations,”IEEE Robotics and Automation Letters, 2025

    2025

  54. [62]

    Sim- to-real of soft robots with learned residual physics,

    J. Gao, M. Y . Michelis, A. Spielberg, and R. K. Katzschmann, “Sim- to-real of soft robots with learned residual physics,”IEEE Robotics and Automation Letters, 2024

    2024

  55. [63]

    Control barrier function based quadratic programs for safety critical systems,

    A. D. Ames, X. Xu, J. W. Grizzle, and P. Tabuada, “Control barrier function based quadratic programs for safety critical systems,”IEEE Transactions on Automatic Control, vol. 62, no. 8, pp. 3861–3876, 2016

    2016

  56. [64]

    Safe autonomous environmental contact for soft robots using control barrier functions,

    A. K. Dickson, J. C. Pacheco Garcia, M. L. Anderson, R. Jing, S. Alizadeh-Shabdiz, A. X. Wang, C. DeLorey, Z. J. Patterson, and A. P. Sabelhaus, “Safe autonomous environmental contact for soft robots using control barrier functions,”IEEE Robotics and Automation Letters, 2025, ...

    2025

  57. [65]

    R. S. Sutton and A. G. Barto,Reinforcement Learning: An Introduction, 2nd ed. Cambridge, MA: MIT Press, 2018

    2018

  58. [66]

    Design and control of a bipedal robotic character,

    R. Grandia, E. Knoop, M. A. Hopkins, G. Wiedebach, J. Bishop, S. Pickles, D. M ¨uller, and M. B ¨acher, “Design and control of a bipedal robotic character,” inRobotics: Science and Systems (RSS), 2024

    2024

  59. [67]

    Prox- imal policy optimization algorithms,

    J. Schulman, F. Wolski, P. Dhariwal, A. Radford, and O. Klimov, “Prox- imal policy optimization algorithms,”arXiv preprint arXiv:1707.06347, 2017

    2017 arXiv

  60. [68]

    An overview of soft robotics,

    O. Yasa, Y . Toshimitsu, M. Y . Michelis, L. S. Jones, M. Filippi, T. Buchner, and R. K. Katzschmann, “An overview of soft robotics,” Annual Review of Control, Robotics, and Autonomous Systems, vol. 6, no. 1, pp. 1–29, 2023

    2023

  61. [69]

    Which is the best pid variant for pneumatic soft robots an experimental study,

    A. H. Khan, Z. Shao, S. Li, Q. Wang, and N. Guan, “Which is the best pid variant for pneumatic soft robots an experimental study,”IEEE/CAA journal of automatica sinica, vol. 7, no. 2, p. 451, 2020

    2020

  62. [70]

    Pd control with computed feedforward of robot manipulators: A design procedure,

    R. Kelly and R. Salgado, “Pd control with computed feedforward of robot manipulators: A design procedure,”IEEE Transactions on Robotics and Automation, vol. 10, no. 4, pp. 566–571, 1994

    1994

  63. [71]

    Optimistix: modular optimisation in jax and equinox,

    J. Rader, T. Lyons, and P. Kidger, “Optimistix: modular optimisation in jax and equinox,”arXiv:2402.09983, 2024

    2024 arXiv

  64. [72]

    The DeepMind JAX Ecosystem,

    DeepMind, I. Babuschkin, K. Baumli, A. Bell, S. Bhupatiraju, J. Bruce, P. Buchlovsky, D. Budden, T. Cai, A. Clark, I. Danihelka, A. Dedieu, C. Fantacci, J. Godwin, C. Jones, R. Hemsley, T. Hennigan, M. Hessel, S. Hou, S. Kapturowski, T. Keck, I. Kemaev, M. King, M. Kunesch, L....

  65. [73]

    Adaptive methods for nonconvex optimization,

    M. Zaheer, S. Reddi, D. Sachan, S. Kale, and S. Kumar, “Adaptive methods for nonconvex optimization,”Advances in neural information processing systems, vol. 31, 2018

    2018

  66. [74]

    A closed-form clf-cbf controller for whole-body continuum soft robot collision avoidance,

    K. Wong, M. St ¨olzle, W. Xiao, and D. Rus, “A closed-form clf-cbf controller for whole-body continuum soft robot collision avoidance,” in 2026 IEEE 9th International Conference on Soft Robotics (RoboSoft). IEEE, 2026, pp. 1171–1178

    2026

  67. [75]

    Stable-baselines3: Reliable reinforcement learning implementa- tions,

    A. Raffin, A. Hill, A. Gleave, A. Kanervisto, M. Ernestus, and N. Dor- mann, “Stable-baselines3: Reliable reinforcement learning implementa- tions,”Journal of machine learning research, vol. 22, no. 268, pp. 1–8, 2021

    2021

  68. [76]

    Dynamics for variable length multisection continuum arms,

    I. S. Godage, G. A. Medrano-Cerda, D. T. Branson, E. Guglielmino, and D. G. Caldwell, “Dynamics for variable length multisection continuum arms,”The International Journal of Robotics Research, vol. 35, no. 6, pp. 695–722, 2016

    2016

  69. [77]

    Modeling and control for a class of tendon-driven continuum mechanisms,

    B. Deutschmann, “Modeling and control for a class of tendon-driven continuum mechanisms,” Ph.D. dissertation, Leibniz Universit ¨at Han- nover, 2020

    2020

  70. [78]

    Design, modeling, and control strategies for soft robots,

    B. J. Caasenbrood, “Design, modeling, and control strategies for soft robots,” 2024

    2024

  71. [79]

    Foundations of robotics,

    B. Siciliano, L. Villani, G. Oriolo, A. De Lucaet al., “Foundations of robotics,”Advanced Textbooks in Control and Signal Processing, pp. 1–719, 2025

    2025

  72. [80]

    Physics-informed neural networks to model and control robots: A theoretical and experimental investiga- tion,

    J. Liu, P. Borja, and C. Della Santina, “Physics-informed neural networks to model and control robots: A theoretical and experimental investiga- tion,”Advanced Intelligent Systems, vol. 6, no. 5, p. 2300385, 2024

    2024

  73. [81]

    Learning low- dimensional strain models of soft robots by looking at the evolution of their shape with application to model-based control,

    R. Valadas, M. St ¨olzle, J. Liu, and C. Della Santina, “Learning low- dimensional strain models of soft robots by looking at the evolution of their shape with application to model-based control,” in2025 IEEE 8th International Conference on Soft Robotics (RoboSoft). IEEE, 2025...

    2025

  74. [84]

    Model uncertainty is high, and baseline regulation is sufficient

    Setpoint Regulation –A ⊤(q) PIDq Feedback authority limited by actuator map A(q); potential forces need to be compensated by integral term; quasi-static objective. Model uncertainty is high, and baseline regulation is sufficient. Potential compensation [5], [50], [70] Setpoint...

  75. [85]

    Fully actuated tracking when current-state gravity/dynamics should be compensated

    Trajectory Tracking A−1(q) M(q) ¨qd +C(q,˙q) ˙qd +G(q) +K(q d) +D˙qd A⊤(q) PIDq Full actuation;A(q)invertible; Coriolis consistency; accurate dynamical model; reliableqestimate. Fully actuated tracking when current-state gravity/dynamics should be compensated. Computed torque

  76. [86]

    Fully actuated tracking with an accurate full model and reliableq,˙q

    Trajectory Tracking A−1(q) M(q) ¨qd +C(q,˙q) ˙q +G(q) +K(q) +D˙q) A−1(q)M(q) PIDq Full actuation;A(q)invertible; accurate dynamical model; reliable state estimates. Fully actuated tracking with an accurate full model and reliableq,˙q. ACTUATION-SPACE CONTROLLERS PID [36], [46]...

    2000

  77. [87]

    Configuration-Space Regulation and Tracking:The configuration-space study isolates the effect of increasingly complete model compensation under identical feedback, ref- erence, and rollout conditions. The control objective is to drive the generalized strain-displacement config...

  78. [88]

    Operational-Space Position and Orientation Tracking: The operational-space study evaluates simultaneous position and orientation tracking using the complete translational and rotational task-space dynamics. The control objective is to track a spherical surface-following figure...

  79. [89]

    If the integrability assumption is met, a transformation exists that enables the change of coordinates ash a(·) :R n →R m, such that∂h a/∂q=A T (q)[46]

    Model-Based Controllers: a) Potential Shaping Actuation Space Controller:It has been demonstrated in literature [46] that it is often possible to reformulate the robot dynamics in actuation coordinates φ= [φ ⊤ a , φ⊤ u ]⊤ ∈R n, whereφ a ∈R m, φu ∈R n−m are denoted as actuated ...

  80. [90]

    getting stuck

    Feedback Control Gain Optimization:We automatically tune the gains of the PID feedback controllers by optimizing the closed-loop behavior using gradient descent, combining the robot models and controllers from the SoRoMoX package with the Optax [72] optimization package. In th...

  81. [91]

    Results:Figure 12 summarizes the results for both gain- optimization cases. The left panels (Figure 12(a),(c)) show the loss over100optimization iterations, while the right panels show the corresponding convergence of the regulated configu- ration variables (Figure 12(b)) and ...

  82. [92]

    The drift and input vector fieldsf 0(y)andf u(y)are obtained directly from the differentiable SoRoMoX forward dynamics

    Approach: a) High-Order CLF-CBF Controller:The soft robot dy- namics are written in control-affine form as ˙y=f 0(y) +f u(y)u,(42) where y= [q ⊤,˙q⊤]⊤, denotes the robot state anduis the vector of tendon-tension inputs. The drift and input vector fieldsf 0(y)andf u(y)are obtai...

  83. [93]

    Each segment has radius36 mm, density 1070 kg/m3, Young’s modulus20 kPa, and shear modulus 20 kPa

    Simulation Setup:The simulation was performed using a tendon-actuated PCS model consisting of two constant-strain segments, each of length0.15 m, resulting in a total backbone length of0.30 m. Each segment has radius36 mm, density 1070 kg/m3, Young’s modulus20 kPa, and shear m...

  84. [94]

    Results:The results are shown in Fig. 13. The pro- posed CLF-CBF controller drives the robot toward the goal while enforcing the prescribed force safety constraint. With the CBF, the goal distance decreases from approximately 0.11 m to 0.03 m over the 8 s simulation horizon. M...

  85. [96]

    Evaluation of Lie Algebra in Special Cases:In our unit tests for these(3)utilities 9, we first verify that the hat operator implements the standard twist–matrix correspondence. Forξ= (ω, v)⊤ ∈R 6 withω, v∈R 3 and[ω] × ∈so(3), we check that bξ= hat SE(3) (ξ) = [ω]× v 01×3 0 ,(7...

  86. [97]

    ThenFK(0 n, s), whereq= 0 n denotes zero strain displacement rather than zero full strain, gives the translationp= [s,0,0] ⊤ and identity rotationR=I 3

    Evaluation of Special Cases of the Kinematics and the Kinematic Loop:First, consider a single-segment PCS soft robot with reference strainξ ref = [0,0,0,1,0,0] ⊤, corre- sponding to a straight, unelongated shape. ThenFK(0 n, s), whereq= 0 n denotes zero strain displacement rat...

  87. [98]

    Consequently, they could in principle be implemented using AD techniques inAD

    Test of Consistency of Kinematics and Dynamics via Auto-Differentiation:Many expressions in (soft) robot models (e.g., Jacobians, Jacobian derivatives, Coriolis matrix, poten- tial forces) can be written as derivatives of other quantities (e.g., forward kinematics, Jacobians, ...

  88. [99]

    We next outline a step-by-step procedure to verify coher- ence between thePlanar PCSandGVSimplementations

    Verification of Inter-System Coherence:For rod-based models, specialized formulations—such as PCC [2], poly- nomial curvature [3], and (Planar) PCS [9], [34], which neglect certain strains and/or use specific (functional) strain bases—can typically be embedded in more general ...

  89. [105]

    All other training hyperparameters are set to their default values as implemented in Stable Baselines3 [75]

    To ensure a fair comparison, all PPO training runs use the same rollout size of 12,800 samples and a fixed PPO minibatch size of 428. All other training hyperparameters are set to their default values as implemented in Stable Baselines3 [75]. As shown earlier in Table IV, PyEl...

  90. [2020]

    Available: http://github.com/google-deepmind

    [Online]. Available: http://github.com/google-deepmind

  91. [2025]

    Available: https://github.com/newton-physics/newton

    [Online]. Available: https://github.com/newton-physics/newton

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.