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A MATLAB Toolbox for Hybrid Rigid Soft Robots Based on the Geometric Variable Strain Approach

Anup Teejo Mathew, Ikhlas Ben Hmida, Costanza Armanini, Frederic Boyer, Federico Renda

arXiv:2107.05494v2cs.ROphysics.app-ph

TL;DR

Soft robots’ compliance and many degrees of freedom make modeling complex, while tools for their modeling and analysis remain limited. SoRoSim is a MATLAB toolbox built on the GVS approach for analyzing soft, rigid, and hybrid robots; its results are compared with published and numerical studies, alongside four application demonstrations.

Abstract

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Soft robotics has been a trending topic within the robotics community for almost two decades. However, available tools for the modeling and analysis of soft robots are still limited. This paper introduces a user-friendly MATLAB toolbox, Soft Robot Simulator (SoRoSim), that integrates the Geometric Variable Strain (GVS) model of Cosserat rods to facilitate the static and dynamic analysis of soft, rigid, or hybrid robotic systems. We present a brief overview of the design and structure of the toolbox and validate it by comparing its results with those published in the literature. To highlight the toolbox's potential to efficiently model, simulate, optimize, and control various robotic systems, we demonstrate four sample applications. The demonstrated applications explore different actuator and external loading conditions of single-, branched-, open-, and closed-chain robotic systems. We think that the soft-robotics research community will significantly benefit from the SoRoSim toolbox for a wide variety of applications.

Introduction

Soft robots offer adaptability and light weight, but their compliance and many degrees of freedom make modeling complex. SoRoSim addresses the need for accessible modeling tools with a MATLAB toolbox based on the Geometric Variable Strain approach.

Introduction

  • Soft robots’ compliance and effectively infinite degrees of freedom increase modeling complexity despite their adaptability and light weight.Source: Introduction · p. 1
  • SoRoSim combines Cosserat-rod fidelity with the GVS approach’s reduced DoF to model soft and hybrid systems accurately and computationally efficiently.The toolbox also implements a nested Gaussian quadrature scheme to solve the GVS formulation.Source: Introduction · p. 2
  • Most existing soft-robotics toolboxes use FEM or lumped-mass models, while many DER-based simulators target computer graphics rather than mechanical systems.TMTDyn is geometrically exact but parametrizes positions and orientations rather than strains.Source: Introduction · p. 2
  • SoRoSim provides GUIs and MATLAB files for defining links, assembling systems, assigning constraints, and applying customized forces and actuation.Users can also combine it with MATLAB functions, add-ons, and their own code for analysis and control.Source: Introduction · p. 2
  • The paper validates toolbox results against published data and commercial software, then demonstrates four applications involving robotic arms, an underwater locomotor, optimization, and inverse-dynamic control.Source: Introduction · p. 2

Governing Equations

SoRoSim models interconnected rigid and soft bodies with a GVS formulation that parameterizes soft-link strain. The resulting generalized dynamics support static and dynamic analysis, while the toolbox’s classes provide link construction, assembly, and analysis workflows.

Governing Equations

  • The GVS model represents soft links as Cosserat rods with six deformation modes, which users can selectively disable to reproduce other beam theories.For example, enabling rotations and shear along y and z produces a Timoshenko beam.Source: Governing Equations · p. 2
  • Soft and rigid bodies are represented relative to their predecessor in the chain, with body-frame mappings used to describe their configurations.Source: Governing Equations · p. 3
  • Each soft-link strain field is parameterized by a finite basis and generalized coordinates, with reference strain representing constrained nonzero strains such as inextensibility.Source: Governing Equations · pp. 3–4
  • The model projects the floating hybrid chain’s dynamics onto generalized coordinates, yielding equations with mass, Coriolis, damping, stiffness, actuation, and external-force terms.The stiffness and damping matrices are associated with linear elastic models; nonlinear strain-dependent constitutive laws are also supported.Source: Governing Equations · p. 4
  • SorosimLink defines links, Twist selects joint degrees of freedom and soft-link deformation modes, and SorosimLinkage assembles systems and supports analysis.Linkage methods solve static and dynamic GVS models and compute parameters including Jacobian, mass, and Coriolis matrices.Source: Governing Equations · pp. 4, 6

Test 1: Fixed–Free Beam With a Follower Tip Force

The fixed–free follower-force beam test compares SoRoSim’s deflection and tip displacements with published results. The toolbox matches the reported results with fewer degrees of freedom and substantially shorter computation time.

Test 1: Fixed–Free Beam With a Follower Tip Force

  • The test models a 100-m cylindrical cantilever beam with a fourth-order bending strain and 15 Gauss quadrature points.Its Young’s modulus is 6.75 GPa and its diameter is 57 cm.Source: Test 1: Fixed–Free Beam With a… · p. 6
  • 42 ms per loading step in SoRoSim versus 16.4 s per loading step reported in, with the same results as using 5 rather than 10 DoF.The beam was modeled under follower tip forces from 0 N to 130 kN; Figure 4 compares the deflected profile and horizontal and vertical tip displacements.Source: Test 1: Fixed–Free Beam With a… · pp. 6–7, Figure 4 · p. 6
  • Normalizing each soft division to unit length avoids poorly scaled matrices and supports faster static solutions and more stable dynamic simulations.The toolbox scales physical quantities by the division’s original length and restores the actual dimensions after simulation.Source: Test 1: Fixed–Free Beam With a… · p. 7

Test 3: Dynamics of a Flexible Flying Rod

SoRoSim reproduces the published flying-rod dynamics and completes the 7-second simulation in less than a second. The example uses a 15-DoF system with time-dependent tip forces and moments.

Test 3: Dynamics of a Flexible Flying Rod

  • The 15-DoF model uses a 10-m rod, an inextensible Kirchhoff model with a quadratic polynomial basis, and time-dependent tip forces and moments.The free joint contributes six DoF; the M2 moment is a triangular pulse peaking at 200 N·m at 2.5 s and returning to zero at 5 s.Source: Test 3: Dynamics of a Flexible Flying… · p. 7
  • The rod’s position, orientation, and deformation matched the published results over the 7-s analysis.The example also combines a free lumped joint with a distributed soft body in one framework.Source: Test 3: Dynamics of a Flexible Flying… · p. 7
  • Less than 1 s of computation solved a 7-s simulation, three orders faster than the previously reported computational time.The prior report averaged 9 min of computation per simulated second; the SoRoSim run was also 7× faster than real time.Source: Test 3: Dynamics of a Flexible Flying… · p. 7

Test 4: Energy Balance in SoRoSim Dynamics

The cantilever-beam example compares damped and undamped dynamics under gravity. Its total-energy behavior follows the expected distinction: decay with damping and conservation without damping.

Test 4: Energy Balance in SoRoSim Dynamics

  • The beam starts from rest with initial bending of 1 rad/m about the y-axis and is simulated for 5 s.It is a 0.5-m tapered link under gravity; all three angular deformation modes are enabled with cubic strain approximation and 11.2 KPa elastic damping in the damped case.Source: Test 4: Energy Balance in SoRoSim… · pp. 8–9
  • The total energy combines kinetic, gravitational potential, and elastic potential energy.The equation’s terms use the cross-sectional screw inertia, area, gravity, vertical coordinate, strain, reference strain, and stiffness as defined in the text.Source: Test 4: Energy Balance in SoRoSim… · p. 9
  • Damped total energy decays, while undamped total energy remains a nonzero positive constant, as expected.Figure 7(a) plots the energy change; the accompanying panels show the damped and undamped motion.Source: Test 4: Energy Balance in SoRoSim… · p. 9, Figure 7 · p. 9
  • Energy values for this example can be computed in separate MATLAB code for postprocessing.Source: Test 4: Energy Balance in SoRoSim… · p. 9

Modeling Applications

The section demonstrates static-equilibrium and dynamic simulations across soft-robotics systems, including custom external-force functions for contact and fluid interactions.

Modeling Applications

  • The toolbox examples cover static equilibrium and dynamic simulations, with custom functions modeling external forces that include contact and fluid interactions.Source: Modeling Applications · p. 9

Hybrid Manipulators

The toolbox models a hybrid robotic arm under gravity, point loading, joint actuation, and cable actuation, and demonstrates analysis of contact and object manipulation. The examples combine rigid and soft links, multiple joint types, and open-, branched-, and closed-chain components.

Case 1: Static Equilibrium

  • The hybrid arm combines rigid joints, a three-link soft buffer, and cable-actuated gripper fingers.The buffer connects rigid disks through three parallel soft links; its grippers use flexible thin sections and rigid thick sections.Source: Hybrid Manipulators · pp. 9–11
  • 1.2 s was the average static-equilibrium simulation time for the 39-DoF arm with seven actuation inputs.The simulated configurations include rigid-joint motion and gripper actuation under gravity and a 2 N follower point force.Source: Hybrid Manipulators · p. 11
  • The toolbox also supports rigid joints controlled by joint coordinates, forces, or moments, with optional joint stiffness.A prismatic joint with positive stiffness is equivalent to a linear spring.Source: Hybrid Manipulators · p. 11

Case 2: Contact Dynamics

  • SoRoSim can analyze robotic systems with collision and contact events, including a three-fingered gripper manipulating a spherical target.The contact model approximates rigid finger sections as spheres, neglects friction, and uses a linear spring and damper for normal force.Source: Hybrid Manipulators · p. 11

Underwater Robotics

The examples apply SoRoSim to underwater propulsion and to design optimization of a cable-actuated soft manipulator. They illustrate modeling fluid–robot interaction and optimizing cable path and tension to meet target positions.

Underwater Robotics

  • The underwater model represents the shell, shaft, hook, and three filaments as a six-link rigid–soft system, with fluid forces implemented through a custom MATLAB function.The custom function accommodates buoyancy, drag, lift, and added-mass forces not included among the default force inputs.Source: Underwater Robotics · p. 12
  • Helical filament deformation generated thrust that propelled the simulated underwater robot forward under actuation and environmental interaction.The dynamic response was simulated for five seconds, with motion shown from rest to 1.5 s.Source: Underwater Robotics · p. 12, Figure 9 · p. 12

Design Optimization

  • Cable-path optimization minimizes error at the manipulator’s midpoint and endpoint using seven variables: path-polynomial coefficients and cable tension.The bounded optimization uses MATLAB’s patternsearch algorithm.Source: Underwater Robotics · pp. 12–13
  • The optimized cable path matched the desired tip position but left about 14% error at the midpoint.The authors attribute the midpoint error to possible geometric constraints and suggest testing different initial values.Source: Underwater Robotics · pp. 12–13
  • The toolbox can be combined with MATLAB packages for optimization, including problems that use dynamic simulation to estimate design parameters.The paper presents the optimization example as using toolbox analysis results outside the package.Source: Underwater Robotics · p. 13

Inverse Dynamic Control

The paper demonstrates inverse-dynamics control of a soft manipulator for trajectory tracking and tip-pose regulation under gravity. The PD controller reduces tracking error faster than the illustrated open-loop inputs, while full-state convergence is not assured for underactuated systems.

Control method

  • The task-space PD law computes actuator strength from manipulator dynamics and tip tracking quantities.The control setup uses six linearly independent cables and constrains cable tension to positive values below 50 N.Source: Inverse Dynamic Control · pp. 13, 15

Control limitation

  • For underactuated systems, the proposed control law may not ensure full-state convergence at steady state; task-space controller issues are outside the article’s scope.Source: Inverse Dynamic Control · p. 15

Case 1: Tip Pose Trajectory Tracking

  • Fewer than 4 s were needed for tip position and orientation error to converge to zero during circular trajectory tracking.The controller’s actuator-strength oscillation is shown during the initial cancellation of system dynamics.Source: Figure 11 · p. 14, Inverse Dynamic Control · p. 15

Control comparison

  • The quasi-static actuator-strength input initially reduced error but then produced persistent oscillations in the dynamic response.This comparison uses actuator strengths corresponding to a quasi-static solution matching tip position and orientation.Source: Inverse Dynamic Control · p. 15

Case 2: Tip Pose Regulation Under Gravity

  • Fewer than 5 s were needed for the PD controller to reduce tip-pose error to zero under gravity, while ramped open-loop tensions reduced error slowly.The feedback controller’s cable tensions converged to constant values corresponding to the static-equilibrium case.Source: Figure 11 · p. 14, Inverse Dynamic Control · p. 15

Discussion and Conclusion

SoRoSim provides a unified GVS framework for modeling, analyzing, and controlling soft, rigid, and hybrid robots, with results compared against literature and numerical studies and four sample applications. The authors report real-time-or-faster performance for most examples, while limiting applicability to systems with soft links modeled as Cosserat rods.

Discussion and Conclusion

  • Table 2 summarizes performance parameters across analyses, including link and degree-of-freedom counts, strain order, collocation order, equation evaluations, solvers, and runtime.The table covers analyses discussed in the toolbox validation and modeling applications sections.Source: Discussion and Conclusion · p. 15, Table 2 · p. 16
  • 2.5× faster-than-real-time actuator-strength computation could be used for model-based inverse dynamic control; most examples ran in real time or faster.The authors attribute performance to the geometrically exact formulation's low number of degrees of freedom and identify an implicit integrator and code optimization as possible future improvements.Source: Discussion and Conclusion · p. 16
  • SoRoSim is suitable only for systems whose soft links can be modeled as Cosserat rods, although its geometrically exact approach supports large deformations and open-, branched-, and closed-chain structures.The authors describe these capabilities as addressing the majority of soft-robotics applications.Source: Discussion and Conclusion · p. 16
  • SoRoSim uses GVS to provide a unified MATLAB framework for modeling, analysis, and control of soft, rigid, and hybrid robots, with user-adjustable models, forces, actuation paths, and functions.The toolbox's simulation results were compared with existing literature and numerical studies, and four examples demonstrated its application in soft robotics.Source: Discussion and Conclusion · p. 16

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