Source-linked AI summary
Discrete Cosserat Approach for Multi-Section Soft Robots Dynamics
Federico Renda, Frederic Boyer, Jorge Dias, Lakmal Seneviratne
TL;DR
Soft robotics lacks a unified modeling framework that is both computationally inexpensive and sufficiently accurate for kinematics and dynamics. The paper develops a discrete Cosserat PCS model for multi-section soft robots, incorporating shear and torsion while retaining Cosserat and rigid-robotics structure. Simulations and experiments show comparable or better performance than the continuous Cosserat model, including average errors of 5.7%, 4.2%, and 4.3% for the discrete model versus 5.1%, 5.2%, and 5.4% for the continuous model.
Problem
Soft robotics needs a unified modeling framework that combines computational efficiency, sufficient accuracy, and integrated mathematical submodels for infinite-DoF robots.
Method
The paper develops a piece-wise constant strain model by discretizing the continuous Cosserat model for multi-section soft-robot dynamics.
Results
The PCS model is validated through plane and out-of-plane simulations and experiments, with performance comparable to or better than the continuous Cosserat model.
Takeaways & Limitations
The PCS framework provides a soft-robot model closely related to rigid-robot dynamics while accommodating shear, torsion, and varied external loads.
Abstract
from arXiv · showhide
In spite of recent progress, soft robotics still suffers from a lack of unified modeling framework. Nowadays, the most adopted model for the design and control of soft robots is the piece-wise constant curvature model, with its consolidated benefits and drawbacks. In this work, an alternative model for multisection soft robots dynamics is presented based on a discrete Cosserat approach, which, not only takes into account shear and torsional deformations, essentials to cope with out-of-plane external loads, but also inherits the geometrical and mechanical properties of the continuous Cosserat model, making it the natural soft robotics counterpart of the traditional rigid robotics dynamics model. The soundness of the model is demonstrated through extensive simulation and experimental results for both plane and out-of-plane motions.
I. INTRODUCTION
Soft robotics needs a unified model that is accurate, computationally manageable, and compatible with design and control. The proposed PCS model discretizes Cosserat mechanics to retain shear, torsion, and rigid-robot-like geometric structure.
- I. INTRODUCTION: Existing soft-robot models include PCC, continuum Cosserat, and FEM approaches, each balancing simplicity, fidelity, and computational demands.PCC reduces variables through constant-curvature arcs, while continuum Cosserat models are accurate but computationally demanding; FEM is limited to quasi-static conditions.
- I. INTRODUCTION: The PCC constant-curvature assumption is not always valid under non-negligible external loads, including gravity.
- I. INTRODUCTION: The paper discretizes the continuous Cosserat model using piece-wise constant strains, yielding a finite strain-vector description analogous to joint coordinates.The discretization analytically integrates the continuum model under the piece-wise constant strain assumption.
- I. INTRODUCTION: PCS includes curvature, elongation, torsion, and shear, enabling treatment of out-of-plane external loads while preserving an SE(3)-based relation to rigid-robot geometry.
- I. INTRODUCTION: The work extends an earlier conference paper with full multi-section dynamics, a recursive coefficient algorithm, and extensive simulations and experiments, including operation in water.
B. Continuous Dynamics
The continuous Cosserat dynamics is formulated through a Lie-group variational framework with distributed actuation, external loads, and internal constitutive forces. The model accommodates cable-driven or fluidic actuation and dense-medium effects such as gravity, buoyancy, drag, added mass, and point loads.
- Variational formulation: Cosserat beam dynamics is derived from a Lie-group variational calculus using a Lagrangian density formed by kinetic and potential-energy terms.This formulation handles configuration spaces whose structure requires the SE(3) Lie group.
- Model components: The dynamics includes distributed actuation loads, distributed external wrenches, and screw inertia expressed through angular and linear force components.The screw inertia matrix uses rotational inertias and mass-related terms in a local micro-body frame.
- Actuation and constitutive forces: Cable-driven and fluidic actuation are modeled within the same framework, using cable tension and path geometry or concentrated pressure loads at section tips.The internal passive forces use a linear Kelvin–Voigt visco-elastic constitutive model.
- Actuation and constitutive forces: Elastic behavior is represented by constant screw stiffness and viscosity matrices parameterized by Young’s modulus, shear modulus, and shear viscosity.The constitutive model is the only stated assumption needed to describe the arm’s elastic behavior.
- External loading: For underwater operation, external loading includes gravity, buoyancy, drag, added mass, and concentrated loads or contacts.The formulation uses water density, a gravity twist, a drag coefficient, added-mass terms, and a Dirac distribution for point loads.
- External loading: Replacing the screw inertia matrix with an added-mass matrix models inertial hydrodynamic forces along the arm, while setting water density and associated coefficients to zero represents operation in air.The framework therefore distinguishes dense- and sparse-medium cases through its fluid-dynamics parameters.
III. DISCRETE COSSERAT MODEL
The discrete Cosserat model replaces continuous strain fields with constant section-wise twists, enabling analytic kinematics for multi-section soft robots. It extends PCC by incorporating shear and torsion while retaining a rigid-robotics-like geometric structure.
- Piece-wise Constant Strain Kinematics: Piece-wise constant strains replace the continuous field with N twist vectors that act like joint vectors in traditional rigid robotics.Under this assumption, the kinematic differential equation has constant coefficients and can be solved analytically with matrix exponentials.
- Piece-wise Constant Strain Kinematics: The model analytically computes position and orientation recursively from the section strain set ξn.Each section transformation is obtained through an exponential representation of the corresponding constant strain.
- Piece-wise Constant Strain Kinematics: Velocity is obtained by piece-wise integration of the continuum velocity model under constant strains and strain rates.The resulting recursive formulation computes each micro-solid's velocity from the section-wise quantities.
- Piece-wise Constant Strain Kinematics: Acceleration is similarly obtained by integrating the continuum acceleration equation, accounting for the non-constant coefficient term involving strain-rate and velocity.The variation-of-parameters solution yields acceleration using strains, strain rates, and strain accelerations.
- Comparison with PCC: The resulting geometric body Jacobian directly maps the section strain vector to body velocity, while the spatial Jacobian follows through the adjoint transformation.This avoids an intermediate arc-parameter map required by PCC and preserves the motion's geometric structure.
- Comparison with PCC: Unlike PCC, the discrete Cosserat approach handles curvature, elongation, shear, and torsion, enabling kinematic modeling under strong environmental interactions.The section transformations form screw arcs whose parameters are determined by the constant strains.
B. Piece-wise Constant Strain Dynamics
The paper derives multi-section PCS dynamics from the continuous Cosserat model using virtual work and the geometric Jacobian. The resulting equation has the structure of a Lagrangian rigid-serial-manipulator model and supports reconstruction of soft-robot motion.
- Piece-wise Constant Strain Dynamics: The continuous Cosserat dynamics are rewritten in weak virtual-work form so they can be transferred to the piece-wise discrete formulation.Substituting the Jacobian variation relation converts continuous virtual strains into the discrete strain coordinates.
- Piece-wise Constant Strain Dynamics: Analytic integration of section-wise contributions produces the PCS dynamic equation with mass, Coriolis, drag, gravitational-buoyancy, actuation, and external-load terms.The formulation assumes constant elastic and actuation loads along each section where specified.
- Piece-wise Constant Strain Dynamics: The resulting dynamic equation recognizes the structure of the Lagrangian model for rigid serial manipulators.Its coefficient matrices are assembled in the discrete strain coordinates of the multi-section soft arm.
- Coefficient Matrices: The mass matrix is symmetric and positive definite, with block elements computed from Jacobian quantities over the sections.The block structure exploits vanishing Jacobian elements outside the relevant section ranges.
- Coefficient Matrices: After computing the joint dynamics, the model reconstructs the soft manipulator's shape, velocity, and acceleration using the discrete kinematic equations.This connects dynamic strain evolution to the robot's spatial motion.
IV. SIMULATION RESULTS
The PCS dynamic model is evaluated through plane and out-of-plane simulations, including torsional motion unavailable to PCC and comparison against a continuous Cosserat cantilever model under non-constant loading.
- Simulation Results: The simulations first evaluate plane motion of a three-section manipulator and then out-of-plane motion involving torsion in all three sections.The out-of-plane test targets a motion identified as impossible with the PCC model.
- Simulation Results: The model is also compared with a cantilever beam simulated using the continuous Cosserat model to assess behavior under non-constant external loads.An efficient recursive coefficient-matrix algorithm is presented before this comparison.
A. Recursive Algorithm
The recursive algorithm exploits the localized block contributions of each section to compute the PCS dynamic coefficient matrices efficiently.
- Recursive Algorithm: Each section contributes nonzero blocks to specific regions of the mass, Coriolis, drag, and gravitational matrices.For the mass, first Coriolis, and drag matrices, section n affects the square block region spanning rows and columns 1 through n.
- Recursive Algorithm: The algorithm reuses Jacobian and adjoint quantities from preceding sections because most components remain inherited and unchanged within the current section.Only one Jacobian component depends directly on the running material coordinate in the described interval.
B. Plane & Out-of-Plane Motion
The PCS model is tested in plane and out-of-plane simulations for a three-section soft manipulator, including torsional motion and screw-based configuration descriptions.
- Simulation setup: The simulated manipulator has three cylindrical sections, each 250 mm long and 10 mm in radius, with specified elastic, viscous, Poisson, and density parameters.The model uses E = 110 kPa, shear viscosity modulus υ = 0.3 kPasec, Poisson modulus 0.5, and ρ = 1080 kg/m3.
- Simulation setup: The out-of-plane load values are expressed in 10−3Nm, while gravity is neglected.
- Out-of-plane motion: Out-of-plane motion is also simulated, with torsional screw axes and translations shown for the final configuration.The screw axes are represented by colored arrows, while black arrows show translation along each screw direction.
- Out-of-plane motion: The reported final configuration uses h1 = 8 mm, h2 = −45 mm, h3 = −683 mm and m1 = 3.9, m2 = 1, m3 = 0.3.
C. Cantilever Beam Comparison
The cantilever comparison evaluates how discrete section count affects the approximation of continuously varying deformation under a non-constant vertical tip load.
- Load handling: The discrete model maps distributed loads through section-wise transformations and integrates their re-mapped contributions over each section.This integration corresponds to the section length multiplied by the mean load over that section.
- Load handling: Wider averaging intervals produce larger discrepancies from the continuously varying load distribution.
- Test case: A cylindrical cantilever with L = 250 mm is tested under a vertical tip load in simulations of continuous and discrete Cosserat models.The beam has radius 10 mm, E = 110 kPa, shear viscosity modulus υ = 0.3 kPasec, Poisson modulus 0, and ρ = 2000 kg/m3.
- Dynamic comparison: The one-section model oscillates faster than the continuous cantilever, while increasing the number of sections progressively slows the oscillation toward the continuous value and reduces tip error.
- Steady-state comparison: Steady-state errors decrease with section count: e1 errors are 5.55%, 1.58%, and 1.42%, while e2 errors are 1.29%, 0.48%, and 0.07% for one, two, and three sections.
V. EXPERIMENTAL RESULTS
Experimental evaluation compares PCS dynamics against measurements from a cable-actuated conical silicone manipulator under single-bending, in-plane multi-bending, and out-of-plane multi-bending conditions.
- Experimental setup: The PCS model is evaluated against experimental data using the same prototype, parameters, and measurements previously used for the continuous Cosserat model.
- Experimental setup: The prototype is a single conical silicone piece actuated by 12 embedded cables arranged at three lengths and 90-degree relative angles.
- Test conditions: Experiments include single bending from cable 11, in-plane multi-bending from cables 9, 11, 1, 3, and out-of-plane multi-bending from cables 11, 5, and 2.
- Modeling conditions: The fluid-dynamics model uses drag and added-mass matrices parameterized by arm radius, fluid density, and coefficients Cx, Cy, Cz, By, and Bz.
A. Comparison
The experimental comparison models the manipulator as four constant-strain cylindrical sections and evaluates PCS predictions against real marker trajectories and continuous-Cosserat results.
- Discrete model: The soft manipulator is modeled as four cylindrical constant-strain sections with lengths L1, L2, L3, L4 and radii equal to the prototype’s mean section radii.
- Comparison procedure: Figure 7 compares continuous-cantilever curvature with discrete curvatures for one, two, and three sections, alongside tip-position errors in e1 and e2.
- Experimental platform: The real prototype and its schematic define the experimental manipulator used for the comparison.
- Comparison procedure: The recursive algorithm is applied to the three experimental load conditions, and the results are compared with continuous-Cosserat predictions.
B. Discussion
The PCS discrete model matches or outperforms the continuous Cosserat model in reported multi-bending experiments, while remaining improvable through friction modeling and finer discretization.
- 5.7%, 4.2% and 4.3% average errors for the discrete model compare with 5.1%, 5.2% and 5.4% for the continuous model across three cases.The discrete model is comparable or better overall, with lower errors in the second and third cases.
- Cable friction should be included to model the hysteresis observed during load-unload cycles, particularly after cable relaxation.Plane-bending experiments show error increases drastically after relaxation, when cable friction mainly drives the load.
- Increasing the number of discrete sections could better capture strain variation caused by external loads.
- Accounting for spatial variation in mass, stiffness, and viscosity matrices could improve modeling for the manipulator’s conical geometry.
- The PCS model is supported by simulation and experiments for plane and out-of-plane multi-bending, with performance comparable to or better than the continuous model.The conclusion frames the method as inheriting continuous Cosserat geometric and mechanical properties.