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Reduced Cartesian Kinetostatics for Tendon-Driven Continuum Robots: Residual-Stabilized Full-Shape Propagation

Ke Wu, Fangju Yang, Xiaohui Zhang, Zhengqiang Zhang, Jingang Yi, Jian S. Dai

arXiv:2609.31771v1cs.RO

TL;DR

The paper addresses the need to predict complete Cartesian backbone geometry for tendon-driven continuum-robot planning and control. It develops a reduced Cartesian model with analytic residual derivatives and residual-corrected propagation along prescribed actuation paths. Simulations and displacement-driven experiments report close shape agreement, faster updates than pointwise GVS solves, and low measured position errors.

  • Problem

    Planning and control tasks for TDCRs require the complete Cartesian backbone geometry, motivating a model that propagates full equilibrium shapes along prescribed actuation paths.

    Source: §I · p. 1, B. Objective · p. 2

  • Method

    The method combines a Cartesian backbone representation and offline moment reduction with analytically differentiated, residual-corrected rate systems for force- and displacement-driven updates.

    Source: §I · pp. 1–2, 1) Tendon-Force-Driven Kinetostatics · p. 4

  • Results

    0.508 ms per update on average, approximately 11 times faster than pointwise GVS equilibrium solves, while displacement-driven experiments report a maximum normalized mean backbone position error of 1.02%.

    Source: §VI · p. 8

  • Takeaways & Limitations

    Across four simulated cases, propagated Cartesian configurations and distributed strains closely match GVS, and residual correction suppresses drift observed with uncorrected Euler.

    Source: 2) Residual Control and Computational… · p. 6, §VI · p. 8

Abstract

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Many planning and control tasks for tendon-driven continuum robots (TDCRs) require the complete Cartesian backbone geometry. We present a reduced Cartesian framework for planar, axially compressible TDCRs that propagates equilibrium configurations along prescribed tendon-force and tendon-displacement trajectories. The backbone is represented by two global position fields. Following exact variation, a Taylor-Galerkin reduction condenses prescribed spatial properties and distributed loads into offline moment vectors, yielding analytic reduced residuals and Jacobians without online spatial quadrature or numerical differentiation. Analytical differentiation and residual correction yield first-order rate systems requiring one fixed-dimensional linear solve per rate evaluation after initial equilibrium alignment on a regular branch. Across four simulated cases covering variable tendon routing, nonuniform geometry, axial compression, and their combined effects, the propagated Cartesian shapes and distributed strains closely match pointwise geometrically variable-strain (GVS) equilibrium solutions. Residual correction suppresses propagation drift across the tested step sizes while adding only about 0.98% to the mean update time of uncorrected Euler. The proposed method requires 0.508 ms per update on average, approximately 11 times faster than pointwise GVS solves. Displacement-driven experiments yield a maximum normalized mean backbone position error of 1.02% and a maximum end-effector position error of 0.28%. These results support efficient and accurate Cartesian full-shape prediction along prescribed actuation paths.

I. INTRODUCTION

The paper shifts TDCR kinetostatics toward direct Cartesian full-shape tracking, motivated by the need for complete backbone geometry in planning and control. It formulates a planar, axially compressible robot model and seeks efficient propagation under force- or displacement-driven actuation.

A. Cartesian Variational Formulation

  • Cartesian position fields make task-level geometry direct equilibrium unknowns, whereas strain coordinates align with rod mechanics but require spatial integration to recover the backbone.The paper represents the backbone with two global position fields and derives strains from their spatial derivatives.Source: §I · p. 1, A. Cartesian Variational Formulation · p. 2

I. INTRODUCTION

  • The proposed formulation tracks the complete Cartesian backbone along actuation-parameterized equilibrium branches rather than solving successive nonlinear equilibria.Its contributions include a Cartesian representation, offline-precomputable reduction, and residual-stabilized propagation.Source: §I · pp. 1–2

A. The studied Manipulator

  • The studied planar robot has two opposite-side tendons and may have spatially varying stiffness and routing, axial compression, and prescribed distributed loads.Tendon friction is neglected; actuation inputs are uniform tendon tensions or tendon displacements.Source: A. The studied Manipulator · p. 2

B. Objective

  • The objective is efficient continuous propagation of the complete equilibrium configuration along prescribed tendon-force or tendon-displacement trajectories while accounting for compression, bending, loads, and varying properties.These inputs and effects are in scope for the proposed Cartesian model.Source: B. Objective · p. 2

B. Post-Variational Galerkin Reduction

The post-variational reduction makes the spatial equilibrium terms polynomial so their spatial contributions can be precomputed and assembled into reduced Galerkin residuals.

B. Post-Variational Galerkin Reduction

  • Taylor truncation around the stress-free metric and a Cartesian polynomial ansatz make the variational terms polynomial in the spatial coordinate, enabling coefficient separation and spatial precomputation.The finite-order approximation becomes less accurate as compression increases, and the continuation is restricted to 0 < λ(s) ≤ 1.Source: B. Post-Variational Galerkin Reduction · p. 3
  • Prescribed structural properties and tendon-routing terms are collected into offline moment vectors, while prescribed distributed loads likewise contribute precomputed Galerkin moments.The required vectors are computed once for prescribed robot properties.Source: B. Post-Variational Galerkin Reduction · p. 4
  • Substituting polynomial Cartesian variations into the post-variational virtual-work form yields a finite set of Bubnov–Galerkin residual equations.The coefficient functions are taken outside the spatial integrals.Source: B. Post-Variational Galerkin Reduction · p. 4

C. Explicit Residual-Stabilized Kinetostatic Propagation

The propagation model treats actuation as a differentiable prescribed path and excludes inertial and rate-dependent effects.

C. Explicit Residual-Stabilized Kinetostatic Propagation

1) Tendon-Force-Driven Kinetostatics:

For tendon-force actuation, analytic differentiation of the reduced residual yields a corrected rate system for tracking equilibria along a regular branch. After initial alignment, the coefficients are updated and mapped back to the Cartesian backbone.

1) Tendon-Force-Driven Kinetostatics:

  • Analytic differentiation of the reduced residual gives derivatives with respect to coefficients and tendon forces without online spatial quadrature or numerical differentiation.Precomputed moment vectors are independent of the coefficients and tendon forces.Source: 1) Tendon-Force-Driven Kinetostatics · p. 4
  • The force-driven rate system combines the residual's total derivative with the correction condition ˙R + γR = 0 on a regular branch with nonsingular Jacobian.The residual correction acts when the propagated state has nonzero residual and vanishes on the equilibrium manifold.Source: 1) Tendon-Force-Driven Kinetostatics · p. 4
  • Initial coefficients are aligned to the complete loading condition, then updated coefficients are substituted into the Cartesian approximation to recover the equilibrium shape.For initially unloaded, stress-free conditions, the initialization is c(0) = 0.Source: 1) Tendon-Force-Driven Kinetostatics · p. 4

2) Tendon-Displacement-Driven Kinetostatics:

For prescribed tendon-displacement trajectories, the formulation imposes displacement constraints through auxiliary multipliers and propagates the constrained equilibrium along a regular branch. Its Taylor–Galerkin reduction reuses offline moments without adding online spatial quadrature or precomputation.

2) Tendon-Displacement-Driven Kinetostatics:

  • Displacement actuation removes the tendon-force potential and imposes global tendon-displacement constraints through auxiliary multipliers in the constrained functional.Exact variation yields mechanical virtual work and the constraints before Taylor truncation and Cartesian polynomial approximation.Source: 2) Tendon-Displacement-Driven… · pp. 4–5
  • The reduced displacement constraints require no additional Galerkin weighting, and their moments are already included in the existing offline moment vectors.The formulation therefore adds neither online spatial quadrature nor a new precomputation structure.Source: 2) Tendon-Displacement-Driven… · p. 5
  • Analytic residual rates stabilize the mechanical and displacement residuals together, yielding rate propagation when the constrained tangent system is nonsingular.The regular branch requires rank H(c) = rank B(c) = 2; initial coefficients and multipliers are solved jointly.Source: 2) Tendon-Displacement-Driven… · p. 5
  • Physical admissibility requires η_i ≤ 0 because −η_i is tendon tension; positive multipliers imply compressive tendon action outside the formulation.Updated coefficients recover the displacement-driven equilibrium configuration under the regularity condition.Source: 2) Tendon-Displacement-Driven… · p. 5

IV. NUMERICAL SIMULATION AND MODEL VALIDATION

The numerical study evaluates the proposed Cartesian formulation against pointwise GVS equilibrium solves, covering shape and strain accuracy, residual control, and online computation.

IV. NUMERICAL SIMULATION AND MODEL VALIDATION

  • The validation compares Cartesian equilibrium shapes, distributed strains, residual control, and online computation against pointwise GVS equilibrium solves.Table I identifies the simulation parameters, numerical settings, and validation cases.Source: §IV · p. 5, Table I · p. 6
  • The study assesses Cartesian equilibrium-shape accuracy and distributed-strain accuracy along the propagation path.The reference is the pointwise geometric variable-strain method (GVS).Source: §IV · p. 5
  • Residual control and online computational performance are also evaluated against the pointwise GVS reference.Source: §IV · p. 5

A. Simulation Setup

Four simulated cases use matched physical parameters, gravity, force samples, and initial equilibria for the proposed model and GVS reference. Cases 3 and 4 reduce effective axial stiffness.

A. Simulation Setup

  • Both models use identical physical parameters, gravitational loading, force samples, and initial equilibria.Source: A. Simulation Setup · p. 6
  • Cases 3 and 4 use effective axial stiffness KA(s) = 0.007EA(s) instead of EA(s).Source: A. Simulation Setup · p. 6
  • The GVS reference uses fifth-order polynomial strain bases, a 10^-7 residual threshold, and preceding-step warm starts.The warm start is used to improve numerical robustness.Source: A. Simulation Setup · p. 6

1) Cartesian Shape and Distributed Strain Accuracy:

Across four cases, propagated Cartesian backbones closely agree with pointwise GVS solutions, and the distributed strain errors are reported along the backbone and force path.

1) Cartesian Shape and Distributed Strain Accuracy:

  • The propagated Cartesian backbones closely agree with pointwise GVS solutions across all four cases.The first row of Fig. 2 shows the backbone comparisons.Source: 1) Cartesian Shape and Distributed… · p. 6
  • Maximum absolute bending-strain errors in Cases 1–4 are 0.969, 0.662, 0.244, and 0.221 m^-1, respectively.The signed bending-strain error is the propagated curvature minus the GVS curvature.Source: 1) Cartesian Shape and Distributed… · p. 6
  • Maximum absolute axial-stretch-ratio errors in Cases 3 and 4 are 5.303 × 10^-3 and 3.033 × 10^-3, respectively.The authors report that these results accurately capture GVS baseline responses in strain space.Source: 1) Cartesian Shape and Distributed… · p. 6

2) Residual Control and Computational Performance:

Residual correction limits propagation drift while adding little update cost, and the experimental validation examines displacement-driven backbone and end-effector prediction.

2) Residual Control and Computational Performance:

  • 0.508 ms per update versus 5.586 ms for pointwise GVS solves, with residual correction adding about 0.98% to uncorrected Euler’s mean update cost.The update time corresponds to an approximately 11-fold acceleration over pointwise GVS equilibrium solves.Source: 2) Residual Control and Computational… · p. 6
  • Residual correction suppresses accumulated equilibrium-residual drift across the tested step sizes, while uncorrected explicit Euler shows larger, step-size-dependent drift.The tested steps are 5.0×10^-4, 1.0×10^-3, and 5.0×10^-3 s.Source: 2) Residual Control and Computational… · p. 6

V. EXPERIMENTAL VALIDATION

  • The displacement-driven experiment evaluates complete backbone-shape trajectories, distributed Cartesian errors, and corresponding end-effector trajectories along a linear tendon-displacement path.The passage describes the experimental evaluation scope, not its numerical outcomes.Source: §V · p. 6

A. Experimental Setup

Displacement-driven experiments show close agreement between predicted and measured backbone shapes and tip trajectories. The paper presents these results within a Cartesian propagation framework for planar TDCRs.

A. Experimental Setup

  • The experimental platform uses two antagonistic motor-driven tendons and an RGB-D camera that tracks seven backbone markers.Axial deformation is negligible over the tested range, with λ(s) ≈ 1.Source: A. Experimental Setup · p. 6

B. Experimental Results and Analysis

  • 1.02% is the maximum normalized mean backbone error; maximum normalized Euclidean tip errors are 0.062% in x and 0.092% in y.The measured and predicted backbone shapes and tip trajectories agree closely along the linear tendon-displacement path.Source: B. Experimental Results and Analysis · p. 7, Figure 4 · p. 8

VI. CONCLUSION

  • Offline spatial moments and analytic residual derivatives support force- and displacement-driven updates with one fixed-dimensional linear solve per rate evaluation after initial alignment on a regular branch.Across four simulated mechanical cases, propagated Cartesian configurations and distributed strains closely match GVS; residual correction suppresses drift, with an average update time of 0.508 ms.Source: §VI · p. 8
  • The reported framework covers planar robots; future work will address three-dimensional deformation, tendon friction, and material hysteresis.Source: §VI · p. 8
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