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Real-Time Whole-Body Safe Motion Generation for Multi-Segment Tendon-Driven Continuum Robots

Fangju Yang, Siyi Ma, Tonghao Guan, Tingcong Liu, Hang Yang, Zhengqiang Zhang, Jian S. Dai, Ke Wu

arXiv:2610.00220v1cs.RO

TL;DR

Tendon-driven continuum robots require accurate nonuniform-curvature models and collision avoidance along the whole body. This paper combines an actuation-space variable-curvature model with multipoint CBF-QP control; in 100 trials, it reports higher collision-free success than a non-CBF baseline in both static and dynamic scenarios.

  • Problem

    Real-time control of tendon-driven continuum robots must handle spatially varying curvature and collision risks distributed along the backbone, which tip-only control cannot ensure.

    Source: §I · p. 1

  • Method

    An energy-based actuation-space variable-curvature model supplies analytical backbone Jacobians for multipoint CBF-QP whole-body safety control.

    Source: §I · p. 2, §II · p. 2, §III · p. 5

  • Results

    96% static and 100% dynamic collision-free success over 100 trials, versus approximately 60% and 80%, respectively, for the non-CBF baseline.

    Source: 2) Whole-Body Collision Avoidance · pp. 7–8

  • Takeaways & Limitations

    With 600 monitoring points, mean control-step time is 6.66 ms, supporting real-time whole-body collision avoidance with dense backbone sampling.

    Source: 3) Computation Time Evaluation · p. 8

Abstract

from arXiv · show

Real-time motion generation for tendon-driven continuum robots requires accurate modeling of nonuniform bending and whole-body collision avoidance. This paper presents a unified actuation-space framework for planar multi-segment tendon-driven continuum robots. An energy-based variable-curvature model captures spatially varying tendon spacing and bending stiffness and provides analytical Jacobians for differential inverse kinematics and safety monitoring. A multipoint CBF-QP enforces backbone clearance under obstacle motion and actuation-velocity bounds, while its decision dimension depends only on the number of independently actuated segments. The model closely agrees with GVS references, with a maximum curvature error of $5.223 \times 10^{-2}\,\mathrm{m}^{-1}$. Over 100 MuJoCo trials, the proposed method achieves collision-free success rates of 96% and 100% in static and dynamic scenarios, respectively, compared with approximately 60% and 80% without CBF constraints. Hole-traversal tests further demonstrate safe motion in constrained environments. With 600 backbone monitoring points, the mean control-step time is 6.66 ms, demonstrating real-time whole-body safe motion generation.

I. INTRODUCTION

Tendon-driven continuum robots need accurate models of spatially varying bending and safety constraints along the entire backbone. The paper develops an actuation-space framework that connects variable-curvature modeling with whole-body safe motion generation.

I. INTRODUCTION

  • Spatially varying tendon spacing and stiffness can produce nonuniform curvature, while collision risk extends along the backbone, making tip-only control insufficient for whole-body safety.The introduction identifies real-time modeling and safety control together as a central challenge for tendon-driven continuum robots in confined and dynamic environments.Source: §I · p. 1
  • Prior kinematic approaches trade accuracy against computational cost, and existing safety methods lack a unified real-time framework for nonuniform mechanics and explicit whole-body constraints.The identified gap includes analytical mappings at arbitrary backbone locations and safety constraints that account for dynamic obstacles and actuation-velocity bounds.Source: §I · pp. 1–2
  • The proposed framework combines energy-based variable-curvature modeling, analytical backbone Jacobians, and multipoint CBF-QP safety control in actuation space.Its reported contributions include curvature errors below 5.3 × 10−2 m−1 against GVS references and 96% and 100% collision-free success in static and dynamic scenarios, respectively, over 100 trials.Source: §I · p. 2

II. VARIABLE-CURVATURE KINEMATIC MODELING IN ACTUATION SPACE

  • The model represents a planar robot as serial flexible segments with independent tendon-displacement inputs and potentially varying geometric and mechanical properties.For each segment, the tendon-length constraint relates actuation to curvature while allowing tendon spacing to vary along the backbone.Source: §II · p. 2, A. Geometric Relations and Energy… · p. 2

A. Geometric Relations and Energy Functional

  • The geometric constraint is paired with Euler–Bernoulli bending energy in a constrained variational formulation for the backbone configuration.The figure caption identifies the related overview as robot kinematics, whole-body collision avoidance, and the real-time CBF-QP framework.Source: A. Geometric Relations and Energy… · p. 2, Figure 1 · p. 3

B. Variable-Curvature Forward Kinematic Mapping

The forward model derives the backbone’s variable curvature from prescribed tendon displacements, then integrates curvature to obtain segment and whole-robot orientation and position.

B. Variable-Curvature Forward Kinematic Mapping

  • For prescribed tendon-length variations, constrained minimum potential energy determines curvature subject to each segment’s tendon-length relation.The derivation introduces a time-varying Lagrange multiplier for each segment’s differential tendon-length constraint.Source: B. Variable-Curvature Forward… · p. 3
  • Interface continuity, a fixed-base condition, and a tip condition of zero external moment constrain the segment-wise variational equilibrium.The resulting interface condition preserves tangent-angle continuity between adjacent segments.Source: B. Variable-Curvature Forward… · pp. 3–4
  • The resulting local curvature depends jointly on actuation and the spatial distributions of tendon spacing and bending stiffness.The derivation defines an actuation-compliance coefficient and uses the tendon-length constraint to determine curvature for each segment.Source: B. Variable-Curvature Forward… · p. 4
  • Integrating curvature yields each segment’s relative rotation, the global tangent angle, and the forward mapping from actuation space to Cartesian position.The global angle includes the rotations of preceding segments and the local rotation of the current segment.Source: B. Variable-Curvature Forward… · p. 4

C. Residual-Based Variable-Curvature Inverse Kinematics

The inverse-kinematics method uses differential actuation-to-position mappings to regulate tip tracking error and generate tendon velocities for a desired trajectory.

C. Residual-Based Variable-Curvature Inverse Kinematics

  • The point Jacobian maps multi-segment actuation rates to Cartesian velocity at any backbone position.At the tip, the method uses this differential mapping to relate actuation to desired trajectory tracking.Source: C. Residual-Based Variable-Curvature… · pp. 4–5
  • First-order tip-residual dynamics provide smooth residual regulation without excessively aggressive actuation near equilibrium.The inverse-kinematics problem seeks actuation inputs that make the tip position match a desired trajectory.Source: C. Residual-Based Variable-Curvature… · p. 5
  • For robots with more than two independently actuated segments, a pseudoinverse gives nominal actuation velocity, with damping available near kinematic singularities.The tip task is two-dimensional, so the actuation mapping is redundant when n > 2.Source: C. Residual-Based Variable-Curvature… · p. 5
  • With a positive residual-convergence gain, the tip residual converges asymptotically to zero and the tip tracks its desired trajectory.The actuation influence of a segment extends to that segment and all downstream backbone positions, but not to preceding backbone portions.Source: C. Residual-Based Variable-Curvature… · pp. 4–5

III. REAL-TIME WHOLE-BODY COLLISION-AVOIDANCE FRAMEWORK

The framework enforces whole-body safety by monitoring multiple backbone positions and using their actuation-space point Jacobians in a quadratic program that also tracks the tip trajectory.

III. REAL-TIME WHOLE-BODY COLLISION-AVOIDANCE FRAMEWORK

  • Multiple backbone monitoring positions generate CBF constraints through actuation-space point Jacobians, unifying tip tracking and whole-body collision avoidance in one quadratic program.The framework is designed for real-time safety control rather than imposing collision-avoidance constraints only at the tip.Source: §III · p. 5

A. Actuation-Space System and Whole-Body Safe Set

The section represents tendon-length variations as the robot’s actuation state and their rates as inputs, then uses actuation-space kinematics to define a whole-body safe set from monitored backbone positions and obstacle clearances.

A. Actuation-Space System and Whole-Body Safe Set

  • q(t) is the actuation state and u(t) is the corresponding actuation-velocity input.The system is expressed directly in actuation space.Source: A. Actuation-Space System and… · p. 5
  • Forward kinematics express any backbone position as a function of the actuation state q(t).The model supports positions at arbitrary arc lengths.Source: A. Actuation-Space System and… · p. 5
  • Safety monitoring samples the backbone at selected arc lengths, with each point assigned a position and Jacobian for velocity evaluation.The point velocities are obtained from the actuation input through their Jacobians.Source: A. Actuation-Space System and… · p. 5
  • The whole-body safe set comprises actuation states for which all safety-monitoring positions maintain the prescribed minimum clearance from obstacles.Each monitored point is evaluated relative to obstacle positions.Source: A. Actuation-Space System and… · pp. 5–6

B. Multipoint Control Barrier Function Constraints

The multipoint CBF constraints account for obstacle motion and convert clearance requirements at each monitored backbone point into affine inequalities on actuation velocity.

B. Multipoint Control Barrier Function Constraints

  • Differentiating each point–obstacle safety function incorporates both robot-point velocity and obstacle velocity, which is zero for stationary obstacles.The resulting condition is applied to ensure forward invariance of the safe set.Source: B. Multipoint Control Barrier Function… · p. 6
  • For every safety-point–obstacle pair, the first-order CBF condition yields a linear safety constraint on the actuation-velocity input.The condition uses a positive CBF gain.Source: B. Multipoint Control Barrier Function… · p. 6
  • Enforcing all NαNo inequalities maps local clearance requirements across multiple backbone positions into a unified whole-body constraint set in actuation space.The stacked constraint matrices collect the inequalities for all safety-point–obstacle pairs.Source: B. Multipoint Control Barrier Function… · p. 6

C. Real-Time Whole-Body Safe Motion Control via CBF-QP

The controller minimally adjusts nominal inverse-kinematic actuation velocities through an online CBF-QP, subject to whole-body safety and tendon-velocity bounds. Given a safe initial state and continued feasibility, the constraints preserve the time-varying safe set.

C. Real-Time Whole-Body Safe Motion Control via CBF-QP

  • The CBF-QP selects actual actuation velocity to minimally modify the nominal inverse-kinematic input while enforcing whole-body safety and tendon-velocity bounds.The optimization variable is n-dimensional actuation velocity.Source: C. Real-Time Whole-Body Safe Motion… · p. 6
  • If the initial state is safe and the constraints remain feasible, the CBF conditions guarantee forward invariance of the time-varying safe set.Selected backbone monitoring positions therefore maintain prescribed clearance throughout motion.Source: C. Real-Time Whole-Body Safe Motion… · p. 6
  • The online procedure evaluates kinematics and nominal input, updates pointwise safety constraints, solves the QP, and advances the actuation state.Algorithm 1 also specifies initialization and desired-trajectory inputs.Source: C. Real-Time Whole-Body Safe Motion… · p. 6

IV. NUMERICAL VALIDATION

The numerical validation is organized around kinematic accuracy, whole-body collision avoidance, and computational efficiency. The supplied figure captions identify kinematic comparison and MuJoCo obstacle-avoidance and hole-traversal evaluations.

IV. NUMERICAL VALIDATION

  • The simulations evaluate kinematic accuracy, whole-body collision avoidance, and computational efficiency, including static obstacles and moving-obstacle trajectory tracking.Kinematic predictions are compared with GVS results.Source: §IV · p. 6
  • Fig. 2 presents a GVS shape comparison and segment-wise curvature errors, while Fig. 3 covers MuJoCo static and dynamic obstacle avoidance and hole traversal.The captions specify evaluation scope, not numerical outcomes.Source: Figure 2 · p. 7, Figure 3 · p. 7
  • Table I is identified as containing simulation parameters.The supplied table caption does not specify individual parameter values.Source: Table I · p. 7

A. Simulation Setup

The control tests use a three-segment robot and evaluate MuJoCo motion with static and moving obstacles as well as hole traversal, comparing CBF constraints enabled and disabled under identical task settings.

A. Simulation Setup

  • Control tests use three segments with a total robot length of 0.75 m and tendon spacings of [16.5, 23.5, 30.5] mm.The segment lengths are [0.30, 0.25, 0.20] m.Source: A. Simulation Setup · p. 7
  • MuJoCo validation covers whole-body collision avoidance with static and moving obstacles, plus hole traversal.Source: A. Simulation Setup · p. 7
  • The comparisons enable or disable hard CBF constraints under identical task settings.Source: A. Simulation Setup · p. 7

1) Variable-Curvature Model Accuracy:

The model closely matches GVS curvature references, while CBF-constrained control improves collision-free success over the non-CBF baseline in static and dynamic scenarios. Dense backbone monitoring remains computationally feasible, and hardware validation and improved task completion in tightly constrained passages remain future work.

1) Variable-Curvature Model Accuracy:

  • 5.223 × 10^-2 m^-1 is the maximum absolute curvature error for S3 over the 20-s trajectory, with close agreement to GVS solutions.Errors for S1 and S2 are 2.109 × 10^-3 and 4.929 × 10^-2 m^-1, respectively.Source: 1) Variable-Curvature Model Accuracy · p. 7

2) Whole-Body Collision Avoidance:

3) Computation Time Evaluation:

V. CONCLUSION

  • Future work will address hardware validation and improved task completion in tightly constrained passages.Source: §V · p. 8
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