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

Problem Statement

As shown in Fig. 1 , we consider a planar, axially compressible tendon-driven continuum robot with a straight and stress-free reference configuration. Let $s\in[0,L]$ denote the reference arc-length coordinate of the backbone. Two tendons are routed longitudinally on opposite sides of the backbone.

With tendon friction neglected, the prescribed actuation inputs are either the spatially uniform tendon tensions $F_{1}(t),F_{2}(t)\geq 0$ or the tendon displacements $\overline{\Delta l}_{1}(t),\overline{\Delta l}_{2}(t)$.

Methodology

Let $\mathcal{F}=\{O;\hat{\mathbf{x}},\hat{\mathbf{y}}\}$ denote a fixed global Cartesian frame. The origin $O$ is placed at the robot base, $\hat{\mathbf{x}}$ is aligned with the stress-free backbone, and $\hat{\mathbf{y}}$ defines the transverse direction. The backbone centerline expressed in $\mathcal{F}$ is $[x(s),y(s)]^{\mathbf{T}}$.

The clamped-base conditions are A subscript $s$ denotes differentiation with respect to $s$. Admissible configurations satisfy $x_{s}^{2}+y_{s}^{2}>0$ over $s\in[0,L]$, with $x_{s}(0)>0$.

Numerical Simulation and Model Validation

The numerical study evaluates the Cartesian equilibrium-shape accuracy, distributed strain accuracy along the propagation path, residual control, and online computational performance of the proposed formulation against pointwise geometric variable-strain method (GVS) equilibrium solves [ 21 ] . Table I lists the four cases, numerical settings, and tendon-force profiles over $0\leq t\leq T$.

Both models use identical physical parameters, gravitational loading $q_{y}(s)=-\rho gA(s)$, force samples, and initial equilibria.

Experimental Validation

The experiment validates displacement-driven prediction of the backbone shape and end-effector motion along a linear tendon-displacement path, considering the complete backbone-shape trajectory, its distributed Cartesian error, and the corresponding end-effector trajectory. The platform in Fig. 3 uses two antagonistic motor-driven tendons and an RGB-D camera tracking seven backbone markers.

Table III lists the robot parameters and linear input profile.

Sources and Demonstrations

Figures are reproduced from Wu et al. See the full paper for experimental details and the project page, when the paper links one, for demonstrations and videos.