Tendon-Driven Continuum Robots

• Tendon-driven continuum robots are flexible manipulators actuated by tendons routed along compliant backbones that produce continuous bending and stiffness modulation.
• They leverage advanced modeling techniques such as piecewise constant curvature, Clarke-manifold transformations, and Cosserat rod formulations to address nonlinearities, friction, and hysteresis.
• These systems enable applications in minimally invasive surgery, confined-space inspection, and contact-rich manipulation, while integrating robust control strategies like backstepping, MPC, and visual servoing.

Searching arXiv for recent relevant papers on tendon-driven continuum robots and related modeling/control topics. I’m querying arXiv for “tendon-driven continuum robots” and specific recent work on modeling, control, and design. Tendon-driven continuum robots (TDCRs) are continuum manipulators whose shape is generated by tendon actuation rather than by a chain of discrete revolute joints. In the reported architectures, a compliant backbone is routed by tendons through spacer disks, grooves, tubes, or comparable guides, and differential tendon length or tension produces distributed bending, and in some designs also extension, compression, torsion, locomotion, or stiffness modulation. Their compliance, compactness, and miniaturization capability make them attractive for confined-space tasks, including minimally invasive surgery, industrial inspection, narrow-tube locomotion, and contact-rich manipulation; however, the same properties make modeling, sensing, and control difficult because the mechanics are strongly nonlinear, distributed, and often affected by friction, hysteresis, tendon slack, and external contact (Shentu et al., 2024, Danesh et al., 19 Feb 2025, Cho et al., 2024).

1. Mechanical architectures and actuation topologies

A standard TDCR segment is actuated by tendons arranged around a flexible backbone. For almost all tendon-driven continuum robots, a segment is actuated by three or four tendons constrained by its mechanical design, and in many experimental platforms those tendons are symmetrically spaced around the backbone (Grassmann et al., 2024). Representative implementations include a single-section robot with four Kevlar tendons at $90^\circ$ separation on a 500 mm spring-steel backbone, a single-section large-deflection platform with four Kevlar tendons offset by 2.0 cm on a 0.5 m 304 stainless steel backbone, and a two-segment system with eight tendons total, four per segment, arranged at $90^\circ$ intervals around a compliant backbone (Danesh et al., 19 Feb 2025, Danesh et al., 2024, Jabari et al., 6 Feb 2026).

Actuation topology is not limited to parallel routing. One current line of work replaces the usual parallel-tendon assumption with non-parallel, helical, or otherwise general three-dimensional routing. A universal-jointed TDCR discretizes the body into rigid links joined by synovial universal joints and supports arbitrary tendon routing through geometrically exact waypoint descriptions, including helical routing, while ExoNav uses a routed tendon inside a helically notched nitinol tube to generate a helical backbone shape (Shentu et al., 2024, Moradkhani et al., 19 Jan 2026). This broadens the reachable shape set beyond analytically simple constant-curvature motions.

Several architectures couple tendon actuation with additional mechanical functions. Antagonistic tendon pairs can regulate both position and stiffness through co-contraction; a layer-jamming arm uses tendon actuation for motion and vacuum-driven layer jamming for tunable stiffness; a millimeter-scale wrist uses three peripheral tendons for bending and a fourth central tendon for an integrated gripper; and a spring-based confined-space robot uses four tendons to produce coupled bending and axial length change in a compression spring backbone (Yi et al., 2023, Ouyang, 2019, Leavitt et al., 2023, Hu et al., 10 Mar 2026). These examples show that “tendon-driven continuum robot” denotes a family of actuation principles rather than a single morphology.

Segment count and actuator redundancy vary widely. A three-segment extensible TDCR has 12 actuators, with three tendon actuators and one backbone actuator per segment, while another task-space control study considers a six-tendon, three-output arm, explicitly treating the robot as over-actuated (Hachen et al., 2024, Maghooli et al., 2024). At the opposite extreme, a single-segment long TDCR can be deliberately kept mechanically simple and rely on environmental contact to realize multiple effective curvatures during motion planning (Rao et al., 2024).

2. State representations and configuration manifolds

A defining modeling issue in TDCRs is that tendon coordinates are mechanically coupled. For symmetrically arranged tendon sets, the physically admissible tendon-displacement vectors do not span all of $\mathbb{R}^n$; instead they lie on a lower-dimensional set governed by tendon displacement balance,

$\sum \rho_i = 0.$

For an $n$-tendon segment, the admissible set is written as

$$\boldsymbol{Q} = \Set{\boldsymbol{\rho} \in \mathbb{R}^n \mid \rho_i = \rho_\mathrm{Re}\cos{\psi_i} + \rho_\mathrm{Im}\sin{\psi_i}\,\wedge\,\sum \rho_i = 0},$$

with tendon angles

$\psi_i = 2\pi(i - 1)/n.$

This expresses the central fact that the motion-relevant degrees of freedom are two even when the tendon count is larger (Grassmann et al., 2024).

The Clarke transform provides a linear representation of that two-dimensional manifold. Using the generalized matrix

$$\boldsymbol{M}_\mathcal{P} = \dfrac{2}{n} \begin{bmatrix} \cos\left(0\right) & \cos\left(2\pi\dfrac{1}{n}\right) & \cdots & \cos\left(2\pi\dfrac{n-1}{n}\right)\ \sin\left(0\right) & \sin\left(2\pi\dfrac{1}{n}\right) & \cdots & \sin\left(2\pi\dfrac{n-1}{n}\right) \end{bmatrix},$$

the tendon vector is mapped to the Clarke coordinates

$$\boldsymbol{M}_\mathcal{P}\boldsymbol{\rho} = \begin{bmatrix} \rho_\mathrm{Re}& \rho_\mathrm{Im} \end{bmatrix}^T,$$

and each tendon displacement is reconstructed as

$$\rho_i = \rho_\mathrm{Re}\cos{\psi_i} + \rho_\mathrm{Im}\sin{\psi_i}.$$

In this form, the Clarke coordinates are local coordinates of a two-dimensional manifold embedded in the $90^\circ$0-dimensional tendon space (Grassmann et al., 2024).

The same coordinates connect directly to constant-curvature arc parameters: $90^\circ$1 This means the tendon manifold is linearly related to the common arc representation $90^\circ$2, which avoids the singularities associated with polar curvature-angle coordinates and gives a closed-form bridge between tendon space, arc space, and pose variables $90^\circ$3 (Grassmann et al., 2024).

The same idea has been extended to dynamic modeling and control. In a dynamic Clarke-manifold formulation, each segment with $90^\circ$4 tendons is represented by $90^\circ$5, with

$90^\circ$6

The resulting manifold-space dynamics are controlled directly by linear PD or PID laws. A practical difficulty is that mapped tendon forces can become negative; the reported “shifting” procedure adds a uniform offset equal to the minimum tendon force so that all tendon forces become nonnegative while the manifold torque remains unchanged because the cosine and sine sums vanish over a symmetric tendon set (Muhmann et al., 26 Mar 2025). This suggests that configuration-manifold formulations are not only coordinate changes but also enforcement mechanisms for actuation feasibility.

3. Kinematics, statics, and mechanics

TDCR mechanics are modeled by several distinct but overlapping paradigms. The most common simplification is piecewise constant curvature (PCC), in which each segment or subsegment is approximated as a constant-curvature arc. PCC is used for real-time task-space control, static workspace optimization, dynamic Clarke-manifold modeling, and open-loop kinematics of spring-based robots because it is computationally efficient and exposes low-dimensional arc variables (Hachen et al., 2024, Jabari et al., 6 Feb 2026, Muhmann et al., 26 Mar 2025, Hu et al., 10 Mar 2026).

Model family	Core state or assumption	Reported scope
PCC	Constant-curvature arcs	Real-time control, static optimization, simplified inverse kinematics
Universal-jointed optimization model	$90^\circ$7 with tendon waypoints	Arbitrary routing, friction, multi-tendon shape estimation
Cosserat or Kirchhoff rod	$90^\circ$8	Large deflections, external loads, dynamic rod mechanics
Hysteresis-aware deep decoder	$90^\circ$9	Full-shape prediction with configuration history

Within PCC-based formulations, the robot is often modeled quasi-statically as an actuator-space integrator with nonlinear forward kinematics $\mathbb{R}^n$0, or statically through per-vertebra equilibrium. A feasible static workspace study defines the workspace under load as the set of tip positions satisfying static equilibrium under gravity, tendon friction, and external wrench, and optimizes tendon tensions by maximizing the Euclidean norm $\mathbb{R}^n$1 of tip position. In the reported two-segment, eight-tendon robot, genetic-algorithm optimization converged quickly, with the best tradeoff obtained at population size 50 and convergence in about 5 iterations (Jabari et al., 6 Feb 2026).

When tendon routing is general rather than parallel, PCC alone is usually insufficient. The universal-jointed TDCR introduces a mechanically discretized finite-dimensional design space: for $\mathbb{R}^n$2 universal joints the state is

$\mathbb{R}^n$3

and each tendon is described by a geometrically exact waypoint matrix

$\mathbb{R}^n$4

Friction is propagated by the Capstan law,

$\mathbb{R}^n$5

and shape estimation is posed as constrained optimization matching normalized moment patterns to normalized joint angles. On a 170 mm prototype with helical tendons, the reported average tip error in the two-helical-tendon case was $\mathbb{R}^n$6 mm, or $\mathbb{R}^n$7 of length (Shentu et al., 2024).

Cosserat-rod formulations are used when large deflections, distributed loads, or nontrivial tendon-body coupling must be retained. A large-deflection dynamic study models the backbone by the full Cosserat PDE system in $\mathbb{R}^n$8, $\mathbb{R}^n$9, strain variables, and internal forces and moments, and solves the equations by BDF-$\sum \rho_i = 0.$0 time discretization, forward Euler in $\sum \rho_i = 0.$1, and a shooting method (Danesh et al., 2024). ExoNav adapts Cosserat statics to a helically notched tube by introducing an effective area

$\sum \rho_i = 0.$2

an offset neutral axis

$\sum \rho_i = 0.$3

and an undeformed helical neutral axis $\sum \rho_i = 0.$4, then solves the boundary-value problem by shooting (Moradkhani et al., 19 Jan 2026). A general tendon-actuated concentric-tube framework extends this line further by modeling multiple nested extensible, shearable tubes with arbitrary tendon routing, shared centerline bending, relative twist, and relative dilation; across two-tube, three-tube, helical-routing, and external-platform validations, reported tip prediction errors remain around $\sum \rho_i = 0.$5 of total length in the main setups and around $\sum \rho_i = 0.$6 on external robots (Kheradmand et al., 28 Oct 2025).

A persistent mechanical theme is that simplified models omit effects that remain experimentally visible. Across the reported literature, friction, hysteresis, tendon slack, tendon stretch, sensor placement, and external loads repeatedly appear as calibration variables, disturbances, or explicit limitations (Shentu et al., 2024, Cho et al., 2024, Hachen et al., 2024).

4. Control, planning, and stiffness regulation

Model-based TDCR control spans tension-space, tendon-space, arc-space, and task-space formulations. In large-deflection dynamics, a Cosserat-rod-based backstepping controller has been reported for a single-section TDCR executing a bend of about $\sum \rho_i = 0.$7. The desired tip trajectory is

$\sum \rho_i = 0.$8

and the controller is compared with sliding mode control. In experiments, backstepping achieved smoother trajectories, lower overshoot, and shorter settling time, with TPL $\sum \rho_i = 0.$9 mm versus $n$0 mm, settling time $n$1 versus $n$2 iterations, and overshoot $n$3 versus $n$4 (Danesh et al., 2024).

For underactuated antagonistic tendon pairs, simultaneous position-and-stiffness control has been formalized in a passivity-based framework. The central actuation constraint is nonnegative tension, $n$5, and an input transformation separates a positioning channel from a nonnegative stiffening channel. The reported closed-loop stiffness around the target equilibrium is

$n$6

while open-loop co-contraction $n$7 increases transverse stiffness approximately linearly with $n$8. Experimental stiffening measurements reported correlation coefficient $n$9, and position regulation typically converged in under 1.5 s (Yi et al., 2023).

Task-space safety and whole-body collision avoidance motivate optimization-based control. A nonlinear MPC controller for a three-segment extensible TDCR uses the cost

$$\boldsymbol{Q} = \Set{\boldsymbol{\rho} \in \mathbb{R}^n \mid \rho_i = \rho_\mathrm{Re}\cos{\psi_i} + \rho_\mathrm{Im}\sin{\psi_i}\,\wedge\,\sum \rho_i = 0},$$0

subject to actuator bounds, a tendon consistency constraint

$$\boldsymbol{Q} = \Set{\boldsymbol{\rho} \in \mathbb{R}^n \mid \rho_i = \rho_\mathrm{Re}\cos{\psi_i} + \rho_\mathrm{Im}\sin{\psi_i}\,\wedge\,\sum \rho_i = 0},$$1

curvature limits, and signed-distance shape constraints on all disk positions. With prediction horizon $$\boldsymbol{Q} = \Set{\boldsymbol{\rho} \in \mathbb{R}^n \mid \rho_i = \rho_\mathrm{Re}\cos{\psi_i} + \rho_\mathrm{Im}\sin{\psi_i}\,\wedge\,\sum \rho_i = 0},$$2, the controller runs at 30 Hz with average iteration time about 30 ms and, in hardware, kept the robot within a 5.5 mm safety margin with overshoot up to 2.5 mm toward the boundary for an infeasible target (Hachen et al., 2024).

Intentional contact can itself be a control resource. A search-based contact-aided planner for a planar single-segment long TDCR treats environmental contact as a mechanism for realizing multiple effective curvatures without adding extra robot segments. Over 525 queries in convex and non-convex environments, the proposed planner achieved success rates of $$\boldsymbol{Q} = \Set{\boldsymbol{\rho} \in \mathbb{R}^n \mid \rho_i = \rho_\mathrm{Re}\cos{\psi_i} + \rho_\mathrm{Im}\sin{\psi_i}\,\wedge\,\sum \rho_i = 0},$$3, $$\boldsymbol{Q} = \Set{\boldsymbol{\rho} \in \mathbb{R}^n \mid \rho_i = \rho_\mathrm{Re}\cos{\psi_i} + \rho_\mathrm{Im}\sin{\psi_i}\,\wedge\,\sum \rho_i = 0},$$4, and $$\boldsymbol{Q} = \Set{\boldsymbol{\rho} \in \mathbb{R}^n \mid \rho_i = \rho_\mathrm{Re}\cos{\psi_i} + \rho_\mathrm{Im}\sin{\psi_i}\,\wedge\,\sum \rho_i = 0},$$5 in the three tested workspaces, while the reported baselines stayed below $$\boldsymbol{Q} = \Set{\boldsymbol{\rho} \in \mathbb{R}^n \mid \rho_i = \rho_\mathrm{Re}\cos{\psi_i} + \rho_\mathrm{Im}\sin{\psi_i}\,\wedge\,\sum \rho_i = 0},$$6 (Rao et al., 2024). This directly contradicts the common tip-only, contact-avoidance assumption often inherited from rigid-robot planning.

Clarke-manifold dynamics add a further control layer for arbitrary tendon count. In simulation, the manifold “shifting” method yielded $$\boldsymbol{Q} = \Set{\boldsymbol{\rho} \in \mathbb{R}^n \mid \rho_i = \rho_\mathrm{Re}\cos{\psi_i} + \rho_\mathrm{Im}\sin{\psi_i}\,\wedge\,\sum \rho_i = 0},$$7 lower RMSE on average than clipping across two segments, and on a physical one-segment, five-tendon prototype the reported PD controller achieved $$\boldsymbol{Q} = \Set{\boldsymbol{\rho} \in \mathbb{R}^n \mid \rho_i = \rho_\mathrm{Re}\cos{\psi_i} + \rho_\mathrm{Im}\sin{\psi_i}\,\wedge\,\sum \rho_i = 0},$$8 lower RMSE than PID while operating in real time at 1 kHz (Muhmann et al., 26 Mar 2025). This indicates that control design can benefit from state reduction even when the mechanical platform has more than the customary three or four tendons.

5. Perception, visual servoing, and learning-based control

Because onboard sensing is often bulky or incompatible with slender, sterilizable, or biocompatible designs, vision-based and intrinsic-signal-based methods occupy a central place in TDCR control. A deep direct visual servoing method uses an eye-in-hand camera and a modified VGG-16 network to map the target image $$\boldsymbol{Q} = \Set{\boldsymbol{\rho} \in \mathbb{R}^n \mid \rho_i = \rho_\mathrm{Re}\cos{\psi_i} + \rho_\mathrm{Im}\sin{\psi_i}\,\wedge\,\sum \rho_i = 0},$$9 and current image $\psi_i = 2\pi(i - 1)/n.$0 directly to tendon commands,

$\psi_i = 2\pi(i - 1)/n.$1

Training data are generated in Blender under the constant-curvature assumption, including lighting changes and occlusions; the dataset contains 5000 RGB images at $\psi_i = 2\pi(i - 1)/n.$2, final training MSE is $\psi_i = 2\pi(i - 1)/n.$3, and inference takes about 15 ms per frame. The reported controller remains effective when less than $\psi_i = 2\pi(i - 1)/n.$4 of the target image is visible at the start and under occlusions of more than $\psi_i = 2\pi(i - 1)/n.$5 in some cases (Abdulhafiz et al., 2021).

A subsequent hybrid visual servoing framework combines deep-learning-based visual servoing (DLBVS) with image-based visual servoing (IBVS) under supervisory switching driven by Sum of Absolute Differences (SAD) and feature availability. The system uses DLBVS when the image is far from the target or features are lost, and switches to IBVS for final convergence when features are available. In the reported comparison from initial condition $\psi_i = 2\pi(i - 1)/n.$6 mm, HVS reduced average iteration time from $\psi_i = 2\pi(i - 1)/n.$7 s to $\psi_i = 2\pi(i - 1)/n.$8 s, reduced convergence iterations to SAD threshold 0.06 from $\psi_i = 2\pi(i - 1)/n.$9 to $$\boldsymbol{M}_\mathcal{P} = \dfrac{2}{n} \begin{bmatrix} \cos\left(0\right) & \cos\left(2\pi\dfrac{1}{n}\right) & \cdots & \cos\left(2\pi\dfrac{n-1}{n}\right)\ \sin\left(0\right) & \sin\left(2\pi\dfrac{1}{n}\right) & \cdots & \sin\left(2\pi\dfrac{n-1}{n}\right) \end{bmatrix},$$0, and reduced final SAD from $$\boldsymbol{M}_\mathcal{P} = \dfrac{2}{n} \begin{bmatrix} \cos\left(0\right) & \cos\left(2\pi\dfrac{1}{n}\right) & \cdots & \cos\left(2\pi\dfrac{n-1}{n}\right)\ \sin\left(0\right) & \sin\left(2\pi\dfrac{1}{n}\right) & \cdots & \sin\left(2\pi\dfrac{n-1}{n}\right) \end{bmatrix},$$1 to $$\boldsymbol{M}_\mathcal{P} = \dfrac{2}{n} \begin{bmatrix} \cos\left(0\right) & \cos\left(2\pi\dfrac{1}{n}\right) & \cdots & \cos\left(2\pi\dfrac{n-1}{n}\right)\ \sin\left(0\right) & \sin\left(2\pi\dfrac{1}{n}\right) & \cdots & \sin\left(2\pi\dfrac{n-1}{n}\right) \end{bmatrix},$$2, while preserving robustness to occlusions, lighting changes, actuator noise, and physical impacts (Danesh et al., 19 Feb 2025).

Learning can also tune analytical controllers rather than replace them. A deep deterministic policy gradient (DDPG) method tunes the gains of a Modified Transpose Jacobian controller,

$$\boldsymbol{M}_\mathcal{P} = \dfrac{2}{n} \begin{bmatrix} \cos\left(0\right) & \cos\left(2\pi\dfrac{1}{n}\right) & \cdots & \cos\left(2\pi\dfrac{n-1}{n}\right)\ \sin\left(0\right) & \sin\left(2\pi\dfrac{1}{n}\right) & \cdots & \sin\left(2\pi\dfrac{n-1}{n}\right) \end{bmatrix},$$3

for a six-tendon task-space TDCR. In simulation, the reported error norm RMSE decreases from $$\boldsymbol{M}_\mathcal{P} = \dfrac{2}{n} \begin{bmatrix} \cos\left(0\right) & \cos\left(2\pi\dfrac{1}{n}\right) & \cdots & \cos\left(2\pi\dfrac{n-1}{n}\right)\ \sin\left(0\right) & \sin\left(2\pi\dfrac{1}{n}\right) & \cdots & \sin\left(2\pi\dfrac{n-1}{n}\right) \end{bmatrix},$$4 with fuzzy-MTJ to $$\boldsymbol{M}_\mathcal{P} = \dfrac{2}{n} \begin{bmatrix} \cos\left(0\right) & \cos\left(2\pi\dfrac{1}{n}\right) & \cdots & \cos\left(2\pi\dfrac{n-1}{n}\right)\ \sin\left(0\right) & \sin\left(2\pi\dfrac{1}{n}\right) & \cdots & \sin\left(2\pi\dfrac{n-1}{n}\right) \end{bmatrix},$$5 with DRL-MTJ, and in real experiments from $$\boldsymbol{M}_\mathcal{P} = \dfrac{2}{n} \begin{bmatrix} \cos\left(0\right) & \cos\left(2\pi\dfrac{1}{n}\right) & \cdots & \cos\left(2\pi\dfrac{n-1}{n}\right)\ \sin\left(0\right) & \sin\left(2\pi\dfrac{1}{n}\right) & \cdots & \sin\left(2\pi\dfrac{n-1}{n}\right) \end{bmatrix},$$6 m to $$\boldsymbol{M}_\mathcal{P} = \dfrac{2}{n} \begin{bmatrix} \cos\left(0\right) & \cos\left(2\pi\dfrac{1}{n}\right) & \cdots & \cos\left(2\pi\dfrac{n-1}{n}\right)\ \sin\left(0\right) & \sin\left(2\pi\dfrac{1}{n}\right) & \cdots & \sin\left(2\pi\dfrac{n-1}{n}\right) \end{bmatrix},$$7 m (Maghooli et al., 2024). The method remains structurally model-based, since it still depends on forward kinematics, Jacobian evaluation, null-space projection, and an inner tension-control loop.

Shape estimation under hysteresis has produced a distinct learning line. A hysteresis-aware deep decoder conditions on

$$\boldsymbol{M}_\mathcal{P} = \dfrac{2}{n} \begin{bmatrix} \cos\left(0\right) & \cos\left(2\pi\dfrac{1}{n}\right) & \cdots & \cos\left(2\pi\dfrac{n-1}{n}\right)\ \sin\left(0\right) & \sin\left(2\pi\dfrac{1}{n}\right) & \cdots & \sin\left(2\pi\dfrac{n-1}{n}\right) \end{bmatrix},$$8

and outputs a point cloud

$$\boldsymbol{M}_\mathcal{P} = \dfrac{2}{n} \begin{bmatrix} \cos\left(0\right) & \cos\left(2\pi\dfrac{1}{n}\right) & \cdots & \cos\left(2\pi\dfrac{n-1}{n}\right)\ \sin\left(0\right) & \sin\left(2\pi\dfrac{1}{n}\right) & \cdots & \sin\left(2\pi\dfrac{n-1}{n}\right) \end{bmatrix},$$9

for full-shape prediction. Using a combined Chamfer-plus-EMD loss, the reported test Chamfer distance is $$\boldsymbol{M}_\mathcal{P}\boldsymbol{\rho} = \begin{bmatrix} \rho_\mathrm{Re}& \rho_\mathrm{Im} \end{bmatrix}^T,$$0 m, compared with $$\boldsymbol{M}_\mathcal{P}\boldsymbol{\rho} = \begin{bmatrix} \rho_\mathrm{Re}& \rho_\mathrm{Im} \end{bmatrix}^T,$$1 m for a non-hysteresis learned model and $$\boldsymbol{M}_\mathcal{P}\boldsymbol{\rho} = \begin{bmatrix} \rho_\mathrm{Re}& \rho_\mathrm{Im} \end{bmatrix}^T,$$2 m for the physics-based baseline, while runtime is $$\boldsymbol{M}_\mathcal{P}\boldsymbol{\rho} = \begin{bmatrix} \rho_\mathrm{Re}& \rho_\mathrm{Im} \end{bmatrix}^T,$$3 ms versus $$\boldsymbol{M}_\mathcal{P}\boldsymbol{\rho} = \begin{bmatrix} \rho_\mathrm{Re}& \rho_\mathrm{Im} \end{bmatrix}^T,$$4 ms for the physics-based pipeline (Cho et al., 2024). This establishes hysteresis as a first-class forward-kinematics variable rather than a residual nuisance.

Intrinsic motor signals have also been recast as sensing channels. A unified multi-dynamics framework links motor electrical dynamics, motor-winch transmission, tendon force, continuum deformation, and reflected current. Under a constant tendon force of $$\boldsymbol{M}_\mathcal{P}\boldsymbol{\rho} = \begin{bmatrix} \rho_\mathrm{Re}& \rho_\mathrm{Im} \end{bmatrix}^T,$$5, the real robot exhibits about 52.4 ms delay and the full electromechanical model reproduces that lag; near self-contact at motion limits, current rises toward saturation near 5 A while tendon velocity drops. The same framework supports passive contact detection, active contact sensing with current-rise thresholds, and object size estimation, with reported hardware MAE $$\boldsymbol{M}_\mathcal{P}\boldsymbol{\rho} = \begin{bmatrix} \rho_\mathrm{Re}& \rho_\mathrm{Im} \end{bmatrix}^T,$$6 mm and $$\boldsymbol{M}_\mathcal{P}\boldsymbol{\rho} = \begin{bmatrix} \rho_\mathrm{Re}& \rho_\mathrm{Im} \end{bmatrix}^T,$$7 after direct transfer from simulation (Alsarraj et al., 22 Nov 2025). A plausible implication is that TDCR perception need not be external-sensor-centric if the actuation chain itself is modeled as part of the robot.

6. Embodied variants, simulation infrastructures, and application domains

TDCR embodiments span macro-scale, meso-scale, and millimeter-scale systems. At the small-scale end, a two-photon-polymerized microsurgical wrist has square cross section $$\boldsymbol{M}_\mathcal{P}\boldsymbol{\rho} = \begin{bmatrix} \rho_\mathrm{Re}& \rho_\mathrm{Im} \end{bmatrix}^T,$$8, wrist length 3.75 mm, three peripheral tendons for bending in arbitrary planes, and a fourth central tendon for an integrated gripper; the reported wrist achieves greater than $$\boldsymbol{M}_\mathcal{P}\boldsymbol{\rho} = \begin{bmatrix} \rho_\mathrm{Re}& \rho_\mathrm{Im} \end{bmatrix}^T,$$9 bending in all four directions and can grasp while bent (Leavitt et al., 2023). At larger scale, a modular layer-jamming arm combines four tendons per segment with vacuum-actuated stiffness modulation, and the reported two-segment prototype reaches a stiffness ratio of 17.5 at 12.5 psi with load capacity rising from 0.2 N to 2.7 N (Ouyang, 2019).

Medical and endoluminal navigation provide a major application cluster. ExoNav is a tendon-driven helically notched tube for spinal cord stimulation lead placement; across four prototypes, reported tip-position RMSEs are 1.76 mm, 2.33 mm, 2.18 mm, and 1.33 mm, and in follow-the-leader experiments the maximum RMSE is 3.75 mm (Moradkhani et al., 19 Jan 2026). A separate shape-aware whole-body control framework for a three-segment cable-driven continuum robot combines a physics-informed backbone model, residual learning via an Augmented Neural ODE, and MPPI control; in a bronchoscopy phantom, the reported framework reduces wall contacts from $$\rho_i = \rho_\mathrm{Re}\cos{\psi_i} + \rho_\mathrm{Im}\sin{\psi_i}.$$0 to $$\rho_i = \rho_\mathrm{Re}\cos{\psi_i} + \rho_\mathrm{Im}\sin{\psi_i}.$$1 per trial, raises target reach from 12/18 to 18/18, and reduces completion time from $$\rho_i = \rho_\mathrm{Re}\cos{\psi_i} + \rho_\mathrm{Im}\sin{\psi_i}.$$2 s to $$\rho_i = \rho_\mathrm{Re}\cos{\psi_i} + \rho_\mathrm{Im}\sin{\psi_i}.$$3 s (Kasaei et al., 14 Oct 2025).

Confined-space locomotion and exploration motivate different embodiments. The universal-jointed TDCR has been configured for helical rolling in a narrow tube; the reported locomotion demo uses a 15 mm diameter robot in a 21 mm inner-diameter tube at 0.33 Hz (Shentu et al., 2024). A caterpillar-inspired spring-based compressive TDCR mounted on a commercial arm integrates a bristle contact sensor for surface perception and obstacle detection; across 145 targets the reported mean position error is 4.32 mm with standard deviation 2.73 mm, and the system reconstructs rough surface point clouds of scanned objects using contact-triggered probing (Hu et al., 10 Mar 2026).

Contact-rich manipulation has recently required new simulation infrastructure. A continuum-mechanics-informed discretization places a TDCR natively inside MuJoCo by representing the backbone as rigid links joined by elastic 3-DoF spherical joints with analytically derived stiffnesses

$$\rho_i = \rho_\mathrm{Re}\cos{\psi_i} + \rho_\mathrm{Im}\sin{\psi_i}.$$4

The static approximation error is shown to scale as $$\rho_i = \rho_\mathrm{Re}\cos{\psi_i} + \rho_\mathrm{Im}\sin{\psi_i}.$$5. On a 3-segment TDCR mounted on a 7-DoF Franka arm, system identification yields mean tip tracking error about 7.7 mm, or $$\rho_i = \rho_\mathrm{Re}\cos{\psi_i} + \rho_\mathrm{Im}\sin{\psi_i}.$$6 of robot length, and zero-shot state-based imitation policies transferred from simulation achieve $$\rho_i = \rho_\mathrm{Re}\cos{\psi_i} + \rho_\mathrm{Im}\sin{\psi_i}.$$7 real-world success for whole-body grasping and $$\rho_i = \rho_\mathrm{Re}\cos{\psi_i} + \rho_\mathrm{Im}\sin{\psi_i}.$$8 for switch flipping (Shentu et al., 21 Jun 2026). This suggests that physically grounded rigid-body approximations can be sufficient for learning contact-rich whole-body behaviors when tendon routing and contact are represented in the same physics engine.

7. Methodological tensions and open problems

A persistent methodological divide runs between analytically convenient constant-curvature models and more general routing-aware or rod-theoretic models. PCC enables real-time control, closed-form mappings, and low-dimensional optimization, but general tendon routing, frictional force propagation, helical backbones, and external loading often require optimization-based or Cosserat-style formulations (Hachen et al., 2024, Shentu et al., 2024, Moradkhani et al., 19 Jan 2026). This suggests that the central modeling question is not whether PCC is “correct” or “incorrect,” but which physical effects must remain explicit for a given task.

A second tension concerns state choice. Case-specific tendon coordinates are sufficient for three- and four-tendon segments, but Clarke-manifold formulations argue that arbitrary $$\rho_i = \rho_\mathrm{Re}\cos{\psi_i} + \rho_\mathrm{Im}\sin{\psi_i}.$$9-tendon segments should be treated as systems evolving on a two-dimensional manifold satisfying the balance condition $90^\circ$00 (Grassmann et al., 2024, Muhmann et al., 26 Mar 2025). The practical consequence is substantial: control, inverse mapping, and improved state representations can then be expressed in a robot-count-agnostic form rather than as special-purpose decompositions.

A third issue is the location of sensing and intelligence. Visual servoing studies show that feature-based IBVS is precise near the goal but fragile under occlusion and lighting changes, whereas DLBVS and hybrid switching enlarge the operating region and retain robustness (Danesh et al., 19 Feb 2025, Abdulhafiz et al., 2021). Intrinsic-sensing work argues, by contrast, that motor current and winch angle already encode interaction signatures if the electromechanical chain is modeled coherently (Alsarraj et al., 22 Nov 2025). Neither view eliminates the other; rather, the data indicate that TDCR perception is becoming multimodal and model-distributed.

Finally, hysteresis, friction, tendon slack, backlash, and self-contact remain unresolved across architectures. Hysteresis can make identical current tendon configurations produce different shapes depending on prior configuration, learned whole-body models still depend on calibration and distributional coverage, and hardware MPC studies explicitly report loss of controllability when tendons slacken at high curvature (Cho et al., 2024, Hachen et al., 2024). Rigid-body simulation approximations for soft robots also remain conditional on slenderness and on the validity of Kirchhoff assumptions (Shentu et al., 21 Jun 2026). A plausible implication is that future TDCR systems will continue to combine physics-based structure, learned residuals, and task-specific sensing rather than converging to a purely analytical or purely data-driven paradigm.