Synchronous Tendon Routing in Robotics

• Synchronous tendon routing is a strategy that mechanically couples joint motions via fixed geometric constraints, ensuring predictable actuation ratios.
• It employs designated tendon paths, pulley radii, and linkage ratios to convert a single actuator displacement into precise joint rotations.
• This method underpins advanced underactuated designs in robotic hands and continuum robots, achieving dexterous manipulation with reduced complexity.

Synchronous tendon routing is a mechanical and control strategy for tendon-driven robotic systems in which all joints within a serial chain or coordinated multi-chain structure are mechanically coupled so that joint motions occur in a fixed ratio, typically with a minimal number of actuators. By leveraging designated routing paths and geometric constraints—often implemented via pulley radii, linkage ratios, or spur gear trains—synchronous tendon routing guarantees that a single actuator displacement produces instantaneous joint rotations in a precise and predictable pattern. This methodology is foundational in modern underactuated robotic hands and continuum robots, enabling compact, lightweight, and robust designs suitable for dexterous manipulation while maintaining high stiffness and compliance.

1. Foundational Mechanical Principles

Synchronous tendon routing employs a configuration in which each actuating tendon travels through a sequence of guiding elements (cylinders or pulleys) whose radii enforce predetermined angular velocity ratios between the joints. For an $n$-joint serial manipulator, let $R_i$ denote the radius of the tendon wrapping path at joint $i$. The coupling is expressed by constant-length constraints on intermediate tendon segments. For a 3-joint finger:

• The routing enforces

$$R_1\,\Delta\theta_1 + R_2\,\Delta\theta_2 = 0,\quad R_2\,\Delta\theta_2 + R_3\,\Delta\theta_3 = 0$$

which yields the constant angular ratios

$$\frac{\Delta\theta_2}{\Delta\theta_1} = -\frac{R_1}{R_2},\qquad \frac{\Delta\theta_3}{\Delta\theta_2} = -\frac{R_2}{R_3}$$

and, for a net tendon displacement $q$,

$\theta_i = \frac{q}{R_i}$

• With all pulleys locked via inelastic (non-elastic) tendons, joint angles remain strictly proportional throughout the actuation range (Yuan et al., 11 Dec 2025).

Alternative implementations (e.g., the BRL/Pisa/IIT SoftHand) achieve near-synchronous or synergy-based coupling across multiple fingers or joints using nonuniform gear trains, linkage bar constraints, or empirical angular couplings (Li et al., 2022).

2. Kinematic and Static Modeling

The kinematics of a synchronous tendon-routed finger are defined by the joint-angle-to-tendon-displacement mapping: $\theta_i(q) = \frac{q}{R_i}$ for $i=1,2,3$. Fingertip position in a fixed base frame is given by: $$x(q) = L_1\cos\theta_1 + L_2\cos(\theta_1+\theta_2) + L_3\cos(\theta_1+\theta_2+\theta_3)$$

$$y(q) = L_1\sin\theta_1 + L_2\sin(\theta_1+\theta_2) + L_3\sin(\theta_1+\theta_2+\theta_3)$$

The Jacobian relating tendon velocity to fingertip velocity is: $$J(q) = \begin{bmatrix} -\sum_{i=1}^3 \frac{L_i}{R_i} \sin\left(\sum_{j=1}^i \frac{q}{R_j}\right) \ \sum_{i=1}^3 \frac{L_i}{R_i} \cos\left(\sum_{j=1}^i \frac{q}{R_j}\right) \end{bmatrix}$$ (Yuan et al., 11 Dec 2025)

Tendon elasticity is modeled via a linear spring law: $$k_{t,i} = \frac{E_i A_i}{L_{t,i}},\qquad T_i = k_{t,i}\Delta\ell_i$$ where $E_i$ is Young’s modulus, $A_i$ is cross-sectional area, $L_{t,i}$ is the tendon’s rest length, and $\Delta\ell_i$ is its elongation. Elongation perturbs the mapping: $\theta_i' = \theta_i - \frac{\Delta\ell_i}{R_i}$ Iterative schemes guarantee force-elongation equilibrium under load.

Structural stiffness in the tendon direction is: $$K_q = \sum_{i=1}^3 k_{t,i} \left(\frac{\partial \ell_i}{\partial q}\right)^2$$ and the fingertip Cartesian stiffness matrix is: $K_x = J(q) K_q J(q)^\top$ (Yuan et al., 11 Dec 2025)

3. Synchronous Routing in Multi-Finger and Adaptive Hands

Synchronous tendon routing extends naturally to multi-finger hands through coupling tendons or functionally partitioned routing. In the UTRF-RoboHand, five identical fingers with three joints each are controlled by five actuators—one per finger—rather than fifteen, yielding substantial reductions in total actuator count and mechanical complexity. Each finger contains only a routed pair of tendons, further simplifying wiring and improving reliability (Yuan et al., 11 Dec 2025).

In the BRL/Pisa/IIT SoftHand, a single actuator drives three tendons in the palm drum, each serving a set of digits (thumb/index, middle/ring, little). Within each finger, linkages, gear pairs, and elastic bands integrate both "adaptive synergy" (across digits) and "soft synergy" (compliance at joints). Empirical relations (e.g., $\theta_2 = \alpha \theta_1$, $\theta_3 = \beta \theta_2$ with $\alpha\approx 1.2$–$1.3$, $\beta\approx 0.7$–$0.8$) enforce near-synchronous angular coupling, as validated by both simulation and experiment (Li et al., 2022).

4. General Modeling and Design for Underactuated Chains

Formally, a synchronous tendon network for an $N$-link chain actuated by $M<N$ winches can be represented as: $R^F\,q = R^A\,u$ where $R^F$ is the $M \times N$ flexion–radius matrix, $R^A$ is the winch–radius matrix, $q \in \mathbb{R}^N$ are joint angles, and $u \in \mathbb{R}^M$ are actuator rotations. Synchronous routing corresponds to rows of $R^F$ that are nearly uniform—i.e., each actuator excites an equal linear combination of joint motions.

Torque mapping follows as: $\tau_q = (R^F)^T(R^A)^{-1}\tau_u$ which sets joint torques given actuator torques. The structure of $R^F$ strongly influences the accessible submanifold of joint space, the achievable dexterity, and the transmission stiffness (Islam et al., 23 May 2024).

Co-optimization methods such as policy-gradient reinforcement learning or evolutionary search are employed to tune the set of routing parameters ($R^F$, $R^A$, preloads) for task-specific end-effector reachability and force capabilities (Islam et al., 23 May 2024). Nearly synchronous routing (identical rows in $R^F$) enables all joints to flex (or extend) nearly in synchrony with one actuator input, at the cost of reducing independent configuration space.

5. Synchronous Routing in Continuum and Soft Manipulators

In soft continuum robots—such as inflated-beam “vine” or tentacle manipulators—synchronous routing involves wrapping one or more tendons helically or longitudinally along the backbone, each imposing a geometric constraint matching actuator pull to local backbone curvature and torsion. For a backbone segment parameterized by draw angle $\theta$ and contraction ratio $\lambda$: $$\cos(2\theta) = \frac{R_i R_o + b^2}{\sqrt{(b^2+R_i^2)(b^2+R_o^2)}}$$ where $R_i,R_o$ are the inner and outer helical path radii, and $b$ is the normalized pitch (Blumenschein et al., 2020).

With $M$ tendons placed symmetrically (same $\theta$, equally spaced in $\phi$), uniform synchronous pull generates pure bending or torsion as desired. Forward shape prediction involves integrating local curvature and torsion maps segment by segment, using homogeneous transforms or Cosserat-rod formulations.

For inverse design, desired centerline shapes can be fit by optimizing $\theta_j, \lambda_j$ profiles over discrete backbone segments, enforcing geometric feasibility and collision avoidance. Synchronous multi-tendon activation is essential for producing predictable planar or spatial bends and twists, with task-space reach determined by tendon-routed geometry (Blumenschein et al., 2020).

6. Performance Metrics and Validation

Prototypical implementations demonstrate the practical benefits and precision of synchronous tendon routing. In the UTRF robotic finger:

• Static loading tests yielded a mean deflection prediction error of $0.545\,\mathrm{mm}$ ($0.322\%$ of finger length) and a measured stiffness of $1.2\times 10^3\,\mathrm{N/m}$ under a $3\,\mathrm{kg}$ tip load.
• Simulation-experiment error bands were within $\pm 0.9\,\mathrm{mm}$ across all weights tested.
• In the five-finger UTRF-RoboHand, reliable grasping—including for fragile objects—was achieved using only five actuators, with system mass ca. $2\,\mathrm{kg}$ (Yuan et al., 11 Dec 2025).

The BPI SoftHand demonstrated $>80\%$ grasp success on a diverse object set, with steady holding forces exceeding $19\,\mathrm{N}$ for a single actuator, and experimentally validated fingertip kinematics within $5^\circ$ of simulation predictions (Li et al., 2022).

7. Practical Guidelines and Implementation Considerations

Designing an effective synchronous tendon routing involves:

• Selection and geometric placement of guiding pulleys or helix parameters to ensure precise fixed-ratio coupling.
• Modeling tendon elasticity, frictional losses, and joint compliance to ensure structural stiffness and grasp reliability.
• Determining tradeoffs between pure synchronous motion (high stiffness, low dexterity) and partial synchronization (increased adaptability).
• Employing empirical or co-optimization approaches to tune routing patterns for specific tasks or object classes.
• For continuum robots, simultaneously optimizing tendon draw angles and contraction profiles to enable target 3D centerline shapes under practical geometric constraints.

The key advantage of synchronous routing is that it enables minimal-actuator, robust, and reproducible motion of complex robotic structures, with closed-form analytical models supporting design and analysis across both rigid and soft morphologies (Yuan et al., 11 Dec 2025, Li et al., 2022, Islam et al., 23 May 2024, Blumenschein et al., 2020).