Design and Validation of an Under-actuated Robotic Finger with Synchronous Tendon Routing

Tendon-driven under-actuated robotic fingers provide advantages for dexterous manipulation through reduced actuator requirements and simplified mechanical design. However, achieving both high load capacity and adaptive compliance in a compact form remains challenging. This paper presents an under-actuated tendon-driven robotic finger (UTRF) featuring a synchronous tendon routing that mechanically couples all joints with fixed angular velocity ratios, enabling the entire finger to be actuated by a single actuator. This approach significantly reduces the number of actuators required in multi-finger hands, resulting in a lighter and more compact structure without sacrificing stiffness or compliance. The kinematic and static models of the finger are derived, incorporating tendon elasticity to predict structural stiffness. A single-finger prototype was fabricated and tested under static loading, showing an average deflection prediction error of 1.0 mm (0.322% of total finger length) and a measured stiffness of 1.2x10^3 N/m under a 3 kg tip load. Integration into a five-finger robotic hand (UTRF-RoboHand) demonstrates effective object manipulation across diverse scenarios, confirming that the proposed routing achieves predictable stiffness and reliable grasping performance with a minimal actuator count.

💡 Research Summary

The paper introduces a novel under‑actuated robotic finger (UTRF) that achieves simultaneous actuation of three joints with a single motor by employing a synchronous tendon routing scheme. Traditional under‑actuated fingers often require multiple tendons with complex routing, leading to increased actuator count, weight, and assembly difficulty. In contrast, the proposed design uses six tendons: two actuation tendons (one for flexion, one for extension) and four coupling tendons that wrap around cylindrical guiding surfaces on each link. By carefully selecting the radii (R₁, R₂, R₃) and wrap angles, the coupling tendons enforce fixed angular velocity ratios between adjacent joints (R₁/R₂ and R₂/R₃), causing distal joints to move proportionally when the proximal joint rotates. This deterministic coupling enables the entire finger to be driven by a single actuator in either direction, dramatically reducing the number of motors required for a multi‑finger hand.

A complete kinematic model is derived. Joint angles θᵢ are expressed as θᵢ = q / Rᵢ, where q is the linear displacement of the actuation tendon. The fingertip position (x, y) is obtained by summing the contributions of each link, and a Jacobian matrix is formulated to relate fingertip velocity to tendon speed, facilitating real‑time control. The workspace is computed by iterating over feasible link lengths and joint angles, revealing the reachable area for each link and the overall finger.

The static behavior under external loads is analyzed through a three‑step equilibrium approach. First, the tension in the distal coupling tendon (T₃) is solved from the moment balance on joint 3, accounting for fingertip force, gravity on link 3, and the tendon’s moment arm. Next, the tension in the middle coupling tendon (T₂) is obtained by considering the combined assembly of links 2 and 3, including the moment generated by T₃ transmitted through the wrap angle α₂. Finally, the proximal tendon tension (T₁) is solved by analyzing the full three‑link assembly. Tendon elasticity is incorporated using Hooke’s law (L_Ti′ = L_Ti·(1 + T_i·E_i·A_i)), where E_i and A_i are Young’s modulus and cross‑sectional area. An iterative algorithm updates tendon lengths, joint angles, and tensions until fingertip vertical displacement converges below a predefined threshold, ensuring that elastic elongation is fully accounted for in the static solution.

A physical prototype was fabricated using 3‑D printing for the links and stainless‑steel 1 mm‑diameter tendons (E = 200 GPa). In static loading tests, tip masses ranging from 0.5 kg to 3 kg were applied, and fingertip deflection was measured with an electromagnetic sensor. The measured deflections matched the model predictions closely, with a maximum error of 0.890 mm (0.520 % of total finger length) and an average error of 0.545 mm (0.322 %). Under a 3 kg load, the finger exhibited a deflection of 24.386 mm, corresponding to a stiffness of 1.2 × 10³ N/m, demonstrating that the design provides substantial load‑bearing capability despite its under‑actuated nature.

To validate system‑level performance, five UTRF fingers were integrated into a robotic hand (UTRF‑RoboHand) using only five actuators (one per finger). The hand successfully performed a variety of grasping tasks involving objects of different sizes, shapes, and weights, confirming that the synchronous tendon routing yields predictable stiffness and sufficient compliance for adaptive manipulation.

In summary, the authors present a compact, lightweight, and mechanically simple under‑actuated finger that achieves high stiffness, adaptive compliance, and deterministic motion with a single actuator per finger. The comprehensive kinematic and static models, validated experimentally, provide a solid foundation for model‑based control and future extensions such as tendon material optimization, multi‑finger coordination, and real‑time feedback‑enhanced grasping.