Under-Actuated Tendon-Driven Robotic Finger

• Under-actuated tendon-driven robotic fingers are mechanisms that achieve multi-degree motion using fewer actuators via mechanical coupling and compliant elements.
• They employ diverse tendon routing schemes, kinematic configurations, and integrated sensing strategies to enable adaptive grasping and reliable force distribution.
• Experimental validations report >90% grasp success and fingertip stiffness up to 1.2×10³ N/m, illustrating their robust performance in anthropomorphic robotic applications.

An under-actuated tendon-driven robotic finger (UTRF) is a robotic finger mechanism in which the number of actively controlled inputs (actuators or driven tendons) is strictly less than the number of kinematic degrees of freedom (DOF). This architecture leverages mechanical coupling, compliant elements, and adaptive synergy to realize finger flexion, extension, and adaptive grasping with minimal actuation, resulting in lighter, simpler, and often more robust robotic hands. UTRFs are foundational in modern anthropomorphic robotic hands, tactile-adaptive grippers, and high-strength robot end-effectors, and are characterized by diverse tendon routing schemes, kinematic configurations, and integrated sensing strategies.

1. Kinematic Architectures and Tendon Routing

A canonical UTRF implements a serial chain of phalanges, each connected by rotary (revolute) joints mimicking anatomical MCP, PIP, and DIP joints. Actuation is achieved by routing one or more tendons through U-groove pulleys or material-defined paths running over the joint axes, terminating at various locations along the distal phalanx or fingertip. Variations documented include:

• Single-tendon mechanisms: A single tendon wraps in sequence over each joint, with local moment arms $r_i$ determining joint torque allocation; flexion movement propagates until external resistance is encountered, at which point remaining tendon displacement drives subsequent joints (Makino et al., 26 Mar 2024, Hundhausen et al., 2020, Li et al., 2022).
• Antagonistic dual-tendon schemes: A pair of tendons (flexor and extensor/antagonist), each routed in parallel, provide both active flexion and controlled extension. In Tactile SoftHand-A, tendon endpoints and routing encode different joint isolation behaviors (distal or proximal joint locking) via differential torque distribution (Li et al., 18 Jun 2024).
• Synchronous/coupling tendon systems: Fixed angular velocity ratios between successive joints are achieved through wrapping tendons over cylinders of specified radii, enforcing proportional movement across joints as the actuating tendon is displaced (Yuan et al., 11 Dec 2025).
• Spatial (parallel and serial hybrid) architectures: Some UTRFs expand to 4-DOF per finger and reconfigurable spatial kinematics (e.g., Hamon et al.), integrating parallel-spherical bases to enable transitions between cylindrical and spherical grasps, with tendon transmission matrices mapping input tension to distributed joint torques (Hamon et al., 2021).
• Soft, compliant backbones: Flexure-based joints embedded in elastomeric material combine compliance and geometric constraint, with tendon tension directly setting local flexion angles modulated by the flexure’s stiffness (Hundhausen et al., 2020).

2. Actuation: Underactuation Ratios and Mechanical Synergy

Characteristic of UTRFs is a high underactuation ratio, typically ranging from 2 actuators for 15 DOF (as in Tactile SoftHand-A (Li et al., 18 Jun 2024)) down to a single actuator for up to 3 or 4 joints per finger (Yuan et al., 11 Dec 2025, Makino et al., 26 Mar 2024). Mechanical synergies are introduced by configuring:

• Moment arm allocation (pulley radii): Selecting $r_i$ for each joint sets the torque distribution, often optimized to fit desired posture or effort manifolds (Chen et al., 2019).
• Compliance and joint springs: Passive elastic elements at joints (torsion springs, elastic bands, flexure elements) provide restorative torques, enable bidirectional actuation via antagonistic routing, and dictate the sequence of joint engagement during flexion (Makino et al., 26 Mar 2024, Hundhausen et al., 2020).
• Differential and split-pulley mechanisms: Allow multiple fingers to share a common actuator while accommodating asynchronous object contact and individual finger flexion (Makino et al., 26 Mar 2024).
• Adaptive synergy matrices: Integrated across multi-fingered robotic hands, these define the mapping from a small number of motor commands or tendon excursions to the kinematic subspace of attainable postures, empirically characterized by ratios such as θ₂ = k₁₂ θ₁, θ₃ = k₂₃ θ₂ within a finger (Li et al., 2022).

3. Analytical Modeling and Optimization

A unified analytical framework for UTRF mechanics comprises:

• Kinematic mapping: Joint angles θ relate to tendon excursions $\Delta s = R^T \Delta q$, with R the moment-arm matrix. For synchronous routing, $\theta_i = q / R_i$ (Yuan et al., 11 Dec 2025). Forward kinematics yields fingertip position as a function of actuator displacement (Li et al., 2022, Yuan et al., 11 Dec 2025).
• Static equilibrium: Tendon tension T generates torques τi = r_i T at joint i; joint equilibrium (including external load f{ext}) enforces τ₁ + τ₂ + … = f_{ext}·L, neglecting inertial effects in quasi-static grasps (Makino et al., 26 Mar 2024, Li et al., 2022).
• Compliance and stiffness: Lumped compliance from tendon elasticity, joint springs, and flexures is modeled; e.g., Hookean relation for tendon stretch, τ{dip} = k{dip} θ_{dip} (Yuan et al., 11 Dec 2025, Li et al., 18 Jun 2024).
• Control-oriented constraints: Geometric constraints, such as θ₁ - (r₂/r₁) θ₂ = 0, can impose holonomic constraints, reducing the system to a controllable single-DOF subspace amenable to nonlinear control (e.g., backstepping) (Narkhede et al., 2018).
• Optimization: Design of pulleys, spring constants, and linkage geometry is formalized as a constrained optimization problem targeting specific posture/torque manifolds, exploiting tools from convex programming and black-box optimization (Chen et al., 2019, Ko, 2020).

4. Control, Sensing, and Feedback Mechanisms

Modern UTRFs integrate diverse control strategies and sensing modalities:

• Open-loop synergy-based control: Simple direct mapping of actuator commands to synergy-coordinated joint movements, effective for reaching typical grasp postures (Li et al., 18 Jun 2024, Li et al., 2022).
• Selective joint targeting: Via antagonistic dual-tendon actuation and locking, schemes such as in Tactile SoftHand-A realize independent control of the DIP by differentially programming flexor and extensor excursions (Li et al., 18 Jun 2024).
• Tactile integration and feedback: In-hand, 3D-printed optical tactile sensors (e.g., TacTip) provide contact region and slippage detection through marker tracking and blob-density maps, feeding closed-loop slippage correction to actuators (Li et al., 18 Jun 2024).
• Proprioceptive feedback: Series elastic actuators (SEA) with embedded tension and displacement sensing (via film potentiometers and integrated springs) supply direct tendon force and configuration feedback, enabling detection of contact phases, estimation of joint angles, and object compliance classification without external sensors (Lee et al., 16 Sep 2025).
• Physics-based simulation-driven optimization: End-to-end grasping trials within high-fidelity simulators inform black-box design optimization, explicitly accounting for dynamics and unpredictable grasp failure modes (Ko, 2020).

5. Experimental Validation and Performance Metrics

Benchmarking of UTRFs spans mechanical, control, and manipulation criteria:

• Grasp success and adaptability: High shape-conformance and >90% success rates in multi-object grasping (Tactile SoftHand-A) (Li et al., 18 Jun 2024); adaptive synergy hands robustly envelop diverse objects (Li et al., 2022); effective thin-sheet, book, and object pickup with passively switchable surfaces (Ko, 2020).
• Force output: Static fingertip stiffness up to 1.2 × 10³ N/m under 3 kg tip load (Yuan et al., 11 Dec 2025); individual finger closure force up to ~11.6 N per finger (KIT SoftHand) (Hundhausen et al., 2020); hand-level grasp force supporting >37 kg loads (machined-spring UTRF) (Makino et al., 26 Mar 2024).
• Response and adaptation speed: Slip detection and corrective actuation within 0.5 s (Tactile SoftHand-A) (Li et al., 18 Jun 2024); contact classifying latency ~30–40 ms via SEAs (Lee et al., 16 Sep 2025).
• Compliance and durability: Mechanical closure exceeding 15,000 cycles (KIT SoftHand) (Hundhausen et al., 2020); structural resilience to hammer blows due to elastic machined joints (Makino et al., 26 Mar 2024).
• Model validation: Kinematic and static elastic models predict deflection within 1 mm or 0.3% error of finger length (Yuan et al., 11 Dec 2025); synergy manifold optimization errors in joint torque below 0.13 (normalized units) (Chen et al., 2019).

6. Limitations, Tradeoffs, and Future Directions

While UTRFs achieve substantial reductions in actuator count, wiring, and mass, inherent limitations include:

• Lack of independent joint control: Synchronous/coupled motion prohibits full in-hand manipulation and precise posture control for each joint (Yuan et al., 11 Dec 2025, Li et al., 2022).
• Force/torque estimation: In many designs, grasp force is only indirectly inferred from tendon tension; direct fingertip force measurement remains rare, though tactile/force sensors are being integrated in more recent work (Li et al., 18 Jun 2024, Makino et al., 26 Mar 2024).
• Compliance-stiffness tradeoff: Increasing compliance (for safety or adaptability) reduces maximal fingertip force; optimal tuning of tendon routing, spring constants, or wrap radii is non-trivial and task-dependent (Yuan et al., 11 Dec 2025, Li et al., 2022).
• Extension to spatial manipulation: Extending underactuated design to non-planar, spatial linkages increases kinematic and control complexity and requires additional actuators or sensors for reliable posture classification (Hamon et al., 2021).
• Active sensing for feedback: Incorporation of variable stiffness elements, in-hand tactile/force sensors, and advanced proprioceptive modules (e.g., SEAs), as well as further integration with visual or haptic feedback, is a key future direction (Lee et al., 16 Sep 2025, Li et al., 18 Jun 2024).

7. Application Domains and Open-Source Ecosystems

UTRFs are deployed across:

• Low-cost, open-source anthropomorphic hands: 3D-printed architectures, such as Tactile SoftHand-A and BRL/Pisa/IIT SoftHand, have disseminated through open repositories, enabling rapid prototyping and experimental validation at scale (Li et al., 18 Jun 2024, Li et al., 2022).
• Service and humanoid robotics: Mechanisms supporting full-body self-weight, high-force grasping, and adaptive manipulation in humanoid platforms (e.g., Kengoro hand) (Makino et al., 26 Mar 2024).
• Research in synergetic and compliant actuation: UTRFs serve as testbeds for investigating the fundamental role of mechanical and postural synergies, contact-driven kinematic reconfiguration, and closed-loop adaptation in dexterous manipulation (Chen et al., 2019, Lee et al., 16 Sep 2025, Li et al., 18 Jun 2024).

A plausible implication is that as UTRF designs become more integrated with multimodal sensing and actively controlled compliance, anthropomorphic hands will achieve dexterity and robustness approaching biological counterparts with substantially reduced actuation and control complexity. Open-source designs, multi-material 3D printing, and modular proprioceptive/tactile units are instrumental in accelerating progress in this domain (Li et al., 18 Jun 2024, Li et al., 2022, Lee et al., 16 Sep 2025).