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A Compact Robotic Finger with 2-DoF MCP Joint Embedding DoF-Selective Passive Continuously Variable Transmission for Wide Force-Speed Operating Range

JaeHyung Jang, Jee-Hwan Ryu

arXiv:2609.00769v1cs.RO

TL;DR

Compact robotic fingers must balance force and speed within constrained multi-DoF mechanisms. This paper introduces STAND-MCP, which embeds a force-responsive passive CVT selectively in flexion while preserving direct abduction–adduction transmission. Experiments report 4.19-fold maximum and 3.63-fold mean output-force amplification across tested flexion angles, while ball rolling preserves coordinated 2-DoF motion.

  • Problem

    Compact multi-DoF robotic fingers face a force–speed tradeoff, while extending variable transmission across DoFs increases mechanical and packaging burdens.

  • Method

    STAND-MCP embeds an output-side passive CVT in the flexion pathway and retains direct transmission for abduction–adduction within the MCP joint.

  • Results

    4.19-fold maximum and 3.63-fold mean output-force amplification were demonstrated across five flexion angles from 15° to 75°, while coordinated 2-DoF MCP motion was preserved during passive reconfiguration.

  • Takeaways & Limitations

    The study demonstrates selective transmission-ratio reconfiguration at the transmission-geometry level as a design direction for compact and scalable robotic hands.

  • Takeaways & Limitations

    Human-level force–speed performance with fully integrated compact actuators, higher-DoF workspace and dynamics, and long-term durability remain unevaluated.

Abstract

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This letter presents a compact two-degree-of-freedom (DoF) robotic finger with a flexion-selective passive continuously variable transmission (CVT) to achieve a wide force-speed operating range. Inspired by the functional differentiation of the human metacarpophalangeal (MCP) joint, the proposed mechanism realizes DoF-specific transmission differentiation by selectively assigning passive CVT to the flexion-extension DoF while preserving direct transmission for abduction-adduction. For a wide force-speed operating range, a force-responsive passive CVT is embedded in the flexion pathway, while direct transmission is preserved for the abduction-adduction pathway. To selectively realize transmission adaptation within a multi-DoF MCP mechanism, an output-side passive CVT employing a moving-pulley-inspired wire-routing structure is introduced. The resultant force generated by the wire tensions acting on the pulley that passively increases the flexion moment arm and transmission ratio according to the applied load without additional actuators, sensors, or control. Experimental results demonstrate a maximum output-force amplification of 4.19-fold and a mean amplification of 3.6-fold across the tested flexion angles ranging from 15 degrees to 75 degrees through moment-arm adaptation, thereby substantially expanding the achievable force-speed operating range. Furthermore, dexterous ball-rolling experiments verify that passive transmission adaptation can be achieved while preserving abduction-adduction functionality. These results demonstrate a scalable transmission design strategy for compact multi-DoF robotic hands.

I. INTRODUCTION

Compact multi-DoF robotic fingers face a force–speed tradeoff and packaging burden when transmission adaptation is added across DoFs. STAND-MCP addresses this by selectively applying passive CVT to flexion while retaining direct abduction–adduction transmission within an MCP-integrated mechanism.

  • Motivation: Compact in-finger architectures struggle to combine high grasping force and rapid joint motion because limited volume constrains actuator output.
  • Prior approaches: Tendon-driven architectures increase available actuator output but can introduce wrist–finger motion coupling and friction-related transmission losses.
  • Prior approaches: Active and passive variable-transmission mechanisms can incur additional actuation, reconfiguration, and installation burdens when adaptation is extended across multiple DoFs.
  • Design rationale: The human MCP joint motivates DoF-specific transmission design because its flexion–extension and abduction–adduction DoFs have different functional requirements.
  • Proposed mechanism: STAND-MCP embeds passive variable transmission in the MCP joint, adapting flexion transmission while retaining direct abduction–adduction transmission.
  • Proposed mechanism: The proposed moving-pulley implementation varies the flexion moment arm and transmission ratio according to wire tension without additional sensing or active mode control.

II. CONCEPT OF THE STAND-MCP

The STAND-MCP embeds a force-responsive passive CVT selectively in flexion–extension while retaining direct abduction–adduction transmission, matching the distinct roles of the MCP joint’s two DoFs. Its moving-pulley wire routing passively adapts the flexion moment arm and transmission ratio as wire tension changes, within an underactuated joint whose behavior is mechanically determined.

  • Functional differentiation: The MCP joint’s FE and AA DoFs have distinct requirements: FE needs a wide force-speed range, whereas AA prioritizes compactness and positional controllability.FE supports rapid motion before contact and strong grasping after contact; AA governs inter-finger spacing and object alignment.
  • Selective transmission: The STAND-MCP applies passive variable transmission only to FE and retains direct transmission for AA within a compact multi-DoF MCP architecture.This avoids adding a variable-transmission element to the AA pathway.
  • Passive CVT principle: The moving-pulley CVT converts wire-tension forces into a push that moves the adaptive anchor away from the joint center, increasing the flexion moment arm under resistance.Direct routing would decrease the flexion moment arm because its radial tension component acts toward the joint center.
  • Underactuation: Three DoFs—parallel FE and AA motions plus the flexion-specific CVT—are controlled by two actuators, making the mechanism mechanically programmed.The decision logic depends on system kinematics, kinetics, an extension spring, and a restoring spring.
  • Wire kinematics: The bilateral wire-routing geometry varies with FE and AA posture, so AA Jacobians can become asymmetric at nonzero AA angles while recovering symmetry at the neutral configuration.The neutral reference configuration is also used to isolate flexion moment-arm adaptation when characterizing the flexion-specific transmission.
  • Passive adaptation: When wire-generated transmission force exceeds restoring and friction forces, the slider moves in the positive R direction, increasing the flexion moment arm and transmission ratio.The resulting equilibrium relation is generally implicit because transmission and friction forces depend on instantaneous routing geometry and wire tension.

B. Kinematics and Kinetics of Passive Extension

The passive extension model derives geometry-dependent restoring torque from the spring’s routed geometry, stiffness, and preload. A rotation matrix defines the spring attachment geometry and resulting spring length and force.

  • The fixed spring anchor and flexion-dependent attachment point define the extension-spring geometry in the global frame.
  • The planar rotation matrix determines the spring direction vector and instantaneous spring length as flexion changes.
  • The natural spring length is defined at the neutral configuration, θ_FE = 0.
  • A linear spring model uses stiffness k_E and preload force F_E,0 to determine spring-force magnitude.
  • The spring force acts along the spring direction, producing a corresponding force vector and scalar extension torque about the FE axis.
  • The passive extension torque is governed jointly by spring-routing geometry, stiffness, and preload.

C. Condition-Dependent Motion Generation

STAND-MCP motion is determined by force balance among bilateral wire inputs, flexion-extension and abduction-adduction outputs, and passive restoring elements. These conditions mechanically select AA motion, direct FE motion, or passive CVT reconfiguration.

  • The underactuated mechanism combines two wire inputs with FE, AA, and passive slider coordinates whose force balance determines motion.
  • The mode-transition conditions are formulated under an idealized quasi-static assumption that neglects slider friction.
  • When FE torque is below extension restoring torque, FE remains constrained while differential bilateral actuation can generate AA motion.
  • When FE torque exceeds extension restoring torque, flexion is actuated while the passive slider remains in its current configuration.
  • When wire-generated transmission force exceeds slider restoring force, the slider reconfigures, increasing the flexion moment arm and passively adapting transmission ratio.

A. Overall Design and Components

The STAND-MCP integrates a force-responsive passive CVT within the MCP mechanism. Its slider, guide, spring, and adaptive wire anchor reconfigure the flexion moment arm according to wire tension while preserving the joint’s two-DoF design.

  • The mechanism comprises a force-responsive transmission slider, linear guide, restoring spring, and adaptive wire anchor.
  • The design analysis examines mean and maximum transmission-ratio ranges as functions of geometric parameters h_q and w_q.
  • The passive CVT reconfigures the moment arm geometrically according to wire tension.
  • Increasing wire tension translates the slider and continuously adjusts the adaptive wire anchor, modulating transmission ratio without actuators, sensors, or active control.
  • The linear guide constrains slider motion to one translational axis, while the restoring spring determines the slider displacement–tension relationship.

B. Geometric Parameter Selection

The geometric study evaluates transmission-ratio modulation across the flexion workspace and selects h_q and w_q by balancing transmission performance against compactness. The parameters affect maximum and mean transmission ranges differently.

  • The transmission-ratio range R_TR quantifies modulation produced by flexion moment arm R at a fixed flexion angle θ_FE.
  • R_TR(θ_FE) is evaluated over R ∈ [5, 25] mm, with R_0 = 5 mm, and its maximum and mean are computed over θ_FE ∈ [5°, 85°].
  • Figure 3 compares flexion kinematics, repeatability, force-responsive moment-arm adaptation, theoretical predictions, and output force across fixed and passive-CVT configurations.
  • The contour maps cover feasible geometric ranges w_q ∈ [12, 18] mm and h_q ∈ [8, 15] mm.
  • Increasing h_q monotonically increases both maximum and mean transmission-ratio ranges but enlarges the mechanism envelope; h_q was selected as 10 mm.
  • Increasing w_q improves mean transmission-ratio range while reducing maximum range, so w_q was selected as 15 mm under spatial constraints.

C. Extension Spring Stiffness Selection

The extension spring was sized from conservative gravitational and inertial torque estimates for stable finger extension. Selected parameters keep extension torque above the required threshold throughout the flexion–extension range.

  • The finger was modeled as a 43.2 g rigid aluminum block with rotational inertia I_f ≈ 2.45 × 10^-5 kg·m^2 about the flexion–extension axis.
  • 8.58 N·mm was estimated as the minimum required extension torque from gravitational and inertial loading.The estimate used a worst-case fully extended horizontal posture and a maximum angular velocity of 200 deg/s.
  • k_E = 0.11 N/mm and F_E,pre = 1 N were selected to maintain extension torque above the requirement across the entire θ_FE range.The geometric parameters were selected as a_E = 10 mm, b_E = 3 mm, w_E = 8 mm, and h_E = 8 mm.

D. Restoring Spring Stiffness Selection

The restoring spring was selected to prevent unintended variation of the flexion moment arm while providing sufficient restoring force over the operating range. Its design also accommodates more than 20 mm of deflection.

  • The restoring spring was designed to ensure stable flexion behavior without unintended variation of the flexion moment arm R.
  • The restoring spring must overcome extension torque so flexion input does not induce undesired displacement of R.The maximum extension torque was approximately 16 N·mm, motivating a sufficiently large initial restoring force.
  • The spring must accommodate more than 20 mm of deflection while remaining functional throughout the operating range.
  • k_res = 0.6 N/mm and F_res,0 = 0.9 N were selected for the restoring spring.The stiffness determines the restoring-force profile over the operating range.

V. EXPERIMENTAL VALIDATION

Experiments verified that changing the flexion moment arm continuously changes transmission speed and force. The passive CVT increased its moment arm with input force, producing force amplification while retaining the tested transmission behavior across flexion angles.

  • Average flexion angular velocity decreased from 21.7 deg/s at R = 5 mm to 6.5 deg/s at R = 25 mm.This approximately 3.3-fold decrease corresponds to a proportional increase in effective transmission ratio under identical actuator input.
  • The passive CVT increased R as input force increased, with force sensitivity becoming greater at larger flexion angles.Theoretical loading and unloading curves generally agreed with measurements except at θ_FE = 15°.
  • 4.19-fold maximum and 3.63-fold mean output-force amplification were measured across flexion angles from 15° to 75°.The comparison used fixed-radius systems with R = 25 mm relative to R = 5 mm.
  • As input force increased, the passive CVT progressively shifted from characteristics like R = 5 mm toward those of R = 25 mm.The expanding moment arm continuously increased transmission ratio and output force.
  • Ball-rolling demonstrations included non-contact abduction/adduction and flexed rolling at 0.52 N and 4.71 N contact forces.

VI. DEMONSTRATION

The STAND-MCP demonstrates selective passive transmission adaptation in a compact two-DoF MCP mechanism. Flexion receives load-responsive CVT behavior while abduction–adduction retains direct transmission, and demonstrations preserved coordinated motion.

  • VI. DEMONSTRATION: The demonstrated rolling conditions included low contact force of approximately 0.52 N and high contact force of approximately 4.71 N.Both rolling conditions used a flexed posture, while non-contact abduction/adduction provided a kinematic baseline.
  • VI. DEMONSTRATION: The STAND-MCP assigns passive CVT adaptation to flexion while preserving direct transmission for abduction–adduction.The transmission components are integrated within the MCP joint without additional actuators, sensors, or control.
  • VI. DEMONSTRATION: Coordinated 2-DoF MCP motion was preserved during passive transmission-ratio reconfiguration of the flexion DoF.
  • VI. DEMONSTRATION: Human-level force–speed performance has not been evaluated with compact actuators fully integrated within the hand.
  • VI. DEMONSTRATION: AA performance, higher-DoF extension, workspace, dynamic behavior, and long-term durability remain unevaluated or require further investigation.The spring-dependent extension mechanism may affect control performance during high-bandwidth operation.
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