# Force-Sensing Actuator with a Compliant Flexure-Type Joint for a Robotic Manipulator

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## Abstract

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## 1. Introduction

- High-performance mechanics improve the level of dexterity related to the mechanical structure; control-wise and minimum friction leads to an enhanced stability and fine manipulation capability [16].
- Joint torques are transmitted back as a resistive torque to the DC motor shaft, which can be measured by current sensing [17]. This permits one to estimate external forces exploiting the actuators as sensors, thus improving the dexterity related to the sensory apparatus.
- The resulting mathematical model of the physical behavior of the finger enables alternative strategies to control the mechanical compliance of the finger.
- As additional advantages of the self-sensing approach, the absence of force sensors means wiring simplification, avoidance of sensor drift and reduction of the number of components.

## 2. Design Rationale

**Figure 1.**Finger prototype integrating the inter-phalangeal rotational pivot and the tendon-driven backdrivable transmission.

- the set of directions along which a force applied to the finger can be balanced by the cables (working only under tensile strength and not compression);
- plus, for each of those directions, the maximum value of external force that can be balanced taking into account the output of the actuators used to pull the cables.

## 3. Motor-to-Joint Transmission

#### 3.1. Tendon-Driven Actuation Principle

#### 3.2. Monolithic Frictionless Anti-Rotation System

**Figure 3.**Monolithic frictionless anti-rotation system of the self-sensing linear actuator: prevented rotation and allowed displacement of the nut.

**Figure 4.**Monolithic frictionless anti-rotation system of the self-sensing linear actuator: canceling of the effects caused by the angular oscillations and the hyper-static constraints.

- For each position of the nut in the ball-screw axis, the theoretical configuration of minimum elastic energy of the structure is calculated (Figure 5c);
- For the complete travel of the nut along the ball-screw axis, the previous step permits one to find the successive mechanism configurations, and the total angular displacement of each notch joint is calculated (Figure 5d);
- Parameter values (${\alpha}_{i}$ and ${L}_{i}$ for $i=1,\dots ,3$) minimizing the largest angular displacement among the three joints ($min\left(\right)open="("\; close=")">{\varphi}_{{1}_{max}},{\varphi}_{{2}_{max}},{\varphi}_{{3}_{max}}$) are chosen;
- Once the angular displacements in the articulations are minimized and known, the parameters of the notch joint (Figure 5b) and the material selection (through its Young’s modulus E and its yield strength ${S}_{y}$) are specified, taking into account practical criteria regarding the integration of the actuator in a minimum package.

**Figure 5.**Basic principle of the compliant anti-rotation mechanism: (

**a**) kinematics of the flexure-based three-bar linkage and its design parameters for optimization; (

**b**) elementary circular flexure hinge joint; (

**c**) configuration of minimum energy for a position of the nut; (

**d**) successive mechanism configurations for the travel of the nut.

## 4. Inter-Phalangeal Rotational Joint

**Figure 7.**On the left: flexible pivot ${X}^{2}$ as an interphalangeal pivot joint for the finger design with multiple degrees of freedom. On the right: monolithic structure prototype ${X}^{2}$ obtained by aluminum electrical discharge machining (${S}_{y}$ = 580 MPa, E = 73 GPa, 9 mm × 7 mm × 5 mm).

#### 4.1. Concept

**Figure 8.**Design and rotational motion produced by the ${X}^{2}$ pivot between two successive phalanges of a gripper finger.

#### 4.2. Design Optimization

**Figure 9.**Theoretical trajectories of the vertices for the leaf-type isosceles-trapezoidal flexural (LITF), cartwheel and ${X}^{2}$ pivots.

- the two cartwheels that constitute the ${X}^{2}$ joint are arranged in a manner that makes their two respective center shift vectors $\u03f5\left(\theta \right)$ compensate for one another;
- for a flexible pivot, the value of ϵ increases rapidly with the angular displacement θ; therefore, as the overall displacement θ is distributed into smaller displacements over the two cartwheels, a large reduction of the center shift is achieved for each one.

Geometry | Performances | ||
---|---|---|---|

H | 7 mm | ${\theta}_{max}$ | $\pm {60}^{\circ}$ |

φ | ${65}^{\circ}$ | $\u03f5(\theta =\pm {10}^{\circ})$ | $14\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}$m |

t | 0.15 mm | $\u03f5\left({\theta}_{max}\right)$ | $467\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}$m |

w | 9 mm | ${K}_{\theta}$ | 0.02 Nm/rad |

**Figure 11.**Set of pairs $\left(\right)open="("\; close=")">\phi ,t$ satisfying the objective $\left(\right)open="|"\; close="|">{\theta}_{max}\ge \left(\right)open="|"\; close="|">{\theta}_{design}$ (aluminum material, dimensions $H=7$ mm, $w=9$ mm).

**Figure 12.**Set of pairs $\left(\right)open="("\; close=")">\phi ,t$ satisfying the objective ${K}_{\theta}\ge {K}_{{\theta}_{design}}=0.02$ Nm/rad (aluminum material, dimensions $H=7$ mm, $w=9$ mm).

#### 4.3. Experimental Performances

#### 4.3.1. Accuracy of Motion

**Figure 14.**Observed trajectory of a target point on the tested ${X}^{2}$ pivot and circle fitted to the collected data points.

#### 4.3.2. Identification of Pivot Stiffness

## 5. Modeling and Self-Sensing Strategy

#### 5.1. Modeling of the Tendon-Based Actuation Principle

**Figure 16.**Mechatronic model of the self-sensing actuated joint, including the anti-rotation system and the ${X}^{2}$ pivot.

Parameters | Description |
---|---|

${\theta}_{0f}$ | Joint angle for relaxed flexible pivot |

${\theta}_{0s}$ | Joint angle for relaxed extension spring |

J | Inertia of flexible pivot and finger |

m | Mass of flexible pivot and finger |

g | Gravitational acceleration |

r | Normal distance between tendon and pivot center |

${d}_{G}$ | Distance between the gravity center and pivot center |

d | Distance between the effort application (F) and pivot center |

${K}_{\theta}$ | Flexible pivot stiffness |

${K}_{s}$ | Antagonist spring stiffness |

p | Screw pitch |

η | Ball-screw efficiency |

R | DC motor resistance |

L | DC motor inductance |

${K}_{em}$ | DC motor constant |

${J}_{rotor}$ | DC motor inertia |

f | DC motor viscous friction |

#### 5.2. Self-Sensing of External Forces Applied to the Finger

**Figure 17.**Experimental characterization of the self-sensing capability of the actuated joint (interpolated experimental mean values m along with its confidence intervals defined by its standard deviation σ for all of the angles ranging from ${0}^{\circ}$ to ${70}^{\circ}$ with an average step of ${10}^{\circ}$).

## 6. Conclusions

**Figure 18.**Perspective of integration of the proposed transmission within the context of an anthropomorphic robot hand.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Grossard, M.; Martin, J.; Huard, B.
Force-Sensing Actuator with a Compliant Flexure-Type Joint for a Robotic Manipulator. *Actuators* **2015**, *4*, 281-300.
https://doi.org/10.3390/act4040281

**AMA Style**

Grossard M, Martin J, Huard B.
Force-Sensing Actuator with a Compliant Flexure-Type Joint for a Robotic Manipulator. *Actuators*. 2015; 4(4):281-300.
https://doi.org/10.3390/act4040281

**Chicago/Turabian Style**

Grossard, Mathieu, Javier Martin, and Benoît Huard.
2015. "Force-Sensing Actuator with a Compliant Flexure-Type Joint for a Robotic Manipulator" *Actuators* 4, no. 4: 281-300.
https://doi.org/10.3390/act4040281