# Design and stability analysis of an underactuated hand with passively rotating fingers

Léonie Plancoulaine¹, Franck Plestan¹, Sylvain Guegan³, Damien Chablat¹

¹Nantes Université, École Centrale Nantes, LS2N, CNRS, UMR 6004, 44000 Nantes, France

²Université Rennes, INSA Rennes, LGCGM, F-35000 Rennes, France

contact : leonie.plancoulaine@ls2n.fr

## Introduction

* Hand : high dexterity → difficult to reproduce robotically.

* Several types of robotic hands: soft grippers, totally actuated hands, underactuated ones.

* Most of gripper finger → planar mobility or complex kinematic.

**Problem: trade-off → grasp diversity vs. kinematic complexity**

## Objectives

* Develop a simple passive hand with spatial mobility.

* Allow "classic" type of grasp : spherical - cylindrical, parallel-enveloping / "asymmetric" grasp.

* Analyze fingers stability.

## Hand architecture

**Underactuated hand:** fewer actuators than degrees of freedom (passive elements : springs)
→ self-adaptation, simplified design, durability, limited cost.

<figure><img src="image_1.png" /><figcaption>Finger position</figcaption></figure>

<figure><img src="image_2.png" /></figure>

<figure><img src="image_3.png" /><figcaption>Differential mechanism</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Parallel
grasp</figcaption></figure>

<figure><img src="image_5.png" /></figure>

<figure><img src="image_6.png" /><figcaption>Enveloping
grasp</figcaption></figure>

<figure><img src="image_7.png" /></figure>

<figure><img src="image_8.png" /></figure>

## Method

### Condition for a stable finger grasp:

* Contact forces remain within the physical boundaries of the phalanges.

* Finger rotates in the appropriate direction.

Else → object may be ejected.

1

### Parallel grasp

By applying the fundamental principle of statics, the moment around $z_0$ is :

$$M_z = f_2((-k_2\eta - m_2\mu)c\psi - l_1\eta c\theta_2 + m_2s\psi)$$

2

### Enveloping grasp

By applying the fundamental principle of statics, the moment around $z_0$ is :

$$M_z = f_1((-k_1\eta - m_1\mu)\sin\theta_2 + m_1\cos\theta_2) + f_2((-k_2\eta - m_2\mu)\cos(\psi - \theta_3) - l_1\eta\cos\theta_2 + m_2\sin(\psi - \theta_3))$$

<figure><img src="image_9.png" /></figure>

## Results

1

### Parallel grasp

<figure><img src="image_10.png" /></figure>

* Friction parameters → materials in contact at the object-finger interface:

* $\mu \to$ negligible;

* $\eta \to$ equilibrium limit depend on its value:

$$-\eta_{static} \le \eta \le \eta_{static}$$

* Finger rotates until reaching a stable position along the phalanx.

* Condition of stability on distal phalanx width:

$$n_2 \ge \frac{l_2\eta_{static}c\psi + l_1\eta_{static}c\theta_2}{s\psi - \mu c\psi}$$

2

### Enveloping grasp

* Force distribution between fingers:

* not influenced by the finger rotation → does not alter the kinematic configuration of the enveloping grasp;

* Use of Birglen's optimal geometric parameters [1] in plane.

* Stability in rotation:

* area of unstability for some contact force position depending on the phalanges flexions → location of stable contact point outside the distal phalanx.

<figure><img src="image_11.png" /></figure>

<figure><img src="image_12.png" /></figure>

## Conclusions

* Underactuation within and between fingers:

* simplification of the mechanical design;

* passive adaptability;

* multiple grasps;

* stability.

* Well-suited for industrial or agricultural applications.

<figure><img src="image_13.png" /></figure>

## Perspectives

* Integration of an electric motor and development of control strategies.

* Investigation of the influence of the springs on finger behavior.

* Evaluation of the influence of phalanx form on the rotation of the finger.

## References

[1] L. Birglen, T. Laliberté, et C. Gosselin, *Underactuated Robotic Hands*, vol. 40. in Springer Tracts in Advanced Robotics, vol. 40. Berlin, Heidelberg: Springer, 2008. doi: 10.1007/978-3-540-77459-4.