# Soft Driving Epicyclical Mechanism for Robotic Finger

^{*}

## Abstract

**:**

## 1. Introduction

- Under-actuated;
- Fully actuated;
- Over-actuated.

- Have an active flexion in the range of $[{60}^{\circ},{90}^{\circ}]$ in the MCP, PIP, and DIP joints (see Figure 1) of the finger;
- Perform force in the interval $[4.78\phantom{\rule{3.33333pt}{0ex}}$N$,\phantom{\rule{0.277778em}{0ex}}6.70\phantom{\rule{3.33333pt}{0ex}}$N];
- Achieve operating frequency in the interval $[8.89\phantom{\rule{3.33333pt}{0ex}}$Hz$,\phantom{\rule{0.277778em}{0ex}}22.2\phantom{\rule{3.33333pt}{0ex}}$Hz];
- Propose an actuation system based on viscoelastic behavior of human muscle.

- $CE$, the contractile element;
- ${k}_{pee}$, the parallel elastic element of the muscle;
- ${k}_{see}$, the muscle’s serial elastic element;
- ${k}_{Tee}$, the tendon’s elastic element;
- ${b}_{Tde}$, the tendon’s damping element;
- ${\alpha}_{m}$, the pennation angle.

## 2. Soft Epicyclical Mechanism

- The most used drive mechanism is based on tendons;
- The mass of a prosthetic hand must be under 600 g;
- The number of actuators must be reduced, but the number of the degree of freedom (DoF) must be as high as necessary to perform the prehension;
- An electric actuator is a right approach, but needs to be complemented with soft elements to achieve the desired behavior.

- Higher stiffness into the MCP joints;
- A fixed mechanical relation between the proximal and metacarpal joints, and between the distal and metacarpal joints.

- The finger’s phalanges replace the carrier;
- The gears are replaced by two slotted pulleys;
- The mechanical link is guaranteed by two crossed flexible wires, henceforth, tendons.

## 3. Materials and Methods

- A rotation ${\alpha}_{i}$ around ${x}_{i-1}$;
- A translation ${a}_{i}$ along ${x}_{i-1}$;
- A rotation ${\theta}_{i}$ around ${z}_{i}$;
- A translation ${d}_{i}$ along ${z}_{i}$.

- Vector ${\overrightarrow{r}}_{1}$ between the MCP and PIP joints;
- Vector ${\overrightarrow{r}}_{2}$ between the PIP and DIP joints;
- Vector ${\overrightarrow{r}}_{3}$ between the DIP joint and fingertip.

- ${H}_{0}=2\phantom{\rule{0.277778em}{0ex}}{{l}_{1}}^{2}{\ddot{\theta}}_{1}sin{\left({\theta}_{1}\right)}^{2}\left[\left({m}_{1}+{m}_{2}\right){l}_{2}sin\left({\theta}_{2}\right)-4\phantom{\rule{0.277778em}{0ex}}{m}_{1}-6\phantom{\rule{0.277778em}{0ex}}{m}_{2}\right]$;
- ${H}_{1}=(1/4)\phantom{\rule{0.277778em}{0ex}}{l}_{1}{l}_{2}{m}_{2}{\ddot{\theta}}_{1}cos\left({\theta}_{1}\right)cos\left({\theta}_{2}\right)-(1/2)\phantom{\rule{0.277778em}{0ex}}{{l}_{2}}^{2}{m}_{2}{\ddot{\theta}}_{2}sin{\left({\theta}_{2}\right)}^{2}-4\phantom{\rule{0.277778em}{0ex}}g{m}_{1}$;
- ${H}_{2}=(1/4)\phantom{\rule{0.277778em}{0ex}}\left({m}_{2}+2{m}_{3}\right){{l}_{2}}^{2}{\ddot{\theta}}_{2}cos{\left({\theta}_{2}\right)}^{2}+2\phantom{\rule{0.277778em}{0ex}}{l}_{3}{m}_{3}{\ddot{\theta}}_{3}sin\left({\theta}_{3}\right)+2{\phantom{\rule{0.277778em}{0ex}}}_{2}{\ddot{\theta}}_{2}$;
- ${H}_{3}=(1/2)\phantom{\rule{0.277778em}{0ex}}{l}_{2}cos\left({\theta}_{2}\right)\left({l}_{3}{m}_{3}{\ddot{\theta}}_{3}cos\left({\theta}_{3}\right)+{m}_{2}{\ddot{{y}^{\prime}}}_{1}+2{m}_{3}{\ddot{{y}^{\prime}}}_{2}\right)$;
- ${H}_{4}={l}_{2}sin\left({\theta}_{2}\right)\left(g{m}_{1}-2{F}_{+}\left(2g+3{\ddot{\theta}}_{2}+{\ddot{{x}^{\prime}}}_{1}\right){m}_{2}+\left(g+2{\ddot{\theta}}_{2}\right){m}_{3}\right)$;
- ${H}_{5}={H}_{1}+{H}_{2}+{H}_{3}+{H}_{4}+8{F}_{+}\left(-6g-6{\ddot{{x}^{\prime}}}_{1}\right){m}_{2}+\left(-4g-4{\ddot{{x}^{\prime}}}_{2}\right){m}_{3}$;
- ${H}_{6}=(1/4)\phantom{\rule{0.277778em}{0ex}}\left({m}_{1}+3/2{m}_{2}\right){{l}_{1}}^{2}{\ddot{\theta}}_{1}cos{\left({\theta}_{1}\right)}^{2}{+}_{1}{\ddot{\theta}}_{1}$;
- ${H}_{7}=\left({m}_{2}+2/3{m}_{3}\right){l}_{2}{\ddot{\theta}}_{2}cos\left({\theta}_{2}\right)+2{m}_{2}{\ddot{{y}^{\prime}}}_{1}+4/3{m}_{3}{\ddot{{y}^{\prime}}}_{2}$;
- ${H}_{8}=(3/8)\phantom{\rule{0.277778em}{0ex}}{l}_{1}cos\left({\theta}_{1}\right)\left((2/3)\phantom{\rule{0.277778em}{0ex}}{l}_{3}{m}_{3}{\ddot{\theta}}_{3}cos\left({\theta}_{3}\right)+{H}_{7}\right)$.

## 4. Discussion

^{©}actuator—which is a trade name for nickel–titanium SMA wires—with 0.38 mm diameter, bought at Dynallow Inc. [34]. Figure 21 shows the implementation in the new robotic finger.

## 5. Conclusions

## 6. Patents

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

SMA | Shape Memory Alloy |

MCP | Metacarpophalangeal joint |

PIP | Proximal Interphalangeal joint |

DIP | Distal Interphalangeal joint |

FDS | Flexor Digitorum Superficialis muscle |

M | metacarpus |

PP | Proximal phalanges |

MP | Medial phalangesnd |

DP | Distal phalanges |

$CE$ | The contractile element |

${k}_{pee}$ | The parallel elastic element of the muscle |

${k}_{see}$ | The muscle’s serial elastic element |

${k}_{Tee}$ | The tendon’s elastic element |

${b}_{Tde}$ | The tendon’s damping element |

${\alpha}_{m}$ | The pennation angle |

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**Figure 1.**Main joints and bones of human upper limb, and Flexor Digitorum Superficialis (FDS) muscle, acting over a finger (M: Metacarpus, PP: Proximal phalanges, MP: Medial phalanges, and DP: Distal phalanges).

**Figure 3.**Retained Hill-based model proposed by to describe the behavior of Flexor Digitorum Superficialis (FDS) muscle.

**Figure 6.**(

**a**) Epicyclical mechanism, left side is the initial state and right side is the final state; and (

**b**) soft epicyclical mechanism, left side is the initial state of the section A-A and right side is the final state.

**Figure 8.**(

**a**) Input mechanism of Promain-I finger and (

**b**) real view of the input mechanism of Promain-I finger.

**Figure 9.**Assembly of the finger, yellow represents chassis, blue represents PP phalanx, green represents MP phalanx, and gray represents DP phalanx. (

**a**) Assembly of the PP and MP phalanges, in which PP is the carrier and MP is the driven and (

**b**) Assembly of the MP and DP pahalanges, in which MP is the carrier and DP is the driven.

**Figure 10.**Finger movement: (

**a**) Movement of medial phalanx (MP) with respect to proximal phalanx (PP), left side is the initial position and right side is the final position; and (

**b**) movement of MP phalanx respect to PP phalanx, left side is the initial position and right side is the final position.

**Figure 13.**Automatic detection of finger joints an fingertip position: (

**a**) Crop image; (

**b**) gray scale; (

**c**) binary image; (

**d**) image edges; and (

**e**) Hough transform.

**Figure 18.**Probability density function of: (

**a**) ${\theta}_{2}$ absolute error; and (

**b**) ${\theta}_{3}$ absolute error.

**Figure 20.**Schematic representation of the soft epicyclical tendon-driven actuation system based on Shape Memory Alloy (SMA).

**Figure 21.**Physical implementation of the soft epicyclical tendon-driven actuation system based on SMA.

Type of Actuation | Name | Source |
---|---|---|

over-actuated | UTAH | [3,4,5] |

DextrousHand | [6] | |

AwiwiRobotHand | [7] | |

TheRobonautHand2 | [8,9] | |

fully-actuated | OkadaHand | [10] |

KeioHand | [11] | |

UBHandIV | [12] | |

SensorSpeed | [13] | |

(OttoBock) | ||

ROBIOSS | [14] | |

under-actuated | i-limb Ultra | [15] |

Université Laval | [16] | |

Rutgers Hand | [17] | |

i-HY Hand | [18] | |

Michelangelo | [19] | |

Gifu Hand III | [20] | |

MPL Hand | [21] |

Phase | Measured Force [N] | Standard Deviation [N] | Young’s Modulus [GPa] |
---|---|---|---|

Austenite | 16.33 | 0.77 | ${E}^{A}=31.8$ |

Martensite | 6.00 | 0.39 | ${E}^{M}=11.7$ |

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**MDPI and ACS Style**

Ramirez, J.; Rubiano, A.; Castiblanco, P. Soft Driving Epicyclical Mechanism for Robotic Finger. *Actuators* **2019**, *8*, 58.
https://doi.org/10.3390/act8030058

**AMA Style**

Ramirez J, Rubiano A, Castiblanco P. Soft Driving Epicyclical Mechanism for Robotic Finger. *Actuators*. 2019; 8(3):58.
https://doi.org/10.3390/act8030058

**Chicago/Turabian Style**

Ramirez, Jose, Astrid Rubiano, and Paola Castiblanco. 2019. "Soft Driving Epicyclical Mechanism for Robotic Finger" *Actuators* 8, no. 3: 58.
https://doi.org/10.3390/act8030058