Redundancy understanding and theory for robotics teaching: Application on a human finger model

Med Amine Laribi; Saïd Zeghloul

doi:10.3934/steme.2021002

Article Contents

2021, Volume 1, Issue 1: 17-31. Doi: 10.3934/steme.2021002 open access

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Redundancy understanding and theory for robotics teaching: Application on a human finger model

Med Amine Laribi^1, , and
Saïd Zeghloul^1,

1.
Dept. GMSC, Prime Institute, CNRS - University of Poitiers - ENSMA - UPR 3346, Poitiers, France

* Correspondence: med.amine.laribi@univ-poitiers.fr; Tel: +5-49-496552
* Correspondence: med.amine.laribi@univ-poitiers.fr; Tel: +5-49-496552

Academic Editor: Giuseppe Carbone

Received: September 30, 2020

Revised: December 31, 2020

Published: February 2021

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Abstract

This paper introduces the concept of redundancy in robotics to students in master degree based on a didactic approach. The definition as well as theoretical description related to redundancy are presented. The example of a human finger is considered to illustrate the redundancy with biomechanical point of view. At the same time, the finger is used to facilitate the comprehension and apply theoretical development to solve direct and inverse kinematics problems. Three different tasks are considered with different degree of redundancy. All developments are implemented under Matlab and validated in simulation on CAD software.

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Mathematics Subject Classification: Research article.

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Figure 1. Schematic representation of the hand

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Figure 2. Kinematic diagram of a single finger

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Figure 3. Task1 - Trajectory of the fingertip and corresponding joint angles.

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Figure 4. Task2 - Trajectory of the fingertip and corresponding joint angles.

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Figure 5. Task3 - Trajectory of the fingertip and corresponding joint angles.

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Figure 6. Task 1 - Numerical validation of the pseudo-inverse method: (a) Graphical finger construction (b) Trajectory of the fingertip (c) Computed joint angles of the finger.

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Figure 7. Task 1 - CAD simulation of the finger motion using the pseudo-inverse method.

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Figure 8. Task 2 - Numerical validation of the pseudo-inverse method: (a) Graphical finger construction (b) Trajectory of the fingertip (c) Computed joint angles of the finger.

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Figure 9. Task 2 - CAD simulation of the finger motion using the pseudo-inverse method.

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Figure 10. Task 3 - Numerical validation of the pseudo-inverse method: (a) Graphical finger construction (b) Trajectory of the fingertip (c) Computed joint angles of the finger.

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Figure 11. Numerical validation of the pseudo-inverse method with a criterion on the joint limits: (a) $ a = 1 $, $ b = 1 $ and $ c = 1 $ (b) $ a = 30 $, $ b = 1 $ and $ c = 1 $ (c) $ a = 100 $, $ b = 1 $ and $ c = 1 $.

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Table 1. The three tasks of the finger in the $ ({x}_{1}, {z}_{1}) $ plane.

Task	Initial Configuration	End-effector displacement
T1	$ {q}_{2}=45°, {q}_{3}=90°, {q}_{4}=30° $	60 mm along $ {x}_{1} $
T2	$ {q}_{2}=45°, {q}_{3}=45°, {q}_{4}=45° $	40 mm along $ {z}_{1} $
T3	$ {q}_{2}=0°, {q}_{3}=45°, {q}_{4}=45° $	30 mm along $ {-x}_{1} $ et 20 mm along $ {z}_{1} $

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Table 2. Degree of redundancy

Task	Degree of redundancy $ (\mathit{n}-\mathit{m}) $
T1	2
T2	2
T3	1

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Table 3. Solutions of the IKM with an additional constraint.

Joint variable	Solution 1	Solution 2
$ \mathit{\boldsymbol{q}}_{2} $	$ {q}_{2}=atan2\left(\mathrm{sin}{q}_{2}, \mathrm{cos}{q}_{2}\right) $$ \mathrm{cos}{q}_{2}=\frac{-{\overline{x}}_{1}B-{\overline{z}}_{1}A}{{A}^{2}+{B}^{2}} $; $ \mathrm{sin}{q}_{2}=\frac{{\overline{z}}_{1}B-{\overline{x}}_{1}A}{{A}^{2}+{B}^{2}} $
$ \mathit{\boldsymbol{q}}_{3} $	$ {q}_{3}^{1}=arcos\left(\frac{{\overline{x}}^{2}+{\overline{z}}^{2}-\left({l}_{3}^{2}+{l}_{2}^{2}\right)}{2{l}_{2}{l}_{3}}\right) $	$ {q}_{3}^{2}=-{q}_{3}^{1} $
$ \mathit{\boldsymbol{q}}_{4} $	$ {q}_{4}=\alpha -{q}_{3}-{q}_{2} $

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Table 4. The limit values of the joint angles.

Joint angle	$ \mathit{\boldsymbol{q}}_{2} $	$ \mathit{\boldsymbol{q}}_{3} $	$ \mathit{\boldsymbol{q}}_{4} $
Maximum value	$ 90° $	$ 120° $	$ 70° $
Minimum value	$ 0° $	$ 0 $	$ 0 $

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Table 5. The phalangeal lengths of the finger.

	$ \mathit{\boldsymbol{L}}_{1} $	$ \mathit{\boldsymbol{L}}_{2} $	$ \mathit{\boldsymbol{L}}_{3} $	$ \mathit{\boldsymbol{L}}_{4} $
Phalange length [mm]	152	45	35	32

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Table 6. Orientations of the last phalange.

	Task 1	Task 2	Task 3
$ \mathit{\boldsymbol{\alpha}}=\mathit{\boldsymbol{q}}_{2}+\mathit{\boldsymbol{q}}_{3}+\mathit{\boldsymbol{q}}_{4} $	165°	135°	90°

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