# American Institute of Mathematical Sciences

February  2021, 1(1): 17-31. doi: 10.3934/steme.2021002

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

 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

Received  October 2020 Revised  January 2021 Published  February 2021

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.

Citation: Med Amine Laribi, Saïd Zeghloul. Redundancy understanding and theory for robotics teaching: Application on a human finger model. STEM Education, 2021, 1 (1) : 17-31. doi: 10.3934/steme.2021002
##### References:
 [1] Nof, S.Y. (ed.) (1985) Handbook of Industrial Robotics. John Wiley & Sons, New York. [2] Angeles, J. (2002) Fundamentals of Robotic Mechanical Systems (2nd ed.). Springer Verlag, New York. [3] Chiaverini S., Oriolo G., Maciejewski A.A. (2016) Redundant Robots. In: Siciliano B., Khatib O. (eds) Springer Handbook of Robotics. Springer Handbooks. Springer. [4] E.S. Conkur and R. Buckingham, Clarifying the definition of redundancy as used in robotics, Robotica, 15 (1997), 583-586.  doi: 10.1017/S0263574797000672. [5] C.A. Nelson, M.A. Laribi and S. Zeghloul, Multi-robot system optimization based on redundant serial spherical mechanism for robotic minimally invasive surgery, Robotica, 37 (2019), 1202-1213.  doi: 10.1017/S0263574718000681. [6] H. Saafi, M.A. Laribi and S. Zeghloul, Optimal torque distribution for a redundant 3-RRR spherical parallel manipulator used as a haptic medical device, Robotics and Autonomous Systems, 89 (2017), 40-50. [7] de Wit, C.C., Siciliano, B., Bastin, G. (1996) Theory of Robot Control. Springer‐Verlag, London. [8] Angeles, J. (2006). Fundamentals of Robotic Mechanical Systems: Theory, Methods, and Algorithms (3rd ed.). Springer‐Verlag, New York. [9] M.L. Latash and V.M. Zatsiorsky, Multi-finger prehension: Control of a redundant mechanical system, Advances in Experimental Medicine and Biology, 629 (2009), 597-618.  doi: 10.1007/978-0-387-77064-2_32. [10] Towell, C., Howard, M., Vijayakumar, S. (2010) Learning nullspace policies. The IEEE/RSJ International Conference on Intelligent Robots and Systems, Taipei, pp. 241-248, doi: 10.1109/IROS.2010.5650663. [11] C. Mizera, M.A. Laribi, D. Degez, J.P. Gazeau, P. Vulliez and S. Zeghloul, Architecture choice of a robotic hand for deep-sea exploration based on the expert gestures movements analysis, Mechanisms and Machine Science, 72 (2019), 1-19. [12] Hu, D., Ren, L., Howad, D., Zong, C. (2014) Biomechanical analysis of force distribution in human finger extensor mechanisms. BioMed Research International, 2014: Article ID 743460, https: //doi.org/10.1155/2014/743460. doi: 10.1155/2014/743460.

show all references

##### References:
 [1] Nof, S.Y. (ed.) (1985) Handbook of Industrial Robotics. John Wiley & Sons, New York. [2] Angeles, J. (2002) Fundamentals of Robotic Mechanical Systems (2nd ed.). Springer Verlag, New York. [3] Chiaverini S., Oriolo G., Maciejewski A.A. (2016) Redundant Robots. In: Siciliano B., Khatib O. (eds) Springer Handbook of Robotics. Springer Handbooks. Springer. [4] E.S. Conkur and R. Buckingham, Clarifying the definition of redundancy as used in robotics, Robotica, 15 (1997), 583-586.  doi: 10.1017/S0263574797000672. [5] C.A. Nelson, M.A. Laribi and S. Zeghloul, Multi-robot system optimization based on redundant serial spherical mechanism for robotic minimally invasive surgery, Robotica, 37 (2019), 1202-1213.  doi: 10.1017/S0263574718000681. [6] H. Saafi, M.A. Laribi and S. Zeghloul, Optimal torque distribution for a redundant 3-RRR spherical parallel manipulator used as a haptic medical device, Robotics and Autonomous Systems, 89 (2017), 40-50. [7] de Wit, C.C., Siciliano, B., Bastin, G. (1996) Theory of Robot Control. Springer‐Verlag, London. [8] Angeles, J. (2006). Fundamentals of Robotic Mechanical Systems: Theory, Methods, and Algorithms (3rd ed.). Springer‐Verlag, New York. [9] M.L. Latash and V.M. Zatsiorsky, Multi-finger prehension: Control of a redundant mechanical system, Advances in Experimental Medicine and Biology, 629 (2009), 597-618.  doi: 10.1007/978-0-387-77064-2_32. [10] Towell, C., Howard, M., Vijayakumar, S. (2010) Learning nullspace policies. The IEEE/RSJ International Conference on Intelligent Robots and Systems, Taipei, pp. 241-248, doi: 10.1109/IROS.2010.5650663. [11] C. Mizera, M.A. Laribi, D. Degez, J.P. Gazeau, P. Vulliez and S. Zeghloul, Architecture choice of a robotic hand for deep-sea exploration based on the expert gestures movements analysis, Mechanisms and Machine Science, 72 (2019), 1-19. [12] Hu, D., Ren, L., Howad, D., Zong, C. (2014) Biomechanical analysis of force distribution in human finger extensor mechanisms. BioMed Research International, 2014: Article ID 743460, https: //doi.org/10.1155/2014/743460. doi: 10.1155/2014/743460.
Schematic representation of the hand
Kinematic diagram of a single finger
Task1 - Trajectory of the fingertip and corresponding joint angles.

Task2 - Trajectory of the fingertip and corresponding joint angles.

Task3 - Trajectory of the fingertip and corresponding joint angles.

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.

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

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.

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

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.

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 $c = 1$.
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 ${z}_{1}$
 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 ${z}_{1}$
Degree of redundancy
 Task Degree of redundancy $(\mathit{n}-\mathit{m})$ T1 2 T2 2 T3 1
 Task Degree of redundancy $(\mathit{n}-\mathit{m})$ T1 2 T2 2 T3 1
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) $$\mathrm{cos}{q}_{2}=\frac{-{\overline{x}}_{1}B-{\overline{z}}_{1}A}{{A}^{2}+{B}^{2}} ; \mathrm{sin}{q}_{2}=\frac{{\overline{z}}_{1}B-{\overline{x}}_{1}A}{{A}^{2}+{B}^{2}} \mathit{\boldsymbol{q}}_{3} {q}_{3}^{1}=arcos\left(\frac{{\overline{x}}^{2}+{\overline{z}}^{2}-\left({l}_{3}^{2}+{l}_{2}^{2}\right)}{2{l}_{2}{l}_{3}}\right) {q}_{3}^{2}=-{q}_{3}^{1} \mathit{\boldsymbol{q}}_{4} {q}_{4}=\alpha -{q}_{3}-{q}_{2}  Joint variable Solution 1 Solution 2 \mathit{\boldsymbol{q}}_{2} {q}_{2}=atan2\left(\mathrm{sin}{q}_{2}, \mathrm{cos}{q}_{2}\right)$$ \mathrm{cos}{q}_{2}=\frac{-{\overline{x}}_{1}B-{\overline{z}}_{1}A}{{A}^{2}+{B}^{2}}$; $\mathrm{sin}{q}_{2}=\frac{{\overline{z}}_{1}B-{\overline{x}}_{1}A}{{A}^{2}+{B}^{2}}$ $\mathit{\boldsymbol{q}}_{3}$ ${q}_{3}^{1}=arcos\left(\frac{{\overline{x}}^{2}+{\overline{z}}^{2}-\left({l}_{3}^{2}+{l}_{2}^{2}\right)}{2{l}_{2}{l}_{3}}\right)$ ${q}_{3}^{2}=-{q}_{3}^{1}$ $\mathit{\boldsymbol{q}}_{4}$ ${q}_{4}=\alpha -{q}_{3}-{q}_{2}$
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 $0°$ $0$ $0$
 Joint angle $\mathit{\boldsymbol{q}}_{2}$ $\mathit{\boldsymbol{q}}_{3}$ $\mathit{\boldsymbol{q}}_{4}$ Maximum value $90°$ $120°$ $70°$ Minimum value $0°$ $0$ $0$
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
 $\mathit{\boldsymbol{L}}_{1}$ $\mathit{\boldsymbol{L}}_{2}$ $\mathit{\boldsymbol{L}}_{3}$ $\mathit{\boldsymbol{L}}_{4}$ Phalange length [mm] 152 45 35 32
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°
 Task 1 Task 2 Task 3 $\mathit{\boldsymbol{\alpha}}=\mathit{\boldsymbol{q}}_{2}+\mathit{\boldsymbol{q}}_{3}+\mathit{\boldsymbol{q}}_{4}$ 165° 135° 90°
 [1] María Barbero-Liñán, Miguel C. Muñoz-Lecanda. Strict abnormal extremals in nonholonomic and kinematic control systems. Discrete and Continuous Dynamical Systems - S, 2010, 3 (1) : 1-17. doi: 10.3934/dcdss.2010.3.1 [2] María Barbero-Liñán, Manuel de León, David Martín de Diego, Juan C. Marrero, Miguel C. Muñoz-Lecanda. Kinematic reduction and the Hamilton-Jacobi equation. Journal of Geometric Mechanics, 2012, 4 (3) : 207-237. doi: 10.3934/jgm.2012.4.207 [3] Marc-Auréle Lagache, Ulysse Serres, Vincent Andrieu. Minimal time synthesis for a kinematic drone model. Mathematical Control and Related Fields, 2017, 7 (2) : 259-288. doi: 10.3934/mcrf.2017009 [4] Lotfi Romdhane, Mohammad A. Jaradat. Interactive MATLAB based project learning in a robotics course: Challenges and achievements. STEM Education, 2021, 1 (1) : 32-46. doi: 10.3934/steme.2021003 [5] Jun Lai, Ming Li, Peijun Li, Wei Li. A fast direct imaging method for the inverse obstacle scattering problem with nonlinear point scatterers. Inverse Problems and Imaging, 2018, 12 (3) : 635-665. doi: 10.3934/ipi.2018027 [6] Zhiming Chen, Shaofeng Fang, Guanghui Huang. A direct imaging method for the half-space inverse scattering problem with phaseless data. Inverse Problems and Imaging, 2017, 11 (5) : 901-916. doi: 10.3934/ipi.2017042 [7] David Melching, Ulisse Stefanelli. Well-posedness of a one-dimensional nonlinear kinematic hardening model. Discrete and Continuous Dynamical Systems - S, 2020, 13 (8) : 2271-2284. doi: 10.3934/dcdss.2020188 [8] Carsten Collon, Joachim Rudolph, Frank Woittennek. Invariant feedback design for control systems with lie symmetries - A kinematic car example. Conference Publications, 2011, 2011 (Special) : 312-321. doi: 10.3934/proc.2011.2011.312 [9] Victor Isakov, Joseph Myers. On the inverse doping profile problem. Inverse Problems and Imaging, 2012, 6 (3) : 465-486. doi: 10.3934/ipi.2012.6.465 [10] Raimund Bürger, Kenneth H. Karlsen, John D. Towers. On some difference schemes and entropy conditions for a class of multi-species kinematic flow models with discontinuous flux. Networks and Heterogeneous Media, 2010, 5 (3) : 461-485. doi: 10.3934/nhm.2010.5.461 [11] Raimund Bürger, Antonio García, Kenneth H. Karlsen, John D. Towers. Difference schemes, entropy solutions, and speedup impulse for an inhomogeneous kinematic traffic flow model. Networks and Heterogeneous Media, 2008, 3 (1) : 1-41. doi: 10.3934/nhm.2008.3.1 [12] Mouhamadou Samsidy Goudiaby, Ababacar Diagne, Leon Matar Tine. Longtime behavior of a second order finite element scheme simulating the kinematic effects in liquid crystal dynamics. Communications on Pure and Applied Analysis, 2021, 20 (10) : 3499-3514. doi: 10.3934/cpaa.2021116 [13] Alexandr Mikhaylov, Victor Mikhaylov. Dynamic inverse problem for Jacobi matrices. Inverse Problems and Imaging, 2019, 13 (3) : 431-447. doi: 10.3934/ipi.2019021 [14] Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems and Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271 [15] Nguyen Huy Tuan, Mokhtar Kirane, Long Dinh Le, Van Thinh Nguyen. On an inverse problem for fractional evolution equation. Evolution Equations and Control Theory, 2017, 6 (1) : 111-134. doi: 10.3934/eect.2017007 [16] Russell Johnson, Luca Zampogni. On the inverse Sturm-Liouville problem. Discrete and Continuous Dynamical Systems, 2007, 18 (2&3) : 405-428. doi: 10.3934/dcds.2007.18.405 [17] Mikko Orispää, Markku Lehtinen. Fortran linear inverse problem solver. Inverse Problems and Imaging, 2010, 4 (3) : 485-503. doi: 10.3934/ipi.2010.4.485 [18] A. Doubov, Enrique Fernández-Cara, Manuel González-Burgos, J. H. Ortega. A geometric inverse problem for the Boussinesq system. Discrete and Continuous Dynamical Systems - B, 2006, 6 (6) : 1213-1238. doi: 10.3934/dcdsb.2006.6.1213 [19] Ian Knowles, Ajay Mahato. The inverse volatility problem for American options. Discrete and Continuous Dynamical Systems - S, 2020, 13 (12) : 3473-3489. doi: 10.3934/dcdss.2020235 [20] Marcus Wagner. A direct method for the solution of an optimal control problem arising from image registration. Numerical Algebra, Control and Optimization, 2012, 2 (3) : 487-510. doi: 10.3934/naco.2012.2.487

Impact Factor: