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

Academic Editor: Giuseppe Carbone

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  • 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.

    Mathematics Subject Classification: Research article.

    Citation:

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  • Figure 1.  Schematic representation of the hand

    Figure 2.  Kinematic diagram of a single finger

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

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

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

    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.

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

    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.

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

    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.

    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 $.

    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} $
     | Show Table
    DownLoad: CSV

    Table 2.  Degree of redundancy

    Task Degree of redundancy $ (\mathit{n}-\mathit{m}) $
    T1 2
    T2 2
    T3 1
     | Show Table
    DownLoad: CSV

    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} $
     | Show Table
    DownLoad: CSV

    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 $
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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°
     | Show Table
    DownLoad: CSV
  • [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. NelsonM.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. SaafiM.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. MizeraM.A. LaribiD. DegezJ.P. GazeauP. 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.
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