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

• 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: • • 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 ${z}_{1}$

Table 2.  Degree of redundancy

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

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

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 $0°$ $0$ $0$

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

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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