First, second and third springs are connected between, AB and the palm; AB and AC; and BD and BH. Initially even though the torque is applied to the AC, angle between AB and AC (θ) is kept constant due to the torsion spring between AB and AC. Then, angular velocity of AB becomes zero when its motion is restricted by the object. Initially, θ is known. Thus, β = 90 − (α + θ) for any α value. Assume that AC, AB, CD, DB, EF, BF, FG, BH, GH, HI, DE are l_{1} to l_{11} respectively. Considering ABDC four-bar mechanism;

${l}_{1}\mathrm{sin}\alpha +{l}_{3}\mathrm{sin}\gamma ={l}_{2}\mathrm{cos}\beta +{l}_{4}\mathrm{sin}\delta $ (2)

${l}_{1}\mathrm{cos}\alpha +{l}_{3}\mathrm{cos}\gamma ={l}_{2}\mathrm{sin}\beta +{l}_{4}\mathrm{cos}\delta $ (3)

From (2),

${\mathrm{sin}}^{2}\delta =\frac{{\left({l}_{1}\mathrm{sin}\alpha +{l}_{3}\mathrm{sin}\gamma -{l}_{2}\mathrm{cos}\beta \right)}^{2}}{{\left({l}_{4}\right)}^{2}}$

Figure 4. Kinematic diagram of the index finger [25] .

Figure 5. 3D model of the index finger.

From (3),

${\mathrm{cos}}^{2}\delta =\frac{{\left({l}_{1}\mathrm{cos}\alpha +{l}_{3}\mathrm{cos}\gamma -{l}_{2}\mathrm{sin}\beta \right)}^{2}}{{\left({l}_{4}\right)}^{2}}$

γ can be found for the given α and β from the equation given below.

$\begin{array}{l}{l}_{1}\mathrm{cos}\left(\gamma -\alpha \right)-{l}_{2}\mathrm{sin}\left(\gamma +\beta \right)\\ =\frac{1}{2\left({l}_{3}\right)}\left[{\left({l}_{4}\right)}^{2}-{\left({l}_{1}\right)}^{2}-{\left({l}_{2}\right)}^{2}-{\left({l}_{3}\right)}^{2}+2{l}_{1}{l}_{2}\mathrm{sin}\left(\alpha +\beta \right)\right]\end{array}$

Similarly, solving (2) and (3),

$\begin{array}{l}{l}_{2}\mathrm{sin}\left(\delta +\beta \right)-{l}_{1}\mathrm{cos}\left(\delta -\alpha \right)\\ =\frac{1}{2\left({l}_{4}\right)}\left[{\left({l}_{3}\right)}^{2}-{\left({l}_{2}\right)}^{2}-{\left({l}_{1}\right)}^{2}-{\left({l}_{4}\right)}^{2}+2{l}_{1}{l}_{2}\mathrm{sin}\left(\alpha +\beta \right)\right]\end{array}$

The angle (δ) can be found from the above equation for the given α and β. Therefore, 𝜇 can be found using 𝛿 and angle 𝛾. Consider DE, EF, BF and BD.

${l}_{11}\mathrm{cos}\gamma -{l}_{5}\mathrm{cos}\lambda ={l}_{4}\mathrm{cos}\delta -{l}_{6}\mathrm{cos}\mu $ (4)

${l}_{11}\mathrm{sin}\gamma -{l}_{5}\mathrm{sin}\lambda ={l}_{4}\mathrm{sin}\delta -{l}_{6}\mathrm{sin}\mu $ (5)

Solving the above equations, λ can be found from the below equation.

$\begin{array}{l}{l}_{4}\mathrm{cos}\left(\lambda -\delta \right)-{l}_{11}\mathrm{cos}\left(\lambda -\gamma \right)\\ =\frac{1}{2\left({l}_{5}\right)}\left[{\left({l}_{6}\right)}^{2}-{\left({l}_{4}\right)}^{2}-{\left({l}_{5}\right)}^{2}-{\left({l}_{11}\right)}^{2}+2{l}_{4}{l}_{11}\mathrm{cos}\left(\gamma -\delta \right)\right]\end{array}$

µ also can be found similarly from the below equation.

$\begin{array}{l}{l}_{11}\mathrm{cos}\left(\mu -\gamma \right)-{l}_{4}\mathrm{cos}\left(\mu -\delta \right)\\ =\frac{1}{2\left({l}_{6}\right)}\left[{\left({l}_{5}\right)}^{2}-{\left({l}_{4}\right)}^{2}-{\left({l}_{6}\right)}^{2}-{\left({l}_{11}\right)}^{2}+2{l}_{4}{l}_{11}\mathrm{cos}\left(\gamma -\delta \right)\right]\end{array}$

Therefore, φ can be found using µ and η. Considering BFGH four-bar mechanism;

${l}_{8}\mathrm{sin}\eta ={l}_{6}\mathrm{sin}\mu +{l}_{7}\mathrm{sin}\phi +{l}_{9}\mathrm{sin}\varphi $ (6)

${l}_{8}\mathrm{cos}\eta =-{l}_{6}\mathrm{cos}\mu -{l}_{7}\mathrm{cos}\phi +{l}_{9}\mathrm{cos}\varphi $ (7)

From (6) and (7), φ can be found as below.

$\begin{array}{l}{l}_{8}\mathrm{cos}\left(\varphi -\eta \right)+{l}_{6}\mathrm{cos}\left(\varphi +\mu \right)\\ =\frac{1}{2{l}_{9}}\left[{\left({l}_{8}\right)}^{2}+{\left({l}_{6}\right)}^{2}+{\left({l}_{9}\right)}^{2}-{\left({l}_{7}\right)}^{2}+2{l}_{6}{l}_{8}\mathrm{cos}\left(\eta +\mu \right)\right]\end{array}$

Similarly, φ can be found from the equation given below.

$\begin{array}{l}{l}_{8}\mathrm{cos}\left(\psi +\eta \right)+{l}_{6}\mathrm{cos}\left(\psi -\mu \right)\\ =\frac{1}{2{l}_{7}}\left[{\left({l}_{9}\right)}^{2}-{\left({l}_{8}\right)}^{2}+{\left({l}_{6}\right)}^{2}-{\left({l}_{7}\right)}^{2}+2{l}_{6}{l}_{8}\mathrm{cos}\left(\eta +\mu \right)\right]\end{array}$

Considering joints of a finger fingertip “I”, position, (x, y) and orientation, (70+φ) with respected to the motor shaft A (0, 0) can be found.

$x={l}_{2}\mathrm{cos}\beta +{l}_{8}\mathrm{sin}\eta +{l}_{10}\mathrm{sin}\left(70+\varphi \right)$ (8)

$y={l}_{2}\mathrm{sin}\beta -{l}_{8}\mathrm{cos}\eta +{l}_{10}\mathrm{cos}\left(70+\varphi \right)$ (9)

(8) and (9) can be used to derived the position and orientation of the fingertip relative to the palm.

3. Proposed Hand Prosthesis

The proposed finger and thumb is used to introduce a multi-functional hand prosthesis shown in Figure 6. Table 1 shows summary of specification of the proposed hand prosthesis.

Mechanical Design and Mechanism

The hand prosthesis consists of four main units: first finger unit, second finger unit, third finger unit and a palm [refer Figure 6]. Since the index finger and the thumb play an important role than the other fingers in most of daily grasping activities [2] those two are taken as separate finger units and the middle, ring and little fingers together are taken as a separate unit for the actuation. First finger unit consists of the index finger, a motor and worm and wheel gears (reduction ratio 35:1) as shown in Figure 7(a). Second finger unit consists of three

Table 1. Specification of the proposed hand prosthesis.

Figure 6. 3D model of Proposed Hand Prosthesis.

Figure 7. Finger units of the proposed hand prosthesis. (a) First finger unit. This consists of three phalanxes, worm and wheel, and a motor; (b) Second finger unit. Middle, ring and little fingers; and a motor are available in this unit; (c) Third finger unit. This includes thumb with proximal phalanx and distal phalanx, and 2 motors.

fingers, worm and wheel gears (reduction ratio 35:1), and a motor. These three fingers correspond to the middle, ring and little fingers of the human hand [refer Figure 7(b)] and they are actuated together by a single motor using a single shaft as shown in Figure 7(b). Third finger unit consists of thumb, its worm and wheel gears (reduction ratio 35:1) and two DC motors [refer Figure 7(c)] which are perpendicular to each other. All four motors of the prototype of hand prosthesis have the same specification shown in Table 2. Finger structures, shafts and gears are fabricated from Al7075, stainless steel and Nylon 101 respectively using CNC machine.

All finger units are attached on the palm as shown in Figure 6. The proposed finger mechanism shown in Figure 1 is used to each finger in the first and second finger units. Motors of first and second finger units are connected to the MCP joint. Thumb can generate flexion/extension of the MCP and IP joints using motor −1 and opposition/apposition of thumb are generated using the motor −2 [refer Figure 6 and Figure 7(c)]. The hand prosthesis assists user to generate cylindrical grasp, hook grasp, lateral pinch and tip pinch and palmar pinch shown in Figure 8.

4. Experiment and Results

Simulations and experiments are carried out to compare and verify the motion generation of the proposed finger. Furthermore, experiments are carried out to verify the adaptation ability of the finger and hand prosthesis. The kinematic model is simulated in MATLAB/Simulink environment to achieve the fingertip motion. Index finger motion of prosthesis is captured using a camera by placing passive markers to each joint and captured data is used to derive joint angles.

The experimental set-up is shown in the Figure 9. As the controller, ATmega 2560 (Atmel) is used. The selected motor driver is a dual H-bridge motor driver (L298N). PD control is applied in the joint space to generate the torque command for the MCP joint. As the desired motion MCP motion shown in Figure 11 which is generated from the simulation is used.

Simulation and Experimental Results

Figure 10 shows the trajectory of the tip of the index finger derived from simulation. Origin of the coordinate system (0, 0) is located at the center of the MCP joint. MCP angle variation of index finger is shown in Figure 11. Figure 12

Table 2. Specification of the motor.

(a) (b) (c) (d) (e)

Figure 8. Achievable grasping of the hand prosthesis [18] . (a) Cylindrical grasp; (b) Hook grasp; (c) Lateral pinch; (d) Tip pinch; (e) Palmar pinch.

Figure 9. Experimental set-up.

Figure 10. Simulation results of fingertip trajectory.

displays the angle of PIP with respective to the proximal phalanx. The angle of DIP with respect to the middle phalanx is shown in Figure 13.

Figure 11. MCP angle of index finger.

Figure 12. PIP angle of index finger.

Figure 13. DIP angle of index finger.

Fingertip trajectory for human hand [26] , MATLAB simulation and the experimental values are compared in Figure 14. Minor deviation of the simulation and experimental results are caused by frictional losses, dimensional tolerances associated to fabrication process, differences in spring constants and errors of motion capturing. The fingertip trajectory in Cartesian space for fabricated hand prosthesis is almost similar to actual hand.

Snapshots shown in Figure 15 and Figure 16 illustrate the adaptation ability of middle phalanx of index finger and distal phalanx of index finger respectively. Cylindrical grasping of the hand prosthesis is shown from the sequence of snapshots in Figure 17 and Figure 18 shows sequence of snapshots for hook grasp generation. Figure 15, Figure 16 and Figure 17 have verified the adaptation ability of the hand prosthesis.

Figure 14. Comparison of finger-tip trajectory.

Figure 15. Sequence of images for adaptation of middle phalanx of index finger.

Figure 16. Sequence of images for adaptation of distal phalanx of index finger.

Figure 17. Snapshots of cylindrical grasping.

Table 3 shows motion ranges of MCP, PIP and DIP joints of the proposed index finger which are obtained from the simulation and experiments. Motion ranges of MCP, PIP and DIP joints of human index finger is also given in the

Figure 18. Snapshots of hook grasping.

Table 3. Movable ranges.

table for the comparison. MCP joint of the hand prosthesis have the same movable range as the human hand. However, motion range of PIP and DIP joints has slight variation. The fabricated proposed hand prosthesis is unable to achieve the exact ranges of the 3D model. These slight deviations cause due to friction between links and joints, dimensional accuracy and tolerances associated to fabrication process and actual spring constant ratios are different from calculated values.

5. Discussion and Conclusions

Table 4 shows a comparison of under-actuation of the proposed prosthesis with the prosthesis available in the literature. DOF and number of actuators of the hand of each prosthesis are given for the comparison. It is evident from the table that the proposed prosthesis and Vanderbilt Multigrasp Hand [20] are with the highest DOF by utilising the minimum number of actuators.

An under-actuated and self-adaptive finger was proposed together with a hand prosthesis. The finger consisted of mainly two four-bar mechanisms. Modified mechanism of the finger was used as the thumb mechanism. Furthermore, a hand prosthesis with the proposed fingers and thumb was introduced in the paper. The finger was capable of generating different passive angles for a PIP joint and a DIP joint for each flexion angle of MCP joint. In addition, DIP joint was capable of generating different angles for the same angle of PIP joint. Thumb mechanism allowed for powered articulated thumb opposition/apposition. The weight of prototype hand prosthesis is about 250 g. Kinematic analysis and computer simulations displayed that the finger mechanism was capable of performing required motions. Simulations were used to validate the movable ranges of the joint angles of finger. The movable ranges obtained from the experiments are 0˚ - 90˚, 5˚ - 90˚ and 2˚ - 88˚ for MCP, PIP and DIP respectively. Joint angle variation for MCP, PIP and DIP joints of the finger was obtained using simulations

Table 4. Comparison of under-actuation.

and experiments. The developed hand prosthesis offers a grasp adaptation using four actuators.

The hand prosthesis can be used to substitute a lost hand part of an amputee or can be used a terminal device for an arm prosthesis such as trans-radial prosthesis of trans-humeral prosthesis. Electromyography signals or electroencephalography signals of the wearer can be used to identify the motion intentions of the user to control the hand prosthesis, accordingly.

Acknowledgements

The authors would like to acknowledge the support given by Senate Research Council of University of Moratuwa, Sri Lanka (grant no: SRC/LT/2012/07).

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

Cite this paper

Gopura, R. and Bandara, D. (2018) A Hand Prosthesis with an Under-Actuated and Self-Adaptive Finger Mechanism.*Engineering*, **10**, 448-463. doi: 10.4236/eng.2018.107031.

Gopura, R. and Bandara, D. (2018) A Hand Prosthesis with an Under-Actuated and Self-Adaptive Finger Mechanism.

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