Structural numerical analysis of a three fingers prosthetic hand prototype

1 Sección de Estudios de Posgrado e Investigación, Escuela Superior de Ingeniería Mecánica y Eléctrica, Instituto Politécnico Nacional, Unidad Azcapotzalco, Av. de las granjas No. 682 col. Sta. Catarina, delegación Azcapotzalco C.P. 02550, México D.F. 2 Escuela Superior de Ingeniería Mecánica y Eléctrica, Instituto Politécnico Nacional, Unidad Profesional “Adolfo López Mateos”, edificio 5, 2do piso, col. Lindavista, delegación Gustavo A. Madero. C.P. 07738, México D.F. 3 Instituto Universitario de Investigación del Automóvil, Escuela Técnica Superior de Ingenieros Industriales, Universidad Politécnica de Madrid, Carretera de Valencia, km.7, 28031, Madrid, España.


INTRODUCTION
Human hand is able to realize a wide range of sophisticated movements that provides the ability to interact with the environment and communicate with other people (Clement et al., 2011).The loss of the upper limb may cause serious physical and psychological disorders in the life of an amputee (El Kady et al., 2010).
During the last 30 years, several prosthetic hands have been developed (Carrozza et al., 2005).There are mainly three kind of upper limb prosthesis available for people who lacks an extremity due to congenic factors or because they suffered an amputation (Watve et al., 2011).Otto Bock hand® is a popular and robust prosthesis that allows to exert a maximum force of 100 N but only has one degree of freedom (DOF) and weighs 600 g (Carrozza et al., 2004).
The human hand has 22 DOF and a maximum prehensile force of 500 N however most daily life activities require prehensile forces under 70 N (O' Toole, 2007), and an average weight of 500 g (Carozza et al., 2006).Many prosthetic hand prototypes been *Corresponding author.E-mail: ctorress@ipn.mx,manuel_carbajal@hotmail.com.
developed like the one that is shown in Cipriani et al. (2011) which has 16 DOF and is actuated by 4 electric motors, weighs 530 g and is able to lift a briefcase of 10 kg.Different kinds of actuators and transmissions have been used to produce the prosthesis movement like in Kargov et al. (2007) which presents a prosthesis that use an hydraulic pump and valves as actuators (Zhang et al., 2008); a rack-pinion transmission was used for its design.
The advancements in anatomy, material design and computer technology have made significant contributions to the evolution of the design of prosthetic hands during the last decades, and this has allowed a better approach to imitate the movement and mechanical behavior of a healthy hand, as in Pérez et al. (2012) that presents an anthropometric prosthetic hand prototype designed using computer tomography scan obtained from the healthy hand of a man.The use of analytical and experimental models in biomechanics generates important information that permits to understand the behavior of different systems subjected to mechanical loads (Naaji and Gherghel, 2009).Using the finite element method (FEM), it is possible to determine the stress distribution and strains, which could be difficult to obtain through experimental or analytical methods.
Examples of using FEM in biomechanics can be seen in Bougherara et al. (2009) and Omasta et al. (2012) where the stress distribution was analyzed in a knee prosthesis and in a trans-tibial prosthesis, respectively; these analysis were made using Ansys ® Workbench.Geng et al. (2008) shows the use of FEM in dentistry.
In Stanciu and Stanciu (2009), an analysis was presented using the finite element method in a prosthetic hand prototype that uses hydraulic actuators.The analysis is carried out using cosmosworks ® and was performed in two positions, one for the hand wide open and the other for an intermediate position.A 10 N load was considered and applied at the fingertip of the middle finger.All the fingers have the same structure so the analysis was not realized for the other fingers.
The mechanical design and theoretical analysis of a hand prosthesis actuated by muscle wires is shown in O'Toole (2007).The device that holds the wire was analyzed and the optimum wall thickness of the finger was determined through FEM.
The design of an end effector with anthropomorphic shape is presented in Ohol and Kajale (2009).Using ANSYS ® the FEM only for the critical parts was made, positioning the mechanism in extreme positions.
A static analysis in order to determine the prehensile force exerted by a prosthetic hand is presented in Jung and Moon (2008).Kargov et al. (2004) shows a comparison of the prehensile force distribution between a human hand, a non-adaptative hand prosthesis, and an adaptative prosthetic hand.
Despite the importance that the finite element method has in mechanical design, there are just a few research works in the prosthetic hands design area that uses this tool and there is no existence of any work in which all the mechanical components of the prosthesis are numerically analyzed during a grasping task.Taking into account these aspects, in this work the numerical simulation of the stress distribution in the components of a prosthetic hand prototype both in cylindrical and tip grasp was developed.

MATERIALS AND METHODS
Due to the computing capacity needed to perform the analysis, this work was divided in two phases.The first phase consisted the analysis of the finger mechanism operation and for this objective was used a hand simulator.Once the mechanism operation was assimilated; the second phase is aimed to perform the structural analysis of the prototype doing a cylindrical grasp and a tip grasp.
In order to apply the FEM to the prosthesis prototype, the following steps were considered: 1. Define a function that allows to know the geometrical position of the prototype elements knowing the actuator position, 2. Perform a static analysis of the hand simulator in order to determinate a set of equations that predicts the force that can be exerted at the tip as a function of the finger position, 3. Solve for different positions within the operating range of the mechanism the equations obtained in the static analysis, 4. Perform a FEM analysis of the hand simulator in ANSYS Workbench 2.0, 5. Determinate the areas with the highest magnitude of mechanical stress in the hand simulator, 6. Realize a static analysis of the prototype in order to determine the forces that take place during the cylindrical grasp, which is a grasp used to grab objects with a continues form like bars, and the tip grasp, 7. Execute a FEM analysis of the hand prosthesis prototype doing a cylindrical, and a tip grasp.
All this previous steps were followed in the mentioned sequence in order to perform the analysis.
In this work SolidWorks® Premium 2012 x64 Edition was used to handle the CAD design and fix the prototype in the desired position; Matlab R2010a was used to solve the set of equations and ANSYS Workbench 2.0 Framework Release 13.0.0 in order to realize the FEM analysis.

Fingertip force analysis
The prototype proposed has 3 fingers, the thumb, index and middle finger (Figure 1).Index and middle finger has 1 DOF and the thumb has 2 degrees of freedom, one for the open-close movement and the other for the opposition movement.
The hand simulator structure is the four-bar linkage shown in Figure 2. The movement of the finger is given by a crank and slider mechanism, which is driven by a servomotor.
Figure 3 shows the components that form the hand simulator.The lengths of the components are shown in Table 1.These lengths were used for the calculation of the prostheses elements position and for the estimation of the forces that take place during the grasp.
Figure 4 shows the angles of the hand simulator elements.Within the operating range of the mechanism, the elements do not present any interference, and do not align, so the mechanism does not present a critical position, this is why it is proposed to perform a   Proximal phalange (Bar 1) 7.
Distal phalange static analysis instead of a dynamic one.The first step to make the static analysis of the hand simulator, is obtain the geometric positions for each one of the elements.
The angle θ2 can be obtained through geometric relations, so θ2 is given by the following equation: Elements L2, L3 and L4 can be analyzed as a four-bar linkage, then  by using the Freudenstein equation (Cardona and Clos, 2001)we have: Where: H2 is the vertical distance between servomotor´s shaft and the bottom of element L3, which can be obtained by the following equation: H and X are the vertical and horizontal distance between the servomotor's shaft and the beginning of the distal phalange, respectively, with values H=38.15 mm, X=11.05 mm.
Once θ3 is obtained, the horizontal and vertical projection of element L4 are calculated in order to obtain θ4.Knowing that the angle between θ4 and θ5 is constant, with a value of 26.5651°, θ5 is obtained as follows: The elements L5, L6 and L7 are analyzed as a four-bar linkage, and using the Freudestein equation we have: Where: Calculating the horizontal and vertical projections of element L6 in order to get θ6: We can proceed to describe the position of all the components; a static analysis of the finger was done in order to obtain the fingertip force.This force is the resultant of the pressure exerted by the finger on the manipulated object and this force is used as a simplification in the calculations.Figure 5 shows the forces considered for the analysis.
Assuming a concentrated force at the finger tip (Figure 6), this can be represented by the following non-linear equation: In the free body diagram it is possible to observe that the force transmitted by the crank to the connecting rod is given by the following equation: This force is transmitted to the slider.As the horizontal force is the only one that produces work, it is considered just this force for the next equation: The force transmitted to element 5 is represented by: In order to obtain the fingertip force, it is necessary to analyze simultaneously the elements 6, 7 and 8.For equilibrium, for the element 6 we obtain: Analogously, for the element 7 we get: The corresponding equations for element 8 are: The previous equation 9 form a 9×9 linear system of equations that can be represented in a matrix form as follows: Where: A program was made using MATLAB® to solve Equation 20.In an iterative way, a torque of 1 N-m was chosen to represent the torque the motor develops.
Within the range of -60° to 60°, the fingertip force was calculated for every 10° (Graph 1), and in the position where the mechanism transmits the highest force, a structural static analysis was performed.

FEM analysis of the hand simulator
After the critical position of the mechanism was calculated, a FEM analysis was performed in order to assure that the hand simulator will be able to withstand the loads produced during its operation.
In order to carry out a FEM simulation, the next steps should be followed: 1. Build a geometric model, 2. Create a finite element mesh for the prototype, 3. Define loading conditions, 4. Apply boundary conditions, 5. Solve the analysis, 6. Generate the results analysis.
During the discretization process, tetrahedral elements were used due to the complexity of the prototype surface, and a mesh refinement was used in the regions on high stress, resulting in 67872 elements.
The material employed in the simulation, because of its low density and high strength, was a titanium alloy with the following properties: The simulation was performed for the position θ1=-60° as in this position the force transmit is the greatest, with a torque of 1 N-m applied on the crank.

Grasping force analysis
Once this simulator analysis has been fully understood, a second analysis for the hand prosthesis prototype was made in order to determine the stress field during the grasping task.

Cylindrical grasp
Movement of the index and middle finger of the prosthesis through a four-bar linkage is shown in Figure 7, while the thumb is moved through a crank-slider mechanism.
For the cylindrical grasp, it was considered the handling of a cylindrical body with a diameter of 57 mm which represents a 600 ml water bottle like the one used in Kargov et al. (2004).
For the analysis is assumed that each finger touches the body in just one point and that the forces are perpendicular to the surface on the contact point.Figure 8 shows the forces involved in the cylindrical grasp, neglecting gravity.Since the entire system is at static equilibrium, each individual part showed is in equilibrium.The analysis was performed for each   Connecting Rod 4. Slider 5. Linkage 6.
Distal phalange  finger in the same way as in the hand simulator.Figure 9 shows the components of the thumb.The lengths of the components are given in Table 2.
Figure 10 shows the reference angles of the thumb elements, these angles were calculated using the same methodology employed for the hand simulator.
From the free body diagram shown in Figure 11, we can derive the next equation: This force is transmitted to the slider.As the vertical force does not produce any work, it is neglected and the horizontal force is the only one considered, this one is calculated by the equation: The force over the element 5 is defined by: In a vector form, the force applied by the thumb is given by the equation: We now proceed to calculate the forces of the index and middle finger.These fingers have the same structure (Figure 12), and in order to know the position of its elements a set of reference angles were defined (Figure 13).Elements l1, l2 and l3 are modeled as a four-bar linkage using the Freudestein equation (Table 3), θ3 can be obtained by the next equation: θ2 is obtained from the following equation:  The remaining reference angles are obtained the same way as with the hand simulator and the force exerted by the finger is calculated.
Figure 14 shows the forces acting on the finger elements.
From the crank free body diagram the next equation can be derived: This force is transmitted to the proximal phalange and a set of equations can be obtained in order to calculate the force at index and middle fingertips.
Where: In a vector form, the force exerted by the finger is given by: To determine the torques used in the simulation, it was considered that the 3 fingers exert a 10 N prehensile force on the palm.So it was assumed that the force applied by the index and middle finger has a magnitude of 2.5 N in the X direction.Using Equation 50and knowing that for this position θ1 = 164.9°and θ8 = 123.08°,we can obtain the index finger force FIndex = 4.58 N. The linear system equation that represents the index finger force is solved in an iterative way in order to obtain the torque that has to be applied, its magnitude is MIndex = 0.398 N-m.
With this applied torque, the index and middle finger exert a force of 3.84 N in the vertical direction which should be compensated by the thumb in order to achieve static equilibrium.From Equation 40, the vertical force exerted by the thumb is F = FThumb•sinθ9, so the force applied is For this position θ1= 27.08° and θ9= 45.93°.Solving for Equation 51 FThumb = 10.68 N. Using the set of equations that models the thumb force, the torque was calculated being its magnitude MThumb = 1.88 N-m.With this torque, the horizontal force exerted by the thumb is 7.41 N.
Using the calculated values of the torques, the total force in the palm direction has a value of 12.41 N; this value is greater than the 10 N that was assumed.Recalculating the torques but taking into account the ratio FIndex/FThumb = 0.337, it was obtained that MThumb = 1.51 N-m and MIndex = 0.32 N-m, so the force in the palm has a magnitude of 9.96 N.These are the values used in the simulation.

Tip grasp
For the tip grasp analysis (Figure 15) it was considered a cylindrical body of 7 mm diameter which represents a pen.For this grasp it is considered that only acts the index finger and the thumb so the only set of different parameters are the positions of the cranks and the point where the force is applied, this makes it possible to use the same equations developed previously.
In order to determine the torques needed for the simulation, a 10 N grasping force was considered.The vertical force exerted by the thumb and the index was established to be of 10 N, and the force in the horizontal direction neglected.

FEM analysis of the prosthetic hand prototype
Once the torques have been calculated, the FEM analysis was performed with the goal of assure that the prototype would be able to support the loads that are involved both in the cylindrical and tip grasp.
Based on the geometry model, a FE mesh was carried out using the ANSYS automeshing techniques.Hexahedral elements were used in the primitive geometries and tetrahedral elements in complex regions.A mesh refinement was used in the regions of nonlinear contact in order to obtain convergence in the simulation.With this parameters established, the total finite elements in the model were 171,284 for the cylindrical grasp and 83,467 for the tip grasp (Figure 16).
The material used in the simulations was a titanium alloy, the same as the one used in the simulation of the hand simulator.The base of the prosthesis was fixed restraining the movement of the bottom; also the movement along the symmetry axis of the grasped objects was constrained in order to prevent sliding (Figure 17).
The regions where exist contact with bolts, like the one shown in Figure 18, were defined as a frictionless contact, resulting in a total of 28 nonlinear contacts for the cylindrical grasp simulation and 19 for the tip grasp analysis.In the tip grasp analysis, in order to decrease the computing capacity required, the middle finger was suppressed because is not under any load.Due to the high non linearity of the simulation the load was applied in sub-steps

RESULTS
Graph 1 shows the fingertip force applied by the hand simulator.The maximum force occurs for the position θ 1 = -60° and has a magnitude of 49.95 N, and the minimum force within the operating range is 10 N.This last value could be used as in Jung and Moon (2008), De Laurentis and Mavroidis (2002), Yang et al. (2009), Huang et al. (2006), Kawasaki et al. (2002), andCarozza et al. (2004) through static analyses it is shown that these designs can exert 10 N or less at the fingertip.
As a result of the hand simulator analysis it was obtained the stress distribution of the mechanism (Figure 19).The force reaction obtained at the fingertip was 54 N, which is a value close to the one calculated analytically.
Observing the field stress for the analyzed position, the greatest value obtained, based on the von Mises yield criterion, is located in the element showed in Figure 20, which is used to link the servomotor to the crank, and has a magnitude of 249.46 MPa.
As the greatest stress in the simulator is lower than the yield stress, the design of the elements will not fail.The minimum safety factor obtained is: The area where it is located the highest stress is a zone that has an abrupt change of direction, acting this as a stress concentrator, so adding a fillet would reduce this value.
All the remaining elements present lower stress values so its mechanical integrity is assured.A set of modifications were done in the component at zones where there were located the maximum stresses, adding radius in the corners in order to eliminate the stress concentrators and the hand simulator was analyzed again.With these modifications, the maximum stress was located in the same component but with a magnitude of 227.01 MPa.With this maximum stress, the minimum safety factor obtained is 4.09.With the cylindrical grasp simulation, a maximum stress with a value of 151.9 MPa was obtained (Figure 21).This stress was located in the crank of the thumb mechanism (Figure 22), which is an expected result as for this element, it applied the largest torque considered in the simulation.A safety factor of 6.12 is obtained with this maximum stress.
The applied force in the horizontal direction by the thumb has a magnitude of 6.05 N and the index and  middle finger exert a force of 1.85 N in the simulation, while analytically values of 5.97 N and 2.01 N was calculated respectively, hand the prehensile force has a magnitude of 9.75 in the horizontal direction (Figure 23).For the tip grasp simulation, it was obtained a maximum stress of 73.7 MPa, and the minimum safety factor was 12.6 (Figure 24).
For this case, the maximum stress was located at the crank of the index finger mechanism, while the crank of the thumb presented a maximum stress of 72.11 MPa.
The most important result is that the highest stress at the exterior of the prosthetic hand prototype has a magnitude of 28.19 MPa, and this makes it possible to use another material instead of titanium and this would benefit the cost of the prosthesis as the cost of this material is high.

DISCUSSION
In this work, static analysis was carried out in order to predict the loads in the components of a prosthetic hand prototype, and this load were used in a FEM analysis in order to determine the mechanical stress in the prototype during the cylindrical and tip grasp whilst Jung and Moon (2008) realizes static analysis in order to determine the fingertip force and the results are compared with experimental results.The analytical and experimental results agreed, so it is valid to use the static analysis in order to know the loads in the components of the prototype.
Ohol and Kajale (2009) realizes a FEM analysis for the critical parts of an end effector with anthropomorphic shape but it is unclear the loads applied in the simulation on the other hand, in this work, the FEM analysis was realized for the entire prototype and the loads applied were the maximum loads that are developed on the elements of the prototype during its operation.
Through the use of FEM analysis, O'Toole (2007) determines the material for a hand prosthesis, it also determined optimal wall thickness for some elements of the prosthesis, and in this work FEA is used in order to know the stress distribution in the components of the prototype that will allow to determine the optimal dimensions of the components and in this way reduce the total weight of the prosthesis.Stanciu and Stanciu (2009) in its analysis applies a 10 N load and considers two positions, one for the hand wide open and other for an intermediate position.The first one is a position that would make difficult to grab things with the hand and the second one does not assure that in this position the maximum loads will develop while in this work it is considered only positions where maximum loads take place.

Conclusion
It has been shown that, in order to have a grasping force of 10 N, it is necessary to apply a torque with a magnitude of 1.51 N-m.Also, it is notorious as well that the friction at the unions was neglected in the calculations, so the real torque in order to achieve the force with the required magnitude would be higher.
The actual actuators used in this prototype have a torque with a lower magnitude than the needed.Taking into account this, it is proposed to study the option of using a different servomotor with a higher torque in order to transmit the movement to the fingers.It must be noted that the fabrication of this prototype is based in a stereolithography technic and it is necessary to generate the development through the use of advanced manufacturing techniques.

Figure 1 .
Figure 1.Design of the prosthetic hand prototype.

Figure 3 .
Figure 3. CAD design of the hand simulator.
Finally we can get θ9 using the next equation:

Figure 6 .
Figure 6.Position of the analysis.

Figure
Figure 8. Forces involved in the cylindrical grasp.

Figure 9
Figure9shows the components of the thumb.

Figure
Figure 9. Thumb elements
steps as with the hand simulator, a set of equations is obtained in order to calculate the fingertip force of the thumb

Figure 11 .
Figure 11.Free body diagram for the thumb elements.

Figure 14 .
Figure 14.Free body diagram of the index and middle finger.

Figure 15 .
Figure 15.Position for the tip grasp.

Figure 16 .
Figure 16.Mesh of the prototype.
) the force of the index is obtained and has a magnitude of FIndex = 10.65 N. From the set of equations for the index finger force the torque has a magnitude of MIndex = 0.691 N-m.Considering that for the thumb θ1 = 156.45°and θ9 = 96.32°,and solving Equation (40) FThumb = 10.06N. Using the set of equations for the thumb finger force the torque has a magnitude of MThumb = 0.668 N-m.

Figure 20 .
Figure 20.Element with the greatest stress.

Table 1 .
Lengths of the components.

Table 2 .
Lengths and nomenclature of the elements.

Table 3 .
Lengths of the elements of the index and middle finger.
Figure 19.Hand simulator stress distribution.