# Optimization of Thumb Prosthesis Design by Using Five Performance Criteria

Authors: Nestor Tsamo; Denis Tcheukam Toko; Pierre Kisito Talla
Journal: Applied Bionics and Biomechanics
 (2022)

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## Abstract

The thumb prosthesis mechanism is optimally designed by using five performance criteria including the following: least square structural error, mechanical manufacturing imprecision error, driving optimal torque, mechanical strength reliability, and production cost of the thumb mechanism. This paper was devoted to the optimization of the thumb prosthesis’s mechanism by taking into consideration the manufacturing cost model based on machining cost theory which took into detail the shape of the workpieces and the strength reliability of all the parts composing the entire mechanism. Every optimization problem displays a particular set of an independent vector of optimal parameters, showing the impact of each objective function on the configuration of the prosthetic device. The multiobjective optimization showed that the mechanical reliability and the production cost included in any combination of the simultaneous optimization enabled the achievement of the same optimum variables design, though with some exceptions. With the inclusion of the labor charges, the depreciation rate of the equipment, and production assets in the mathematical’s manufacturing cost model, the optimal manufacturing cost generated from the numerical simulation was 501.0021 USD. Therefore, the global manufacturing cost and the mechanical strength reliability of the whole prosthesis mechanism have a real impact on the customization of the structure, due to the stochastic nature of the trajectory of the cutting tools during the manufacturing processes.


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## Body

## 1. Introduction

The invention of new prosthetic devices over the past decades has observed advances as a consequence of recent technological development, advancing in the aspect of greater dexterous hand devices. The use of a mechanical prosthesis to replace a missing limb is one of the most fascinating challenges faced by prosthetists in the prosthetic rehabilitation protocol [1–4]. The level of autonomy and the capabilities of performing social activities, trade, and daily tasks negatively influenced amputees. However, the state-of-the-art prosthesis failed to coalesce smooth functioning, reliability, suitable personalized aesthetic, adequate maintenance, affordability, and sometimes adequate grasping [5, 6]. Most prostheses fail to closely match the user-specific requirement and have poor design, resulting in the abandonment of the artificial device by the amputees [7]. One of the most popular works on prosthesis optimization in the literature has been done by Vinet et al. [8]. They obtained all the bending angles of the hand morphology using 3D graphical software CATIA on four planar bar mechanisms.The thumb plays a paramount role in the various grasping modes. Liken to unfavorable conditions such as sickness, accident, or congenital malformation, the rehabilitation process of the thumb is greatly affected especially by its contribution to six grasping modes retained by many prosthetists [9, 10]. The design of a thumb prosthetic device is critical since forty percent of the entire functionality of the human depends on the hand. In housekeeping and laundry, the grasping mode is dominated at sixty percent by the contribution of the thumb. Many prosthetists faced the challenge to choose a simple, accurate, and less mobility or degree of freedom, which allows equal force distribution. The least square structural error on the second phalanx of the thumb and the trajectory of the model is determined by the end reflector which is the fingertip [11, 12]. To obtain the safe, smooth, accurate, and affordable functionality of the thumb, one of the most interesting solutions is the use of the planar articulated crossed four-bar mechanism to design the prosthetic mechanism as shown in Figure 1. The synthesis followed by the design methodology of mechanisms has been the subject of several investigations [1–4, 8, 13].Figure 1
Schematic diagram of the thumb finger fit out with its driving system.The main objective of this paper is to introduce into upper limb optimization processes the mechanical reliability of each element of the basic mechanism, used to design the artificial device. Their applications are numerous and their synthesis is generally done by reference ton input positions and n output positions of the considered parameters [14]. The four-link articulated planar mechanism proposed for synthesis finds application in a hand prosthesis that can perform various types of grasping daily objects. Its optimal synthesis will be made by reference to n positions of the coupling and output members. For this purpose, two performance criteria are used: the least square structural error on the angle of flexion of the second phalanx of a thumb and the maximum motor torque necessary to balance a given force at the tip of this finger. This paper will also integrate the global manufacturing cost model taking into account the stochastic nature of the trajectory of the cutting tools during the machining process on the numerical control machine center. This cost will also integrate the social charges, the labor charges, and the depreciation rate of the equipment and productivity asset [15]. All these parameters were adjustable according to the economical standard of the area of the production. In order to study the virtual optimal mechanism, the mathematical model of the objective function relative to the various performance criteria is established [16]. Assuming that in the biomechanics of artificial limb, all the fingers contribute actively to accomplish if necessary each of the six grasping object modes retained by Pelletier and Vinet, Ruben, and Carrozza et al., taking into account the fact that the humanoid thumb mobility in particular among other fingers, the optimal design will be carried out on the base of minimization or maximization of single or a set of objectives function submitted to the various constraints [17–19]. Finally, the entire optimization process will be necessary to appreciate the influence of the said criteria on the prosthesis’s human hand design. After the analysis of the mechanism consisting of the development of all the mathematical relations necessary to be summited to the optimization processes, for its synthesis, we will make the optimal synthesis of the said mechanism by reference to seven positions. Using the data relating to the thumb, the results obtained will be compared and discussed according to the state of art [20].To mechanically simulate the different grasping mode of the object by an artificial hand, planar articulated mechanisms with one simple degree of freedom are generally recommended in hand prosthetic applications [21–25]. Based on this configuration, 1DOF four-crossed bar mechanism is considered in this study to design the thumb which is usually optimized by the means of transmission angle and energy consumed which have a significant effect on the mechanism design. To improve the use of these devices for the patients in need, new additional criteria will be modeled and integrated with the optimization process as objective functions. The first is the manufacturing cost, which would impact the affordability of the prosthesis by the amputees, and the second is the reliability of the setting mechanism and its effect on the prosthesis design. In the first approach, each criterion will be optimized individually and the three first criteria will be optimized simultaneously. Strength reliability and manufacturing cost will be considered individually, and finally, we will carry out the optimization of all the five criteria simultaneously [20].


## 2. Material and Methods

The thumb’s mechanism description, its performance criteria, and its objective functions are presented in this section.


### 2.1. Thumb Finger Description

The thumb finger associated with the four-bar mechanism fit out with its driving system is depicted in Figure1.Figure1 is made of a cable connected both to the driving link at the point TxT,yT and to the driving pulley at the point CxC,yC whose connection angle is ρ. The structure of thumb mechanism is added in the revised manuscript as Figure 2.Figure 2
Structure of thumb finger mechanism without its driving system.Table1 defines the different parameters shown in Figure 1.Table 1
Designations of different parameters shown in Figures1 and 2 [8, 20, 24–27].
ParametersDesignationsPThe applied force with an angle ofθP to the axis of the corresponding phalanxFTmaxMaximum tension developed in the cableP1, P2, P3Lengths of proximal, middle, and distal phalanges (P3=0 for the thumb), respectivelyP3’Effective length of the distal phalanx,P3’=0 for the thumbQM, AM, BA, and QBFixed, driving, junction, and input bar, respectivelyTP1, φ, and ψBending angles of phalanges with respect to the plane of the palmTP2, TP3Bending angles ofP2 and P3 were measured from one phalange to another, respectivelyθPAngle between force at the tip of the finger with the axis of the phalanxθ1, θ2, θ3, and θ4Angles of the phalanges of the mechanism with respect to a plane parallel to the plane of the palmQTrapezometacarpal articulationM1Thumb metacarpalP2’Operative length of the second phalanxP2TP1Bending angle ofM1 according to palmar planTHMPBending angle ofP1 according to M1 (fixed angle)θAThe angle between the second phalanx axis and the various linksθBThe angle betweenM1 and the various linksr1Length of the input bar,r1=QBr2Length of junction bar,r1=BAr3Length of driving bar,r1=AMr4Length of the fixed bar,r1=QMDP1, DP2, and WR3Diameters of phalangesP1 and P2 and width of links, respectivelyEXCEccentricitydPulley diametertmBar thicknessFreudenstein’s relation [23–25, 28] between θ1i and θ2i at any mechanism position i is given by
(1)k1cosθ1i−θ4+k2cosθ2i−θ4−k3=cosθ1i−θ2i,where k1=r4/r2, k2=r4/r1, k3=r12+r22−r32+r42/2r1r2, r1=P12+EXC2, θ1i=TP1i−arctgEXC/P1, θ2i=θA+TP1i+TP2i, r1=P12+EXC2.sinTHMP−arctgEXC/P1/θB, and θB=arctgP12+EXC2.sinTHMP−arctgEXC/P1/M1+P12+EXC2.cosTHMP−arctgEXC/P1.The values of some parameters in Equation (1) are similar to those used in [24] and are given in Table 2.Table 2
Dimension values of thumb’s mechanism [20, 24].
ParametersValuesDP115 mmDP213 mmWR35 mmP143.5 mmP229 mmP3’0 mmEXC2 mmθP90°TP1118°TP1229°TP1340°TP1451°TP1561°TP1673°TP1784°TP2114.274°TP1218.027°TP2324.022°TP2432.142°TP2541.416°TP2655.649°TP2775.760°TP330°P45 NFTmax400 NThe design parameter value bounds of the thumb’s mechanism are taken in [14] and are given in Table 3.Table 3
Design parameter value bounds of thumb’s mechanism [14].
Design variable’s designationθA (degree Celsius)r2 (mm)r3 (mm)r4 (mm)θ4 (degree Celsius)tm (mm)xT (mm)yT (mm)d (mm)Vector of design variablex1x2x3x4x5x6x7x8x9Min87.115572.55053600.79410-510Max82.0631574.9947.53801.58820517The vectorX of the independent variables of the design is defined as
(2)X=x1,x2,x3,x4,x5,x6,x7,x8,x9T=θA,r2,r3,r4,θ4,tm,xT,yT,dT,since the length of r1 is fixed and θA=θ2i−TP2i+TP1i+THMP.In this study, the weight of the thumb mechanism was assumed to be negligible due to the dimensionless bars constituting the four-crossed bar mechanism used to model this artificial thumb [20].


### 2.2. Performance Criteria and Objective Functions of the Thumb Prosthesis Mechanism

The thumb prosthesis design is optimized by using the following five performance criteria:(i)
Least square structural error on the bending angle of the second phalanx should be minimized to ensure that the prosthetic finger should align at the prescribed positions(ii)
Mechanical error on the bending angle of the second phalanx, due to dimensional tolerances and clearances on the articulations, should be minimized(iii)
Maximum shaft driving torque should be minimized and able to counterbalance a grasping force applied at the end of the thumb finger(iv)
Strength reliability of the mechanism should be maximized to ensure the mechanical strength during grasping, holding, or pinching operations(v)
Manufacturing cost of the mechanism should be minimized to evaluate its effect on the design variablesThe objective function associated with the least square structural error on bending angleTP2 is defined as
(3)f1x=16∑i=17θ2ic−θ2id2,where θ2id=x1+TP1i+TP2i is the desired angle and θ2ic is the computed angle. It is also called mean square error, which describes the error, given in the form of least square function, describing the error observed between the computed value θ2ic and the given value θ2id of the bending angle between the palmer plan and the second phalanx TP2 of the thumb [14].The objective function associated with the mechanical error on bending angleTP2 is defined as
(4)f2x=16∑i=170.0032∂θ2i∂r1r12+∑j=19∂θ2i∂xjxj22.By assuming that the frictional forces at joints are negligible, the objective function associated with the maximum shaft driving torque is defined as(5)f3x=maxMP=maxx9MM2x7sinβ+x8cosβ,where β is the traction angle between the flexible cable on the driving system and the driving bar, MP is the driving torque applied to the shaft, and MM is torque reduced to point M of the driving bar. The objective function associated with the strength reliability of the mechanism is defined as
(6)f4x=R1.R2.R3,whereR1, R2, and R3 are the reliability of driven, junction, and driving bars, respectively [28, 29]. Since the prostheses are manufactured in very small sets, the manufacturing computer-aided design is appropriate for this purpose [30]. The manufacturing unitary cost of the jth manufacturing operation on the ith bar of the mechanism is
(7)Cuij=Cp+CTm−CSN+πLijDijCm6×104×V×f+CS103×f×C−KV−K−1,where Cp=Ts/N.Ap/60, CT=Cm/60.πLij.Dij/103V.f, CTm=Cm/60∑j=13Tj+Ttm, and Co=CSπLij.Dij/103f.C−KV−K−1−1/Nare the sum of the expenses related directly to the preparation, cutting duration, out of cutting time, and the cutting tool, respectively; Cs is the cost cutting tools; Ap is the global rate of exploitation of the preparation section, general expenses, and labor; Cm=A+F+R+L+E+S is hourly machine cost; A, F, R, L, E, and S are technical amortization, financial expenses, maintenance expenses and repair, expenses on local or on clutter, expenses on energy, and wage costs and social, respectively; V, N, f, and k are linear speed, mass production set, feed rate in turn/tooth, and Taylor constant function of material of the cutting tool, respectively; Dij are the cutting tools used relatively at ith bar and jth manufacturing operation (milling cutter 2 cuttings ∅20, cutter 2 cuttings ∅6, drill ∅2, drill ∅4, piloting milling cutter to counter boring ∅2×6 and machining tapping M2); Lij=Lsij=Lcij=Lpij=Llij=Ltij are the tool trajectories relative to the facing milling, lateral milling, drilling, counter boring, and tapping of the ith bar of the mechanism and the jth manufacturing operation, respectively; T1, T2, and T3 are the holes 1, 2, and 3, respectively, and Ts is execution duration allowed to the preparation of a set of workpiece. The manufacturing cost of the junction bar is
(8)C3=4CF+πLs31D31+Lc32D32+Lp33D33+Ll34D34Cm6×104V.f+CS103f.C−KV−K−1,where Ls31 is the facing milling, Lc32 is the lateral milling, Lp33 is the drilling, and Ll34 is the counterboring. The manufacturing cost of the driving bar is
(9)C1=4CF+πLs11D11+Lc12D12+Lp13D33+Ll14D14Cm6×104V.f+CS103f.C−KV−K−1,where Ls11 is the facing milling, Lc12 is the lateral milling, Lp13 is the drilling, and Ll14 is the counterboring. The manufacturing cost of the driven bar is
(10)C2=5CF+πLs21D21+Lc22D22+Lp23D33+Ll24D24+Ll25D25Cm6×104V.f+CS103f.C−KV−K−1,where Ls21 is the facing milling, Lc22 is the lateral milling, Lp23 is the drilling, Ll24 is the counterboring, and Lt25 is the tapping. The global manufacturing cost of the mechanism is
(11)f5x=C1+C2+C3=13CF+πM.LD,with
(12)LD=Ls11D11+Lc12D12+Lp13D13+Ll14D14+Ls21D21+Lc22D22+Lp23D23+Ll24D24+Ll25D25+Ls31D31+Lc32D32+Lp33D33+Ll34D34,where f5x is in USD and the dimensions of all mechanism pieces are in millimeter [26]. The objective functions related to individual and multiobjective optimization are
(13a)F1=f1x,(13b)F2=f2x,(13c)F3=f3x,(13d)F4=f4x,(13e)F5=f5x,(13f)f6x=f1x,f2x,f3x,(13g)f7x=f1x,f2x,f3x,f4x,(13h)f8x=f1x,f2x,f3x,f5x,(13i)f9x=f1x,f2x,f3x,f4x,f5x.Each of these multiobjective functions consists of a set of objective functions which are simultaneously optimized. The above functions expressed in terms of design variables are equality constraintshix=0 and inequality constraints gix≤0 according to the MATLAB optimization toolbox. The convergence criteria on the design variables xi, the objective functions fix, and the constraints gix are, respectively, equal to 10−7, 10−9, and 10−13.


## 2.1. Thumb Finger Description

The thumb finger associated with the four-bar mechanism fit out with its driving system is depicted in Figure1.Figure1 is made of a cable connected both to the driving link at the point TxT,yT and to the driving pulley at the point CxC,yC whose connection angle is ρ. The structure of thumb mechanism is added in the revised manuscript as Figure 2.Figure 2
Structure of thumb finger mechanism without its driving system.Table1 defines the different parameters shown in Figure 1.Table 1
Designations of different parameters shown in Figures1 and 2 [8, 20, 24–27].
ParametersDesignationsPThe applied force with an angle ofθP to the axis of the corresponding phalanxFTmaxMaximum tension developed in the cableP1, P2, P3Lengths of proximal, middle, and distal phalanges (P3=0 for the thumb), respectivelyP3’Effective length of the distal phalanx,P3’=0 for the thumbQM, AM, BA, and QBFixed, driving, junction, and input bar, respectivelyTP1, φ, and ψBending angles of phalanges with respect to the plane of the palmTP2, TP3Bending angles ofP2 and P3 were measured from one phalange to another, respectivelyθPAngle between force at the tip of the finger with the axis of the phalanxθ1, θ2, θ3, and θ4Angles of the phalanges of the mechanism with respect to a plane parallel to the plane of the palmQTrapezometacarpal articulationM1Thumb metacarpalP2’Operative length of the second phalanxP2TP1Bending angle ofM1 according to palmar planTHMPBending angle ofP1 according to M1 (fixed angle)θAThe angle between the second phalanx axis and the various linksθBThe angle betweenM1 and the various linksr1Length of the input bar,r1=QBr2Length of junction bar,r1=BAr3Length of driving bar,r1=AMr4Length of the fixed bar,r1=QMDP1, DP2, and WR3Diameters of phalangesP1 and P2 and width of links, respectivelyEXCEccentricitydPulley diametertmBar thicknessFreudenstein’s relation [23–25, 28] between θ1i and θ2i at any mechanism position i is given by
(1)k1cosθ1i−θ4+k2cosθ2i−θ4−k3=cosθ1i−θ2i,where k1=r4/r2, k2=r4/r1, k3=r12+r22−r32+r42/2r1r2, r1=P12+EXC2, θ1i=TP1i−arctgEXC/P1, θ2i=θA+TP1i+TP2i, r1=P12+EXC2.sinTHMP−arctgEXC/P1/θB, and θB=arctgP12+EXC2.sinTHMP−arctgEXC/P1/M1+P12+EXC2.cosTHMP−arctgEXC/P1.The values of some parameters in Equation (1) are similar to those used in [24] and are given in Table 2.Table 2
Dimension values of thumb’s mechanism [20, 24].
ParametersValuesDP115 mmDP213 mmWR35 mmP143.5 mmP229 mmP3’0 mmEXC2 mmθP90°TP1118°TP1229°TP1340°TP1451°TP1561°TP1673°TP1784°TP2114.274°TP1218.027°TP2324.022°TP2432.142°TP2541.416°TP2655.649°TP2775.760°TP330°P45 NFTmax400 NThe design parameter value bounds of the thumb’s mechanism are taken in [14] and are given in Table 3.Table 3
Design parameter value bounds of thumb’s mechanism [14].
Design variable’s designationθA (degree Celsius)r2 (mm)r3 (mm)r4 (mm)θ4 (degree Celsius)tm (mm)xT (mm)yT (mm)d (mm)Vector of design variablex1x2x3x4x5x6x7x8x9Min87.115572.55053600.79410-510Max82.0631574.9947.53801.58820517The vectorX of the independent variables of the design is defined as
(2)X=x1,x2,x3,x4,x5,x6,x7,x8,x9T=θA,r2,r3,r4,θ4,tm,xT,yT,dT,since the length of r1 is fixed and θA=θ2i−TP2i+TP1i+THMP.In this study, the weight of the thumb mechanism was assumed to be negligible due to the dimensionless bars constituting the four-crossed bar mechanism used to model this artificial thumb [20].


## 2.2. Performance Criteria and Objective Functions of the Thumb Prosthesis Mechanism

The thumb prosthesis design is optimized by using the following five performance criteria:(i)
Least square structural error on the bending angle of the second phalanx should be minimized to ensure that the prosthetic finger should align at the prescribed positions(ii)
Mechanical error on the bending angle of the second phalanx, due to dimensional tolerances and clearances on the articulations, should be minimized(iii)
Maximum shaft driving torque should be minimized and able to counterbalance a grasping force applied at the end of the thumb finger(iv)
Strength reliability of the mechanism should be maximized to ensure the mechanical strength during grasping, holding, or pinching operations(v)
Manufacturing cost of the mechanism should be minimized to evaluate its effect on the design variablesThe objective function associated with the least square structural error on bending angleTP2 is defined as
(3)f1x=16∑i=17θ2ic−θ2id2,where θ2id=x1+TP1i+TP2i is the desired angle and θ2ic is the computed angle. It is also called mean square error, which describes the error, given in the form of least square function, describing the error observed between the computed value θ2ic and the given value θ2id of the bending angle between the palmer plan and the second phalanx TP2 of the thumb [14].The objective function associated with the mechanical error on bending angleTP2 is defined as
(4)f2x=16∑i=170.0032∂θ2i∂r1r12+∑j=19∂θ2i∂xjxj22.By assuming that the frictional forces at joints are negligible, the objective function associated with the maximum shaft driving torque is defined as(5)f3x=maxMP=maxx9MM2x7sinβ+x8cosβ,where β is the traction angle between the flexible cable on the driving system and the driving bar, MP is the driving torque applied to the shaft, and MM is torque reduced to point M of the driving bar. The objective function associated with the strength reliability of the mechanism is defined as
(6)f4x=R1.R2.R3,whereR1, R2, and R3 are the reliability of driven, junction, and driving bars, respectively [28, 29]. Since the prostheses are manufactured in very small sets, the manufacturing computer-aided design is appropriate for this purpose [30]. The manufacturing unitary cost of the jth manufacturing operation on the ith bar of the mechanism is
(7)Cuij=Cp+CTm−CSN+πLijDijCm6×104×V×f+CS103×f×C−KV−K−1,where Cp=Ts/N.Ap/60, CT=Cm/60.πLij.Dij/103V.f, CTm=Cm/60∑j=13Tj+Ttm, and Co=CSπLij.Dij/103f.C−KV−K−1−1/Nare the sum of the expenses related directly to the preparation, cutting duration, out of cutting time, and the cutting tool, respectively; Cs is the cost cutting tools; Ap is the global rate of exploitation of the preparation section, general expenses, and labor; Cm=A+F+R+L+E+S is hourly machine cost; A, F, R, L, E, and S are technical amortization, financial expenses, maintenance expenses and repair, expenses on local or on clutter, expenses on energy, and wage costs and social, respectively; V, N, f, and k are linear speed, mass production set, feed rate in turn/tooth, and Taylor constant function of material of the cutting tool, respectively; Dij are the cutting tools used relatively at ith bar and jth manufacturing operation (milling cutter 2 cuttings ∅20, cutter 2 cuttings ∅6, drill ∅2, drill ∅4, piloting milling cutter to counter boring ∅2×6 and machining tapping M2); Lij=Lsij=Lcij=Lpij=Llij=Ltij are the tool trajectories relative to the facing milling, lateral milling, drilling, counter boring, and tapping of the ith bar of the mechanism and the jth manufacturing operation, respectively; T1, T2, and T3 are the holes 1, 2, and 3, respectively, and Ts is execution duration allowed to the preparation of a set of workpiece. The manufacturing cost of the junction bar is
(8)C3=4CF+πLs31D31+Lc32D32+Lp33D33+Ll34D34Cm6×104V.f+CS103f.C−KV−K−1,where Ls31 is the facing milling, Lc32 is the lateral milling, Lp33 is the drilling, and Ll34 is the counterboring. The manufacturing cost of the driving bar is
(9)C1=4CF+πLs11D11+Lc12D12+Lp13D33+Ll14D14Cm6×104V.f+CS103f.C−KV−K−1,where Ls11 is the facing milling, Lc12 is the lateral milling, Lp13 is the drilling, and Ll14 is the counterboring. The manufacturing cost of the driven bar is
(10)C2=5CF+πLs21D21+Lc22D22+Lp23D33+Ll24D24+Ll25D25Cm6×104V.f+CS103f.C−KV−K−1,where Ls21 is the facing milling, Lc22 is the lateral milling, Lp23 is the drilling, Ll24 is the counterboring, and Lt25 is the tapping. The global manufacturing cost of the mechanism is
(11)f5x=C1+C2+C3=13CF+πM.LD,with
(12)LD=Ls11D11+Lc12D12+Lp13D13+Ll14D14+Ls21D21+Lc22D22+Lp23D23+Ll24D24+Ll25D25+Ls31D31+Lc32D32+Lp33D33+Ll34D34,where f5x is in USD and the dimensions of all mechanism pieces are in millimeter [26]. The objective functions related to individual and multiobjective optimization are
(13a)F1=f1x,(13b)F2=f2x,(13c)F3=f3x,(13d)F4=f4x,(13e)F5=f5x,(13f)f6x=f1x,f2x,f3x,(13g)f7x=f1x,f2x,f3x,f4x,(13h)f8x=f1x,f2x,f3x,f5x,(13i)f9x=f1x,f2x,f3x,f4x,f5x.Each of these multiobjective functions consists of a set of objective functions which are simultaneously optimized. The above functions expressed in terms of design variables are equality constraintshix=0 and inequality constraints gix≤0 according to the MATLAB optimization toolbox. The convergence criteria on the design variables xi, the objective functions fix, and the constraints gix are, respectively, equal to 10−7, 10−9, and 10−13.


## 3. Results and Discussions

Three arbitrary points are chosen to conduct the optimization process to validate the results generated during the numerical simulations. Each objective function (known as least square structural error, mechanical error and strength reliability are dimensionless, maximum driving torque is expressed in N.mm, and manufacturing cost is expressed in USD) which is subjected to three equalities and eighteen inequality constraints is optimized by taking into consideration the design variable bounds in Table3. Table 3 also gives the unit relative to each design variable. By using the optimum absolute value of every single function, the multiobjective optimizations F6x, F7x, F8x, and F9x subjected to the same constraints are conducted. The optimization problem solved in this paper has a complex nature. We have five nonlinear objective functions subjected to all type of constraints and must be minimize simultaneously. There is no unique solution to the problem of multiobjective optimization of several objective functions expressed by mathematical expressions which are performance criteria. The selection of optimal solutions is based on Pareto-optimality conditions [31, 32]. A solution is called “pareto-optimal” if there is no other solution for which an improvement of an objective function does not lead to a degradation of at least one of the other functions. It is therefore important, for a problem of simultaneous minimization of several criteria, to generate the “pareto-optimal” solutions. It is what we achieve with “fgoalattain,” which solve multiobjective goal attainment problems. This method is implemented in “MATLAB” software using the “fgoalattain” minimization function. We have chosen the set of vector of the weighting coefficients W=absolutegoal where absolute (goal) is the absolute value of the vector of the objectives set from individual optimization [33]. This condition guarantees the same percentage of achievement of the objectives set for all active objective functions [20]. The optimal design variables of the thumb mechanism are presented in Table 4.Table 4
Thumb optimum design variables versus optimization problem.
F1ΔTP2 10-6(°)F2ΔTP2Mech (°)F3MMAX (N.mm)F4MCOST (US $)F5F6F7F8F9x1 (°)65.035165.035456.992361.219487.314263.385582.152882.063078.1932x2 (mm)7.54937.54929.34957.27668.39445.99645.09425.06035.0603x3 (mm)73.569475.593777.654775.42327 1.664674.994336.987537.035437.0354x4 (mm)11.815611.815412.880211.298311.28429.51216.05776.01976.0197x5 (°)352.4660354.6590360.000035 1.668346.5512344.8354375.7396375.6806375.6806x6 (mm)1 .06241.11571.27841.14141.08241.13620.84850.85170.8517x7 (mm)17.071817.071415.753815.027818.292315.001220.000320.000120.0001x8 (mm)5.00005.0000-4.98735.0000-0.2794-4.98275.00025.00015.0001x9 (mm)15.000015.287417.000017.000014.594512.633610.000010.000010.0000Table4 contains all the generated values of the design variables resulting from the optimization of each objective function and a set of coupled functions according to the optimization strategy adopted. The vectors of transversal, longitudinal, or angular dimensions of the virtual mechanisms are proposed at each optimum obtained. Individual optimization is represented by F1 to F5, and multiobjective optimization is represented by F6 to F9. The units of the various objective function are indicated in Table 5 as follows: f1x and f2x are in degree Celsius, f4x do not have unit, f3x is expressed in N.mm, and f5x is given in USD. The set of optimal design variable validated at each optimization problem corresponds to the situation where an objective function reaches a feasible solution on the computing solver, where all the constraints are satisfied with the lower and upper bounds. Figures 3–5 are plotted with the unit of i=3, 4, and 5 in degree Celsius and with i=1, 2, 6, 7, 8, and 9 in millimeters on y-axis. The optimal design variables versus individual and multiobjective optimization of thumb are presented in Figure 3.Table 5
Thumb optimization problem versus optimum objective function values.
Fixf1ΔTP210−6 (°)f2ΔTP210Mech (°)f3MMax (N.mm)F4F5Mcost (USD)F13133.4298————F2—347.5135———F3——1255.7342—F4———0.9999—F5————499.1203F63133.412521.61222172.2236——F73515.446316.21931821.11161.0000—F81656.955121.61622498.5664—494.8459F93133.401812.49232943.75011.0000495.2544Figure 3
Optimal design variables versus individual and multiobjective optimization of thumb.Figure 4
Optimal design variables versus individual optimization of thumb.Figure 5
Optimal design variables versus multiobjective optimization of thumb.Figure3 helps to interpret the results relating to the optimal design variable, and it is associated with the optimization problem Fi. The y-axis shows the numerical values of the optimal design variable xi, and the abscissa axis is associated with the independent design variable. Table 5 presents the optimal values taken by the objective functions at the pareto-optimal solution for multiobjective optimization.The other nonoptimal values calculated earlier when obtaining the optimal values of the functions to be optimized are removed from Table5 for the reasons of clarity. Figure 4 is plotted by using the partial data from Table 4 from the singular optimization.Figure4 facilitates the interpretation of the results relating to the optimal design variables relative to individual optimization. In essence, these variables are expressed as a function of the objective functions fi relating to the various optimization problems Fi to be solved. Figures 3–5 are associated with a function Fi, the y-axis shows the numerical values of the optimal design variable xi with the unit of i=3, 4, and 5 in degree Celsius and with i=1, 2, 6, 7, 8, and 9 in millimeters as indicated in Table 3. Figure 6 is associated with optimization problem Fi and the abscissa axis is associated with the functional transversal and longitudinal dimensions of the mechanism (xi). The units of the various objective functions are indicated in Table 5 as follows: f1x and f2x are in degree Celsius, and f4x do not have unit, while f3x is expressed in N.mm and f5x is given in USD. The optimal design variables versus multiobjective optimization of the thumb are presented in Figure 5.Figure 6
Behavior of each objective function versus optimization problem.Figure5 is associated with the objective functions fi optimization problem Fi, the y-axis shows the numerical values of the optimal design variable xi, and the abscissa axis is associated with the variable design parameters.Figure5 is plotted based on the partial data from Table 4. The ordinate axis represents the optimal numerical values given by the design variables at the end of the process of simultaneous optimization of a set of coupled objective functions according to the strategy adopted. Moreover, Figure 3 helps to identify the behavior of individual design variables with respect to each simultaneous optimization problem. Figure 4 which presents the behavior of each objective function versus the optimization problem is plotted by using Table 5.Figure6 demonstrates a proper understanding of the behavior of all the objective functions fi relating to the optimization problems during the optimization process of each of the objective functions Fi relating to the different problems to be solved. The ordinate axis gives the values of the function fi considered, and the abscissa axis shows all the functions Fi for all the resulting optimization strategies. Figure 6 also gives a panoramic idea of the behavior of each objective function according to the optimization problem. The y-axis represents the optimal values taken by objective functions when an optimum is obtained at the end of the optimization process of the different optimization problems. Figures 4–6 show simultaneous optimization Fi of all the criteria confer on the thumb, with the same length of the fixed bar r1, the same thickness x6=tm of all the bars, and about the same diameter of the driving pulley. Two observations can be highlighted from the beginning of the manufacturing cost and reliability relating to the optimization process: the first is related to the reliability regarding all criteria being appreciably equal to the unit and the second observation is related to the minimum manufacturing cost, from which the largest design variable values are obtained. The manufacturing constraints such as geometrical and dimensional tolerances, accuracy (0.1 μm), production set (N), cutting tool life span, and the shape of the workpiece guide the designer to choose the suitable manufacturing model cost parameters. The optimization process was conducted by varying three times the value of the stopping criteria as indicated at the end of the last paragraph of Section 2, to ensure the accuracy provided by the CNC machining process. The final results displayed at the end of optimization process are the optimal output, respecting all the constraints including manufacturing ones. The optimal objective function values from individual optimization were used as the weighting absolute goal to be achieved by the multiobjective “fgoalattain” solver. The steps where the system stop with the “nonfeasible” answer of the algorithm are cancelled and not validated for this study. The accuracy configures in the computing solver for design variable, objective function, and constraints are indicated in Section 2.2. From there, the precision observed on the results of Table 4 is justified and imposed to the manufacturer the used of computing-aided manufacturing to produce the part of this artificial mechanism.The value of the optimal function obtained during the entire optimization process relative to the reliability of the whole mechanism is 0.9999, compared to 1 obtained by Ngale et al. [28], when only the driving bar was taken in consideration during the optimization processes. This difference of 0.1111 justifies the contribution of each bar in the reliability mathematical model. Furthermore, the multiobjective optimization showed that the mechanical reliability and the production cost included in any combination of the simultaneous optimization enabled the achievement of the same optimum variables design, though with some exceptions.Earlier works from literature realized by Ngale et al. in 2003 and 2016, Lim et al. in 2018, and Tsamo et al. in 2020 on the prosthesis mechanism’s optimization show that optimal design variables obtained from one single optimization to multiobjective optimization must be different to justify the contribution of the said performance criteria on the improvement of the design [24, 26, 27]. In this paper, the global manufacturing cost and the strength reliability of the whole mechanism are included in all combinations of the simultaneous optimization necessary to obtain the random optimum design variables. The optimal manufacturing cost generated from the numerical simulation is 501.0021 USD, for the thumb versus 961 USD for the thumb proposed by Ventimiglia [34] and 553 USD proposed by Choi et al., for one compliant finger, obtained by 3D printing [16]. The difference is 459.9979 USD with Ventimiglia price and 51.9979 USD with Choi et al. price. This difference is due to the inclusion of the power system, actuation, and miscellaneous component cost by Ventimiglia. The particularity of the cost generate on this study is that the design of the mechanism is based directly on the objective function modelled. The gap observed at the level of the price can be affected to numerous factors such as human factor error while evaluating empirical quantities, machine imprecision, labor charges, the depreciation rate of the equipment, and productivity assets which are included in the mathematical’s manufacturing cost model of this study [35–37]. Otherwise, the simultaneous optimization Fi of all the criteria made it possible to reduce the standard deviation between all design variables. One notices that the virtual mechanisms of the fingers and the thumb are sensitive to each of the new performance criteria such as manufacturing cost and the strength reliability of the whole mechanism. The value of the optimal function obtained during the entire optimization process relative to the reliability F4 is quite different with some exceptions of those obtained by Ngale and Vinet and Ngale who focused their study only on the reliability of the driving bar of the four-crossed bar mechanism [14, 20].


## 4. Conclusion

The paramount goal of this study was to optimize the thumb prosthesis’s mechanism by taking into consideration the manufacturing cost model based on machining cost theory which took into detail the shape of the workpieces and the strength reliability of all the parts composing the entire mechanism. These criteria were added to those choosing from the state-of-the-art based on their popular contribution relative to the structure’s design optimization. The virtual mechanisms of the thumb are sensitive to each of the new performance criteria employed. The value of the optimal function obtained during the entire optimization process relative to the reliability of the whole mechanism is 0.9999, compared to 1 obtained when only the driving bar was taken in consideration during the optimization processes. This difference of 0.1111 justifies the contribution of each bar in the reliability mathematical model. Furthermore, the multiobjective optimization showed that the mechanical reliability and the production cost included in any combination of the simultaneous optimization enabled the achievement of the same optimum variables design, though with some exceptions. It was found that the mechanical error has little influence on the design variables resulting from the simultaneous optimization. For the simultaneous optimization of all the criteria where it was integrated, its influence was reduced considerably because all the optimal values of the objective functions are lower than the value resulting from the individual optimization. The optimal manufacturing cost generated from the numerical simulation was 501.0021 USD, for the thumb versus 961 USD, and 553 USD for one compliant finger, obtained by 3D printing. The difference is 459.9979 USD and 51.9979 USD more than the price generated on this study. This difference is due to the inclusion of the power system, actuation, and miscellaneous component cost by Ventimiglia. The particularity of the cost generate on this study is that the design of the mechanism is based directly to the objective function modelled. The gap observed at the level of the price can be affected to numerous factors such as human factor error while evaluating empirical quantities, machine imprecision, labor charges, the depreciation rate of the equipment, and productivity assets which are included in the mathematical’s manufacturing cost model of this study. Therefore, further studies in the biomechanics field could integrate the weight and the tribological phenomenon in the optimization processes of the mechanism to achieve the real design of these artificial devices.



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*Source: 9647956-2022-09-05.xml*