Open access peer-reviewed chapter

To Design a Small Pneumatic Actuator Driven Parallel Link Mechanism for Shoulder Prostheses for Daily Living Use

Written By

Masashi Sekine, Kento Sugimori and Wenwei Yu

Submitted: November 17th, 2010 Reviewed: April 16th, 2011 Published: August 29th, 2011

DOI: 10.5772/22050

From the Edited Volume

On Biomimetics

Edited by Assoc. Lilyana D. Pramatarova

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1. Introduction

Only in Japan, there are about 82,000 upper limb amputees (Ministry of Health, Labour and Welfare, 2005). Using upper limb prostheses could restore the function for them, thus improve significantly the quality of their activities of daily living [ADL]. Compared with below-elbow prostheses, shoulder prostheses are left behind in their development, due to high degrees of freedom [DOF] required, which demands a large number of actuators, thusdenotes a large size and a heavy weight, and complicated control mechanism.

Recently, there is a certain body of research on developing roboticdevices that could be used as prostheses for shoulder amputees (Jacobson et al., 1982; Motion Control, Inc., 2006-2011; The Johns Hopkins University Applied Physics Laboratory [APL], 2011; Troncossi et al., 2005, 2009a, 2009b). These research efforts have led to artificial prostheses with high functionality and performance.For example, the prosthetic arm of Defense Advanced Research Projects Agency and APL, has 25 DOFs, individual finger movements, dexterity that approaches that of the human limb, natural control, sensory feedback, and a number of small wireless devices that can be surgically implanted (or injected) to allow access to intramuscular signals(APL, 2011). The Utah Arm 3, a modification of the previous Utah Armthat has been the premier myoelectric arm for above elbow amputees, has two microcontrollers that are programmed for the hand and elbow,accordingly, allowing separate inputs and hence simultaneous control of both, and that is, the wearer can operate the hand and elbow concurrently for natural function (Jacobson et al., 1982; Motion Control, Inc., 2006-2011). The hybrid electric prosthesis for single arm amputee of Tokyo Denki University possesses a ball joint of 3 DOFs in humeral articulation. Patient operates the prosthesis to optional point by pressing a switch with the other healthy limb to free the joint, and releases to fix and hold the prosthetic arm stably (Nasu et al., 2001). Moreover, the electromechanical shoulder articulation with 2 DOFs for upper-limb prosthesis that has two actuated joints embedded harmonic drives, an inverted slider crank mechanism, and ball screw, has been developed (Troncossi et al., 2005, 2009a, 2009b).

These prostheseshave the following characteristics: they are more or less anthropomorphic, basically supported by metal frames or parts, driven by electric motors, therefore, many of them seem to be not suitable for the daily living use: they are not lightweight, not convenient, with a bad portability, and lack of backdrivability which could contribute to the safety use in daily living.

Using pneumatic actuators (Festo AG & Co. KG, 2002-2008; Folgheraiter & Gini, 2005), some researchers have developed sophisticated manipulators having structure similar to human upper limb. Employing pneumatic actuators that could naturally realize backdrivability, ensures safety against collisions or contact between the prosthetic shoulder and its environments around. In the Airic's_arm(Festo AG & Co. KG, 2000-2008), 30 artificial muscles were used to move the artificial bone structure comprising the ulna, radius, the metacarpal bones and the bones of the fingers, as well as the shoulder joint and the shoulder blade. The MaximumOne, a robot arm ofArtificial Intelligence and Robotics Laboratory, Politecnico di Milano, consists of two joints with 4 DOFs in all.The shoulder is made up of a ball joint with 3 DOFs and driven by five actuators, and the elbow is a revolute joint with 1 DOF and driven by two actuators (Folgheraiter & Gini, 2005).However, the manipulators are basically not for prosthetic use, moreover, they are not portable, especially due to the big air compressor.

This study aims to develop a lightweight shoulder prosthesis that could be easily fitted to and carried by amputees, therefore a convenient one.This chapter presents kinematical analysis, procedure for finding optimal configurations for the prosthetic arm, and verification of the design concepts.

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2. Design concepts

A shoulder prosthesis for daily living use should be light-weight, portable, and safe. Consideration to design such a shoulder prosthesis is described as follows.

  1. Using small pneumatic actuators driven by small portable air compressor for weight saving and portability. To meet portability and light-weight requirements, small actuators and compressors are musts for shoulder prostheses. The pneumatic actuators Sik-t, Sik-t Power-Type (Squse Inc.: 1g, 20N; 3g, 130N), air compressors MP-2-C (Squse Inc.: 180g, 0.4 MPa) are products developed recently for robotic application with light-weight and good portability. In this research, these products were employed as actuators and their air sources. The purpose of this research is to design shoulder prostheses with optimal spatial functionality using these actuators.

  2. Employing a parallel link mechanism to enable high rigidity and high torque output. The natural viscoelasticity of pneumatic actuators could contribute to backdrivability, and safety of shoulder prostheses, however, it also affects the payload of the system. Moreover, since small actuators have a limited tensile force, a structure that could exert high torque output is preferable. That is why a parallel link mechanism that could improve structural rigidity was employed. However, the parallel link structure usually has a limited stretch along axial direction. The working space of the prostheses should be adjusted to fit individual users’ expected frequently accessed area [EFAA]. To the best of our knowledge, there are no investigation results reported on how to match working space of end-effector to EFAA of individual users’ hand. This is the main objective of this study.

  3. Using a rubber backbone for the parallel link mechanism to enable trade-off between working space and payloads. Since, the parallel link structure usually has a limited stretch along axial direction. A flexible backbone for the parallel link could give more possibility to deal with the trade-off between payloads and working space, however, this raises one more design variable, which should also be carefully investigated in the design process. This is an on-going research theme, and will be addressed in other papers.

  4. Designing a special backpack that could contain the shoulder prostheses and all accessories, and could be worn by the amputee user himself with minimal effort. This needs the shoulder prosthesis be foldable, and the backpack be designed for conveniently getting the shoulder prosthesis in and out. This will be approached in the next stage and addressed in other papers. The ultimate goal of this study is to build a shoulder prosthesis that could be used in daily living by shoulder amputees. The purpose of this paper is to describe the structure of the prosthesis, and approach to find optimal configurations based on the aforementioned design consideration. Fig. 1 shows an illustration of the shoulder prosthetic system, which is drafted with computer aided design [CAD] software SolidWorks (Dassault Systèmes SolidWorks Corp.) and human body model from HumanWorks software (zetec, Ltd.)

The remainder of this chapter is organized as follows. At first, in section 3, the basic structure of the prosthesis was described and several formulae for kinematics and statics of it were derived for further analysis. After that, the way to achieve the spatial accessibility and manipulability was explained in section 4. Then several experimental results concerning the design of the prosthesis were shown with discussion. Following that, a conclusion was given based on the results and discussion.

Figure 1.

The shoulder prosthetic system.

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3. Kinematics and statics

To decide the physical dimension of the shoulder prosthesis, both the kinematics, statics of the prosthesis and the spatial configuration, expected frequently accessed area [EFAA] of users’ hand should be considered. In this section, the kinematics and statics of the proposed shoulder prosthesis were derived for further analysis. Several estimative indexes were also defined to compare different potential solutions, thus to decide physical dimensions (configuration) of the shoulder prosthesis. As described before, only the arm structure was modelled and analyzed, whereas the prosthesis with hand, the backpack, and the connection between arm structure and backpack were left for further studies.

3.1. Arm structure

The details of the arm structure (in the following explanation, denoted as the Arm) are shown in Fig. 2. The Armis composed of three segments. Segment1 links two disks called the Base1 and the Platform1 with the Backbone1 and three pneumatic actuators, placed equiangularly with respect to the center of the Base1. The Backbone1 is fixed to the center of the Base1, and connected to the center of the Platform1 with two passive revolute joints. To simplify the analysis, the Backbone1 was assumed as a compression spring that can only move along longitudinal direction, but not as a rubber rod as described in the design concept section. By assembling Base1 and Platform1 with a compressed Backbone1, actuators and wires that connect the pneumatic actuators with two disks are constantly loaded. This allows the Platform1 to move along the longitudinal direction of the Backbone1, turn around the joint of Platform1 and Backbone1, as a result of length changes of the three actuators.

The Platform1 disk of Segment1 is also used as the Base2 disk of the Segment2, which has a similar structure with the Segment1, but with a different length. Segment3 contains only a rigid rod (Rod) fixed to the center of outside the Platform2, i.e. the Base3 of the Segment3. For the convenience of description, let h1, h2 and lRbe the initial length of the Backbone1, 2 and Rodrespectively.

Figure 2.

The structure of theArm.

3.2. Kinematics

In order to analyze the behavior of end-effector and working space of the Arm, forward and inverse kinematics model of the parallel link mechanism were derived.

The coordinate system of the Armis shown in Fig. 3. Without loss of the generality, the thicknesses of all disks, and the shaft diameters of Backbone1, 2, Rodwere set to 0. The global coordinate system OB1-XYZis located at the center of the Base1, with the Z-axis directed along the Backbone1. The contact points of three pneumatic actuators to Base1 (B11 , B12 , B13) were aligned equiangularly along the peripheral of a circle with a radius rB, and B11 is on the X-axis.

The local coordinate system OP1-xP1 yP1 zP1 locates at the center of disk Platform1, h1 away from OB1along the Z-axis. The contact points of pneumatic actuators to Platform1 (P11 , P12 , P13) are on radius rP. In turn, the contact points of three pneumatic actuators to Base2 (B21 , B22 , B23) are equiangularly set on circumference of a circle, radius rB, and B21is on the xP1-axis. The local coordinate systemOP2 -xP2 yP2 zP2 is set at the center of Platform2, and the distance from OP1is h2. The contact points (P21 , P22 , P23) are aligned equiangularly along the peripheral of a circle with a radius rP, and P21 is on the xP2-axis.

Finally, the Rodof length lRis fixed up at OP2 along the zP2-axis.

Suppose the actuators are activated, and their lengths change (expressed discretely: ligets to li, i=1,..., 6). Therefore, h1 and h2 are converted to h1 and h2 , two passive joints (Joint1) of Platform1 and the one (Joint2) of Platform2 rotate by α,β, γ, σ(see Fig. 2, 3), respectively.

Figure 3.

Geometry of the parallel link arm.

At first, the position of OP1, P1i,P1i’(i=1, 2, 3)andthe rotation matrix R1 of Joint1 in OP1-xP1 yP1 zP1, i.e. P1 OP1, P1 P1i, P1 P1i’ and P1 R1 can be presented as follows:

OP1P1: (0, 0, 0),   PP111: (rP, 0, 0),   PP112: (-12rP, 32rP, 0),   PP113: (-12rP, -32rP, 0)RP11=(1000cosαsinα0sinαcosα)(cosβ0sinβ010sinβ0cosβ)=(cosβ0sinβsinαsinβcosαsinαcosβcosαsinβsinαcosαcosβ) E1

And,

PP11i'=RP11PP11i(i=1,2,3)E2

Therefore, the coordinate of each element of P1 P1i’ is:

PP111': (rPcosβrPsinαsinβrPcosαsinβ)PP112': (12rPcosβ12rPsinαsinβ+32rPcosα12rPcosαsinβ+32rPsinα )PP113': (12rPcosβ12rPsinαsinβ32rPcosα12rPcosαsinβ32rPsinα)E3

Next, P1i’, B1i’in OB1-XYZ, i.e. B1 P1i’ and B1 B1i’ can be defined as:

BB111: (rB, 0, 0),   BB112: (12rB32rB, 0),   BB113: (12rB32rB, 0)PB11i'=PP11i'+(0, 0, h1')T(i=1,2,3)E4

So, each element of B1 P1i’ can be expressed as:

PB111': (rPcosβrPsinαsinβ, -rPcosαsinβ+h1')PB112': (12rPcosβ12rPsinαsinβ+32rPcosα12rPcosαsinβ+32rPsinα+h1' )PB113': (12rPcosβ12rPsinαsinβ32rPcosα12rPcosαsinβ32rPsinα+h1')E5

Accordingly, the length of each actuator can be defined:

li'=PB11i'BB11i¯(i=1,2,3)E6

By Equation (5), (7) and (8), the equations describing the relation between the lengths of pneumatic actuators, wires attached in Segment1, Backbone1, and the angles of two revolute joints can be defined as follows:

(l1')2=(rPcosβ-rB)2+(rPsinαsinβ)2+(-rPcosαsinβ+h1')2E7
(l2')2=(12rPcosβ+12rB)2+ (12rPsinαsinβ+32rPcosα32rB)2        +(12rPcosαsinβ+32rPsinα+h1')2E8
(l3')2=(12rPcosβ+12rB)2+ (12rPsinαsinβ32rPcosα+32rB)2        +(12rPcosαsinβ32rPsinα+h1')2E9

Equation (9) is a simultaneous equation with three unknown variables, α,β,h1’. In the case that the lengths l1’, l2’, l3’ are given, it is possible to calculate α,β,h1’ by using the Newton method (Ku, 1999; Merlet, 1993; Press et al., 1992). Similarly, for the Segment2 (i.e. in OP1-xP1 yP1 zP1), the following equation can be derived. In the case that the length l4’, l5’, l6’, are given, it is possible to calculate γ, σ,h2’.

(l4')2=(rPcosσrB)2+(rPsinγsinσ)2+(rPcosγsinσ+h2')2(l5')2=(12rPcosσ+12rB)2+ (12rPsinγsinσ+32rPcosγ32rB)2        +(12rPcosγsinσ+32rPsinγ+h2')2(l6')2=(12rPcosσ+12rB)2+ (12rPsinγsinσ32rPcosγ+32rB)2        +(12rPcosγsinσ32rPsinγ+h2')2E10

The position of the Rodend PRE’ in OP2 -xP2 yP2 zP2, i.e. P2 PRE’ can be defined as:

PP2RE'=(0,0,lR)TE11

Let x, y, zbe the coordinate of B1 PRE’, then end position of the Rod, x, y, zin OB1-XYZcan be described as:

(xyz)T=RP11(RP22PP2RE'+OP1P2')+OB1P1'(xyz)=(cosβ0sinβsinαsinβcosαsinαcosβcosαsinβsinαcosαcosβ)[(cosσ0sinσsinγsinσcosγsinγcosσcosγsinσsinγcosγcosσ)(00lR)+(00h2')]+(00h1')E12

3.3. Jacobian matrix

In order to evaluate the motion characteristics of the Arm, it is necessary to develop theJacobian matrix of the Armstructure.

As li’(i=1, ,6), α,β,h1’, γ, σand h2’ can be taken as functions of time t, noticing li’,x, y, zare functions of α,β,h1’, γ, σ, h2’, the Equation (9), (10), (12) can be differentiated with respect to t, to get the following equations. The A, B, Care the matrixes with element aij, bij(i,j=1,2,3), cij(i=1,2,3, j=1, , 6), respectively.

(l1'l2'l3')=(a11a12a13a21a22a23a31a32a33)(αβh1')=A(αβh1'),(l4'l5'l6')=(b11b12b13b21b22b23b31b32b33)(γσh2')=B(γσh2'),E13
(xyz)=(c11c12c13c14c15c16c21c22c23c24c25c26c31c32c33c34c35c36)(αβh1'γσh2')=C(αβh1'γσh2')E14

Next, the orientation of B1 PRE’ in OB1-XYZcan be expressed by Equation (15), where the element of the matrix P1 R1 P2 R2 is presented by rij(i,j=1,2,3).

RP11RP22=(cosβ0sinβsinαsinβcosαsinαcosβcosαsinβsinαcosαcosβ)(cosσ0sinσsinγsinσcosγsinγcosσcosγsinσsinγcosγcosσ)=(r11r12r13r21r22r23r31r32r33)E15

By using Equation (15), the Euler angles (ϕ,θ, ψ) can be acquired (Yoshikawa, 1988).

φ=atan2(r23,r13),θ=atan2((r13)2+(r23)2,r33),ψ=atan2(r32,r31)(0θπ)E16

Equation (17) can be derived by differentiating Equation (16) with regard to time t(Yoshikawa, 1988):

φ=(r23)r13r23(r13)(r23)2+(r13)2,θ=((r13)2+(r23)2)r33(r13)2+(r23)2(r33)(r13)2+(r23)2+(r33)2,ψ=(r32)r31+r32(r31)(r32)2+(r31)2(0θπ)E17

ϕ,θ, ψare functions of α,β, γ, σ. Therefore, by using a matrix Dwith element dij(i=1, 2, 3, j=1, 2, 4, 5), (ϕ,θ, ψ) Tcan be expressed as:

(φθψ)=(d11d120d14d150d21d220d24d250d31d320d34d350)(αβh1'γσh2')=D(αβh1'γσh2')E18

Equation (14) and (18) can be integrated to the following equation.

(xyzφθψ)=(CD)(αβh1'γσh2')E19

Let inverse matrix of Aand Bbe A-1 and B-1, which comprise elements a-1 ij, b-1 ij(i,j=1,2,3), then α,β,h1’, γ, σ, h2’can be derived from Equation (13).

(αβh1')=A1(l1'l2'l3')=(a111a112a113a121a122a123a131a132a133)(l1'l2'l3'),(γσh2')=B1(l4'l5'l6')=(b111b112b113b121b122b123b131b132b133)(l4'l5'l6'),(αβh1'γσh2')=(A100B1)(l1'l2'l3'l4'l5'l6')E20

Therefore, by using Equation (19) and (20), the vector representing the posture of end-effector can be acquired.

(xyzφθψ)=(CD)(αβh1'γσh2')=(CD)(A100B1)(l1'l2'l3'l4'l5'l6')=(c11c12c13c14c15c16c21c22c23c24c25c26c31c32c33c34c35c36d11d120d14d150d21d220d24d250d31d320d34d350)(a111a112a113000a121a122a123000a131a132a133000000b111b112b113000b121b122b123000b131b132b133)(l1'l2'l3'l4'l5'l6')=J1(l1'l2'l3'l4'l5'l6')E21

From the above equation, the Jacobian matrixJ1 can be calculated.

3.4. Static mechanics

Let force generated and virtual displacement of each actuator be τandΔl, and the force generated and virtual displacement of Rodend PRE’ be Fand Δx. From virtual work principle, the relationship between these two pairs is:

FTΔx=τTΔlE22

By using Equation (21)

Δx=J1Δl,J11Δx=Δl,Δl=JΔx(J=J11)E23

Therefore, from Equation (22) and (23), the relation between Fand τis acquired.

FTΔx=τTJΔx,FT=τTJ,(FT)T=(τTJ)TF=JTτE24
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4. Method of analysis

In this section, an evaluation index based on the Jacobian Matrix is presented, and the process to evaluate possible configurations, i.e., the physical dimension of the shoulder prosthesis is described.

4.1. An estimative index of manipulability: condition number

The condition number (Arai, 1992) was employed as evaluation indicator for the motion characteristics of the Armmechanism. The condition number is based on the singular value of the Jacobian matrix. The Equation (24) can be described as the expression for how τis convertedinto F. Furthermore, a singular value decomposition, expressed by Equation (25), can make the property of Jeven clearer.

JT=UΣVTE25

Here, Uand Vare 6x6 orthogonal matrixes, which can be described by Equation (26).

U=(u1,,u6)T,V=(v1,,v6)T,Σ=diag(σ1,,σ6),(σ1σ2σ60)E26

Substituting Equation (25) into (24), the relation between τand Fcan be rewritten as Equation (27).

F=UΣVTτ,UTF=ΣVTτE27

Equation (26) and (27) can be rewritten using the elements of Uand V.

uiTF=σiviTτE28

Considering the function of the manipulator, it is preferable that the forces that could be generated at the end of the Rodin all direction are as uniform as possible. That is, it is the ratio of the maximum singular value to the minimum one, i.e., the condition number, should be close to 1 as much as possible.

Since the condition number of JTreflects the both the forceand torque working at the end of the Rod, in order to conduct proper evaluations, it is necessary to separate the influence of forceand torque. Therefore, in the Equation (24), JTis separated into the part contributing to the force and the one contributing to the torque, and singular value decomposition was conducted at two parts separately. Therefore, JTis separated as follows.

JT=[JfTJmT]E29

Here,JfTandJmTare 3x6 matrixes, so, three singular values σfi, σmi(i=1,2,3) exist in each of JfTand JmT. Thus, we use the following three condition numbers as estimative index:

C=σ1σ6Cf=σf1σf3Cm=σm1σm3E30

4.2. An outline of the evaluation process

The following is an outline of the evaluation process.

  1. Setting up a coordinate space ΣCS1;

  2. Setting up an initial configuration (physical dimension of the Armmechanism), and modelling the Armand human body in ΣCS1;

  3. Defining EFAA (Expected Frequently Accessed Area) and RA (Reachable Area) of the Armin ΣCS1;

  4. For different length of pneumatic actuators, reflecting translational motion of the actuators, numerically calculating andplotting the Rodend positionPRE’;

  5. Calculating the estimative indexes for all the PRE’ in EFAA and RA;

  6. Changing the parameters of the Armmechanism, and going back to recalculating Step 4;

  7. After a certain number of loops of execution (Step 4 to 6), evaluating all the configurations to decide configurations optimal for the spatial accessibility (plot number in EFAA) and manipulability.

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4.3 Modelling the Armand human body in the coordinate space

A 3D human body model software (HumanWorks) was used for the above-mentioned human body. This HumanWorks model, shown in Fig. 4, 167.0 centimeters tall, is a 50th percentile model of Japanese male based on Japanese Industrial Standards.

Fig. 5(a) shows the coordinate system, ΣCS1,for the shoulder prosthetic system. It presents not only the geometry of the Arm, and also the EFAA, and their relationship.

As illustrated in the Fig. 5(a), the point of origin is set at the intersection of the median sagittal plane(Y=0), with the horizontal plane(X=0) and the coronal plane(Z=0) passing through the acrominon. Positions of the acrominon are assumed as (0, ±170, 0). In Fig. 5(c), (d), the size of some parts of human body is presented. Considering the shape and positional relationships of the shoulder, neck, head and the Arm, the Base1 of the Armis set at (-80, 150, 0), the axis direction of OB1-XYZ(see Fig. 3) is set to conform to the Zaxis of the ΣCS1. Moreover, based on the arm size of the 50th percentile model, we estimated the size of the Armsuitable for the human body, and setup initial values for the physical dimension of the Arm(Fig. 5(b)). These initial values, which constitute an initial configuration, are summarized as follows (see Fig. 3 for the meanings of the symbols).

Figure 4.

A 3D human body model of HumanWorks.

h1=100h2=170lR=250rB=50rP=45(mm)E31

Figure 5.

The coordinate systemΣCS1 and the HumanWorks model.

4.4. Important areas in the working space of the Arm

Two important areas in the working space of the Armare defined for the analysis and evaluation of the Arm. One is the area close to the chest and the median sagittal plane, which is expected to be accessed very frequently during most daily living tasks. This area is the EFAA defined before, and expressed as ΣEFAA. Another is the area that represents the reachable area [RA] of the end effector, defined as ΣRA. Geometries of ΣEFAAand ΣRAare illustrated as in Fig. 6. The volumes of ΣEFAAand ΣRAare set to 12000cm3 and 75000cm3, respectively.

Figure 6.

Geometries of the areasΣEFAAandΣRA.

4.5. Calculation and analysis

All the calculation was calculated numerically by using Matlab (MathWorks, Inc.). A simplified Arm, as shown in Fig. 7, is drawn for visualization in Matlab. PRE’ is calculated by substituting the initial values to Equation (12), with variable length of actuators, which stands for the translational motion of the pneumatic actuators. Three different values were set for each li(i=1, ,6) in Fig. 3(b). Supposing that the resting length of the actuators are LWi(i=1, ,6), and the maximum,minimum, middle increment of the actuators are Lmax, Lmin, Lmid, then the three different values are LWi+ Lmax, LWi+ Lmin, LWi+Lmid (Fig. 8). Thus, for each Armconfiguration, a total number of 36=729 sets of calculation were calculated for PRE’, then the configuration could be evaluated.

Figure 7.

Models with Matlab.

Figure 8.

Three different values ofli.

The estimative index described in section 4.1 was adapted as follows, for evaluating different aspects of the system.

  • NEFAA: Number of points plotted in ΣEFAA(Spatial accessibility)

  • NRA: Number of points plotted in ΣRA(Spatial accessibility)

  • Mt(C): 15percent trimmed mean condition number C(Eq. 30) of points plotted in ΣEFAA(Manipulability)

  • Mt(Cf): 15percent trimmed mean condition number Cf(Eq. 31) of points plotted in ΣEFAA(Manipulability)

  • Mt(Cm): 15percent trimmed mean condition number Cm(Eq. 32) of points plotted in ΣEFAA(Manipulability)

After the calculation and evaluation, the physical dimensions of the Armstructure were changed, and the calculation and evaluation for the new configuration were repeated. Note, in this chapter, only the results of changing h2 and lRare to be reported.

From this process, optimal configurations of the Armstructure could be determined.

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5. Results

5.1. Results of the initial configuration

A plot of PRE’ for the Armstructure with the initial configuration is shown in Fig. 9, where points in red, blue and gray stand for the PRE’ located in ΣEFAA, in ΣRAand outside of the both areas, respectively. Basically, the group of points is longitude-axis-symmetric.

Figure 9.

PlottingPRE’ with the initial parameter.

Table 1 shows the values of estimative indexes defined in section 4.5

NEFAANRAMt(C)Mt(Cf)Mt(Cm)
68393852240279.411195.5

Table 1.

Estimative index with the initial parameter.

5.2. Results of other configurations generated by changing parameters

Suppose that the parameters h2 and lRare changed as in Equation (34), where, beginning with 170mm, 250mm (parameters of the initial configuration) h2 and lRincrease or decrease incrementally by 25mm and 50mm, respectively, and iis an integer, taking value 0, 1, 2.

h2=170±25ilR=250±50i(i=0,1,2)(mm)E32

Therefore, a total number of 25 combinations could be made, and identified with No.1-1, , No.1-25, as shown in Table 2. The estimative indexes were calculated for all the combinations. The NEFAAand NRAare shown in Fig. 10. The horizontal axis stands for the ID of combination, and vertical axis represents NEFAA(Fig. 10(a)) or NRA(Fig. 10(b)).

NO.1-12345678910111213
h2(mm)120145170195220120145170195220120145170
lR(mm)150150150150150200200200200200250250250
NO.1-141516171819202122232425
h2(mm)195220120145170195220120145170195220
lR(mm)250250300300300300300350350350350350

Table 2.

Combinations of h2and lR.

Fig. 10(b) shows a tendency that as the Armlength lRgets longer, NRAdecreases almost monotonically, but there is a steep descent after No.1-21.This can be attributed to the fact that a certain Armlength lRwould make the Rodend more likely to go over the ΣRA.

Figure 10.

NEFAAandNRAof 25 combinations ofh2,lR.

Whereas, as shown in Fig. 10(a), there are roughly two stages. In the stage after No. 1-13, NEFAAdecreases while vibrating irregularly and strongly. This can be attributed to the reason same as before: a certain Armlength lRwould make the Rodend more likely to go over the ΣEFAA. A different point is that the value of No. 1-16, with a smallest h2 (h2 = 120mm), shows a large local maximum. It seems h2 affected NEFAAmore than NRA. In the stage before No.1-12, NEFAAgradually increases as the Armlength gets longer, which means that it is necessary to precisely investigate the possibility of the Armwith shorter length, i.e., smaller lRand h2. Thus, the Armwith the parameters shown in Equation (33) was investigated.

h2=100+7ilR=70+20i(i=0,, 9)(mm)E33

A total number of 100 combinations could be made, and identified with No.2-1, , No.2-100, as shown in Table 3. The estimative index was calculated for all the combinations. The results are shown in Table 3 and Fig. 11, 12.

NO.2-1234567891011121314151617181920
h2(mm)100107114121128135142149156163100107114121128135142149156163
lR(mm)7070707070707070707090909090909090909090
NO. 2-2122232425262728293031323334353637383940
h2(mm)100107114121128135142149156163100107114121128135142149156163
lR(mm)110110110110110110110110110110130130130130130130130130130130
NO. 2-4142434445464748495051525354555657585960
h2(mm)100107114121128135142149156163100107114121128135142149156163
lR(mm)150150150150150150150150150150170170170170170170170170170170
NO. 2-6162636465666768697071727374757677787980
h2(mm)100107114121128135142149156163100107114121128135142149156163
lR(mm)190190190190190190190190190190210210210210210210210210210210
NO. 2-81828384858687888990919293949596979899100
h2(mm)100107114121128135142149156163100107114121128135142149156163
lR(mm)230230230230230230230230230230250250250250250250250250250250

Table 3.

Combinations of h2and lR.

As shown in Fig. 11, within the range, as the ArmlRgets longer, NEFAAand NRAchanges in the opposite direction: NEFAAincreases and NRAdecreases. Thus, it is reasonable that, prospective solution for the Armshould be specified within this range.

In Fig. 12, the horizontal axis stands for the ID of combination, and vertical axis represents Mt(C) (Fig. 12(a)), Mt(Cf) (Fig. 12(b))and Mt(Cm) (Fig. 12(c)). There is a steep increase between No. 2-87 and No. 2-88. Since, the smaller value of Mt(C), Mt(Cf) and Mt(Cm) means the better manipulability of the shoulder prosthesis, prospective solutions should be chosen from the combinations before the No. 88.

Apparently, better accessibility requires a bigger value of NEFAA and NRA, however, from the aforementioned results, it is clear that within the range investigated (as shown in Fig. 11), they can not be satisfied simultaneously. That is, NEFAA and NRA should be traded-off depending on which area (EFAA or RA) is more important.

For this purpose, thresholds were determined as follows to reflect different weighting policies and the constraint from manipulability, i.e., Mt(C), Mt(Cf), Mt(Cm). The average μ0and standard deviation σ0 of NEFAA, NRA, Mt(C), Mt(Cf), Mt(Cm) were calculated. For Mt(C), Mt(Cf), Mt(Cm), threshold value was set as μ0+0.5σ0, which stands for the largest mean value that could be allowed. IN the EFAA-favoured policy, NEFAA should be larger than μ0+0.5σ0(upper bound), but NRA should be at least larger than μ0-0.5σ0 (lower bound). Similarly, in the RA-favoured policy, NRA should be larger than μ0+0.5σ0, but, NEFAA should be at least larger than μ0-0.5σ0. Equation (34)-(a) and (b) show the threshold values reflecting the EFAA-favoured and RA-favoured policies, respectively.

Figure 11.

NEFAAandNRAof 100 combinations ofh2,lR.

Figure 12.

Mt(C),Mt(Cf) andMt(Cm) of 100 combinations ofh2,lR.

NEFAA48.43NRA480.16Mt(C)232709.74Mt(Cf)113.18Mt(Cm)363.05}(a)      or      NEFAA28.97NRA559.78Mt(C)232709.74Mt(Cf)113.18Mt(Cm)363.05}(b) E34

By using the Equation (34), 8 configuration candidates were selected. In Table 4, the configurations painted red and blue are selected by Equation (34)-(a) and Equation (34)-(b), respectively.

Furthermore, since Mt(C), Mt(Cf), Mt(Cm) are the mean values for C, Cf, Cm, for the configurations selected by the threshold values shown in the Equation (34), there is a possibility that, at some postures, the manipulability might be extremely bad. Therefore, it is necessary to specify the lower bound for the worst manipulability that could be allowed. The following three additional estimative indexes were defined, which means the configuration with a smaller 3rd quartile (the value of a point that divides a data set into 3/4 and 1/4 of points) would be tolerated. They are only defined for EFAA, because this area requires precise manipulation more than RA.

  • Q3(C): the 3rd quartile of C(Eq. 30) of points in ΣEFAA

  • Q3(Cf): the 3rd quartile of Cf(Eq. 31) of points in ΣEFAA

  • Q3(Cm): the 3rd quartile of Cm(Eq. 32)of points in ΣEFAA

NO. 2-1234567891011121314151617181920
h2(mm)100107114121128135142149156163100107114121128135142149156163
lR(mm)7070707070707070707090909090909090909090
NEFAA335681010131518881112121617172425
NRA666663657643637634622618616614641633627621615614608602600590
Mt(C)5070456269417986180962402804938777898844112960103360452315054660187825039062675898939851015707744880410
Mt(Cf)79.2584.2158.9187.3990.53109.30114.93119.21130.85114.7970.6575.5583.15106.78112.5889.04105.86110.8377.0677.25
Mt(Cm)228.61253.73185.23277.29255.36340.25368.36387.89485.79452.91136.36155.24184.67256.05282.46242.75325.68352.09278.81289.48
NO. 2-2122232425262728293031323334353637383940
h2(mm)100107114121128135142149156163100107114121128135142149156163
lR(mm)110110110110110110110110110110130130130130130130130130130130
NEFAA1214161619222527293224242830303135353941
NRA617615605601601592582578576570599591577571569560552540526518
Mt(C)4885547609628636907177982714386896270028710269844853563588936869270595766358055170878761548635588802
Mt(Cf)66.8761.5876.1780.5186.4772.1767.0765.9864.8392.5565.1969.5873.4672.9977.1278.9063.6466.6477.5878.17
Mt(Cm)113.31114.71156.27173.98215.79201.13196.84207.91212.25311.08115.32129.50164.08171.75188.88201.24175.38190.43225.62232.43
NO. 2-4142434445464748495051525354555657585960
h2(mm)100107114121128135142149156163100107114121128135142149156163
lR(mm)150150150150150150150150150150170170170170170170170170170170
NEFAA3031313333374244464739404143444546475050
NRA562556548544536521521517511509532528522518518515513503495489
Mt(C)897549485268424703807620178502823458510591295879276179465769697598670691159959231009409336894581100830
Mt(Cf)60.5162.7666.8666.8470.6470.6169.0469.8070.5062.7044.0645.9347.5258.0559.9762.1764.2157.0057.0959.57
Mt(Cm)178.42192.94134.79141.66156.08164.88184.89194.87218.37212.98105.68115.80126.23154.23166.32177.94192.08188.33193.69208.67
NO. 2-6162636465666768697071727374757677787980
h2(mm)100107114121128135142149156163100107114121128135142149156163
lR(mm)190190190190190190190190190190210210210210210210210210210210
NEFAA4142434345484849525244454648505050535456
NRA514514504498482475459445439429474460454444446437429425423419
Mt(C)641958238187069935629611785546916499596597942104230126780133450140020117990121270129470137990139560145390149270
Mt(Cf)42.1352.5154.7858.0558.9851.2353.7755.1955.3357.7465.9668.5571.1961.5362.5465.9169.3069.3371.2372.07
Mt(Cm)92.34116.28128.30141.54149.27144.18157.25167.21174.10187.66167.59183.12197.12170.24180.53196.54213.14219.68231.90241.78
NO. 2-81828384858687888990919293949596979899100
h2(mm)100107114121128135142149156163100107114121128135142149156163
lR(mm)230230230230230230230230230230250250250250250250250250250250
NEFAA5858585757626265676767677069717174747573
NRA434434428420424419417415411411422418414414414409405405401393
Mt(C)9089497221103940109450116780149070158490359530357440381760296780325030600120654050690430749300617550665350703810753910
Mt(Cf)44.4847.4250.4054.0457.1169.1872.59141.36132.40139.36118.09127.62231.17242.59251.72268.34218.07231.35241.08254.22
Mt(Cm)107.26118.82131.14139.62152.66199.98216.74487.80502.54546.75280.27321.11672.03770.05845.97951.96783.71868.60942.701034.10

Table 4.

Combinations of h2and lR.

Fig. 13 shows the values of the new estimative indexes Q3(C), Q3(Cf) and Q3(Cm) for 100 configurations listed in Table 4. Q3(Cm) oscillated with a gradually decreasing peak-to-peak value. Q3(C), Q3(Cf) oscillated, with a biggest peak-to-peak value of 20000, and 250 respectively, and turned to stable at a comparatively low level between No.2-50 to No.2-70. Thus it is clear that these new indexes could provide useful information to select optimal Armconfigurations further.

The values of all the estimative indexes for the selected configurations (for both EFAA-favoured and RA-favoured policies) are shown in Table 5, and Fig. 14.

NO. 2-2930343536415960
h2(mm)156163121128135100156163
lR(mm)110110130130130150170170
NEFAA2932303031305050
NRA576570571569560562495489
Mt(C)71026984487059576635805518975494581100830
Mt(Cf)64.8392.5572.9977.1278.9060.5157.0959.57
Mt(Cm)212.25311.08171.75188.88201.24178.42193.69208.67
Q3(C)83723185210183300201290183480148600105920112380
Q3(Cf)32.50163.16213.12226.05186.47






175.1632.2833.70
Q3(Cm)309.56498.23332.77371.69365.79229.55338.79365.39

Table 5.

8 values of the estimative indexes for selected configurations.

Figure 13.

Q3(C),Q3(Cf) andQ3(Cm) of 100 configurations listed inTable 4.

Figure 14.

Plots of values of the estimative indexes of 8 selected configurations.

It is certain that the selected configuration No.2-29, 30, 34, 35, 36, 41 resulted in a larger NRA, and configuration No.2-59, 60 made a larger NEFAA. Considering the NRA, NEFAAand Q3(C), Q3(Cf), Q3(Cm) together, among the RA-favoured configuration, No.2-29, among the EFAA-favoured, No.2-59 are the optimal configurations.

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6. Discussion

Optimal configurations were selected, using the estimative indexes proposed. How the selected configurations meet the requirements of the shoulder prostheses for daily living is the most important concern. Prototyping and experiments in real daily living environment are necessary to give the evaluation. However, on this research stage, preliminary verification could be done by comparing one selected configuration with the initial solution. The comparison was shown in Table 6 and Fig. 15.

Indexh2-lRCombinationUpgrading(%)
InitialOptimal(No.2-59)
h2(mm)170156
lR(mm)250170
NEFAA6850-26.47
NRA39349525.95
Mt(C)8522409458188.90
Mt(Cf)279.4157.0979.57
Mt(Cm)1195.50193.6983.80
Q3 (C)16046010592033.99
Q3 (Cf)57.5132.2843.87
Q3(Cm)324.22338.79-4.49

Table 6.

A comparison between the optimal with the initial configuration.

As shown in Table 6, all the other indexes are improved at a price of 26.47% reduce of NEFAA. This could be improved or compensated by 1) including h1, and disk size rB, rP, as design parameters to enable better combinations; 2) employing a flexible backbone; 3) increasing actuators’ operating range by changing pneumatic actuators, or serially connecting the current actuators.

There is another important clue shown in Fig. 15. There are 2 relative features of the selected configuration: 1) plots of the selected configuration are more compact than the initial solution; 2) the center of the plot distribution locates quite far from the center of the EFAA. Due to the two features, there are fewer points plotted in ΣEFAA. However,if the center of this compact distribution could be directed towards the center of EFAA, much better configuration could be expected. The relocation of the distribution center could be realized by biasing the initial posture of the shoulder prosthesis, i.e., adjusting resting length of actuators.

Considering the fact that the Arm is used as shoulder prostheses, twisting or bending users’ trunk could also contribute to the posture control. However, this is not preferable, since it could result in fatigue damage accumulation in lower back muscles, due to frequent use of upper limb in daily living. That is why the spatial accessibility would be a very important issue in our future research. Moreover, the existence of singular points should be confirmed, and investigation from the viewpoint of mechanics should be done.

Figure 15.

A comparison between the optimal with the initial configuration.

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7. Conclusion

In the research, approach to optimize configuration for shoulder prostheses considering the spatial accessibility and manipulability was proposed. Since for an individual user, the preferable EFAA and RA might be different due to individual difference in daily living style and tasks, and physical constitution, rather than configuration itself, the approach to find the configuration is more important. Thus our research could facilitate the design process of shoulder prostheses with constrained functional elements.

In the near future, estimative indexes to evaluate spatial distribution should be devised, with which the items for further investigation mentioned in the section 6 should be carried out and verified.

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Acknowledgments

This work was supported in part by a Grant-in-Aid for Scientific Research (B), 2011, 23300206, from the Ministry of Education, Culture, Sports, Science and Technology of Japan, and by the Mitsubishi Foundation.

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Written By

Masashi Sekine, Kento Sugimori and Wenwei Yu

Submitted: November 17th, 2010 Reviewed: April 16th, 2011 Published: August 29th, 2011