Optimal synthesis of a planar four-link mechanism used in a hand prosthesis

Mechanism and Machine Theory 36 (2001) 1203±1214

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Optimal synthesis of a planar four-link mechanism used in a hand prosthesis E. Ngale Haulin, A.A. Lakis *, R. Vinet  Department of Mechanical Engineering, Ecole Polytechnique de Montr eal, University of Montreal, CP 6079, Succursale Centre-ville, Montreal, Que., Canada H3C 3A7 Received 19 July 1999; accepted 12 June 2000

Abstract The optimal synthesis of a planar four-link mechanism is carried out with reference to n positions of the output and coupling bars. The e€ects of the number of positions in the optimal dimensions of the mechanism are presented. Through a comparative study, we are able to select the number of points and the position of each which best satisfy the performance criteria. The optimal mechanism obtained is of the balance±balance type, with a minimum acceptable angle of transmission and no dead point. This crossed four-link mechanism is used in the development of a hand prosthesis. Ó 2001 Published by Elsevier Science Ltd.

1. Introduction A considerable amount of research has been carried out on the synthesis of mechanisms with several linkages [1±13]. Such mechanisms can be used in many ways and, in general, synthesis is carried out with reference to n input positions and n output positions [1±4]. The optimisation of these mechanisms highlights several performance criteria, notably: design errors based on the residual error in Freudenstein's equation [1±4]; the mean quadratic error in the angular displacement of one link or in the curvilinear displacement of one point of the mechanism [4±7,9,11]; the quality of transmission [2,4,8,11,13]; and the angle of transmission [1,7]. Our study emphasises the criterion based on the mean quadratic error of the angular displacement of a link, in order to ensure satisfactory coordination of the angles of the mechanism. The four-link mechanism which we propose to synthesize is used in a hand prosthesis which is able to grasp a cylindrical object and to pick up an object with its digits [10,11,13]. Its optimal synthesis *

Corresponding author. Tel.: +1-514-340-4711x4906; fax: +1-514-340-4176. E-mail address: [email protected] (A.A. Lakis).

0094-114X/01/$ - see front matter Ó 2001 Published by Elsevier Science Ltd. PII: S 0 0 9 4 - 1 1 4 X ( 0 1 ) 0 0 0 3 9 - 8

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will be e€ected with reference to n positions of the coupling and output links [13]. To achieve this, Haulin et al. [13] proposed two performance criteria: the mean quadratic error in the angle of the coupling link and the maximum driving torque necessary to counter-balance a force exerted on the end of a ®nger. In this study, the optimal synthesis of the mechanism will be based upon two criteria: the energy consumed over a cycle of that mechanism and the mean quadratic error in the angle of the coupling link or in the bending angle of the second joint of one ®nger. We shall ®rst analyse the mechanism by generating all the mathematical equations necessary for synthesis and, second, we shall carry out the optimal synthesis of the mechanism with reference to at least three positions. Using data relative to the middle ®nger, the results obtained will be compared and discussed. 2. Analysis of the mechanism 2.1. Mechanism description Fig. 1 [12,13] shows a crossed four-link mechanism used in a hand prosthesis. QM is the ®xed link and AM is the driving link. TP1, TP2 and TP3 are, respectively, the angles of phalanges P1, P2 and P3. The coupling link AB is rigidly connected to phalanges P2 and P3. The angles of bars

Fig. 1. A four-link mechanism used in a hand prosthesis.

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h1 , h2 , h3 and h4 are measured relative to a plane parallel to the plane of the palm. The articulation IP between the two ®rst joints leads outwards from EXC towards the back of the hand (point B). The force P exerted on the end of a ®nger makes an angle hP with the axis of phalanx P3 whose P30 is the e€ective length. This mechanism has only one degree of freedom. In e€ect, following Gruebler's formula [13±15] for planar mechanisms, the number of degrees of freedom is as follows: degree of freedom ˆ 3…N 1† 2P1 2P2 with N ˆ total number of bars ˆ 4; P1 ˆ total number of joints with 1 degree of freedom ˆ 4; P2 ˆ total number of joints with 2 degrees of freedom ˆ 0. 2.2. Freudenstein's equation Taking the plane of the palm as a reference and maintaining the clockwise direction of the angles, the vector for the mechanism in Fig. 1 for a given position ``i'' is as follows: ~ R1i ‡ ~ R2i ‡ ~ R3i ‡ ~ R4 ˆ ~ 0

…1†

with Rxi ˆ rx ejhxi ˆ rx ‰cos hxi ‡ j sin hxi Š: In consequence, we obtain: r1 cos h1i ‡ r2 cos h2i ‡ r3 cos h3i r1 sin h1i ‡ r2 sin h2i ‡ r3 sin h3i

r4 cos h4 ˆ 0;

…2†

r4 sin h4 ˆ 0:

To establish the relationship between position h1i of the output link and position h2i of the coupling link, we eliminate h3i from Eq. (2). We thus obtain Freudenstein's equation as follows: k1 cos…h1i

h4 † ‡ k2 cos…h2i

h4 †

k3 ˆ cos…h1i

h2i †

with k1 ˆ

r4 ; r2

k2 ˆ

r4 ; r1

k3 ˆ

r12 ‡ r22 r32 ‡ r42 : 2r1 r2

2.3. Calculation of the angles of the mechanism From Fig. 1, it can be shown that the required or known angle of the output link is: h1i ˆ TP1i ‡ hB with

 hB ˆ

arctan

 EXC : P1

That of the coupling link is: h2i ˆ hA ‡ …TP2i ‡ TP1i † ˆ h2ir

…r ˆ required†;

where hA is the angle between the second phalanx axis and the coupling link.

…3†

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By eliminating h3i from Eq. (2), we obtain the real value of the angle of the coupling link: p ! B  B2 4AC ˆ h2ic …c ˆ calculated† …4† h2i ˆ 2 arctan 2A with A ˆ k1 cos…h1i

h4 †

k2 cos h4

B ˆ 2…k2 sin h4

sin h1i †;

C ˆ k1 cos…h1i

h4 † ‡ k2 cos h4

k3 ‡ cos h1i ; k3

cos h1i :

Since we are dealing with a crossed four-link mechanism, the unique solution is: p ! B ‡ B2 4AC h2i ˆ 2 arctan 2A (cf. [13,15,16]). By eliminating h2i from Eq. (2), we obtain: p ! E E2 4DF h3i ˆ 2 arctan 2D

…5†

…6†

with D ˆ k4 cos…h1i

h4 †

E ˆ 2…k2 sin h4

sin h1i †;

F ˆ k4 cos…h1i

h4 † ‡ k2 cos h4 r12

k2 cos h4

r22

r32

k5 ‡ cos h1i ; k5

cos h1i ;

r42

r4 ‡ ‡ ; k5 ˆ : r3 2r1 r3 The minimum angle of transmission can be expressed as  2  r ‡ r22 r32 r42 ‡ 2r3 r4 cos…h3n h4 † ; lmin ˆ arccos 1 2r1 r2 k4 ˆ

where h3n is the angle of the driving link at position n of the mechanism. The angles corresponding to the dead points of the mechanism are: " ! # 2 2 2 r …r ‡ r † r 2 1 3 4 ‡p ; TH1D1 ˆ h4 cos 1 2…r2 ‡ r1 †r4 ! 2 2 2 r ‡ r …r ‡ r † 2 1 3 4 ; TH3D1 ˆ h4 cos 1 2r3 r4 " ! # r32 …r2 r1 †2 r42 1 ‡p ; cos TH1D2 ˆ h4 2jr2 r1 jr4 ! r32 ‡ r42 …r2 r1 †2 1 TH3D2 ˆ h4 cos ; 2r3 r4

…7†

…8†

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Fig. 2. Finger con®guration and ®ngertip locus.

where …TH1D1; TH3D1† and …TH1D2; TH3D2† are the angles of the output and driving links at the two dead points, (D1) and (D2), respectively. The limiting angles of the mechanism, TH1L1, TH3L1, TH1L2 and TH3L2 are obtained from its initial and ®nal positions as de®ned by Guo et al. [11]. Given lengths of ®rst, second and third phalanges of a human ®nger (Fig. 2), these authors modelled two con®gurations of the middle ®nger using an exoskeletal device for measuring joint angles a1 , a2 and a3 . The model obtained for the pinching con®guration (®nger grip: 5° 6 a1 6 42:5°) is given by a2 ˆ 2:45a1 and a3 ˆ 3:35a1 ; using Figs. 1 and 2, it can be shown that TP2 ˆ a1 and TP2 ‡ TP1 ˆ a2 ; therefore, 5° 6 TP1 6 42:5° and TP2 ˆ 1:45TP1. For the holding con®guration (cylindrical grip: 42:5° 6 a1 6 80° or 42:5° 6 TP1 6 80°), this model is expressed by a2 ˆ 2a1 ‡ 19:125° and a3 ˆ 3:35a1 ; therefore, TP2 ˆ TP1 ‡ 19:125°. We used computer aided design techniques to obtain the morphology design of the hand prosthesis and found that the ¯exion angle TP3 is constant for the two con®gurations. 3. Optimal synthesis of the mechanism 3.1. Synthesis with reference to n positions By applying the least squares technique to Freudenstein's equation, we can de®ne: n X 2 Dˆ ‰k1 cos…h1i h4 † ‡ k2 cos…h2i h4 † k3 cos…h1i h2i †Š

…9†

iˆ1

(cf. [14]), where n is the number of positions. In order to obtain the smallest error between the desired value of the angle of the coupling link and that which is actually obtained, we assume the following equations: oD ˆ 0; ok1

oD ˆ 0; ok2

oD ˆ0 ok3

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and obtain the following system of Eq. (10): Pn Pn 2 2 h4 † h4 † cos…h1i iˆ1 cos …h1i iˆ1 cos…h2i Pn 6 Pn 2 h4 † 4 iˆ1 cos…h1i h4 † cos…h2i h4 † iˆ1 cos …h2i Pn Pn h4 † h4 † iˆ1 cos…h1i iˆ1 cos…h2i 9 8 9 8 Pn > < Piˆ1 cos…h1i h2i † cos…h1i h4 † > = < k1 > = > n  k2 ˆ h2i † cos…h2i h4 † : iˆ1 cos…h1i > > Pn : ; : > ; > cos…h h † k3 1i 2i iˆ1

h4 †

Pn

iˆ1 Pn iˆ1

cos…h1i cos…h2i

h4 †

3

7 h4 † 5

n …10†

By using Cramer's rule, we obtain: k1 , k2 and k3 . Since we know r1 , we can calculate r4 , r2 and r3 . 3.2. Optimal synthesis The optimal dimensions of the mechanism are obtained by varying the angles hA and h4 . In fact, several mechanisms are obtained; the optimal mechanism is that with the smallest value of the energy consumed or the smallest value of the mean quadratic error of the bending angle TP2 of the second joint of one ®nger, that is to say of angle h2 of the coupling link. In addition, this mechanism should have a minimum acceptable transmission angle: lmin P 35° [15] and should not reach a dead point: TH1L1 P TH1D1;

TH1L2 6 TH1D2;

TH3L1 P TH3D1;

TH3L2 6 TH3D2:

Assuming that frictional forces at joints are negligible and as shown in Fig. 1: · The driving torque MM is: 8 u1 ˆ sin…hP ‡ w h03i †; > > > > < u2 ˆ Z r2 sin…hP ‡ w h2i †; u2 u3 MM ˆ r3 Pv with : v ˆ u1 ‡ ; u3 ˆ sin…h03i h1i †; > u4 > u ˆ r2 sin…h2i h1i †; > > : 40 h3i ˆ h3i p

…11†

…12†

and Z ˆ P30 sin hP ‡ P2 sin…hP ‡ wi

ui †

EXC cos…hP ‡ wi

· The energy consumed over a cycle of the mechanism is: Z h037 jMM j dh03 ; Energy ˆ h031

ui †:

…13†

where the absolute value of driving torque jMM j is used in order to prevent negative energy consumption; h03 ˆ h3 p; h031 and h037 are, respectively, the initial and the ®nal positions of the driving link with respect to the horizontal plane. · The mean quadratic error is: n n 1 X 1 X …TP2ic TP2ir †2 ˆ …h2ic h2ir †2 ; …14† Error ˆ n 1 iˆ1 n 1 iˆ1

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where n is the number of known positions and n 1 the number of intervals; TP2ic and h2ic represent the calculated or real values obtained, while TP2ir and h2ir are the required or known values. To guarantee that the links stay crossed during movement of the mechanism, certain constraints apply. If we take xQy as the point of origin, with the positive y-axis oriented towards the bottom and the positive x-axis oriented towards the right and parallel to the plane of the palm, we have the following relations: hQB < hQA and hMB < hMA where hQA , hMB and hMA are de®ned as is   yB yQ 1 hQB : hQB ˆ tan xB xQ is the angle measured in a clockwise direction, from axis x to the link QB. In addition, the following condition must be ful®lled: h1i 6 h2i where h1i and h2i are, respectively, the angle of the output link and that of the coupling link.

4. Calculation and discussion 4.1. Data relative to the middle ®nger Eccentricity EXC ˆ 2 mm [12,13]. Force P ˆ 45 N with hP ˆ 90°. Length of the joints (in mm) [12,13]: P1 ˆ 44:5; P2 ˆ 29:20; P30 ˆ 17:30. Bending angles: TP3 ˆ 30° ˆ constant [12,13]. Finger grip: 5° 6 TP1 6 42:5°, TP2 ˆ 1:45TP1 [11,13]. Cylindrical grip: 42:5° 6 TP1 6 80°, TP2 ˆ TP1 ‡ 19:125° [11,13]. p Fig. 1 shows that we have r1 ˆ P12 ‡ EXC2 and h1i ˆ TP1i ‡ hB with hB ˆ arctan…EXC= P1†. Otherwise, we have: 7:5° 6 hA 6 77:5°; 5 6 r2 6 15; 5 6 r4 6 7:5; 315° 6 h4 6 360°. 4.2. Results and discussion A computer program has been written using Matlab software. By varying the angles h4 and hA by intervals of 1° and 2.5°, respectively, we obtain two optimal mechanisms: one related to the minimum of the energy consumed and the other to the minimum mean quadratic error. In each case, the minimum angle of transmission is calculated. The optimal mechanisms obtained are of the balance±balance type, with an acceptable angle of transmission and no dead point. The optimal variables of the mechanisms are presented in Tables 1, 2, 3, 4, 5, 6 and 7, respectively, for 3, 4, 5, 6, 7, 8 and 9 positions of the output and the coupling links. They are functions on the one hand of the total number of positions Nbpos and, on the other hand, of the ratio Nb1 =Nb2 , where Nb1 is the number of positions for the ®nger grip, and Nb2 is the number of positions for the cylindrical grip. In addition, Nbpos ˆ Nb1 ‡ Nb2 1, since the two grips have a common point for which TP1 ˆ 42:5°.

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Since the three position synthesis leads to a null mean quadratic error, Table 1 shows only optimal variables based on minimum energy consumed. Tables 2±7 show those based on both the minimum energy consumed and, in brackets, the minimum mean quadratic error of the angle of the coupling link. These tables enable us to state that: · Because of their in¯uence on the performance criteria used, the optimum linear and angular dimensions, except the optimum angle h4 based on the minimum mean quadratic error, vary as functions of both the ratio Nb1 =Nb2 and the total number of positions Nbpos . · Since hA and r2 are related to the coupling link and due to their in¯uence on the mean quadratic error which is zero for a three position synthesis, hA and both h4 and r2 are, for a three position, Table 1 Optimal dimensions for minimum energy with three positions of the output and coupling links Optimal variables (ratio Nb1 =Nb2 ˆ 1=1) r2 (mm)

r3 (mm)

R4 (mm)

hA …°†

h4 …°†

7.58

45.30

7.42

10

338

Table 2 Optimal dimensions for minimum energy (minimum error) with four positions of the output and coupling links Ratios Nb1 =Nb2 3/2 2/3

Optimal variables r2 (mm)

r3 (mm)

r4 (mm)

hA …°†

h4 …°†

6.58 (6.44) 6.54 (6.54)

45.11 (45.19) 45.38 (45.38)

7.50 (7.47) 7.37 (7.37)

30 (32.5) 30 (30)

320 (317) 317 (317)

Table 3 Optimal dimensions for minimum energy (minimum error) with ®ve positions of the output and coupling links Optimal variables r2 (mm) r3 (mm) r4 (mm) hA …°† h4 …°†

Ratios Nb1 =Nb2 3/3

4/2

2/4

6.79 (6.37) 44.96 (45.17) 7.47 (7.37) 25.0 (32.5) 326 (317)

6.60 (6.31) 45.02 (45.16) 7.43 (7.35) 27.5 (32.5) 323 (317)

6.74 (6.47) 45.23 (45.36) 7.43 (7.29) 27.5 (30) 321 (317)

Table 4 Optimal dimensions for minimum energy (minimum error) with six positions of the output and coupling links Optimal variables

Ratios Nb1 =Nb2 4/3

5/2

3/4

2/5

r2 (mm) r3 (mm) r4 (mm) hA …°† h4 …°†

6.70 (6.27) 44.95 (45.16) 7.40 (7.29) 25.0 (32.5) 326 (317)

6.73 (6.26) 44.95 (45.16) 7.46 (7.30) 25.0 (32.5) 326 (317)

7.01 (6.47) 44.95 (45.19) 7.45 (7.43) 20.0 (32.5) 331 (317)

6.71 (6.44) 45.22 (45.35) 7.39 (7.25) 27.5 (30) 321 (317)

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Table 5 Optimal dimensions for minimum energy (minimum error) with seven positions of the output and coupling links Optimal variables r2 (mm) r3 (mm) r4 (mm) hA …°† h4 …°†

Ratios Nb1 =Nb2 4/4

5/3

6/2

3/5

2/6

6.77 (6.34) 44.97 (45.18) 7.46 (7.37) 25.0 (32.5) 326 (317)

6.88 (6.34) 44.90 (45.20) 7.48 (7.50) 22.5 (35.0) 329 (315)

6.43 (6.32) 45.11 (45.20) 7.50 (7.50) 32.5 (35.0) 318 (315)

7.07 (6.49) 44.96 (45.21) 7.49 (7.50) 20.0 (32.5) 331 (317)

6.69 (6.42) 45.22 (45.35) 7.37 (7.23) 27.5 (30) 321 (317)

Table 6 Optimal dimensions for minimum energy (minimum error) with eight positions of the output and coupling links Optimal variables

Ratios Nb1 =Nb2 5/4

6/3

7/2

4/5

3/6

2/7

r2 (mm) r3 (mm)

6.74 (6.31) 44.97 (45.18) 7.45 (7.34) 25.0 (32.5) 326 (317)

6.86 (6.31) 44.90 (45.20) 7.47 (7.49) 22.5 (35.0) 329 (315)

6.41 (6.30) 45.10 (45.20) 7.48 (7.48) 32.5 (35.0) 318 (315)

6.67 (6.42) 45.05 (45.21) 7.47 (7.45) 27.5 (32.5) 323 (317)

6.90 (6.35) 45.01 (45.28) 7.43 (7.35) 22.5 (32.5) 328 (316)

6.68 (6.55) 45.21 (45.41) 7.36 (7.49) 27.5 (32.5) 321 (315)

r4 (mm) hA (°) h4 (°)

Table 7 Optimal dimensions for minimum energy (minimum error) with nine positions of the output and coupling links Optimal variables r2 (mm) r3 (mm) r4 (mm) hA (°) h4 (°)

Ratios Nb1 =Nb2 5/5

6/4

7/3

4/6

3/7

6.64 (6.38) 45.05 (45.20) 7.46 (7.43) 27.5 (32.5) 323 (317)

6.72 (6.28) 44.96 (45.17) 7.44 (7.33) 25.0 (32.5) 326 (317)

6.53 (6.29) 45.02 (45.20) 7.48 (7.47) 30.0 (35.0) 321 (315)

6.86 (6.28) 45.01 (45.27) 7.43 (7.31) 22.5 (32.5) 328 (316)

6.94 (6.40) 45.02 (45.29) 7.47 (7.40) 22.5 (32.5) 328 (316)

respectively, smaller and greater than those obtained at other positions. At these positions, the value of hA and that of the driving link r3 are more important for the minimum mean quadratic error than are those relative to the minimum energy consumed. Figs. 3, 4 and 5 represent, respectively, the optimal values of both energy consumed (energy) and mean quadratic error (error), the total number of mechanisms synthesized (Nbmec ) and the minimum transmission angle (Mumin ) related to optimisation criteria as functions of the total number of positions Nbpos and the ratio Nb1 =Nb2 . For each criterion, three sets of ratio are used: 1. Nb1 > Nb2 (Nb1 =Nb2 ˆ 3=2, 4/2, 5/2, 5/3, 6/3, 7/3), respectively, for 4, 5, 6, 7, 8 and 9 positions; 2. Nb1 ˆ Nb2 (Nb1 =Nb2 ˆ 1=1, 3/3, 4/4, 5/5), respectively, for 3, 5, 7 and 9 positions; 3. Nb1 < Nb2 (Nb1 =Nb2 ˆ 2=3, 2/4, 2/5, 3/5, 3/6, 3/7), respectively, for 4, 5, 6, 7, 8 and 9 positions.

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Fig. 3. Optimal values of energy consumed and mean quadratic error.

Fig. 4. Number of mechanisms synthesized during the optimisation process.

Fig. 5. Minimum angle of transmission of optimal mechanisms.

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These ®gures enable us to state that: · Of the 77 mechanisms which were synthesized with reference to three positions, the optimal mechanism has a null mean quadratic error and an energy consumed which is almost double that obtained with the other positions. · For n > 3, the energy consumed varies hardly at all as a function of both the ratio Nb1 =Nb2 and the total number of positions. Otherwise, the number of mechanisms synthesized and the mean quadratic error vary highly as functions of the above mentioned parameters for Nbpos 6 7 and vary very little for Nbpos > 7. In addition, for Nbpos 6 7, the mean quadratic error is proportional to Nbpos when Nb1 < Nb2 and inversely proportional when Nb1 P Nb2 . · The minimum angle of transmission obtained for the energy consumed is greater than that obtained the for the mean quadratic error since the quality of the transmission is in¯uenced by on this error; nevertheless, the values computed are acceptable since they are greater than 35°. For the energy consumed, this angle is proportional to Nbpos when Nbpos 6 7 and inversely proportional when Nbpos > 7 whilst that related to the mean quadratic error is inversely proportional to Nbpos . The optimal linear and angular dimensions obtained in [13] using the maximum driving torque are generally greater than those obtained in this study. Therefore, the optimal mechanism depends highly on the criterion used. 5. Conclusion The crossed four-link mechanism used in a hand prosthesis has been optimally synthesized following two performance criteria, namely the energy consumption and the mean quadratic error of the bending angle of the second joint of the middle ®nger. Therefore, two optimal mechanisms were obtained. This study permits us to con®rm that the three position synthesis leads to a null mean quadratic error. However, the energy consumed obtained is very high. With three and more positions, this study leads to the conclusion that the mean quadratic error is an important factor in the process of optimising the mechanism. The mechanisms parameters such as the mean quadratic error, the energy consumed, the minimum angle of transmission and the total number of mechanisms synthesized vary very little when Nbpos > 7. In addition, the mean quadratic error is inversely proportional to Nbpos when Nbpos 6 7 and Nb1 P Nb2 . The method of synthesizing by the least squares methods has proven e€ective when the total number of positions and the appropriate ratios are carefully selected. Finally, the maximum driving torque [13], the energy consumed and the mean quadratic error used separately in the design of a mechanical ®nger based on a four-bar linkage must be optimised simultaneously because of the in¯uence of each of them on the design.

References [1] S.V. Kulkarni, R.A. Khan, Synthesis of a four-bar mechanism with optimum transmission angle, in: Proceedings of the 6th World Congress Th. Mach. Mech., Vol. 1, Halsted Press, New York, USA, 1983, p. 94.

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