On mechanism design optimization for motion generation

Mechanism and Machine Theory

Mechanism and Machine Theory 42 (2007) 1251–1263

www.elsevier.com/locate/mechmt

On mechanism design optimization for motion generation Peter J. Martin a, Kevin Russell a

b

a,*

, Raj S. Sodhi

b

Armaments Engineering and Technology Center, US Army Research, Development and Engineering Center, Picatinny Arsenal, NJ 07806-5000, USA Department of Mechanical Engineering, New Jersey Institute of Technology, Newark, NJ 07102-1982, USA Received 14 March 2006; received in revised form 17 August 2006; accepted 30 November 2006 Available online 5 February 2007

Abstract A set of Burmester curves can represent an infinite number of planar four-bar motion generator solutions for a given series of prescribed rigid-body poses. Unfortunately, given such a vast number of possible mechanical solutions, it is difficult for designers to arbitrarily select a Burmester curve solution that ensures full link rotatability, produces feasible transmission angles and is as compact as possible. This work presents an algorithm for selecting planar four-bar motion generators with respect to Grashof conditions, transmission angle conditions and having the minimal perimeter value. This algorithm has been codified into MathCAD for enhanced analysis capabilities and ease of use. The example in this work demonstrates the synthesis of a compact planar, four-bar crank rocker motion generator with feasible transmission angles. Ó 2006 Elsevier Ltd. All rights reserved.

1. Introduction 1.1. Background and objective Planar four-bar mechanisms have been, and remain widely implemented in mechanical systems. Due primarily to the joint type, joint axes orientations and overall planar kinematics of this mechanism, it is often very practical to analyze, design, fabricate and implement. In regards to the widespread usefulness of the planar four-bar mechanism, the number of works produced for the design and analysis of this mechanism-both graphical and analytical-make up one of the most extensive sources for any mechanical linkage. The objective in kinematic motion generation is to determine the mechanism parameters required to approximate or precisely achieve a series of prescribed rigid-body poses. Motion generation methods, specifically Burmester curve-based motion generation produce an infinite number of solutions for a prescribed series of rigid-body poses. In other words, the user can select a mechanism solution from an infinite array of possible mechanism solutions. Although all of these solutions may be kinematically feasible, some of them may not

*

Corresponding author. Tel.: +1 973 724 6073; fax: +1 973 724 6027. E-mail address: [email protected] (K. Russell).

0094-114X/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmachtheory.2006.11.009

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represent practical engineering solutions. For example, some mechanisms may produce undesired transmission angles, require large workspaces or not ensure full link rotatability. One method to employ Burmester curve-based motion generation while adhering to specific mechanism design conditions and constraints is through a search and selection algorithm. This work presents an algorithm for selecting planar four-bar motion generators with respect to Grashof conditions, transmission angle conditions and having the minimal perimeter value. This algorithm has been codified into MathCAD for enhanced analysis capabilities. 1.2. Literature review Mechanism synthesis with optimization includes the work of Yao and Angeles [1] who apply the contour method in the approximate synthesis of planar linkages for rigid-body guidance. Cabrera et al. [2] considers solution methods for the optimal synthesis of planar mechanisms. Cossalter et al. [3] presented a numerical method for optimum synthesis of planar mechanisms for motion, path and function generation. The work of Krishnamurty and Turcic [4] applies non-linear goal programming for the optimal synthesis of planar mechanisms. The work of Akhras and Angeles [5] considers unconstrained non-linear least-square techniques in the optimization of planar linkages for rigid-body guidance. Sandgren’s [6] work proposes a design tree structure for the optimization of mechanisms that considers both geometric and topological change. Khare and Dave [7] presented a procedure for the synthesis of planar four-bar double rocker mechanisms for coordinating the prescribed extreme positions. Da Lio et al.’s [8] work deals with the use of natural coordinates for the synthesis of mechanisms using optimization methods. A gradient-based optimization approach for path synthesis problems was proposed by Sancibrian et al. [9] and a global optimum obtaining problem in approximate path synthesis of linkages was considered by Marı´n and Gonza´lez [10]. The work of Avil’es et al. [11] presents a method for the optimum synthesis of planar mechanisms with lower pairs and a configuration of any type. Vasiliu and Yannou [12] proposed a method to synthesize the dimensions of a planar path generator. Rao [13] applies geometric programming to the synthesis of planar four-bar function generators and spherical mechanisms with minimum structural error. The kinematic analysis and design optimization of spatial parallel manipulators were considered in the works of Tsai and Joshi [14] and Carretero and Podhorodeski [15]. Hong and Erdman [16] introduced a new application Burmester curves for adjustable linkages. In the work of Arsenault and Boudreau [17], planar parallel mechanisms are synthesized with respect to workspace, dexterity, stiffness and singularity avoidance. In the work of Smaili et al. [18] a new tabu-gradient search algorithm is incorporated for the optimum synthesis of linkages. 2. Burmester curve synthesis 2.1. Methodology The Burmester curve methodology and algorithm presented in this section are included in the work of Sandor and Erdman [19]. Although there are numerous Burmester curve solution methodologies, the method included in Sandor and Erdman’s work is among those can be codified in modern mathematical analysis software (e.g. MathCAD) with relative ease. A planar four-bar mechanism and left-side dyad WZ are illustrated in Fig. 1. Vector W points from the fixed pivot m to the moving pivot k. Vector Z points from the moving pivot k to the coupler point P. Fig. 2 illustrates the displacement of dyad WZ resulting in a coupler point displacement (from point P1 to Pj) of dj. Eqs. (1) and (2) represent a closed-loop formulation of the displacement of dyad WZ in Fig. 2. For four prescribed coupler positions, there will exist three coupler point displacements (d), three coupler displacement angles (a) and subsequently, the three closed-loop formulations in Eqs. (3)–(5). These three equations form a set in which five unknowns exist (complex unknowns W, Z and angles b2, b3 and b4). Assuming a range of solutions for angles b3 and b4 can be calculated, given a prescribed range of values for angle b2, a range for vectors W and Z could be calculated using any two of Eqs. (3)–(5). A locus of moving pivot locations (variable k1 in Fig. 2) could be produced knowing k1 = R1  Z and a locus of the fixed pivot locations (variable m in Fig. 2) could be produced knowing m = k1  W. Fixed and

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Im Re

P Z

k

W

m

Fig. 1. Planar four-bar mechanism and dyad.

θ1 Pj

θj

Rj R1

δj

Im

Re

P1

Ze iαj Z

αj kj k1

We iβj

W

βj

m Fig. 2. Planar four-bar mechanism dyad in positions ‘‘1’’ and ‘‘j’’.

moving pivot loci are also called circle and center point curves respectively, or more commonly, Burmester curves. Fig. 3 illustrates a pair of Burmester curves produced for four user-specified coupler positions. Each point on the circle point curve has a corresponding center point curve point (or vice-versa). A planar four-bar motion generator can be constructed given two pairs of circle and center point curve points. Weibj þ Zeiaj  dj  Z  W ¼ 0 Wðe

ibj

 1Þ þ Zðe

iaj

 1Þ ¼ dj

ð1Þ ð2Þ

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P.J. Martin et al. / Mechanism and Machine Theory 42 (2007) 1251–1263 1.30 circle point curve 1.20

center point curve

1.10

Imaginary

1.00 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.80

0.90

1.00

1.10

1.20

1.30

1.40

1.50

1.60

1.70

1.80

1.90

2.00

Real Fig. 3. A Burmester curve pair.

Wðeib2  1Þ þ Zðeia2  1Þ ¼ d2 Wðe

ib3

Wðe

ib4

ð3Þ

 1Þ þ Zðe

ia3

 1Þ ¼ d3

ð4Þ

 1Þ þ Zðe

ia4

 1Þ ¼ d4

ð5Þ

2.2. Burmester curve algorithm As mentioned in the previous section, it was assumed that ranges for angles b3 and b4 could be calculated given a prescribed range for angle b2. Table 1 includes an algorithm [19] for calculating b3 and b4 given a prescribed range for b2. Eqs. (3)–(5) can be expressed in matrix form as Eq. (6). The only unknown data in Eq. (6) are rotations b3 and b4. This system can only have a solution if the rank of the augmented matrix of the coefficients is 2. The augmented matrix ‘‘M’’ is formed by adding the right-hand column in Eq. (6) to the coefficient matrix of the Table 1 Computation of b3 and b4 for a given value of b2 D ¼ D1 þ D2 eib2 2

2

2

jD3 j jDj cos h3 ¼ jD4 j 2jD 3 kDj pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi  sin h3 ¼  1  cos2 h3  P 0

0 6 h3 6 p b3 = argD + h3  argD3 2

2

2

jD4 j jDj cos h4 ¼ jD3 j 2jD 4 kDj pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin h4 ¼ j 1  cos2 h4 j P 0 0 6 h4 6 p b4 = argD  h4  argD4

Note: For complete algorithm and description please refer to Ref. [19].

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left side. Therefore it is necessary that the determinant of the augmented matrix of this system be equal to zero (Eqs. (7) and (8)). 2 ib 3 2 3 d2 e 2  1 eib4  1   6 ib3 7 W 6 7 ib4 ¼ 4 d3 5 ð6Þ 4e  1 e  15 Z ib4 ia4 d e 1 e 1 4 2 ib 3 e 2  1 eia2  1 d2 6 7 Det½M ¼ Det4 eib3  1 eia3  1 d3 5 ¼ 0 ð7Þ eib4  1 D2 e

ib2

þ D3 e

ib3

þ D4 e

ib4

eia4  1

d4

þ D1 ¼ 0

ð8Þ

where  ia3   e  1 d3    D2 ¼  ia e 4  1 d4   ia2   e  1 d2   D3 ¼  ia e 4  1 d4   ia2   e  1 d2    D4 ¼  ia e 3  1 d3  D1 ¼ D2  D3  D4 3. Grashof, transmission angle and mechanism perimeter criteria 3.1. Methodology Full link rotatability is a particularly practical design characteristic for planar four-bar mechanisms-especially when the crank link is driven by a drive system that rotates continuously. Grashof criteria predict the rotation behavior (or rotatability) of a four-bar linkage’s inversions based on its link lengths. Table 2 includes the five Grashof classifications for planar four-bar mechanisms. In this table, variables Lmin, Lmax, La and Lb represent the longest link length, shortest link length and intermediate link lengths respectively. With the exception of the Grasof triple rocker, all other Grashof classifications have full link rotatability. An additional practical design characteristic for planar four-bar mechanisms is that it produces feasible transmission angles. The transmission angle is the angle between the coupler link and the follower link. A transmission angle of 90° is ideal while angles of 0° and 180° result in maximum joint loading in the mechanism and subsequently maximum joint wear. Another practical design characteristic for planar four-bar mechanisms is compactness. In this work, a compact mechanism is defined as one in which the sum of the crank, coupler, follower and ground lengths (or the mechanism perimeter) is the smallest possible. In general, compact mechanisms produce smaller workspaces and are more structurally sound than four-bar mechanisms with larger perimeters.

Table 2 Planar four-bar mechanism Grashof classifications Type of mechanism

Shortest link

Relationship between link lengths

Crank rocker Drag link Double rocker Change point Triple rocker

Crank Ground Coupler Any Any

Lmax + Lmin < La + Lb Lmax + Lmin < La + Lb Lmax + Lmin < La + Lb Lmax + Lmin = La + Lb Lmax + Lmin > La + Lb

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Input Generated Burmester Curve

1. Calculate all Possible Mechanism Solutions for the given Burmester Curve

2. Select Mechanism Solutions with Feasible Transmission Angles

3. Select Particular Grashof Mechanism Solutions

4. Select Minimum Perimeter Solution

5. Output Optimized Motion Generator Fig. 4. Diagram of secondary algorithm.

3.2. Algorithm In addition to the Burmester curve algorithm in Section 2, an additional secondary algorithm is incorporated to select the circle and center point pairs of the optimized motion generator. Fig. 4 illustrates a diagram of the secondary algorithm. The generated Burmester curves are the initial input and the parameters of an optimized motion generator are the final output of the secondary algorithm. The first procedure in the secondary algorithm (block #1 in Fig. 4) involves the calculation of every mechanism solution for discrete sets of points for given circle and center point curves. Since Burmester curves are infinite, the exact section of a circle and center point curve pair to consider as well as the precise number of discrete circle and center points to include is at the discretion of the user. This piecewise analysis of circle and center point curves will allow user to consider specific curve regions of interest. Appendix A.1 includes the MathCAD commands to calculate lengths of the crank, coupler, follower and ground links for every mechanism solution for discrete sets of circle and center curve points from the given Burmester curves. The second procedure in the secondary algorithm (block #2 in Fig. 4) involves the selection (from among the possible solutions produced in the first procedure) of all mechanism solutions that satisfies user-defined minimum and maximum transmission angle conditions. Prior to mechanism solution selection, the transmission angles of all mechanism solutions in the first procedure are calculated for a prescribed crank rotation range. Appendix A.2 includes the MathCAD commands for transmission angle calculation and selection of mechanism solutions with feasible transmission angles. The third procedure in the secondary algorithm (block #3 in Fig. 4) involves the selection of all mechanism solutions of a particular Grashof classification from among the possible solutions produced in the second procedure. Table 2 includes all of the Grashof mechanism classifications and conditions. The example in this work demonstrates the synthesis of a compact planar, four-bar crank rocker motion generator. Appendix A.3 includes the MathCAD commands to select Grashof crank rocker solutions. The fourth procedure in the secondary algorithm (block #4 in Fig. 2) involves the selection of the mechanism solution (from among the possible solutions produced in the third procedure) with the minimum

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perimeter. A minimum perimeter condition can help improve the efficiency in selecting compact four-bar motion generator designs from among all possible solutions. In general, compact mechanisms produce smaller workspaces and are more structurally sound than four-bar mechanisms with larger perimeters. Appendix A.4 includes the MathCAD commands to locate the most compact Grashof crank rocker solution. The fifth procedure in the secondary algorithm (block #5 in Fig. 4) involves the determination of the dimensional parameters of the compact planar four-bar (crank rocker) motion generator selected in the fourth procedure. Such parameters include the coordinates of the fixed pivots and moving pivots of the optimized planar four-bar mechanism. The joints of a planar four-bar motion generator are represented by two pairs of fixed and moving pivot points (two m–k1 pairs) from the associated Burmester curves. Appendix A.5 includes the MathCAD commands to locate the coordinates of the fixed and moving pivots of the compact Grashof crank rocker solution. 4. Example Four prescribed rigid-body poses are included in Table 3. For this motion generation application, a Grashof crank rocker with a minimized perimeter and transmission angles between 40° and 140° is a satisfactory solution. Using the Burmester curve generation method in Section 2 with the following prescribed range (in radians): b2 ¼ 0:05; 0:15; . . . ; 0:75 the circle and center pivot curves illustrated in Fig. 5 were generated. From the circle and center point curves in Fig. 5, the line plot in Fig. 6 illustrates all possible crank and follower length solutions and the surface plots in Fig. 7 illustrate all possible coupler and ground length solutions. Although all of these solutions are planar four-bar motion generators that will achieve the prescribed rigid-body poses, an acceptable mechanism solution in this example is a compact Grashof crank rocker with feasible transmission angles. In Fig. 5, there are 30 data points each for the circle and center pivot curves. Each point on the center pivot curve (m) has a corresponding point on the circle pivot curve (k1) or vice-versa. Therefore, there are 30 m–k1 pairs that represent crank or follower links. In Fig. 6, each m–k1 pair is assigned a cell number (for use in Table 3 Prescribed rigid-body poses p

a (deg) i1.9161 i1.7778 i1.6369 i1.4919

3.6292 13.5386 51.8900

3.0 center pivot curve 2.5

Imaginary

1.4159, 1.6345, 1.6356, 0.8147,

circle pivot curve

2.0

1.5

1.0

0.5 1.8

2.0

2.2

2.4

2.6

2.8

3.0

Real

Fig. 5. Burmester curves for prescribed rigid-body poses.

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Crank/follower length

6.0 5.0 4.0 3.0 2.0 1.0 0.0 -1

4

9

14

19

24

29

Cell number

Fig. 6. Crank and follower length solutions for prescribed rigid-body poses.

Ground Length Solutions

Coupler Length Solutions

29

1.2500-1.5000

25

21

1.0000-1.2500

21

0.7500-1.0000

17

17 13

13

0.2500-0.5000

4.0000-5.0000 3.0000-4.0000 2.0000-3.0000 1.0000-2.0000

5

5

0.0000-1.0000

1

1

26

21

16

11

6

0.0000-0.2500

1

26

21

16

11

6

1

0.5000-0.7500

5.0000-6.0000

9

9

Cell number

Cell number

25

Cell number

29

Cell number

Fig. 7. Coupler and ground length solutions for prescribed rigid-body poses.

MathCAD) and the scalar magnitude of each m–k1 pair is the length of the crank/follower link. If any two m– k1 pairs are considered, the scalar magnitudes between the two k1 and m points are the lengths of the coupler and ground links respectively. Fig. 7 illustrates every possible coupler and ground length solution for the prescribed rigid-body poses. These surface plots are the result of measuring the scalar coupler and ground lengths from every possible combination of any two m–k1 pairs in Fig. 5. To arbitrarily select a mechanism solution that meets all of the stated design requirements from this vast number of possible solutions would be a daunting task. Fig. 8 illustrates the compact Grashof crank rocker motion generator solution produced from the algorithm included in the Appendix and Fig. 9 illustrates the transmission angles achieved by this mechanism. Appendix A.5 produced the two m–k1 pairs that the four-bar motion generator in Fig. 8 is comprised of. The lowermost m–k1 pair (or crank link) in Fig. 8 has coordinates of m = (2.5693, i1.0289) and k1 = (2.3082, i0.8245). The uppermost m–k1 pair (or follower link) in Fig. 8 has coordinates of m = (3.1397, i3.8534) and k1 = (2.0298, i1.4117). Since the Burmester curve algorithm calculates algebraic mechanism solutions, the rigid-body poses achieved by the compact crank rocker motion generator are exactly identical to those specified in Table 3. 5. Discussion Using MathCAD and AutoCAD software, all of the data calculated and measured in this work were specified to four decimal places. These software packages can calculate and measure numbers to over eight decimal

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m

p1 Imaginary

fixed pivot locus

k1 m moving pivot locus

Real

k1

Fig. 8. Crank rocker motion generator solution with minimum perimeter.

Transmission angle [deg.]

140

120

100

80

60

40 0

60

120

180

240

300

360

Crank displacement [deg.]

Fig. 9. Transmission angles of optimized motion generator.

places. The Grashof conditions in the secondary algorithm (Appendix A.3) can be modified to enable the user to design compact motion generators of any Grashof type. In Appendix A.1, the data points on the circle and center pivot curves are fed into the secondary algorithm to calculate the lengths of the crank, coupler, follower and ground link solutions. Variable ‘‘end’’ in this Appendix A.1 denote the total number of m–k1 pairs used (in the example problem, 30 pairs were used). Due to the lack of space, an abridged version of the codified algorithm has been included in the Appendix, for the entire codified algorithm, refer to reference [20]. The circle and center point curves illustrated in Figs. 3 and 5 are not sections of the same Burmester curve. Although the Burmester curve method included in the work of Sandor and Erdman [19] is described in this work, any circle and center point curve generation approach is applicable since the algorithm presented by the authors requires only the discrete complex circle and center point coordinates (Fig. 4).

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6. Conclusion A set of Burmester curves can represent an infinite number of planar four-bar motion generator solutions for a given series of prescribed rigid-body poses. Unfortunately, given such a vast number of possible mechanical solutions, it is difficult for designers to arbitrarily select a Burmester curve solution that is ensures full link rotatability, produces feasible transmission angles and is as compact as possible. One method to employ Burmester curve-based motion generation while adhering to specific mechanism design conditions and constraints is through a search and selection algorithm. This work presents an algorithm for selecting planar four-bar motion generators with respect to Grashof conditions, transmission angle conditions and having the minimal perimeter value. This algorithm has been codified into MathCAD for enhanced analysis capabilities and ease of use. The example in this work demonstrates the synthesis of a compact planar, four-bar crank rocker motion generator with feasible transmission angles. Appendix A A.1. The MathCAD commands in this section calculates all mechanism solutions for a given circle and center point curve qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i :¼ j :¼ 0, 1, 2, . . . , end  1 2 2 CRANKi :¼ FOLLOWERi :¼ ðmxi  kxi Þ þ ðmy i  ky i Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi GROUNDi;j :¼ ðmxi  mxj Þ2 þ ðmy i  my j Þ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi COUPLERi;j :¼ ðkxi  kxj Þ2 þ ðky i  ky j Þ2

A.2. The MathCAD commands in this section selects all mechanism solutions from among the solutions produced in Appendix A.1with feasible transmission angles

START := 0 STEP := 10 END := 360 ANGLE := for i 2 0. . .rows(COUPLER)  1 for i 2 0. . .rows(COUPLER)  1 2 3 mxi kxi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ðkx mx Þ þðky my Þ i i i i 4 5 if i 5 j ui my i ky i pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2

mxj mxi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2

3

ðmxj mxi Þ þðmy j my i Þ my j my i

5 if i 5 j pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ðmxj mxi Þ þðmy j my i Þ   u v a cos jui ijjvjj j if i 5 j ANGLESi;j vj

4

ðkxi mxi Þ þðky i my i Þ

ANGLESi,j 0 otherwise var_1 0 for i 2 0..rows(COUPLER)  1 for i 2 0..rows(COUPLER)  1 p p ::END  180 for d 2 START; STEP  180 L (GROUNDi,j)2 + (CRANKi)2  ð2  GROUNDi;j  CRANKi  cosðANGLESi;j þ dÞÞ h i ðCOUPLERi;j Þ2 þðFOLLOWERj Þ2 ðLÞ trans a cos if i 5 j 2COUPLERi;j FOLLOWERj

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trans 0 otherwise var_1 augment(var_1,trans) END ccolumns 1 þ ðSTEP Þ rrows cols(COUPLER)2 M var_10,1 for i 2 2..ccolumns M stack(M,var_10,i) for i 2 1..rrows  1 Z var_10,ccolumnsÆi+1 for j 2 2..ccolumns Z stack(Z,var_10,ccolumnsÆi+j) M augment(M,Z) M MT   1Þ1 for i 2 0; 1:: colsðvar 1 ccolumns END for j 2 0; 1:: STEP Pi,j Mi,j  p p Pi,0 0 if ReðM i;j Þ < ð40  180 Þ _ ReðM i;j Þ > 140  180 Mi,j otherwise Pi,j array ð0 0 0 0Þ array 1 ð0 0 0 0Þ for z 2 0..rows(P)  1 for i 2 0..rows(COUPLER)  1 for j 2 0..rows(COUPLER)  1 for d 2 0 if COUPLERi,j 5 0 ^ FOLLOWER j 5 0 3 2 2 2 ðCOUPLERi;j Þ þ ðFOLLOWERj Þ  2 3 2 2 7 6 ðGROUNDi;j Þ þ ðCRANKi Þ   7 64

5 7 6 2  GROUND  CRANK  i;j i 7 6 7 6 cosðANGLES þ dÞ i;j 7 6 dummy a cos 6 2COUPLERi;j FOLLOWERj 7 7 6 7 6 7 6 5 4

array

augment

CRANKi ; COUPLERi;j ; FOLLOWERj ; GROUNDi;j



if dummy = Pz,0 array_1 stack(array_1,array) if dummy = Pz,0 continue otherwise array_1 CCRANK := ANGLEh0i CCOUPLER := ANGLEh1i h2i FFOLLOWER := ANGLE GGROUND := ANGLE

h3i

A.3. The MathCADÒ commands in this section selects all Grashof crank rocker solutions from among the solutions produced in Appendix A.2

Cell :¼ array ð0 0 0 0Þ for i 2 0..rows(GGROUND)  1

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00

11 CCRANKi BB CCOUPLERi CC B CC v sortB @@ FFOLLOWERi AA GGROUNDi

 CCRANKi ; CCOUPLERi ; FFOLLOWERi ; M1 augment GGROUNDi array stack(array,M1) if CCRANKi = v0 continue otherwise array1 ð0 0 0 0Þ for i 2 0..rows(array)  1 00 11 arrayi;0 BB arrayi;1 CC B CC v sortB @@ arrayi;2 AA arrayi;3 M1 augment(arrayi,0,arrayi,1,arrayi,2 ,arrayi,3) array1 stack(array1,M1) if v0 + v3 < v1 + v2 continue otherwise array1 A.4. The MathCADÒ commands in this section selects the minimum perimeter solution from among the solutions produced in Appendix A.3

Cell2 := perimeter 1 M2 ð0 0 0 0Þ for i 2 1..rows(Cell)  1 Dummy_Perimeter Celli,0 + Celli,1 + Celli,2 + Celli,3 M1 augment(Celli,0, Celli,1, Celli,2, Celli,3) M2 M1 if Dummy_Perimeter < perimeter perimeter Dummy_Perimeter if Dummy_Perimeter < perimeter continue otherwise solution augment(M2, perimeter) A.5. The MathCADÒcommands in this section selects the fixed and moving pivot coordinates of the solution produced in Appendix A.4

Cell3 := for i 2 0..rows(kx)  1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 Dummy Follower ðmxi  kxi Þ ðmy i  my i Þ array augment(mxi, kxi,myi,kyi) coordinates arrayifDummy_Follower = Cell20,0 coordinates_1 arrayifDummy_Follower = Cell20,2 continue otherwise solution stack(coordinates, coordinates_1) References [1] J. Yao, J. Angeles, Computation of all optimum dyads in the approximate synthesis of planar linkages for rigid-body guidance, Mechanism and Machine Theory 35 (8) (2000) 1065–1078.

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