Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 54, 1, pp. 75-85, Warsaw 2016 DOI: 10.15632/jtam-pl.54.1.75 KINEMATIC SYNTHESIS OF SPATIAL LINKAGES WITH SPHERICAL PAIRS Serikbay Kosbolov, Algazy Zhauyt, Serikbol Kosbolov KazakhNational Technical University named afterK.I. Satpayev, Department ofAppliedMechanics andBasics ofMachine Design, Almaty, Kazakhstan; e-mail: kosbolov@mail.ru; ali84jauit@mail.ru; serikbol k@mail.ru A solution to the problem of synthesizing an initial three-dimensional kinematic chain with spherical and rotary kinematic pairs is presented. It is shown that this chain can be used as a structural module for structural-kinematic synthesis of motion of a three-dimensional four-link generating lever mechanisms by preset positions of the input and output links. Keywords: mechanism, four-link, kinematic pairs, kinematic chains, synthesis 1. Introduction Some papers demonstrate that four-link basic kinematic chains (BKC)may be used as a struc- tural module with structural and kinematic synthesis of plain linkage mechanisms. Such an approach to the synthesis of plain mechanisms allows reducing the problem of their structural and kinematic synthesis to solution of the problem of BKC synthesis (Joldasbekov et al., 1987), which is very useful for automation of mechanisms engineering. This paper testifies that a spe- cified approach may be applied to the problem of structural and kinematic synthesis of spatial linkage mechanisms (Kosbolov and Rakhmatulina, 2012b). The solution of the problem of syn- thesis of spatial BKC of RSS type (R – rotational, S - spherical kinematic pairs) is represented, and its use as a structural module with structural and kinematic synthesis of spatial linkage mechanisms through predetermined positions of input and output links is shown (Kosbolov et al., 2005). Amethod of solving the problem of BKC synthesis of RSS type is based on the intro- duction of twomovable bodies invariably associated with the input and output links (Kosbolov and Rakhmatulina, 2013b). 2. On the existence of solution to the problem of initial kinematic chain synthesis with spherical kinematic pairs Problem statement: given N of finite distant positions of two solids Q1 and Q2 Q1(θ 1 i ,ψ 1 i ,φ 1 i) Q2(XDi,YDi,ZDi,θ 2 i ,ψ 2 i ,φ 2 i) i =1,N (2.1) where θ j i , ψ j i , φ j i are fixed axis Eulerian angles OXY Z and XDi, YDi, ZDi are coordinates of the point Di of the solid Q2. It is required to find such points in the fixed axis as A(XA,YA,ZA), of the solid Q1 and C(xC,yC,zC) of the solid Q2, so that distance between the points B and C in all positions of the solids Q1 and Q2 is little different from some constant value R (Fig. 1). Problem solution: Let us introduce a weighted difference for the i-th position of the solids in form ∆qi = | −−→ BiCi| 2−R2 =(XCi−XBi) 2+(YCi−YBi) 2+(ZCi−ZBi) 2−R2 i = −−→ 1,N (2.2) 76 S. Kosbolov et al. Fig. 1. Equivalent four-link kinematic chain ABCD where    XBi YBi ZBi 1    =    XA Ti10 YA ZA 0 0 0 1       xB yB zB 1      XCi YCi ZCi 1   =   XDi Ti20 YDi ZDi 0 0 0 1     xC yC zC 1   (2.3) and T i j0 =   e′i1 e ′ i2 e ′ ic m′i1 m ′ i2 m ′ ic n′i1 n ′ i2 n ′ ic   j =1,2 i =1,N (2.4) where e j i1 =cosψ j i cosφ j i − cosθ j i sinψ j i sinφ j i mi1 =sinψ j i cosφ j i +cosθ j i cosψ j i cosφ j i n j i1 =sinθ j i sinφ j i e j i2 =−cosψ j i sinφ j i − cosθ j i sinψ j i cosφ j i m j i2 =−sinψ j i sinφ j i +cosθ j i cosψ j i sinφ j i n j i2 =sinθ j i cosφ j i e j i3 =sinθ j i sinψ j i m j i3 =−sinθ j i cosψ j i n j i3 =cosφ j i (2.5) It is a function of ten parameters: XA, YA, ZA, xB, yB, zB, R, xC, yC, zC. By grouping these parameters in fours with the common parameter R, let us represent the weighted diffe- rence in three different forms (McCarthy, 1995; Golynski, 1970; Innocenti, 1995; Kosbolov and Rakhmatulina, 2012b, 2013b,c; Kosbolov et al., 2014) ∆(1)qi =(X̃Ai −XA) 2+(ỸAi −YA) 2+(Z̃Ai −ZA) 2−R2 ∆(2)qi =(x̃Bi −xB) 2+(ỹBi −yB) 2+(z̃Bi −ZB) 2−R2 ∆(3)qi =(x̃Ci −xC) 2+(ỹCi −yC) 2+(z̃Ci −ZC) 2−R2 (2.6) Kinematic synthesis of spatial linkages with spherical pairs 77 where   X̃Ai Ỹ ′′Ai Z̃Ai 1   =   0 Ti10 0 0 0 0 0 1     xB yB zB 1   +   XDi Ti20 YDi ZDi 0 0 0 1     xC yC zC 1     x̃Bi ỹBi z̃Bi 1   =   0 Ti01 0 0 0 0 0 1     XDi −XA YDi −YA ZDi −ZA 1   +   0 Ti21 0 0 0 0 0 1     xC yC zC 1     x̃Ci ỹCi z̃Ci 1   =−   0 Ti02 0 0 0 0 0 1     XA−XDi YA−YDi ZA−ZDi 1   +   0 Ti12 0 0 0 0 0 1     xB yB zB 1   (2.7) whereTikj is the transfer matrix from the k coordinate system to the j system determined as T i 01 = [T i 10] T T i 02 = [T i 20] T T i 21 =T i 01×T i 20 T i 12 =T i 02×T i 10 (2.8) The necessary conditions for minimum of the sum of squares of the weighted difference S = N∑ i=1 [∆(k)qi ] 2 k =1,2,3) (2.9) may be written as the following system of equations ∂S ∂XA =0 ∂S ∂YA =0 ∂S ∂ZA =0 ∂S ∂R =0 ∂S ∂xB =0 ∂S ∂yB =0 ∂S ∂zB =0 ∂S ∂R =0 ∂S ∂xC =0 ∂S ∂yC =0 ∂S ∂zC =0 ∂S ∂R =0 (2.10) From (2.10)1, considering (2.6)1 and (2.9), we obtain N∑ i=1 ∆(1)qi (X̃Ai −XA)= 0 N∑ i=1 ∆(1)qi (ỸAi −YA)= 0 N∑ i=1 ∆(1)qi (Z̃Ai −ZA)= 0 N∑ i=1 ∆(1)qi R =0 (2.11) Assume that R 6=0. Then from the last equality of system (2.11), it follows that N∑ i=1 ∆(1)qi =0 (2.12) With provision for (2.12), the system of equations (2.11) takes the form N∑ i=1 ∆(1)qi X̃Ai =0 N∑ i=1 ∆(1)qi ỸAi =0 N∑ i=1 ∆(1)qi Z̃Ai =0 N∑ i=1 ∆(1)qi =0 (2.13) 78 S. Kosbolov et al. By substituting expressions for ∆ (1) qi from (2.6)1 into system (2.13), we obtain N∑ i=1 [ X̃2AiXA+ X̃AiỸAiYA+ Z̃AiX̃AiZA+ 1 2 (R2−X2A−Y 2 A−Z 2 A)X̃Ai ] = 1 2 N∑ i=1 (X̃2Ai+ Ỹ 2 Ai+ Z̃ 2 Ai)X̃Ai N∑ i=1 [ X̃AiỸAiXA+ Ỹ 2 AiYA+ Z̃AiỸAiZA+ 1 2 (R2−X2A−Y 2 A−Z 2 A)ỸAi ] = 1 2 N∑ i=1 (X̃2Ai+Y 2 Ai+Z 2 Ai)ỸAi N∑ i=1 [ Z̃AiX̃AiXA+ ỸAiZ̃AiYA+ Z̃ 2 AiZA+ 1 2 (R2−X2A−Y 2 A −Z 2 A)Z̃Ai ] = 1 2 N∑ i=1 (X̃2Ai+ Ỹ 2 Ai+ Z̃ 2 Ai)Z̃Ai N∑ i=1 [ X̃AiXA+ ỸAiYA+ Z̃AiZA+ 1 2 (R2−X2A−Y 2 A −Z 2 A)X̃Ai ] = 1 2 N∑ i=1 (X̃2Ai+ Ỹ 2 Ai+ Z̃ 2 Ai) (2.14) System (2.14) is linear with respect to the variables XA, YA, ZA and H1 =(R 2−X2A−Y 2 A− Z2A)/2, thus it may be written as   N∑ i=1 X̃2Ai N∑ i=1 X̃AiỸAi N∑ i=1 X̃AiZ̃Ai N∑ i=1 X̃Ai N∑ i=1 X̃AiỸAi N∑ i=1 ỸAi N∑ i=1 ỸAiZ̃Ai N∑ i=1 ỸAi N∑ i=1 X̃AiZ̃Ai N∑ i=1 ỸAiZ̃Ai N∑ i=1 Z̃2Ai N∑ i=1 Z̃Ai N∑ i=1 X̃Ai N∑ i=1 ỸAi N∑ i=1 Z̃Ai N     XA YA ZA H1   = 1 2   N∑ i=1 R2AiX̃Ai N∑ i=1 R2AiỸAi N∑ i=1 R2AiZ̃Ai N∑ i=1 R2Ai   (2.15) where R2Ai = X̃ 2 Ai+ Ỹ 2 Ai+ Z̃ 2 Ai. The solution to this system by Cramer’s rule is as follows (XA,YA,ZA,H1)= 1 D1 (DXA,DYA,DZA,DH1) D1 6=0 (2.16) Similarly, from (2.10)2, considering (2.6)2 and (2.9), we obtain a system of linear equations in the unknowns xB, yB, zB, H2   N∑ i−1 x̃2Bi N∑ i−1 x̃BiyBi N∑ i−1 x̃Biz̃Bi N∑ i−1 x̃Bi N∑ i−1 x̃BiyBi N∑ i−1 ỹ2Bi N∑ i−1 ỹBiz̃Bi N∑ i−1 ỹBi N∑ i−1 x̃Biz̃Bi N∑ i−1 ỹBiz̃Bi N∑ i−1 z̃2Bi N∑ i−1 z̃Bi N∑ i−1 x̃Bi N∑ i−1 ỹBi N∑ i−1 z̃Bi N     xB yB zB H2   = 1 2   N∑ i=1 R2Bix̃Bi N∑ i=1 R2BiỹBi N∑ i=1 R2Biz̃Bi N∑ i=1 R2Bi   (2.17) By solving this system by Cramer’s rule, we obtain (xB,yB,zB,H2)= 1 D2 (DxB,DyB,DzB,DH2) D2 6=0 (2.18) Kinematic synthesis of spatial linkages with spherical pairs 79 From (2.10)3, considering (2.6)3 and (2.10)1, we obtain a system of linear equations in the unknowns xC, yC, zC, H3   N∑ i−1 x̃2Ci N∑ i−1 x̃CiỹCi N∑ i−1 x̃Ciz̃Ci N∑ i−1 x̃Ci N∑ i−1 x̃CiỹCi N∑ i−1 ỹ2Ci N∑ i−1 ỹCiz̃Ci N∑ i−1 ỹCi N∑ i−1 x̃Ciz̃Ci N∑ i−1 ỹCiz̃Ci N∑ i−1 z̃2Ci N∑ i−1 z̃Ci N∑ i−1 x̃Ci N∑ i−1 ỹCi N∑ i−1 z̃Ci N     xC yC zC H3   = 1 2   N∑ i=1 R2Cix̃Ci N∑ i=1 R2CiỹCi N∑ i=1 R2Ciz̃Ci N∑ i=1 R2Ci   (2.19) From which we obtain xC, yC, zC, H3 (xC,yC,zC,H3)= 1 D3 (DxC,DyC ,DzC,DH3) D3 6=0 (2.20) Eliminating the first four unknowns XA, YA, ZA, R, based on formula (2.15), it is possible to bring system (2.10) to a system of six equations with six unknowns xB, yB, zB, xC, yC, zC, which is convenient to be given as N∑ i=1 ∆ (1) qi ∂∆ (2) qi ∂xB =0 N∑ i=1 ∆ (1) qi ∂∆ (3) i ∂xC =0 N∑ i=1 ∆ (1) qi ∂∆ (2) qi ∂yB =0 N∑ i=1 ∆ (1) qi ∂∆ (3) i ∂yC =0 N∑ i=1 ∆ (1) qi ∂∆ (2) qi ∂zB =0 N∑ i=1 ∆ (1) qi ∂∆ (3) i ∂zC =0 (2.21) Apparently, equations of this system are the same as the three equations of the thirteen degree in the three unknown functions given in the work by Kosbolov et al. (2005), though in this case we have a system of six equations in six unknown functions. Solution of system (2.21) is labor-intensive task, so it is more effective to apply a search algorithm for the minimum of the function S stated below: 1. Give arbitrarily reference points B(0) ∈ Q1, C (0) ∈ Q2. 2. Solve the system of linear equations (2.16) and determine X (1) A , Y (1) A , Z (1) A , R (1) 1 . 3. Give points A(1) ∈ Q, C(0) ∈ Q2. 4. Solve the system of equations (2.18) and determine x (1) B , y (1) B , z (1) B , R (1) 2 . 5. Give points A(1) ∈ Q, B(1) ∈ Q1. 6. Solve the system of equations (2.20) and determine x (1) C , y (1) C , z (1) C , R (1) 3 . 7. Check |Xi+1A −X i A| ¬ ε, |Y i+1 A −Y i A| ¬ ε, |Z i+1 A −Z i A| ¬ ε, |R i+1−Ri| ¬ ε. 8. If this condition is satisfied, the iterating is completed. 9. If this condition is not satisfied, proceed to item 1 by replacing the reference points B(0) and C(0) for the found points B(1) and C(0). 10. Then check the accuracy of theprescribed function reproductionbyanalysis of theposition RKC ABCD rD0 =T10T21T32rD3 80 S. Kosbolov et al. 11. The iterating is completed, if the accuracy of reproduction satisfies theprescribed function. If it does not satisfy the prescribed accuracy, it is necessary to proceed to item 1 of the given algorithm. Byapplying thealgorithm,weobtainadecreasing sequenceof values of theobjective function S (1) 1 , S (1) 2 , S (1) 3 , S (2) 1 , S (2) 2 , S (2) 3 which has a limit equal to the value of the function S at the point of local minimum.When satisfying the inequality max ( |R(i)−R(i−1)|, |X (i) A −X (i−1) A |, |Y (i) A −Y (i−1) A |, |Z (i) A −Z (i−1) A | ) ¬ ε where ε is the prescribed calculation accuracy, the iterating is completed. Convergence of the suggested algorithm is proved by theWeierstrass theorem. Weierstrass theorem: For each function f(x) continuous over [a,b] and any real number ε > 0, such a polynomial p(x) may be found that ‖P(x)−f(x)‖ < ε. As a result of the problem solution, the points A(XA,YA,ZA) are determined in the fixed system of coordinates, B(0) ∈ Q1, C (0) ∈ Q2, such that when coinciding the link BC with them, we obtain the desired RKC in form of an open four-link chain ABCD. Then we check the accuracy of the prescribed function reproduction by analysis of the po- sition of RKC ABCD. If the accuracy of reproduction satisfies the prescribed function, the iteration is completed, and if it does not satisfy the prescribed accuracy, it is necessary to proceed to item 1 of the prescribed algorithm. When specifying a part of the desired synthesis parameters in various combinations, we obtain different modifications of RKC (Kosbolov and Rakhmatulina, 2013b). • If the coordinates of point A(XAi,YAi,ZAi) and Eulerian angles θ 1 i , ψ 1 i , φ 1 i of the solid Q1 aswell as the axes of pointDi(XDi,YDi,ZDi) andEulerian angles θ 1 i ,ψ 1 i ,φ 1 i of the solid Q2 are specified, we obtain a three-link open chain ABCD (Fig. 1). The necessary conditions for the minimum of the sum S in this case takes the form ∂S ∂j =0 j = xB,yB,zB,R,xC,yC,zC (2.22) and to find the minimum S, we may use the algorithm given above, considering that the parameters XA,YA,ZA are specified. If the pointsA(XA,YA,ZA) and D(XD,YD,ZD) are fixed, then, as a result of the synthesis of RKC, we obtain a spatial four-link chain ABCD. • Given the coordinates xC = yC = zC = 0 of the point C ∈ Q2, coordinates XDi, YDi, ZDi of the point D of the solid Q2 and Eulerian angles θ 1 i , ψ 1 i , φ 1 i of the solid Q1, and the desired parameters XA, YA, ZA, R, xB, yB, zB. The necessary conditions for the minimum of the sum S takes the form ∂S ∂j =0 j = XA,YA,ZA,R,xB,yB,zB (2.23) To find the minimum of the function S we may use again the algorithm given above, considering that xC = yC = zC =0. • Given coordinates xB,yB,zB =0 of the point B of the solid Q1 andEulerian angles of the solid Q2, θ 2 i , ψ 2 i , φ 2 i . The original problem reduces to the definition of sphere of positions of the fixed point C of the solid Q2 which is the least remote from N (Fig. 1). Kinematic synthesis of spatial linkages with spherical pairs 81 The necessary conditions for the minimum of the sum S is ∂S ∂j =0 j = XA,YA,ZA,R,xC,yC,zC (2.24) This problemwas studied in detail in work byKosbolov et al. (2013c). For its solution we may also use the algorithm given above, assuming xB,yB,zB =0, but in this special case, the algorithm of theminimum search is absolutely coincidingwith the kinematic inversion method. Thus, as we see, the problem of RKC with spherical kinematic pairs is solved, and their modifications may be used as modules of structural and kinematic synthesis of spatial linkage mechanisms through specified positions of the input and output links. 3. Example Suppose that it is necessary to design a six-linkage mechanism with spherical pairs (Fig. 2), approximately reproducing seven body positions specified in Table 1 and the initial data in Table 2. Fig. 2. Kinematic diagram of spatial linkage mechanisms with spherical pairs Table1.Assignedpositionsof thebody for synthesis of a singlemovablemechanism–six-linkage mechanism with N =7 Position XOi YOi ZOi Euler angles [deg] No. (i) θ1i ψ 1 i φ 1 i 1 0.30 0.12 0.01 0 0 0 2 0.35 0.17 0.24 28 35 17 3 0.44 0.21 0.25 34 38 5 4 0.51 0.15 0.32 17 24 12 5 0.50 0.30 0.45 50 50 21 6 0.60 0.25 0.41 45 33 24 7 0.55 0.32 0.35 0 0 0 When N = 7, as known in mobile spatial systems, there are points (not more than 20) with seven positions on one sphere. Furthermore, the points which are in the seven considered positions are approaching the sphere. As noted, the exact spherical points of a movable system correspond to an absolute minimum S = 0 of the sum S = ∑ ∆2qi, because they make all ∆qi 82 S. Kosbolov et al. Table 2. Initial data (N =7) N =7 ZLL=6, ZKK=6, MJ0=−1.2, NJ0=−1.2, ZII= 6, dmj=0.2 KJ0=−1.2 XS=0.3 YS=0.12 ZS=0.01 F=0 P=0 T=0 XS=0.35 YS=0.17 ZS=0.24 F=28 P=35 T=17 XS=0.44 YS=0.21 ZS=0.25 F=34 P=38 T=5 XS=0.51 YS=0.15 ZS=0.32 F=17 P=24 T=12 XS=0.5 YS=0.3 ZS=0.45 F=50 P=50 T=21 XS=0.6 YS=0.25 ZS=0.41 F=45 P=33 T=24 XS=0.55 YS=0.32 ZS=0.35 F=0 P=0 T=0 Table 3.Results of calculation (N =4) LL=3, KK=2, II= 1 MJ=−0.6, NJ=−0.8, KJ=−1 N =1 A =0.488270 C =−2.22103 N =2 A =0.303894 C =3.96393 N =3 A =0.151985 C =−3.85753 N =4 A =0.146845 C =3.00599 R0=0.245293 R01=0.245194 S =4.296551E-0.3 S1=4.295576E-0.3 LL=3, KK=2, II= 2 MJ=−0.6, NJ=−0.8, KJ=−0.8 N =1 A =0.488270 C =−3.42355 N =2 A =0.303894 C =2.28575 N =3 A =0.151985 C =−2.95424 N =4 A =0.146845 C =3.00600 R0=0.245216 R01=0.245194 S =4.292136E-0.3 S1=4.295576E-0.3 LL=3, KK=2, II= 3 MJ=−0.6, NJ=−0.8, KJ=−0.6 N =1 A =0.488174 C =−3.18222 N =2 A =0.303906 C =5.18244 N =3 A =0.152115 C =−2.55608 N =4 A =−0.146845 C =0.3006 R0=0.245195 R01=0.245194 S =4.294402E-0.3 S1=4.295576E-0.3 LL=3, KK=2, II= 4 MJ=−0.6, NJ=−0.8, KJ=−0.6 N =1 A =0.488217 C =−2.74528 N =2 A =0.303942 C =8.77332 N =3 A =0.152078 C =−2.92302 N =4 A =0.146866 C =3.00599 R0=0.245217 R01=0.245194 S =4.297097E-0.3 S1=4.295576E-0.3 (i = 1,2, . . . ,7) vanish. The approximate spherical points of a movable system correspond to the local minima of the sum or are located in the vicinity of the local minimum of the function S = F(xC,yC,zC). Both these and other are common points of the surfaces Gx = 0, Gy = 0, Gz =0, and they are determined based on the solution to system (2.21). We are definitely interested not in all system solutions (2.24), but in those which correspond to theminimumof the sumS.These solutions in this case are anumericalmethod for searchingof theminimum of the sum S, based on the searching algorithm for theminima of the function S. Since we need five spherical points of the movable system for construction of the six-linkage mechanism (Fig. 2) having analysed the results we selected five points of the minimum of the Kinematic synthesis of spatial linkages with spherical pairs 83 surface S = F(xC,yC,zC). The coordinates of these points, the parameters of approaching spheres corresponding to them and the values ∆qi maximum permodule are given in Table 3. 4. Research results Judging from deviations, the points C1, C2, C3 are the exact spherical points (found with the given measure of inaccuracy), and C4 and C5 – are approximate points. To solve this problem, a Visual Basic program has been written. The results of the problem solution are given in Table 3. The calculations have beenmade within the range −1.2 < MJ < 0.6 −1.2 < NJ < 0.6 −1.2 < KJ < 0.6 with a pitch dmj =0.2. In the entire range of calculations, the process of calculation is concurrent. The global mini- mum is equal to Smin =0.0042921 and it is achieved at LL=3 KK=2 II= 2 InTable 3, only a part of results in the neighborhood of the global minimum is shown. Its value is highlighted in bold frame in Table 3. Below, the results of calculation of the objective function in form of carpet plots and 3D plots (Figs. 3 and 4) are given. Fig. 3. LL=1,2,3, Smin =−0.0042921, volume graphics 84 S. Kosbolov et al. Fig. 4. LL=4.5,6, Smin =−0.0042921, volume graphics 5. Discussion Ascanbeconcluded, theuse of oneand the sameobjective functionbeinggenerated for synthesis of BKC and its modification allows automating of the process of synthesis of spatial linkage mechanisms through predetermined positions of the input and output links of the mechanism. 6. Conclusions In summary, in the synthesis of BKC with spherical kinematic pairs through predetermined positions of the input and output links of amechanismwhen two adjacent links of BKC tend to infinity, it is necessary to replace the spherical kinematic pair for a plain or cylindrical one. In such a case, the synthesizedmechanism takes formof a spatial linkmechanismafter determining the required parameters. References 1. 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