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An investigation of properties of the forward displacement analysis of the generalized Stewart platform by means of general optimization methods
Pergamon
Mech, Mach. Theory Vol. 33, No. 3, pp. 245-253, 1998 © 1997 ElsevierScience Ltd. All rights reserved Printed in Great Britain PlI: S01D4-114X(97)00044-X oo94-114x/98 $19.o0+ o.oo
AN INVESTIGATION OF PROPERTIES OF THE F O R W A R D DISPLACEMENT ANALYSIS OF THE GENERALIZED STEWART PLATFORM BY MEANS OF GENERAL OPTIMIZATION METHODS Z. ~IKA Department of Mechanics, Faculty of Mechanical Engineering, Czech Technical University, Prague, Czech Republic
and P. KO(~ANDRLE* Stress Calculation Group, Skoda Turbines Ltd., Pilsen, Czech Republic
and V. STEJSKAL Department of Mechanics, Faculty of Mechanical Engineering, Czech Technical University, Prague, Czech Republic
Abstract--The method for the determination of all solutions of the forward displacement analysis of the generalized Stewart platform is proposed and described. The kinematic problem is transformed to the problem of global minimization of the specially created objective function. The problem is formulated in the domain of complex numbers, the general optimization methods are used for the solution.' The presented method is very easily applicable on the manipulators of arbitrary structure, it has practically unlimited potential of parallelization. © 1997 Elsevier Science Ltd
1. INTRODUCTION The generalized Stewart platform is a fully parallel mechanism with six degrees of freedom. It is used in flight and automotive simulators, robotic end-effectors, and other applications requiring spatial mechanisms with high structural stiffness. The geometry of the Stewart platform is shown in Fig. 1. A great attention has been paid to the forward displacement analysis of this type of manipulator in the recent years. The kinematic problem can be stated as follows: given the lengths of the 6 variable limbs, find all the possible positions of the movable platform (end-effector of the manipulator). For some simplified types of the parallel manipulator, it is possible to determine the number of solutions of the forward displacement analysis (direct kinematic problem) by means of geometric theorems [1]. Some other works (like e.g. [2]) dealt with the reduction of number of constraint equations and coordinates describing position of the moving platform. For many simplified types of platform, the forward displacement analysis was successfully transformed to one polynomial equation for one unknown [3, 4] etc. The maximum number of possible solutions has been found even for the most general type of platform, namely by means of special algebraic procedures (Gr6bner basis computations) [5]. *The work was done during Ph.D. studies at the Department of Mechanics, Faculty of Applied Sciences, University of West Bohemia, Pilsen, Czech Republic. 245
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et al.
The method proposed here differs from the above mentioned approaches. The direct kinematic problem is transformed to the problem of global minimization of the specially created objective function. The aim is to demonstrate the applicability and worthiness of the complete solution of this problem by means of utilization of general optimization algorithms. This method was proposed by the authors in [6], where also the solution for two simplified types of Stewart platform was carried out, the solution of the most general platform is presented here. The approach based on the usage of optimization methods enables to investigate properties of the manipulators of different type and structure, manipulators with special relationships between their individual dimensions, etc. The only condition is the formulation of the problem in the domain of complex numbers and the constitution of the objective function in such a way that the description of the given problem be unique. The whole further effort is then left to the computer. 2. FORMULATION OF THE OPTIMIZATION PROBLEM The basic assumption for the formulation of the optimization problem is, that a polynomial equation of a certain degree corresponds to the forward displacement analysis of the parallel manipulator of the generalized Stewart platform type, as it is for the special type of platform solved, e.g. in [3]. The number of roots of this hypothetic polynomial equation is equal to its degree except for special cases, where some of the roots coincide. If the dimensions of the manipulator and values characterizing drives (lj = AjBj, j = 1. . . . . 6) were generated randomly, the probability of occurrence of these special cases would be uncomparably smaller than the probability of occurrence of the normal situation with full number of roots. However, we have to consider that the roots are in general from the domain of complex numbers. The necessity to formulate the kinematic relationships and consequently the optimization problem also in the domain of complex numbers follows directly from this fact. Further requirements on the optimization problem are as follows: --good numerical properties of the objective function; --small number of optimization variables; --uniqueness of the problem description by the objective function. The simple possibility of the choice of optimization parameters represents the usage of the real and imaginary components of angles tpt, ~p,, tPx~htpw, tpv (see Fig. 2). The number of optimization variables is in this case equal to 10, which is very good. Unfortunately there were complications with the numerical realization of this approach. The complications were probably caused by the very strong non-linearity of the objective function arising from the products of complex goniometric functions containing exponential members (sin(cPRE + iCplM) = sin(q~RE)COsh(q~XM) + i COS(~PRE)Sinh(tpIM)).
The effort was therefore concentrated to formulate the objective function on the basis of cartesian coordinates of points and their mutual distances. Eighteen optimization variables were chosen, particularly real and imaginary components of cartesian coordinates of points B~, B2 and B3 (see Fig. 1). The objective function was created in such a way that its zero values (global minima) correspond to the situation, when the kinematic constraint equations were satisfied. The position of a rigid body is uniquely given by the position of its three non-collinear points (Bt, B2, B3). Then the positions of points B4, Bs, B6, have to be uniquely expressed by means of position of triangle Bj B2B3. If this condition was not satisfied, our objective function would be valid for more different platforms simultaneously. The following relationships hold between the points of basic triangle [BIBzl2 - ~ = 0,1B~B312- ~ = 0,1B2B31' - ~ = 0.
(1)
The unique position of the auxiliary point B with respect to the triangle BI, B2, B3 is given by (see Fig. 3 and Fig. 1) It
= B~ + (B2 -
B,) x (B3 - Bt)/x/d, d2 sin/3,.
(2)
Forward displacement analysis of the generalized Stewart platform •
C
Y Fig. 1. Parallel manipulator of the generalized Stewart platform type.
x
y Fig. 2. Description of platform position by means of five angular coordinates.
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248
I
cl4
~
I
Fig. 3. Position of the auxiliary point.
~/dld2
The division by the constant sin(fit) is introduced because of consistence of physical dimensions. The derivation of the linear dependencies for the positions of points//4, B5 and//6 comes from the procedure used for the description of multibody systems by means of so-called natural coordinates [7]. We shall use the following notation: B, B:, j = 1 , . . . , 6--position vectors of centers of spherical couplings and of auxiliary point in the system x, y, z ~0, B~o,j = 1. . . . . 6---position vectors of above mentioned points in the coordinate system ¢, g, connected with the moving platform S--matrix of directional cosines for the transformation of relative vectors from the system ~, g, ~ to the system x, y, z Then we can write ( B i - Bj) = S(B~ - B~) and for the three vectors [B2
- -
Sl, 63
-
-
BI, ][~
-
-
Bl] = S[B2o - B,o, B3o - B1o, ~ - B,o]
(4)
or
(5)
X = SX0. Owing to the non-complanarity of vectors, we can carry out the inversion
(6)
XXU = S, B , = Bi + S(B40 -
Bt0) = BL + X X ; t ( B 4 o
-
Bt0)
B5 = BI + S(B~o - 91o) = Sl + XXo-l(Sso - B,0) B6 = B, + S(B6o - B,o) = Bm + X X i ' ( B ~
p4 =
Xol(B,o -
Bto)
P5
Xot(Bso
Bl0)
=
-
- B1o),
(7)
(8)
p6 = Xo-a(B6o - - Blo).
Carrying out the multiplication, we obtain the following linear combinations: B4 = Bl(1 - - p41 - - p,2 - - p43) -t- 112p41 q-
B3p42 + Iip43
B5 = B l ( 1 - -
psi -- ps2 -- p53) + B2psj + B3p52 + Bps3
B6 = B l ( 1 -
p61 - p6: - p63) q- B2p6t + B3p62 "Jr"Bp63.
(9)
Forward displacementanalysisof the generalized Stewart platform
249
Now we can summarize the quantities defining our optimization problem: --18 optimization parameters--real and imaginary components of cartesian coordinates of points B~, B: and B3 --6 lengths lj, j = 1. . . . . 6 characterizing the current state of drives --geometric constants--18 cartesian coordinates of points Aj, j = 1. . . . . 6 --lengths of sides of triangle B~B:B3:d~, d2, d3 --three constant vectors p4, ps, 1o6characterizing positions of points B4, B5 and B6 with respect to the triangle B~B2B3 9 complex expressions represent the basis for the creation of the objective function - - 2
- - 2
- - 2
2
- - 2
J~ = B4A4
- - 2
- - 2
- - 2
- - /~4, j8 = B5z45
- - 12, f 9 =
- - 2 B6A 6
--
/2.
(lO)
The objective function is defined as a sum of squares of all real and imaginary parts of these expressions 9
F = ~ (~2E q-fj~u)-
(1 l)
j=l
The global minimum of the objective function (1 l) is reached, when both real and imaginary parts of all above defined expressions (fjRE,fIM,j = 1. . . . . 9) are equal to zero (which means that kinematical constraint equations of the manipulator are satisfied). 3. THE METHOD OF SOLUTION The above described formulation transforms our kinematic problem to the problem of global optimization of sum of squares objective function. The professional optimization software UFO [8] was used for this purpose. This modular system was developed in the Institute of Computer and Information Science of the Academy of Sciences of the Czech Republic. UFO offers a broad group of methods for the solution of our problem, it generates after the specification of optimization problem the source code in the programming language Fortran. The procedure for the evaluation of the objective function was added to the source code. We chose the multi-level method for the numerical realization of the global optimization problem. This modern stochastic method [9] involves a combination of sampling and local search techniques. The Quasi-Newton method was chosen for the performance of individual local optimizations. If the number of global extremes, and thus the maximum possible number of configurations of the given manipulator type, is to be found reliably, it is necessary to perform the whole above described calculation for more different manipulators of different dimensions. If the same number of global extremes is calculated repeatedly (10 times) for the series of manipulators of given type with randomly generated dimensions, then the maximum number of configurations of this type of manipulator can be considered to be reliably determined. The set of manipulators of randomly generated dimensions and states of drives was investigated. We have used the following strategy for the random generation of manipulators. The points A~ and Bi (see Fig. 1) were randomly generated in the 3D cube (x ~ ( - 10, 10); y E ( - 10, 10); z ~ ( - 10, 10)). The dimensions of manipulator described in Section 2 were computed from the random values of the cartesian coordinates of these points. The subsequent global optimization search passes through an 18th dimensional cube in the space of optimization variables. Several different regions of optimization variables were sought through for each of these investigated manipulators. In particular we used (( - 15, 15) x ( - 15, 15) x . . . ); ( ( - 1 7 , 17) x ( - 1 7 , 1 7 ) x . . . ) and ( ( - 1 9 , 19) x ( - 1 9 , 1 9 ) x . . . ) cubes. These cubes do not represent bounds on the optimization variables, many solutions have been found outside them. The cubes serve only as the spaces of random generation of starting points of individual local optimizations.
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Applying the above described procedure, we calculated repeatedly 40 assembly configurations for the individual randomly generated generalized Stewart platforms. The dimensions and calculated results for one of the solved platforms are given in the full extent in the Appendix. We would like to emphasize some of the interesting properties of our objective function. The vector of optimization variables for each of the global extremes (F = 0) corresponds to the solution of kinematic equations. However, the objective function may have in general many non-zero (F :~ 0) extremes. Fortunately this is not the case of the investigated function. The majority of extremes found during our numerical experiments are the zero-ones (for example 40 from 47 etc.). The non-zero (non-solution) extremes are strongly separated from the global solution extremes. For example the highest value of numerical zero extreme is 0.3788 x 10 -~5, whereas the smallest non-zero value is 0.1065 × 106.
5. CONCLUSIONS The proposed method represents an original approach to the solution of kinematic problems for the manipulators of different types and structures. The complete solution of the direct position kinematics of the generalized Stewart platform was successfully carried out in this paper. The solutions for two special simplified cases were done by the authors in [6]. Particularly the special case having six distinct joints in the base and three distinct joints in the platform, and the case for which the joints in the base and in the platform are situated on the sides of the triangle were considered. Therefore the presented method has proven its applicability and worthiness for the solution of this class of kinematic problems. Using the proposed method, the kinematic problem is transformed to the problem of global minimization of sum of squares objective function. The own formulation of the objective function is relatively very simple, especially in comparison with the process of reduction of forward displacement analysis to one polynomial equation, which usually requires difficult looking for specific approaches for individual types of manipulator. The important positive feature of the method is essentially unlimited potential of parallelization of calculations. Each local minimization may run independently, the necessary communication between processors is limited to the operations of global inventory of calculated minima. For the general serial manipulator, the dimensions and position ensuring that all admissible configurations of the manipulator are real have been found [10-12]. However, it is still in question, whether this is possible for the forward displacement analysis of the generalized Stewart platform [5]. To find the answer to this question by means of approach utilizing general optimization methods is the current aim of the authors.
REFERENCES
1. Hunt, K. H. and Primrose, E. J., Mech. Mach. Theory, 1993, 28, 31. 2. Dasgupta Bhaskar and Mruthyunjaya,T. S., Mech. Mach. Theory, 1994, 29, 819. 3. Wen, F. and Liang, C. G., Mech. Mach. Theory, 1994, 29, 547. 4. Yin, J. P. and Liang, C. G., Mech. Mach. Theory, 1994, 29, 1. 5. Faugere, J. C. and Lazard, D., Mech. Mach. Theory, 1995, 30, 765. 6. Sika, Z. and Kocandrle, P., Journal of Czech and Slovak Mech. Engineering, 1995, 46, 84 (in Czech). 7. Garcia De Jalon, J., Unda, J. and Avello, A., Comput. Meths. Appl. Mech. Engrg, 1986, 56, 309. 8. Luksan, L. et al., Interactive system for universalfunctionaloptimization (UFO), Institute of computer @ information science, Czech Republic, Tech. report no 599, 1994. 9. Rinnoy Kan, A. H. (3. and Timmer, G. T., Math. Programming, 1987, 39, 26. 10. Lee, H. Y. and Liang, C. (3., Mech. Mach. Theory, 1988, 23, 209. I1. Lee, H. Y. and Liang, C. G., Mech. Mach. Theory, 1988, 23, 219. 12. Lee, H. Y., Woernle, C. and Hiller, M., Journal of Mech Design, 1991, 113, 481.
Forward displacement analysis o f the generalized Stewart platform
APPENDIX
Manipulator Dimensions Lengths characterizing the current state of drives: 1~ = 6.259228559118601 /2 = 8.500337420021544 /3 = 18.694619163170160 h = 13.552309286827120 /5 = 7.545529525613933 /6 = 9.243871435074205 Geometric constants: Centers of base points: A~ = [-4.040221264696095 - 1.511192141063164 3.387036404127466] r A2 = [6.560928742675682 4.472522401684421 4.423214355684349] T A3 = [9.465313793662773 4.158166984912020 -3.557549847837350] r A~ = [1.050214695824799 8.632713079243020 -5.446276861091519] r A~ = [-6.100164496992252 6.691086296020493E-001 -2.581163426953443] T A~ = [3.823984516765840 -7.30610236763124 - 1.190891745689443E-001] r Lengths of sides of triangle B~B2B3: d~ = 18.072944068579510 d2 = 5.908290767833534 d3 = 13.454540847533640 Vectors characterizing positions o f points B4, Bs, B6 with respect to the triangle B~B2B3: p4 = [2.572455517157682E-001 -3.341579155133697E-001 -4.31420668274632E-001] r p5 = [4.719317551338843E-001 8.329231820236864E-001 -5.831286397187205E-001] r p6 = [7.679307647562370E-001 -9.528239835647631E-001 -2.269314005064037E-001] r Space o f starting points generation: ( ( - 17, 17) x ( - 1 7 , 1 7 ) x . . . )--18th dimensional cube
Computed Results Total number o f computed c o n f i g u r a t i o n s ~ 0 N u m b e r of real configurations--2 N u m b e r of couples o f complex conjugate configurations--19 Configuration n u m b e r !. (real) B~ = [-.5763964830D + 01 -.1093517223D - 09"i; -6881938143D + 01 +.5465420360D - 10*i; .6737841743D + 00 -.3164871704D - 10*i] 132 = [-.6751633855D - 01 - . 5 4 0 2 0 0 6 8 1 2 D - ll*i; .8903735949D + 01 +.1593108719D - 10*i; .7381561403D + 01 -.3651409853 - 10*i] B3 = [-.7181724077D + 01 -.6538060403D - 10*i; -.1539001975D + 01 +.8721238438D - 10*i; .2759663274D + 01 -.9906729758D - 10*i] Configuration n u m b e r 2. (real) B~ = [-.6209165847D + 01 +.7323246699D - 09"i; -.6658250789D + 01 -.3703965741D - 09"i; .5618976747D + 00 +.2437592785D - 09"i] B2 = [-.7885890078D - 01 +.9437269139D - 10*i; .8973296918D + 01 -.1679544167D - 09"i; .7248104400D + 01 +.3269421842D - 09"i] B3 = [-.7447398369D + 01 +.4266606690D - 09"i: -.1169452755D + 01 -.6134656125D - 09"i; .2364056426D + 01 +.6710973978D - 09"i] Configuration n u m b e r 3. B~ = [-,9451889863D + 01 +.5270214099D + 00*i: -.3539226008D + 01 -.8053333983D + 00*i; .7564750091D + 00 -.4633314436D + 00*i] B2 = [.2307911300D + 01 +.1957851825D + 00*i; .1014871390D + 02 -.5506697228D + 00*i; - . 3 7 6 5 9 2 0 3 6 6 D + 00 -.8246967208D + 00*i] B3 = [-.8876326174D + 01 +.2328459545D + 00*i; .2457532602D + 01 -.5550372102D + 00*i; -.1656242979D + 00 +.9808179018D + 00*i] Configuration number 4. B~ = [.6007113275D + 01 -.1831877484D + 02"i; -.1214037278D + 02 +.2547015423D + 02"i; .3115232004D + 02 -.5894144810D + 01*i] B2 = [-.3399834264D + 02 +.138688482D + 02"i; .1766908352D + 02 +.1227096031D + 01*i; -.8726911937D + 01 -.4154461537D + 02'i] B3 = [.8944179083D + 00 -.2395249619D + 02"i; .9007980707D + 01 +.2921329814D + 02"i; .382572280D + 02 -.8297807065D + 01*i] Configuration n u m b e r 5. B~ = [-.1083427441D + 02 -.4916785978D + 00*i; -.1223005784D + 01 +.2491218756D + 01*i; .1392287196D + 01 +.2159393267D + 01*i] B~ = [.4459359788D + 01 +.1675344759D + 01*i; .9830502971D + 01 -.2126558896D + 01*i; - . 2 7 1 1 1 2 3 0 9 4 D + 01 -.2090581485D + 01*i] B3 = [-.9431105500D + 01 +.3701371116D + 00*i; .4184589464D + 01 +.2525789924D + 01*i; -.1025125033D + 01 +.2735531727D + 01*i] Configuration number 6. B~ --- [-.1490044829D + 02 +.5502861366D + 00*i; -.7034713284D + 00 -.6538527852D + 01*i; .1598838707D + 01 - . 6 2 9 6 6 9 6 7 0 6 D + 01*i] 132--- [.5198428741D + 01 +.1745667396D + 01*i; -.1632820683D + 01 +.2291086344D + 01*i; -.2444640002D + 01 -.2383035391D + 01*i] B~ - [-.9004894394D + 01 -.7246150124D + 00*i; .1993815121D + 01 -.5363087517D + 01*i; .3254438616D + 01 -.3668737658D + 01*i]
251
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Configuration number 7. B~ = [.1725988143D + 01 +.3602668539D + 01*i; -.7331502036D + 01 +.3563196618D + 01*i; .3363928227D + 01 +.1507374970D + 01*i] B., = [.21795120016D + 02 +.4552430676D + 02"i; -.1062560736D + 02 -.6067527352D + 01*i; .4943268185D + 02 -.1744384025D + 02"i] B3 = [.9897474655D + 01 -.1253383207D + 02"i; -.1855139172D + 02 +.1668790759D + 01*i; - . 9 8 9 5 3 1 6 6 7 2 D + 01 -.6834289518D + 01*i] Configuration number 8. B~ = [-,1529797650D + 01 -,4454767664D + 00*i; -.7137327340D + 01 +.4010111777D - 01*i; .4880928572D + 01 +.8996293626D + 00*i] B2 = [.1063772761D + 01 +.9177423998D + 00*i; .1065020601D + 02 -.3746138557D + 00*i; .7620404073D + 01 +.2301777389D + 01*i] B3 = [-.4539878093D + 01 -.9939017882D + 00*i; -.2180020305D + 01 +.5709378067D + 00*i; .7185078417D + 01 -.9588970461D + 00*i] Configuration number 9. B~ = [-.1381141353D + 02 -.2240362292D + 01*i; .3445953681D + 01 -.7462582619D + 01*i; .5451003816D + 00 -.5314018131D + 01*i] B2 = [.7131132559D + 01 +.2306466054D + 01*i; -.4051936052D + 01 +.5364182443D + 01*i; -,3587125663D + 01 -.5544283329D + 01*i] B3 = [-.957890905D + 01 +.1048375268D + 01*i; -.1971317539D + 01 -.4847027787D + 01*i; -.1760461949D + 01 -.5422266541D + 01*i] Configuration number 10. B~ = [-.1525994509D + 02 +,3337626774D + 01*i; .4497311983D + 01 .5152017048D + 01 +.1021455397D + 02"i] B2 = [.7125430651D + 01 +,2758815524D + 01*i; -.6026618379D + 01 .7501103798D + 01 -.6114408398D + 01*i] B3 = [-.8471911958D + 01 +.3974709669D + 01*i; .5079964092D + 01 .6239555587D + 01 +.7129672383D + 01*i]
+.3231885257D + 01*i; -.1644145630D + 01*i; +.1567712210D + 01*i;
Configuration number 11. B~ = [.2789222755D + 01 -.4801571140D + 01*i; -.6776665204D + 01 +.8017622621D + 01*i; .1365766390D + 02 +.7303218552D + 01*i] B2 = [.4231502199D + 01 +.9352568190D + 01*i; .1787867021D + 02 -.6589094878D + 00*i; -.1151617740D + 00 -.6746809367D + 01*i] B3 = [-.1554243108D + 01 -.3248077074D + 01*i; .2941252531D + 01 +.9566382059D + 01i; .1461958497D + 02 -,1328641823D + 01*i] Configuration number 12. B~ = [-.1345321519D + 02 -.1837827149D + 01*i; -.2949444001D + 01 +.5939439006D + 01*i; .1280339635D + 01 +.4156766523D + 01*i] B2 = [.5833184649D + 01 -.1870713927D + 01*i; -.1689114283D + 01 -.9970303857D + 00*i; -.1880695760D + 01 +.1190489692D + 01*i] B3 = [-.8342932671D + 01 -.8422732054D + 01*i; -.2699485361D + 00 +.6574806793D + 01*i; .4188710936D + 01 +.1822117222D + 01*i] Configuration number 13. B~ = [-.3293140933D + 01 +.9763202188D + 01*i; -.7223029328D + 01 +.1035297351D + 00*i] B: = [-.2492353085D + 02 -.1604256367D + 02"i; .5590405928D + 02 -.9898511869D + 01*i] B3 = [-.8085496272D + 01 -.5148186801D + 01*i; .2900699428D + 02 -.9410876392D + 00*i]
.3289109251D + 01
-.1290635427D + 01*i;
.4551764164D + 01
+.5669115211D + 02"i;
.2265447641D + 01
+.3154654649D + 02"i;
Configuration number 14. B~ = [.1344315757D + 01 +.5090028260D + 01*i; -.1012059417D + 02 +.6319382226D + 01*i; .8675553056D + 01 +.5105147307D + 01*i] B2 = [-.2577242720D + 01 +.4633607523D + 01*i; .9989128962D + 01 +.5455225823D - 01*i; -.1335270181D + 01 -.7300836953D + 01*i] B3 = [-.3681086599D + 01 +.6324828080D + 01*i; -.1114965472D + 02 +.2002293642D + 01*i; .9774603505D + 00 +.8592180524D + 01*i] Configuration number B~ = [.4210007054D + -.7461362506D + 01 B2 = [.3672352258D + - . 5 5 1 6 8 3 4 7 0 2 D + 00 B3 = [-.1674704025D - . 9 3 8 3 3 2 0 5 5 8 D + 01
15. 00 -.8125291169D + 01*i; -.6594095699D + 01 +.5059679843D + 01*i; -.5712049554D + 01*i] 01 +.2315689153D + 02'i; .2889302086D + 02 +.1891421515D + 01*i; -.4160624231D + 01*i] + 02 -.2924626460D + 01*i; ,1797359931D + 01 +.1832139271D + 02"i; +.5734532427D + 01*i]
Configuration number 16. B~ = [-.8285289931D + 01 +.5652283518D + 01*i; ,4309247489D + 01 -.9916092927D + 00*i; .9832215994D + 01 +.4618324946D + 01*i] B2 = [.4225246476D + 01 +.1107213757D + 01*i; .1017727238D + 02 +.5467105340D + 01*i; - . 4 3 0 4 8 1 6 5 1 6 D + 01 +.3277070163D + 01*i] B3 = [-.6629071114D + 01 +.2222860712D + 01*i; .9482336161D + 01 -.3229790403D + 00*i; .5602823153D + 01 +.4093192597D + 01*i]
Forward displacement analysis o f the generalized Stewart platform Configuration number 17. B~ = [-.3556637302D + 01 +.7168168626D + 01*i; -.1607917006D + 02 -.1420593574D + 01*i; .1222211106D + 01 +.1116098709D + 02"i] B2 = [.2573470877D + 02 +.4770434676D + 01*i; -.1406571105D + 01 +.1160747190D + 02"i; .6176631561D + 01 -.1324605023D + 02"i] B3 = [-.28654521492D + 01 +.1224817908D + 02"i; -.1357078272D + 02 -.1240773922D + 01*i; .8588777870D + 01 +.1062311327D + 02"i] Configuration number 18. B~ = [-.1345673388D + 02 +.3698721171D + 01*i; .8094046470D + 01 +.4531304188D + 01*i; .2551262926D + 01 +.1040377998D + 02"i] B2 = [.7820798960D + 01 +.4518673325D + 00*i; -.6804778504D + 01 +.3502072850D + 01*i; - . 3 4 3 1 9 1 8 7 4 8 D + 00 -.8166453560D + 01*i] B3 = [ - . I 114496837D + 02 +.6307151743D + 00*i; .2301094634D + 01 .4726763007D + 01*i; -.8241695356D + 00 +.7967116212D + 01*i] Configuration number 19. B~ = [ . 2 8 0 5 1 9 1 3 3 3 D + 01 +.3689240411D + 01*i; .3943221810D + 01 -.5665988125D + 01*i; .6725582385D + 01 +.1692434947D + 01*i] B2 = [-.1580794829D + 02 +.3760777139D + 02"i; -.1772084542D + 02 -.3258552677D + 02"i; .4397674547D + 02 +.2984841785D + 01*i] B3 = [-.1643350755D + 02 +.8473728141D + 01*i; .4302675492D + 01 -.2513355924D + 02"i; .1846211273D+02 +.1013147153D+02"i] Configuration number 20. B~ = [,5905035281D + 01 +.7230263269D + 01*i; .1083140092D + 02 -.1721860539D + 02"i; .1837663190D + 02 +.9380867968D + 01*i] B2 = [-.1029852464D + 02 +.1058783125D + 02"i; -.9635024637D + 01 -.2376379718D + 02"i; .2282070706D + 02 -.8519847873D + 01*i] B3 = [.1008145610D + 01 +.1201616324D + 02"i; -.7451844895D + 01 -.3587922502D + 02"i; .3688405614D + 02 -.7787413671D + 01*i] Configuration number 21. B~ = [-.3465809453D + 02 +.8546930510D + 02"i; .1654707158D + 03 -.2955972040D + 02"i; .5310827201D + 02 +.1519034427D + 03"i] B2 = [.1704277160D + 02 +.6633038659D + 02"i; .1052324597D + 03 -.4222999801D + 02"i; .5004549455D + 02 +.7802826268D + 02"i] B3 = [-.1100909176D + 03 +.7205212605D + 02'il .2594449938D + 03 +.7144304361D + 02"i; -.3989710389D + 02 +.2648405435D + 03"i] The configurations 22, 23 . . . . . 40 are complex conjugate to the configurations 3, 4 . . . . . 21, respectively.
MAMT 33/3--B
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Mech, Mach. Theory Vol. 33, No. 3, pp. 245-253, 1998 © 1997 ElsevierScience Ltd. All rights reserved Printed in Great Britain PlI: S01D4-114X(97)00044-X oo94-114x/98 $19.o0+ o.oo
AN INVESTIGATION OF PROPERTIES OF THE F O R W A R D DISPLACEMENT ANALYSIS OF THE GENERALIZED STEWART PLATFORM BY MEANS OF GENERAL OPTIMIZATION METHODS Z. ~IKA Department of Mechanics, Faculty of Mechanical Engineering, Czech Technical University, Prague, Czech Republic
and P. KO(~ANDRLE* Stress Calculation Group, Skoda Turbines Ltd., Pilsen, Czech Republic
and V. STEJSKAL Department of Mechanics, Faculty of Mechanical Engineering, Czech Technical University, Prague, Czech Republic
Abstract--The method for the determination of all solutions of the forward displacement analysis of the generalized Stewart platform is proposed and described. The kinematic problem is transformed to the problem of global minimization of the specially created objective function. The problem is formulated in the domain of complex numbers, the general optimization methods are used for the solution.' The presented method is very easily applicable on the manipulators of arbitrary structure, it has practically unlimited potential of parallelization. © 1997 Elsevier Science Ltd
1. INTRODUCTION The generalized Stewart platform is a fully parallel mechanism with six degrees of freedom. It is used in flight and automotive simulators, robotic end-effectors, and other applications requiring spatial mechanisms with high structural stiffness. The geometry of the Stewart platform is shown in Fig. 1. A great attention has been paid to the forward displacement analysis of this type of manipulator in the recent years. The kinematic problem can be stated as follows: given the lengths of the 6 variable limbs, find all the possible positions of the movable platform (end-effector of the manipulator). For some simplified types of the parallel manipulator, it is possible to determine the number of solutions of the forward displacement analysis (direct kinematic problem) by means of geometric theorems [1]. Some other works (like e.g. [2]) dealt with the reduction of number of constraint equations and coordinates describing position of the moving platform. For many simplified types of platform, the forward displacement analysis was successfully transformed to one polynomial equation for one unknown [3, 4] etc. The maximum number of possible solutions has been found even for the most general type of platform, namely by means of special algebraic procedures (Gr6bner basis computations) [5]. *The work was done during Ph.D. studies at the Department of Mechanics, Faculty of Applied Sciences, University of West Bohemia, Pilsen, Czech Republic. 245
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et al.
The method proposed here differs from the above mentioned approaches. The direct kinematic problem is transformed to the problem of global minimization of the specially created objective function. The aim is to demonstrate the applicability and worthiness of the complete solution of this problem by means of utilization of general optimization algorithms. This method was proposed by the authors in [6], where also the solution for two simplified types of Stewart platform was carried out, the solution of the most general platform is presented here. The approach based on the usage of optimization methods enables to investigate properties of the manipulators of different type and structure, manipulators with special relationships between their individual dimensions, etc. The only condition is the formulation of the problem in the domain of complex numbers and the constitution of the objective function in such a way that the description of the given problem be unique. The whole further effort is then left to the computer. 2. FORMULATION OF THE OPTIMIZATION PROBLEM The basic assumption for the formulation of the optimization problem is, that a polynomial equation of a certain degree corresponds to the forward displacement analysis of the parallel manipulator of the generalized Stewart platform type, as it is for the special type of platform solved, e.g. in [3]. The number of roots of this hypothetic polynomial equation is equal to its degree except for special cases, where some of the roots coincide. If the dimensions of the manipulator and values characterizing drives (lj = AjBj, j = 1. . . . . 6) were generated randomly, the probability of occurrence of these special cases would be uncomparably smaller than the probability of occurrence of the normal situation with full number of roots. However, we have to consider that the roots are in general from the domain of complex numbers. The necessity to formulate the kinematic relationships and consequently the optimization problem also in the domain of complex numbers follows directly from this fact. Further requirements on the optimization problem are as follows: --good numerical properties of the objective function; --small number of optimization variables; --uniqueness of the problem description by the objective function. The simple possibility of the choice of optimization parameters represents the usage of the real and imaginary components of angles tpt, ~p,, tPx~htpw, tpv (see Fig. 2). The number of optimization variables is in this case equal to 10, which is very good. Unfortunately there were complications with the numerical realization of this approach. The complications were probably caused by the very strong non-linearity of the objective function arising from the products of complex goniometric functions containing exponential members (sin(cPRE + iCplM) = sin(q~RE)COsh(q~XM) + i COS(~PRE)Sinh(tpIM)).
The effort was therefore concentrated to formulate the objective function on the basis of cartesian coordinates of points and their mutual distances. Eighteen optimization variables were chosen, particularly real and imaginary components of cartesian coordinates of points B~, B2 and B3 (see Fig. 1). The objective function was created in such a way that its zero values (global minima) correspond to the situation, when the kinematic constraint equations were satisfied. The position of a rigid body is uniquely given by the position of its three non-collinear points (Bt, B2, B3). Then the positions of points B4, Bs, B6, have to be uniquely expressed by means of position of triangle Bj B2B3. If this condition was not satisfied, our objective function would be valid for more different platforms simultaneously. The following relationships hold between the points of basic triangle [BIBzl2 - ~ = 0,1B~B312- ~ = 0,1B2B31' - ~ = 0.
(1)
The unique position of the auxiliary point B with respect to the triangle BI, B2, B3 is given by (see Fig. 3 and Fig. 1) It
= B~ + (B2 -
B,) x (B3 - Bt)/x/d, d2 sin/3,.
(2)
Forward displacement analysis of the generalized Stewart platform •
C
Y Fig. 1. Parallel manipulator of the generalized Stewart platform type.
x
y Fig. 2. Description of platform position by means of five angular coordinates.
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248
I
cl4
~
I
Fig. 3. Position of the auxiliary point.
~/dld2
The division by the constant sin(fit) is introduced because of consistence of physical dimensions. The derivation of the linear dependencies for the positions of points//4, B5 and//6 comes from the procedure used for the description of multibody systems by means of so-called natural coordinates [7]. We shall use the following notation: B, B:, j = 1 , . . . , 6--position vectors of centers of spherical couplings and of auxiliary point in the system x, y, z ~0, B~o,j = 1. . . . . 6---position vectors of above mentioned points in the coordinate system ¢, g, connected with the moving platform S--matrix of directional cosines for the transformation of relative vectors from the system ~, g, ~ to the system x, y, z Then we can write ( B i - Bj) = S(B~ - B~) and for the three vectors [B2
- -
Sl, 63
-
-
BI, ][~
-
-
Bl] = S[B2o - B,o, B3o - B1o, ~ - B,o]
(4)
or
(5)
X = SX0. Owing to the non-complanarity of vectors, we can carry out the inversion
(6)
XXU = S, B , = Bi + S(B40 -
Bt0) = BL + X X ; t ( B 4 o
-
Bt0)
B5 = BI + S(B~o - 91o) = Sl + XXo-l(Sso - B,0) B6 = B, + S(B6o - B,o) = Bm + X X i ' ( B ~
p4 =
Xol(B,o -
Bto)
P5
Xot(Bso
Bl0)
=
-
- B1o),
(7)
(8)
p6 = Xo-a(B6o - - Blo).
Carrying out the multiplication, we obtain the following linear combinations: B4 = Bl(1 - - p41 - - p,2 - - p43) -t- 112p41 q-
B3p42 + Iip43
B5 = B l ( 1 - -
psi -- ps2 -- p53) + B2psj + B3p52 + Bps3
B6 = B l ( 1 -
p61 - p6: - p63) q- B2p6t + B3p62 "Jr"Bp63.
(9)
Forward displacementanalysisof the generalized Stewart platform
249
Now we can summarize the quantities defining our optimization problem: --18 optimization parameters--real and imaginary components of cartesian coordinates of points B~, B: and B3 --6 lengths lj, j = 1. . . . . 6 characterizing the current state of drives --geometric constants--18 cartesian coordinates of points Aj, j = 1. . . . . 6 --lengths of sides of triangle B~B:B3:d~, d2, d3 --three constant vectors p4, ps, 1o6characterizing positions of points B4, B5 and B6 with respect to the triangle B~B2B3 9 complex expressions represent the basis for the creation of the objective function - - 2
- - 2
- - 2
2
- - 2
J~ = B4A4
- - 2
- - 2
- - 2
- - /~4, j8 = B5z45
- - 12, f 9 =
- - 2 B6A 6
--
/2.
(lO)
The objective function is defined as a sum of squares of all real and imaginary parts of these expressions 9
F = ~ (~2E q-fj~u)-
(1 l)
j=l
The global minimum of the objective function (1 l) is reached, when both real and imaginary parts of all above defined expressions (fjRE,fIM,j = 1. . . . . 9) are equal to zero (which means that kinematical constraint equations of the manipulator are satisfied). 3. THE METHOD OF SOLUTION The above described formulation transforms our kinematic problem to the problem of global optimization of sum of squares objective function. The professional optimization software UFO [8] was used for this purpose. This modular system was developed in the Institute of Computer and Information Science of the Academy of Sciences of the Czech Republic. UFO offers a broad group of methods for the solution of our problem, it generates after the specification of optimization problem the source code in the programming language Fortran. The procedure for the evaluation of the objective function was added to the source code. We chose the multi-level method for the numerical realization of the global optimization problem. This modern stochastic method [9] involves a combination of sampling and local search techniques. The Quasi-Newton method was chosen for the performance of individual local optimizations. If the number of global extremes, and thus the maximum possible number of configurations of the given manipulator type, is to be found reliably, it is necessary to perform the whole above described calculation for more different manipulators of different dimensions. If the same number of global extremes is calculated repeatedly (10 times) for the series of manipulators of given type with randomly generated dimensions, then the maximum number of configurations of this type of manipulator can be considered to be reliably determined. The set of manipulators of randomly generated dimensions and states of drives was investigated. We have used the following strategy for the random generation of manipulators. The points A~ and Bi (see Fig. 1) were randomly generated in the 3D cube (x ~ ( - 10, 10); y E ( - 10, 10); z ~ ( - 10, 10)). The dimensions of manipulator described in Section 2 were computed from the random values of the cartesian coordinates of these points. The subsequent global optimization search passes through an 18th dimensional cube in the space of optimization variables. Several different regions of optimization variables were sought through for each of these investigated manipulators. In particular we used (( - 15, 15) x ( - 15, 15) x . . . ); ( ( - 1 7 , 17) x ( - 1 7 , 1 7 ) x . . . ) and ( ( - 1 9 , 19) x ( - 1 9 , 1 9 ) x . . . ) cubes. These cubes do not represent bounds on the optimization variables, many solutions have been found outside them. The cubes serve only as the spaces of random generation of starting points of individual local optimizations.
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Applying the above described procedure, we calculated repeatedly 40 assembly configurations for the individual randomly generated generalized Stewart platforms. The dimensions and calculated results for one of the solved platforms are given in the full extent in the Appendix. We would like to emphasize some of the interesting properties of our objective function. The vector of optimization variables for each of the global extremes (F = 0) corresponds to the solution of kinematic equations. However, the objective function may have in general many non-zero (F :~ 0) extremes. Fortunately this is not the case of the investigated function. The majority of extremes found during our numerical experiments are the zero-ones (for example 40 from 47 etc.). The non-zero (non-solution) extremes are strongly separated from the global solution extremes. For example the highest value of numerical zero extreme is 0.3788 x 10 -~5, whereas the smallest non-zero value is 0.1065 × 106.
5. CONCLUSIONS The proposed method represents an original approach to the solution of kinematic problems for the manipulators of different types and structures. The complete solution of the direct position kinematics of the generalized Stewart platform was successfully carried out in this paper. The solutions for two special simplified cases were done by the authors in [6]. Particularly the special case having six distinct joints in the base and three distinct joints in the platform, and the case for which the joints in the base and in the platform are situated on the sides of the triangle were considered. Therefore the presented method has proven its applicability and worthiness for the solution of this class of kinematic problems. Using the proposed method, the kinematic problem is transformed to the problem of global minimization of sum of squares objective function. The own formulation of the objective function is relatively very simple, especially in comparison with the process of reduction of forward displacement analysis to one polynomial equation, which usually requires difficult looking for specific approaches for individual types of manipulator. The important positive feature of the method is essentially unlimited potential of parallelization of calculations. Each local minimization may run independently, the necessary communication between processors is limited to the operations of global inventory of calculated minima. For the general serial manipulator, the dimensions and position ensuring that all admissible configurations of the manipulator are real have been found [10-12]. However, it is still in question, whether this is possible for the forward displacement analysis of the generalized Stewart platform [5]. To find the answer to this question by means of approach utilizing general optimization methods is the current aim of the authors.
REFERENCES
1. Hunt, K. H. and Primrose, E. J., Mech. Mach. Theory, 1993, 28, 31. 2. Dasgupta Bhaskar and Mruthyunjaya,T. S., Mech. Mach. Theory, 1994, 29, 819. 3. Wen, F. and Liang, C. G., Mech. Mach. Theory, 1994, 29, 547. 4. Yin, J. P. and Liang, C. G., Mech. Mach. Theory, 1994, 29, 1. 5. Faugere, J. C. and Lazard, D., Mech. Mach. Theory, 1995, 30, 765. 6. Sika, Z. and Kocandrle, P., Journal of Czech and Slovak Mech. Engineering, 1995, 46, 84 (in Czech). 7. Garcia De Jalon, J., Unda, J. and Avello, A., Comput. Meths. Appl. Mech. Engrg, 1986, 56, 309. 8. Luksan, L. et al., Interactive system for universalfunctionaloptimization (UFO), Institute of computer @ information science, Czech Republic, Tech. report no 599, 1994. 9. Rinnoy Kan, A. H. (3. and Timmer, G. T., Math. Programming, 1987, 39, 26. 10. Lee, H. Y. and Liang, C. (3., Mech. Mach. Theory, 1988, 23, 209. I1. Lee, H. Y. and Liang, C. G., Mech. Mach. Theory, 1988, 23, 219. 12. Lee, H. Y., Woernle, C. and Hiller, M., Journal of Mech Design, 1991, 113, 481.
Forward displacement analysis o f the generalized Stewart platform
APPENDIX
Manipulator Dimensions Lengths characterizing the current state of drives: 1~ = 6.259228559118601 /2 = 8.500337420021544 /3 = 18.694619163170160 h = 13.552309286827120 /5 = 7.545529525613933 /6 = 9.243871435074205 Geometric constants: Centers of base points: A~ = [-4.040221264696095 - 1.511192141063164 3.387036404127466] r A2 = [6.560928742675682 4.472522401684421 4.423214355684349] T A3 = [9.465313793662773 4.158166984912020 -3.557549847837350] r A~ = [1.050214695824799 8.632713079243020 -5.446276861091519] r A~ = [-6.100164496992252 6.691086296020493E-001 -2.581163426953443] T A~ = [3.823984516765840 -7.30610236763124 - 1.190891745689443E-001] r Lengths of sides of triangle B~B2B3: d~ = 18.072944068579510 d2 = 5.908290767833534 d3 = 13.454540847533640 Vectors characterizing positions o f points B4, Bs, B6 with respect to the triangle B~B2B3: p4 = [2.572455517157682E-001 -3.341579155133697E-001 -4.31420668274632E-001] r p5 = [4.719317551338843E-001 8.329231820236864E-001 -5.831286397187205E-001] r p6 = [7.679307647562370E-001 -9.528239835647631E-001 -2.269314005064037E-001] r Space o f starting points generation: ( ( - 17, 17) x ( - 1 7 , 1 7 ) x . . . )--18th dimensional cube
Computed Results Total number o f computed c o n f i g u r a t i o n s ~ 0 N u m b e r of real configurations--2 N u m b e r of couples o f complex conjugate configurations--19 Configuration n u m b e r !. (real) B~ = [-.5763964830D + 01 -.1093517223D - 09"i; -6881938143D + 01 +.5465420360D - 10*i; .6737841743D + 00 -.3164871704D - 10*i] 132 = [-.6751633855D - 01 - . 5 4 0 2 0 0 6 8 1 2 D - ll*i; .8903735949D + 01 +.1593108719D - 10*i; .7381561403D + 01 -.3651409853 - 10*i] B3 = [-.7181724077D + 01 -.6538060403D - 10*i; -.1539001975D + 01 +.8721238438D - 10*i; .2759663274D + 01 -.9906729758D - 10*i] Configuration n u m b e r 2. (real) B~ = [-.6209165847D + 01 +.7323246699D - 09"i; -.6658250789D + 01 -.3703965741D - 09"i; .5618976747D + 00 +.2437592785D - 09"i] B2 = [-.7885890078D - 01 +.9437269139D - 10*i; .8973296918D + 01 -.1679544167D - 09"i; .7248104400D + 01 +.3269421842D - 09"i] B3 = [-.7447398369D + 01 +.4266606690D - 09"i: -.1169452755D + 01 -.6134656125D - 09"i; .2364056426D + 01 +.6710973978D - 09"i] Configuration n u m b e r 3. B~ = [-,9451889863D + 01 +.5270214099D + 00*i: -.3539226008D + 01 -.8053333983D + 00*i; .7564750091D + 00 -.4633314436D + 00*i] B2 = [.2307911300D + 01 +.1957851825D + 00*i; .1014871390D + 02 -.5506697228D + 00*i; - . 3 7 6 5 9 2 0 3 6 6 D + 00 -.8246967208D + 00*i] B3 = [-.8876326174D + 01 +.2328459545D + 00*i; .2457532602D + 01 -.5550372102D + 00*i; -.1656242979D + 00 +.9808179018D + 00*i] Configuration number 4. B~ = [.6007113275D + 01 -.1831877484D + 02"i; -.1214037278D + 02 +.2547015423D + 02"i; .3115232004D + 02 -.5894144810D + 01*i] B2 = [-.3399834264D + 02 +.138688482D + 02"i; .1766908352D + 02 +.1227096031D + 01*i; -.8726911937D + 01 -.4154461537D + 02'i] B3 = [.8944179083D + 00 -.2395249619D + 02"i; .9007980707D + 01 +.2921329814D + 02"i; .382572280D + 02 -.8297807065D + 01*i] Configuration n u m b e r 5. B~ = [-.1083427441D + 02 -.4916785978D + 00*i; -.1223005784D + 01 +.2491218756D + 01*i; .1392287196D + 01 +.2159393267D + 01*i] B~ = [.4459359788D + 01 +.1675344759D + 01*i; .9830502971D + 01 -.2126558896D + 01*i; - . 2 7 1 1 1 2 3 0 9 4 D + 01 -.2090581485D + 01*i] B3 = [-.9431105500D + 01 +.3701371116D + 00*i; .4184589464D + 01 +.2525789924D + 01*i; -.1025125033D + 01 +.2735531727D + 01*i] Configuration number 6. B~ --- [-.1490044829D + 02 +.5502861366D + 00*i; -.7034713284D + 00 -.6538527852D + 01*i; .1598838707D + 01 - . 6 2 9 6 6 9 6 7 0 6 D + 01*i] 132--- [.5198428741D + 01 +.1745667396D + 01*i; -.1632820683D + 01 +.2291086344D + 01*i; -.2444640002D + 01 -.2383035391D + 01*i] B~ - [-.9004894394D + 01 -.7246150124D + 00*i; .1993815121D + 01 -.5363087517D + 01*i; .3254438616D + 01 -.3668737658D + 01*i]
251
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Configuration number 7. B~ = [.1725988143D + 01 +.3602668539D + 01*i; -.7331502036D + 01 +.3563196618D + 01*i; .3363928227D + 01 +.1507374970D + 01*i] B., = [.21795120016D + 02 +.4552430676D + 02"i; -.1062560736D + 02 -.6067527352D + 01*i; .4943268185D + 02 -.1744384025D + 02"i] B3 = [.9897474655D + 01 -.1253383207D + 02"i; -.1855139172D + 02 +.1668790759D + 01*i; - . 9 8 9 5 3 1 6 6 7 2 D + 01 -.6834289518D + 01*i] Configuration number 8. B~ = [-,1529797650D + 01 -,4454767664D + 00*i; -.7137327340D + 01 +.4010111777D - 01*i; .4880928572D + 01 +.8996293626D + 00*i] B2 = [.1063772761D + 01 +.9177423998D + 00*i; .1065020601D + 02 -.3746138557D + 00*i; .7620404073D + 01 +.2301777389D + 01*i] B3 = [-.4539878093D + 01 -.9939017882D + 00*i; -.2180020305D + 01 +.5709378067D + 00*i; .7185078417D + 01 -.9588970461D + 00*i] Configuration number 9. B~ = [-.1381141353D + 02 -.2240362292D + 01*i; .3445953681D + 01 -.7462582619D + 01*i; .5451003816D + 00 -.5314018131D + 01*i] B2 = [.7131132559D + 01 +.2306466054D + 01*i; -.4051936052D + 01 +.5364182443D + 01*i; -,3587125663D + 01 -.5544283329D + 01*i] B3 = [-.957890905D + 01 +.1048375268D + 01*i; -.1971317539D + 01 -.4847027787D + 01*i; -.1760461949D + 01 -.5422266541D + 01*i] Configuration number 10. B~ = [-.1525994509D + 02 +,3337626774D + 01*i; .4497311983D + 01 .5152017048D + 01 +.1021455397D + 02"i] B2 = [.7125430651D + 01 +,2758815524D + 01*i; -.6026618379D + 01 .7501103798D + 01 -.6114408398D + 01*i] B3 = [-.8471911958D + 01 +.3974709669D + 01*i; .5079964092D + 01 .6239555587D + 01 +.7129672383D + 01*i]
+.3231885257D + 01*i; -.1644145630D + 01*i; +.1567712210D + 01*i;
Configuration number 11. B~ = [.2789222755D + 01 -.4801571140D + 01*i; -.6776665204D + 01 +.8017622621D + 01*i; .1365766390D + 02 +.7303218552D + 01*i] B2 = [.4231502199D + 01 +.9352568190D + 01*i; .1787867021D + 02 -.6589094878D + 00*i; -.1151617740D + 00 -.6746809367D + 01*i] B3 = [-.1554243108D + 01 -.3248077074D + 01*i; .2941252531D + 01 +.9566382059D + 01i; .1461958497D + 02 -,1328641823D + 01*i] Configuration number 12. B~ = [-.1345321519D + 02 -.1837827149D + 01*i; -.2949444001D + 01 +.5939439006D + 01*i; .1280339635D + 01 +.4156766523D + 01*i] B2 = [.5833184649D + 01 -.1870713927D + 01*i; -.1689114283D + 01 -.9970303857D + 00*i; -.1880695760D + 01 +.1190489692D + 01*i] B3 = [-.8342932671D + 01 -.8422732054D + 01*i; -.2699485361D + 00 +.6574806793D + 01*i; .4188710936D + 01 +.1822117222D + 01*i] Configuration number 13. B~ = [-.3293140933D + 01 +.9763202188D + 01*i; -.7223029328D + 01 +.1035297351D + 00*i] B: = [-.2492353085D + 02 -.1604256367D + 02"i; .5590405928D + 02 -.9898511869D + 01*i] B3 = [-.8085496272D + 01 -.5148186801D + 01*i; .2900699428D + 02 -.9410876392D + 00*i]
.3289109251D + 01
-.1290635427D + 01*i;
.4551764164D + 01
+.5669115211D + 02"i;
.2265447641D + 01
+.3154654649D + 02"i;
Configuration number 14. B~ = [.1344315757D + 01 +.5090028260D + 01*i; -.1012059417D + 02 +.6319382226D + 01*i; .8675553056D + 01 +.5105147307D + 01*i] B2 = [-.2577242720D + 01 +.4633607523D + 01*i; .9989128962D + 01 +.5455225823D - 01*i; -.1335270181D + 01 -.7300836953D + 01*i] B3 = [-.3681086599D + 01 +.6324828080D + 01*i; -.1114965472D + 02 +.2002293642D + 01*i; .9774603505D + 00 +.8592180524D + 01*i] Configuration number B~ = [.4210007054D + -.7461362506D + 01 B2 = [.3672352258D + - . 5 5 1 6 8 3 4 7 0 2 D + 00 B3 = [-.1674704025D - . 9 3 8 3 3 2 0 5 5 8 D + 01
15. 00 -.8125291169D + 01*i; -.6594095699D + 01 +.5059679843D + 01*i; -.5712049554D + 01*i] 01 +.2315689153D + 02'i; .2889302086D + 02 +.1891421515D + 01*i; -.4160624231D + 01*i] + 02 -.2924626460D + 01*i; ,1797359931D + 01 +.1832139271D + 02"i; +.5734532427D + 01*i]
Configuration number 16. B~ = [-.8285289931D + 01 +.5652283518D + 01*i; ,4309247489D + 01 -.9916092927D + 00*i; .9832215994D + 01 +.4618324946D + 01*i] B2 = [.4225246476D + 01 +.1107213757D + 01*i; .1017727238D + 02 +.5467105340D + 01*i; - . 4 3 0 4 8 1 6 5 1 6 D + 01 +.3277070163D + 01*i] B3 = [-.6629071114D + 01 +.2222860712D + 01*i; .9482336161D + 01 -.3229790403D + 00*i; .5602823153D + 01 +.4093192597D + 01*i]
Forward displacement analysis o f the generalized Stewart platform Configuration number 17. B~ = [-.3556637302D + 01 +.7168168626D + 01*i; -.1607917006D + 02 -.1420593574D + 01*i; .1222211106D + 01 +.1116098709D + 02"i] B2 = [.2573470877D + 02 +.4770434676D + 01*i; -.1406571105D + 01 +.1160747190D + 02"i; .6176631561D + 01 -.1324605023D + 02"i] B3 = [-.28654521492D + 01 +.1224817908D + 02"i; -.1357078272D + 02 -.1240773922D + 01*i; .8588777870D + 01 +.1062311327D + 02"i] Configuration number 18. B~ = [-.1345673388D + 02 +.3698721171D + 01*i; .8094046470D + 01 +.4531304188D + 01*i; .2551262926D + 01 +.1040377998D + 02"i] B2 = [.7820798960D + 01 +.4518673325D + 00*i; -.6804778504D + 01 +.3502072850D + 01*i; - . 3 4 3 1 9 1 8 7 4 8 D + 00 -.8166453560D + 01*i] B3 = [ - . I 114496837D + 02 +.6307151743D + 00*i; .2301094634D + 01 .4726763007D + 01*i; -.8241695356D + 00 +.7967116212D + 01*i] Configuration number 19. B~ = [ . 2 8 0 5 1 9 1 3 3 3 D + 01 +.3689240411D + 01*i; .3943221810D + 01 -.5665988125D + 01*i; .6725582385D + 01 +.1692434947D + 01*i] B2 = [-.1580794829D + 02 +.3760777139D + 02"i; -.1772084542D + 02 -.3258552677D + 02"i; .4397674547D + 02 +.2984841785D + 01*i] B3 = [-.1643350755D + 02 +.8473728141D + 01*i; .4302675492D + 01 -.2513355924D + 02"i; .1846211273D+02 +.1013147153D+02"i] Configuration number 20. B~ = [,5905035281D + 01 +.7230263269D + 01*i; .1083140092D + 02 -.1721860539D + 02"i; .1837663190D + 02 +.9380867968D + 01*i] B2 = [-.1029852464D + 02 +.1058783125D + 02"i; -.9635024637D + 01 -.2376379718D + 02"i; .2282070706D + 02 -.8519847873D + 01*i] B3 = [.1008145610D + 01 +.1201616324D + 02"i; -.7451844895D + 01 -.3587922502D + 02"i; .3688405614D + 02 -.7787413671D + 01*i] Configuration number 21. B~ = [-.3465809453D + 02 +.8546930510D + 02"i; .1654707158D + 03 -.2955972040D + 02"i; .5310827201D + 02 +.1519034427D + 03"i] B2 = [.1704277160D + 02 +.6633038659D + 02"i; .1052324597D + 03 -.4222999801D + 02"i; .5004549455D + 02 +.7802826268D + 02"i] B3 = [-.1100909176D + 03 +.7205212605D + 02'il .2594449938D + 03 +.7144304361D + 02"i; -.3989710389D + 02 +.2648405435D + 03"i] The configurations 22, 23 . . . . . 40 are complex conjugate to the configurations 3, 4 . . . . . 21, respectively.
MAMT 33/3--B
253