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Four-precision-point synthesis of the spherical four-bar function generator
Jnl. Mechanisms Volume 2, pp.133-139/Pergamon Press 1967/Printed in Great Britain
Four-Precision-Point Synthesis of the Spherical Four-Bar Function Generator Prof. John R. Zimmerman* Received 28 November 1966 Abstract A design algorithm for the spherical four-bar mechanism used as a function generator with three precision points has been presented in detail in a recent text by Hartenberg and Denavit [1]. This brief paper presents an algorithm for a four-precision-point design.
Zusammenfassung Der Konstruktions-Algorithmus fLir ein sph~risches Funktions-Viergelenkgetriebe mit drei Pr~zisions-Punkten wurde von Hartenberg und Denavit ausfLihrlich dargestellt [1 ]. Die vorliegende Ver6ffentlichung gibt einen Algorithmus fOr den Entwurf mit vier Pr~izisions- Punkten. A6erpak"r I"[pOgKTHOOBOqHbI~tanropHTM ~it~ HQHMeHeHHfl Cd~)epHqeCKaFO MeTbIpex3BeHHI,IKa K
eocnpoM3Benenn0o ~yHKun~ C TpeMg roqnbtMH rO~r~aMH nonpo6Ho onncaHb~ XapreH6eproM a 12eHaaaro,~t [1]. HacTosmaa pa6ora zaer anroparM nna npoeKxrlposaHl4~t cdtacplt~lCCKrtXMe×aHrI3MOBC '-/erblbMIlTOtlHblMH TOtIKaMI4.
Displacement Equation THE MECHANISMis shown in Fig. 1, with orthogonal coordinate frames attached to each link in accordance with a scheme developed by Hartenberg and Denavit [2] for matrix analysis of spatial link mechanisms. In Fig. 1 the four revolute joints have been labeled R t , . . . , R4. The z-axes have been Z3
Figure 1. Sketch of the spherical four-bar mechanism with the notation of Hartenberg and Denavit. * Associate Professor of Mechanical Engineering, The Pennsylvania State University, University Park, Pennsylvania 16802, U.S.A.
133
134
placed on the axes of these revolutes, and with arbitrarily selected senses. The x-axes are such that axis x~ is mutally perpendicular to axes z~_t and z~. (The subscripts are treated cyclically: thus xt is perpendicular to z , and zt.) For the present problem, where all coordinate frames share a common origin, the senses of the x-axes are arbitrary. The y-axes are not shown since no measurements will be referred to them. It is understood, however, that each xi, yi, z~ frame is dextral. The angles are measured using the x- and z-axes. Angle 0~ is measured from the x~to the x~+ 1-axis. It is positive if counterclockwise when viewed from the positive z~-axis. Angle 0[~is from the z~- to the z~+ i-axis and is positive if counterclockwise when viewed from the positive x~+ t-axis. By inspection of Fig. 1, and recalling that the x2-axis is perpendicular to axes z t and z 2, which happen to be stationary, it is clear that frame 2 is the fixed one and on the ground link. Similarly, since axis xt is always perpendicular to axes zt and z,, frame 1 must be on the left-hand lever (or crank). Likewise frame 4 is on the coupler, and frame 3 on the righthand lever. In all that follows link 1 is regarded as the input link, whose position is measured as 0t, and link 3 (measured by 02) as the output link. The angular link lengths are cq . . . . . 0[4. Using the matrix methods [3] ofanalysis, it can be shown that the displacement equation for this mechanism, in terms of the notation of Fig. 1, and at the i-th mechanism position, is sin 0[t cot 0[2cos 0] + sin 0[t cot 0[, cos 022-~
COS0[ 3
sin 0[2 sin 0[,
cos 0[t cot 0[2cot 0[,
=sin 0~ sin 0 ~ - c o s cq cos 0~t cos0~
(i)
To prevent the notation getting entirely out of hand later, angles 0~ and 0~, the input and output angles, are henceforth written as ¢~ and ~ . Thus, sincq cot 0[2cos ¢~ + sin 0[t cotct, cos~q+ . cos~3 - cos 0[t cot 0[2cot 014 sln 0[ 2 sin 014 = sin ¢i sin ¢ ~ - cos cq cos ¢i cos $i.
(la)
Equation (la) can be solved explicitly for the output angle, ~k~. The result is ¢i = 2 arc tan._, +A v/(A~ + B~ + C,),? B~+ C~
(2)
where Ai -
- sin ¢i,
B~- sin 0[t cot 0[, + cos ctt cos ¢~, Ci-
cos cq cot 0[2cot :t, - sin cq cot 0[2 COS ~/~i
COS 0[3
sin ~2 sin ~z4"
(3)
Equation (2) is among the many evidences of the very similar behavior of the spherical and planar four-bars. The analogous equation for the planar four-bar can be put into the same general form. And just as the planar four-bar can be connected in an upper branch or in a lower branch configuration, the spherical four-bar can be connected in two ways, in a positive or in a negative branch configuration, as the ( + ) sign before the radical indicates.
135 For computational purposes it is convenient to rewrite equation (2) as .
.)
.
~i = - arc tan
Ai+e,J(A?+B?+C~)
(4)
Bi+Ci
where + i for the positive branch _ 1 for the negative branch
e=
The quantity e serves as a branch indicator. In designing four-bar function generators, planar or spherical, an all too familiar experience is to discover, for some design, that some precision point positions of the mechanism can be assumed by one branch, the others only by the other branch. The mathematics of design by precision point methods show a nice indifference to the physical realities. We are assured only that a given set of links, connected in a given sequence, can exist in each of the configurations corresponding to the several precision points. There is no certainty that the mechanism can in fact move from one position to the next. A design involving a mixing of branches ordinarily will be discarded.
Four Point Synthesis It is assumed that angle a t (the shaft angle) is dictated by functional requirements. The unknown link lengths are ~2, %, ~ . If three pairs of ~b~, Oi values were required, equation (la) could be written for each pair and the set of three equations solved simultaneously for ~.,, %, e,. This is what is done in a three-precision-point design [1]. For four precision points one needs an additional design parameter. The initial position of the output link is used here as a fourth parameter. Let ~ = ~ks+ A~,,
(6)
where ~,~ is the initial position of the output link. Substituting for ~ in equation (la), the equation can be written in a pseudolinear form by defining a set of five variables, KI . . . . . K 5, in terms of the four design variables ~2, %, ~4, @~. And then writing the resulting equation for each of four precision points, one has 5
GtiKj=Fi,
i = 1. . . . .
4,
(7)
j=l
where
Gil ----sin 0q cos ~ i , Gi2 ---sin ~1 cos A@i, Gi3 ~ - sin ~i sin A~, i ,
(8)
Gi4 ~ ] ,
Gis - - sin ~i cos A@l-- cos ~1 cos ~blsin A@~ ; Fi = sin ~blsin A@:-- cos ~ 1 cos ~i cos A@i ;
(9)
136
and Kt -= cot :q/cos ¢,, K 2 = COt ~.~, K 3 -~ Cot z~a.tan ¢~, K~ =cos :q/(sin :~2 sin :~ cos ¢,) - ( c o s :q cot Zz cot :q)/cos ¢~,
(to)
K 5 = tan ¢~
Equation (7) constitutes a system of four equations in five unknowns. A fifth equation is needed. Since the unknowns, K t . . . . . Ks, are not independent, this additional equation ought to express their dependence. By inspection of equation (10),
KzKs-K3=O.
(11)
Summarizing, a system of five algebraic equations (equations (7) and (1 t)), all but one of which are linear, must be solved. A superposition technique is especially convenient here. Let ,;--Ks,
Hi-----Gis,
i=1 .....
4.
(12)
~ GijKi=2Hi+Fi,
i= I.... ,4.
(13)
Kj=2mi+n j,
j=l .... ,4.
(14)
Then equation (7) can be rewritten as 4
j=l
Now set
This is quite legitimate: no matter what values Kj and 2 in fact have, values of mj and n i can always be found to satisfy equation (14). Indeed either mj or n i could be selected arbitrarily in any of the equations. Substituting in equation (13),
2 ~ G&nj+ ~ Gunj=2Hi+F i, j=l
i=l,...,
4.
(15)
j=l
Since the values of either the mj which satisfy
mj or the ni can be selected at will, select those values for
4 j=l
Gi.imi=Hi,
i=1 .... ,4.
(16)
i= I .....
(17)
Then equations (15) and (16) together imply 4
G~jnj= F i ,
4.
j=l
For a given design problem the values of4q, A@i, i = 1. . . . . 4, and o f ~ , are known. Hence the values of Gu, Fi, and Hi, i = 1. . . . . 4 can be calculated using equations (8), (9) and (12). Equations (16) and (17) each comprise a system of four linear equations in four unknowns which can then be solved by some standard technique for linear equations.
137
Now, substituting ;. for
Ks
and 2 m i + n i for K i, j = 1 , . . . , 4, in equation (11), a;. 2 + b;. + c = 0 ,
(18)
where
a=--mz, b-n2-m c=
3,
- n3~.
(19)
In this context an equation such as equation (18) is referred to as a "compatibility equation ;" corresponding to each real root, if any, of this quadratic there is a solution to the design problem. For each real root ~. the design is found by the equations
(/ s = arc tan( K 3/ K 2) , :t~ = arc tan()./K3), ~2 = arc tan[1/(Kl cos @s)], c~3 = arc cos(sin ~2 sin ~4 cos ~k,K4 + cos cq cos ~tz cos ~4),
(20)
which are found readily from equations (10) and (12a). For any design at hand a mixing of branches can be detected by solving equation (4) for the branch indicator, e, at each precision point. The four values of e should be consistent. The accuracy of a feasible function generator can be determined by calculating the output angle ~ for a spectrum of values of ff~, the input angle, utilizing equation (4).
Summary of the Computational Algorithm* Given 1. The function y =y(x) is to be mechanized over the interval (x~, x/). 2. The shaft angle is ctI.
Designer's choices 1. Choose total angular travels, Aq~r and A~br for the input and output links. 2. Choose the initial position, tk,, of the input link. 3. Choose four precision points, xt, i = 1, 2, 3, 4, in the interval (xs, x:). Computations 1. Compute y, =y(xs), y : =y(x:), yz =y(x~), i= 1, 2, 3, 4. 2. Compute ~pi=qb,+(xz-xs)A~r/(xy-x,), i= 1, 2, 3, 4. 3. Compute A~i =(y~-y~)A~/r/(y:-y,), i= I, 2, 3, 4. 4. Compute Gu, Fi, Hi, i,j, =1, 2, 3, 4, using equations (8), (9) and (12b). 5. Solve equation (16) for the m i and equation (17) for the n2. 6. Find a, b, c by equation (19). 7. Find the roots of the compatibility quadratic, equation (18). If the quadratic has complex roots or is inconsistent, change one of the design assumptions (increment ~bs, perhaps) and restart. * A more detailed algorithm, in flow-chart form, is available on request from the author.
138
8. For each distinct real root of the compatibility equation there is a distinct design. For each distinct real root, therefore, compute ~,, ~_,, ~3, :~, using equation (20). 9. For each design check to be sure no mixing of branches has occurred. Find J/~ = ~ + A~q, i = l , 2, 3, 4. Using equation (4), find e at each precision point. If the e values are inconsistent, discard the design. If neither root yields a feasible design, alter one of the assumed parameters (say qSs) and restart. For each feasible design, survey structural error by the following steps. 10. Select a spectrum of values of x i, i = l to n, where n > 4 , in the interval (x,, x:). 11. Compute Yi =y(xl), i = 1 to n. 12. Compute qSi=c~,+(xi-%)AqSr/(X~-X:), i=1 to n. 13. Compute desired values of ~k, ~"s=~/~+(yl-y,)A~Or/(y:-ys), i = t to n. 14, Compute actual values of ~, ~9~ct, i = 1 to n, by using equation (2) repetitively. 15. Compute structural errors, Ei, Ei---~act--@ des, i = 1 to n. 16. Compute root mean square error,
17. Record results.
Sample Design Based on the detailed computational algorithm, the author wrote a F O R T R A N II program. Several hundred designs for a number of functions to be mechanized and for various assumptions for (k,, A~br, and A@T have been computed. Among them, the following is a typical design. Function: y=loglox Interval: (1,10) Shaft Angle (~q): 120 ° Precision Points (Chebyshev spacing) [3] : xl = 1"34254217 Xz = 3.77792460 x3 = 7.22207546 x4 = 9-65745783 Angular Ranges: A~br=75 ° A~kr=75 ° Input Link Initial Position: 4,=40 °
139 Computed Results: ~s-86.7319 o ~_, = 9 0 . 2 4 7 9 ° ~3 = 49.8636 ° ~,, = 77.9285 ° e--- - 1 ERMs=0"3176 °
References
[l]
I"IARTENBERGR. S. and DENAVlT J. Kinematic Synthesis of Linkages. McGraw-Hill (1964). [2] I-IARrENBEROR. S. and DENAvrr J. Analysis of spatial linkages by matrix methods. N.S.F. Report, Contract NSF-G 19704, Northwestern University, Evanston, Illinois (1963). [3] FREUDENSX~t~F. Structural error analysis in plane kinematic synthesis. J. Engng Ind. (1959).
Four-Precision-Point Synthesis of the Spherical Four-Bar Function Generator Prof. John R. Zimmerman* Received 28 November 1966 Abstract A design algorithm for the spherical four-bar mechanism used as a function generator with three precision points has been presented in detail in a recent text by Hartenberg and Denavit [1]. This brief paper presents an algorithm for a four-precision-point design.
Zusammenfassung Der Konstruktions-Algorithmus fLir ein sph~risches Funktions-Viergelenkgetriebe mit drei Pr~zisions-Punkten wurde von Hartenberg und Denavit ausfLihrlich dargestellt [1 ]. Die vorliegende Ver6ffentlichung gibt einen Algorithmus fOr den Entwurf mit vier Pr~izisions- Punkten. A6erpak"r I"[pOgKTHOOBOqHbI~tanropHTM ~it~ HQHMeHeHHfl Cd~)epHqeCKaFO MeTbIpex3BeHHI,IKa K
eocnpoM3Benenn0o ~yHKun~ C TpeMg roqnbtMH rO~r~aMH nonpo6Ho onncaHb~ XapreH6eproM a 12eHaaaro,~t [1]. HacTosmaa pa6ora zaer anroparM nna npoeKxrlposaHl4~t cdtacplt~lCCKrtXMe×aHrI3MOBC '-/erblbMIlTOtlHblMH TOtIKaMI4.
Displacement Equation THE MECHANISMis shown in Fig. 1, with orthogonal coordinate frames attached to each link in accordance with a scheme developed by Hartenberg and Denavit [2] for matrix analysis of spatial link mechanisms. In Fig. 1 the four revolute joints have been labeled R t , . . . , R4. The z-axes have been Z3
Figure 1. Sketch of the spherical four-bar mechanism with the notation of Hartenberg and Denavit. * Associate Professor of Mechanical Engineering, The Pennsylvania State University, University Park, Pennsylvania 16802, U.S.A.
133
134
placed on the axes of these revolutes, and with arbitrarily selected senses. The x-axes are such that axis x~ is mutally perpendicular to axes z~_t and z~. (The subscripts are treated cyclically: thus xt is perpendicular to z , and zt.) For the present problem, where all coordinate frames share a common origin, the senses of the x-axes are arbitrary. The y-axes are not shown since no measurements will be referred to them. It is understood, however, that each xi, yi, z~ frame is dextral. The angles are measured using the x- and z-axes. Angle 0~ is measured from the x~to the x~+ 1-axis. It is positive if counterclockwise when viewed from the positive z~-axis. Angle 0[~is from the z~- to the z~+ i-axis and is positive if counterclockwise when viewed from the positive x~+ t-axis. By inspection of Fig. 1, and recalling that the x2-axis is perpendicular to axes z t and z 2, which happen to be stationary, it is clear that frame 2 is the fixed one and on the ground link. Similarly, since axis xt is always perpendicular to axes zt and z,, frame 1 must be on the left-hand lever (or crank). Likewise frame 4 is on the coupler, and frame 3 on the righthand lever. In all that follows link 1 is regarded as the input link, whose position is measured as 0t, and link 3 (measured by 02) as the output link. The angular link lengths are cq . . . . . 0[4. Using the matrix methods [3] ofanalysis, it can be shown that the displacement equation for this mechanism, in terms of the notation of Fig. 1, and at the i-th mechanism position, is sin 0[t cot 0[2cos 0] + sin 0[t cot 0[, cos 022-~
COS0[ 3
sin 0[2 sin 0[,
cos 0[t cot 0[2cot 0[,
=sin 0~ sin 0 ~ - c o s cq cos 0~t cos0~
(i)
To prevent the notation getting entirely out of hand later, angles 0~ and 0~, the input and output angles, are henceforth written as ¢~ and ~ . Thus, sincq cot 0[2cos ¢~ + sin 0[t cotct, cos~q+ . cos~3 - cos 0[t cot 0[2cot 014 sln 0[ 2 sin 014 = sin ¢i sin ¢ ~ - cos cq cos ¢i cos $i.
(la)
Equation (la) can be solved explicitly for the output angle, ~k~. The result is ¢i = 2 arc tan._, +A v/(A~ + B~ + C,),? B~+ C~
(2)
where Ai -
- sin ¢i,
B~- sin 0[t cot 0[, + cos ctt cos ¢~, Ci-
cos cq cot 0[2cot :t, - sin cq cot 0[2 COS ~/~i
COS 0[3
sin ~2 sin ~z4"
(3)
Equation (2) is among the many evidences of the very similar behavior of the spherical and planar four-bars. The analogous equation for the planar four-bar can be put into the same general form. And just as the planar four-bar can be connected in an upper branch or in a lower branch configuration, the spherical four-bar can be connected in two ways, in a positive or in a negative branch configuration, as the ( + ) sign before the radical indicates.
135 For computational purposes it is convenient to rewrite equation (2) as .
.)
.
~i = - arc tan
Ai+e,J(A?+B?+C~)
(4)
Bi+Ci
where + i for the positive branch _ 1 for the negative branch
e=
The quantity e serves as a branch indicator. In designing four-bar function generators, planar or spherical, an all too familiar experience is to discover, for some design, that some precision point positions of the mechanism can be assumed by one branch, the others only by the other branch. The mathematics of design by precision point methods show a nice indifference to the physical realities. We are assured only that a given set of links, connected in a given sequence, can exist in each of the configurations corresponding to the several precision points. There is no certainty that the mechanism can in fact move from one position to the next. A design involving a mixing of branches ordinarily will be discarded.
Four Point Synthesis It is assumed that angle a t (the shaft angle) is dictated by functional requirements. The unknown link lengths are ~2, %, ~ . If three pairs of ~b~, Oi values were required, equation (la) could be written for each pair and the set of three equations solved simultaneously for ~.,, %, e,. This is what is done in a three-precision-point design [1]. For four precision points one needs an additional design parameter. The initial position of the output link is used here as a fourth parameter. Let ~ = ~ks+ A~,,
(6)
where ~,~ is the initial position of the output link. Substituting for ~ in equation (la), the equation can be written in a pseudolinear form by defining a set of five variables, KI . . . . . K 5, in terms of the four design variables ~2, %, ~4, @~. And then writing the resulting equation for each of four precision points, one has 5
GtiKj=Fi,
i = 1. . . . .
4,
(7)
j=l
where
Gil ----sin 0q cos ~ i , Gi2 ---sin ~1 cos A@i, Gi3 ~ - sin ~i sin A~, i ,
(8)
Gi4 ~ ] ,
Gis - - sin ~i cos A@l-- cos ~1 cos ~blsin A@~ ; Fi = sin ~blsin A@:-- cos ~ 1 cos ~i cos A@i ;
(9)
136
and Kt -= cot :q/cos ¢,, K 2 = COt ~.~, K 3 -~ Cot z~a.tan ¢~, K~ =cos :q/(sin :~2 sin :~ cos ¢,) - ( c o s :q cot Zz cot :q)/cos ¢~,
(to)
K 5 = tan ¢~
Equation (7) constitutes a system of four equations in five unknowns. A fifth equation is needed. Since the unknowns, K t . . . . . Ks, are not independent, this additional equation ought to express their dependence. By inspection of equation (10),
KzKs-K3=O.
(11)
Summarizing, a system of five algebraic equations (equations (7) and (1 t)), all but one of which are linear, must be solved. A superposition technique is especially convenient here. Let ,;--Ks,
Hi-----Gis,
i=1 .....
4.
(12)
~ GijKi=2Hi+Fi,
i= I.... ,4.
(13)
Kj=2mi+n j,
j=l .... ,4.
(14)
Then equation (7) can be rewritten as 4
j=l
Now set
This is quite legitimate: no matter what values Kj and 2 in fact have, values of mj and n i can always be found to satisfy equation (14). Indeed either mj or n i could be selected arbitrarily in any of the equations. Substituting in equation (13),
2 ~ G&nj+ ~ Gunj=2Hi+F i, j=l
i=l,...,
4.
(15)
j=l
Since the values of either the mj which satisfy
mj or the ni can be selected at will, select those values for
4 j=l
Gi.imi=Hi,
i=1 .... ,4.
(16)
i= I .....
(17)
Then equations (15) and (16) together imply 4
G~jnj= F i ,
4.
j=l
For a given design problem the values of4q, A@i, i = 1. . . . . 4, and o f ~ , are known. Hence the values of Gu, Fi, and Hi, i = 1. . . . . 4 can be calculated using equations (8), (9) and (12). Equations (16) and (17) each comprise a system of four linear equations in four unknowns which can then be solved by some standard technique for linear equations.
137
Now, substituting ;. for
Ks
and 2 m i + n i for K i, j = 1 , . . . , 4, in equation (11), a;. 2 + b;. + c = 0 ,
(18)
where
a=--mz, b-n2-m c=
3,
- n3~.
(19)
In this context an equation such as equation (18) is referred to as a "compatibility equation ;" corresponding to each real root, if any, of this quadratic there is a solution to the design problem. For each real root ~. the design is found by the equations
(/ s = arc tan( K 3/ K 2) , :t~ = arc tan()./K3), ~2 = arc tan[1/(Kl cos @s)], c~3 = arc cos(sin ~2 sin ~4 cos ~k,K4 + cos cq cos ~tz cos ~4),
(20)
which are found readily from equations (10) and (12a). For any design at hand a mixing of branches can be detected by solving equation (4) for the branch indicator, e, at each precision point. The four values of e should be consistent. The accuracy of a feasible function generator can be determined by calculating the output angle ~ for a spectrum of values of ff~, the input angle, utilizing equation (4).
Summary of the Computational Algorithm* Given 1. The function y =y(x) is to be mechanized over the interval (x~, x/). 2. The shaft angle is ctI.
Designer's choices 1. Choose total angular travels, Aq~r and A~br for the input and output links. 2. Choose the initial position, tk,, of the input link. 3. Choose four precision points, xt, i = 1, 2, 3, 4, in the interval (xs, x:). Computations 1. Compute y, =y(xs), y : =y(x:), yz =y(x~), i= 1, 2, 3, 4. 2. Compute ~pi=qb,+(xz-xs)A~r/(xy-x,), i= 1, 2, 3, 4. 3. Compute A~i =(y~-y~)A~/r/(y:-y,), i= I, 2, 3, 4. 4. Compute Gu, Fi, Hi, i,j, =1, 2, 3, 4, using equations (8), (9) and (12b). 5. Solve equation (16) for the m i and equation (17) for the n2. 6. Find a, b, c by equation (19). 7. Find the roots of the compatibility quadratic, equation (18). If the quadratic has complex roots or is inconsistent, change one of the design assumptions (increment ~bs, perhaps) and restart. * A more detailed algorithm, in flow-chart form, is available on request from the author.
138
8. For each distinct real root of the compatibility equation there is a distinct design. For each distinct real root, therefore, compute ~,, ~_,, ~3, :~, using equation (20). 9. For each design check to be sure no mixing of branches has occurred. Find J/~ = ~ + A~q, i = l , 2, 3, 4. Using equation (4), find e at each precision point. If the e values are inconsistent, discard the design. If neither root yields a feasible design, alter one of the assumed parameters (say qSs) and restart. For each feasible design, survey structural error by the following steps. 10. Select a spectrum of values of x i, i = l to n, where n > 4 , in the interval (x,, x:). 11. Compute Yi =y(xl), i = 1 to n. 12. Compute qSi=c~,+(xi-%)AqSr/(X~-X:), i=1 to n. 13. Compute desired values of ~k, ~"s=~/~+(yl-y,)A~Or/(y:-ys), i = t to n. 14, Compute actual values of ~, ~9~ct, i = 1 to n, by using equation (2) repetitively. 15. Compute structural errors, Ei, Ei---~act--@ des, i = 1 to n. 16. Compute root mean square error,
17. Record results.
Sample Design Based on the detailed computational algorithm, the author wrote a F O R T R A N II program. Several hundred designs for a number of functions to be mechanized and for various assumptions for (k,, A~br, and A@T have been computed. Among them, the following is a typical design. Function: y=loglox Interval: (1,10) Shaft Angle (~q): 120 ° Precision Points (Chebyshev spacing) [3] : xl = 1"34254217 Xz = 3.77792460 x3 = 7.22207546 x4 = 9-65745783 Angular Ranges: A~br=75 ° A~kr=75 ° Input Link Initial Position: 4,=40 °
139 Computed Results: ~s-86.7319 o ~_, = 9 0 . 2 4 7 9 ° ~3 = 49.8636 ° ~,, = 77.9285 ° e--- - 1 ERMs=0"3176 °
References
[l]
I"IARTENBERGR. S. and DENAVlT J. Kinematic Synthesis of Linkages. McGraw-Hill (1964). [2] I-IARrENBEROR. S. and DENAvrr J. Analysis of spatial linkages by matrix methods. N.S.F. Report, Contract NSF-G 19704, Northwestern University, Evanston, Illinois (1963). [3] FREUDENSX~t~F. Structural error analysis in plane kinematic synthesis. J. Engng Ind. (1959).