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1、Available online at Mechanism# ScienceDirectandMachine TheoryMechanism and Machine Theory xxx (2008) xxxxxxIntroduction of control points in splines for synthesis of optimized cam motion programM. Mandal, T.K. Naskar*Department of Mechanical Engineering, Jadavpur University, Kolkata 700 032, West Be
2、ngal, India Received 8 July 2007; received in revised form 29 December 2007; accepted 22 January 2008AbstractBasic objective of synthesis of cam displacement functions is minimizing the acceleration and jerk of the cam-follower especially in high-speed drives. Here classical splines of 6-, 7- and 8-
3、orders and B-splines of 6- and 8-orders are taken for designing cam displacement functions. Multiple control points are introduced. Acceleration and jerk are minimized by manipulating the control point parameters. A searching procedure is adopted, based on GA and fuzzy membership function. It establ
4、ishes that introduction of control points largely minimizes acceleration and jerk. 2008 Elsevier Ltd. All rights reserved.Keywords: Classical splines; B-splines; Control points; Optimization; Genetic algorithm1. IntroductionOne of the basic objectives of designing cam motion program is to minimize t
5、he kinematics parameters like AP and JP of the follower for smooth and noiseless drive, especially in high-speed machines. Polynomial splines used as cam displacement functions yield good results in lowering AP and JP of the follower 1-8. Higher order polynomials are combined piecewise for construct
6、ing splines with the objective of designing cam displacement functions. A classical spline of order m is a curve consisting of polynomial pieces, each of degree m1, that are tied together at their ends, called knots, in such a way that the curve along with its derivatives, up to and including the de
7、rivative of order m 2, is continuous. For example, if a classical spline of order 5 is used to represent a cam curve, then the displacement will be made up of polynomial pieces of degree 4 and will be continuous. The velocity, acceleration and jerk will be continuous but the fourth derivative, the p
8、ing, will not be continuous. The classical spline always interpolates prescribed values at knots. Many works were done by manipulating the knots in B-splines 10. In these works, the knots were varied rather arbitrarily. Also no general method for varying the APs was proposed. An attempt was made to
9、present a general method for manipulating the knot parameters of 6-order classical splines that yielded satisfactoryCorresponding author. Tel.: +91 33 24146116; fax: +91 33 24146890. E-mail address: tknaskarmech.jdvu.ac.in (T.K. Naskar).0094-114X/S - see front matter 2008 Elsevier Ltd. All rights re
10、served. doi:10.1016/j.mechmachtheory.2008.01.005Please cite this article in press as: M. Mandal, T.K. Naskar, Introduction of control points in splines for synthesis Mech. Mach. Theory (2008), doi:10.1016/j.mechmachtheory.2008.01.005ARTICLE IN PRESS2M. Mandal, T.K. Naskar/Mechanism and Machine Theor
11、y xxx (2008) xxxxxxNomenclaturesfollower displacementAfollower accelerationPfollower ping/Ppeak value of follower jerkJejerk at end position9cam rotation anglehtotal rise or fall of the followerCPcontrol pointFDfollower displacement at CPsVfollower velocityJfollower jerkAPpeak value of follower acce
12、lerationPPpeak value of follower pingPeping at end positionmorder of splineytotal angle of any segmentAPangular position of CPsTPBtruncated power basisresults 9. In 9, the intermediate knots were called the control points (CP) and were characterized by two parameters - AP and FD.In this work classic
13、al splines of orders 6, 7 and 8 are taken; AP and /P are minimized by manipulating the values of AP and FD of each CP; and a comparative study is done on the results thus obtained. It is observed that the acceleration and jerk are so interrelated that lowering of peak value of one causes rise to tha
14、t of the other. In addition to classical splines of orders 6, 7 and 8, CPs are also introduced in 6- and 8-order fi-splines for the same objective.2. Synthesis of cam displacement functions by classical splinesThere is a fundamental principle 11 that guides the synthesis of cam displacement function
15、s. The fundamental principle states:(1) A displacement function must be continuous through the first and second derivatives (i.e. velocity and acceleration) across the entire cycle.(2) The jerk function must be finite across the entire interval.This means that every cam function must have third orde
16、r continuity (function plus two derivatives) at all boundaries. That is, if velocity and acceleration curves are continuous and jerk function gives finite values across the interval, it would be considered a satisfactory cam displacement function.A classical spline of order 6 conform the said fundam
17、ental principles and a number of such splines can be blended together at knots to get a desired cam function as shown in Fig. 1. Here three splines are joined together at two intermediate knots to get a smooth curve. These intermediate knots are CPs 9.Following specifications are considered for the
18、synthesis of a single dwell cam displacement function with 6-order classical splines:Rise h in a cam rotation angle of 2y, fall h in next 2y, dwell at zero displacement for remaining 2(p 2y) cam rotation angle; the angular velocity of the camshaft is taken as constant. Two CPs are introduced at APsP
19、lease cite this article in press as: M. Mandal, T.K. Naskar, Introduction of control points in splines for synthesis Mech. Mach. Theory (2008), doi:10.1016/j.mechmachtheory.2008.01.005ARTICLE IN PRESSM. Mandal, T.K. Naskar I Mechanism and Machine Theory xxx (2008) xxx-xxx 3 SPLINES CONNECTED BY KNOT
20、SCam Rotation Angle.9Fig. 1. Cam displacement diagram: splines and knots.3equal to c and 3c with corresponding FDs of h/2 each; two end knots at cam rotation angles of 0 and 4c with FD of 0 each. These are stated in Table 1. It needs four polynomial pieces for the segments of 0c, c2c, 2c3c and 3c4c.
21、The displacement equations of the above four polynomial pieces are 112e32+f3+f45*1 = ct1S2 = 2S3 a35*4 = fl456-y6-2y6-3yy5+ b355+ /1+ e1n 4( ft 3(ft3246-yy6-yy6-yyc2dS+f2y6-3yyy0-3yyy6-3yyy432e4c46-3yy(1)(2) (3) (4)There are 24 unknown coefficients like a1,. . .,a4, b1,. . . ,b4, c1,. . .,c4, d1,. .
22、 .,d4, e1,. . .,e4, f1,. . .,f4 necessitating 24 equations for solution. Successive derivations of the Eqs. (1)(4) give sets of equations for velocity, acceleration, jerk and ping. Fifteen smoothness equations, three interpolation equations, six boundary condition equations, i.e. a total of 24 equat
23、ions are obtained as described in 11. Splines of orders 7 and 8 are considered for analysis. For the latter the number of unknown coefficients will be 28 and 32, respectively.3. OptimizationFor optimization, GA 12 is adopted here since it is an efficient way to search a highly non-linear multidimens
24、ional space. A good overview of the many practical applications of the GA is found in 13. TheTable 1Specifications of single dwell cam displacement function by 6-order splineCam rotation angleFDV72y 3y 4y0h/2 hh/2 0Please cite this article in press as: M. Mandal, T.K. Naskar, Introduction of control
25、 points in splines for synthesis ., Mech. Mach. Theory (2008), doi:10.1016/j.mechmachtheory.2008.01.005ARTICLE IN PRESS4M. Mandal, T.K. Naskar I Mechanism and Machine Theory xxx (2008) xxx-xxxalgorithm starts from an initial set of candidate individuals called the initial population and, using genet
26、ic operators - crossover, mutation, selection - which try to mimic natural selection laws, simulate the biological evolution producing new populations with better individuals at each iterative step. After a number of iterations, which depends on the complexity of the problem, the algorithm finds the
27、 optimal solution to the problem as the best fit individual 12. Fig. 2 illustrates the steps of a simple GA 14.4. Objective function 15,16As in the hierarchical optimization method, only one objective function f1(x), is first optimized while the second objective function f2(x) is ignored. The optimi
28、zation is carried out taking into account the constraints and using standard methods such as a random search and variable metric combination. The optimal value, referred to as the ideal value for this objective function, is represented by/1min(x1) and the design parameters contained in vector X1 are
29、 referred to as a fuzzy set. This set is then substituted into the second objective functiona2(), to obtain /2max(x1). The second objective is optimized to get its ideal value, /2min(x). The fuzzy set belonging to x2 is substituted into the first objective function to obtain f1maxx). These values de
30、noted as /1min,/2max,/2min and fmax, respectively, are used to form the global objective function.The membership function is expressed in general terms asHf.x) =fi(x)-afi(x)if/,(x) fmin(5)aLax /minif fmin *5 fi to fmaxwhere i = 1, 2.From 15 it is observed that the search space is concave in nature.
31、This is obvious from the ApJp map shown in Fig. 6. Since Ap decreases while Jp increases and vice versa, Ap and Jp cannot be minimum simultaneously. That is why the search space for optimum point is considered to be in between fimin and fimax, where f1 stands for Ap and f2 stands for Jp. From the se
32、t of Eq. (5) the following membership function is formed:/min J i1/to/max /mfor fimin fix c, F(x) 0. In that case/1W/1min and f2Wf2minThe problem is how to select a path which leads to the minimization of bothf1(x) and/2(M simultaneously. The method proposed below is used to solve this problem LetA
33、1 = fmax fmin, A 2 = f2max fminMultiplying Eq. (8) by A1A2 and rearranging, the following equation can be obtained:2(9)f2ix) = -f1(x)+h1wheremiA2 rm(10)1 = J2 + (b c)A2 -/1A1Eq. (9) represents a straight line with gradient A2/A1,which passes through the point (0,h1) in thef1(x),f2(x) coordinate plan
34、e. The objective of optimization is to find an x* which makes the global objective F(x) 0, i.e. b c. From Eq. (9), it can be seen that as b c, the straight line labeled 1 in Fig. 3 moves parallel towards the optimal solution. In this optimization process, the line will pass through the point/*, if t
35、he two-objective function is convex. This will produce the optimal solution since the ideal point f1min, /2min does not lie in the feasible solution space.A further improvement can be added to improve the search for the optimum point if the search space is concave in nature. By adding a multiple to
36、the second term in the global objective function F(x), the search path can be changed to find a better common optimal point. Thus Eq. (5) can be modified toF(x) = (1 X) minJ 2 + x/max fm7H ) J1J2 )fmax minb c(11)andf2ix) = - , f1 (x) + h2A A1(12)wherer mmr/lJ Fig. 3. Two-objective optimization path.
37、Please cite this article in press as: M. Mandal, T.K. Naskar, Introduction of control points in splines for synthesis ., Mech. Mach. Theory (2008), doi:10.1016/j.mechmachtheory.2008.01.005ARTICLE IN PRESS6M. Mandal, T.K. Naskar/Mechanism and Machine Theory xxx (2008) xxxxxx min + (b(13)2 = f2(1 X) A
38、2 m c)A2 -:f1A A1Eq. (11) means that the gradient of the line can be changed. The optimization path can now be searched using a line, which can be translated and rotated by changing the values of the constants b, c and X. This can enable the point on the curve that is closest to the ideal optimum po
39、int to be found.5.Case studyFor y = 45 and h = 40 unit (initial choice) the above curves obtained are shown in Fig. 4. The AP and /P for these values are 94.515 units and 707.690 units, respectively, for 6-order classical spline. In case of 7- and 8-order splines those values are 94.1046 and 118.553
40、7 units for acceleration and 857.3946 units and 475.0003 units for jerk, respectively. The values obtained by optimization program are compared with these initial choice (IC) values. The term unit is omitted for all future cases.6.Optimization of acceleration and jerkAttempt is made to optimize both
41、 the KPs by manipulating the CPs. In this process one of the two characteristics of the CP - the FD and the AP - is varied keeping the other one fixed, and lastly both are varied. The process is described here for 6-order spline and the comparative results among 6, 7 and 8-orders splines are describ
42、ed later.6.1. Optimization of accelerationThis single variable optimization can be performed in several ways. Here, simple numerical search method is followed. First, the FD at AP = y and 3y is varied to find out for what value of the FD the AP reaches the lowest mark. Fig. 5(1) shows that for a fix
43、ed y of 45, the acceleration has the minimum value of 85.178 when FD becomes 18.306. The corresponding /P is 587.808. Both of these values are lower than their initial values.Secondly, for a fixed FD, the AP is varied to find out for what value of the AP the acceleration becomes the lowest. Fig. 5(2
44、) shows that for a fixed FD of 20, AP has the lowest value of 84.916 when AP becomes 47.29. The corresponding /P is 576.898. Both these values are, therefore, further lowered (Table 2).Fig. 4. Case study: SVAJP vs. h.Please cite this article in press as: M. Mandal, T.K. Naskar, Introduction of contr
45、ol points in splines for synthesis Mech. Mach. Theory (2008), doi:10.1016/j.mechmachtheory.2008.01.005ARTICLE IN PRESSM. Mandal, T.K. Naskar/Mechanism and Machine Theory xxx (2008) xxxxxx7Fig. 5. Optimized AP and Jp for 6-order spline: (1) FD varying while AP = 45 and (2) AP varying while FD = 20.6.2. Optimization of jerkFor obtaining the optimum JP the same
