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1、Introduction 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 Bengal, IndiaAbstractBasic objective of synthesis of cam displacement functions is minimizing the acceleratio
2、n and jerk of the cam-follower especially in high-speed drives. Here classical splines of 6-, 7- and 8-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 p
3、oint parameters. A searching procedure is adopted, based on GA and fuzzy membership function. It establishes that introduction of control points largely minimizes acceleration and jerk. 2008 Elsevier Ltd. All rights reserved.Keywords: Classical splines; B-splines; Control points; Optimization; Genet
4、ic algorithm1. IntroductionOne of the basic objectives of designing cam motion program is to minimize the 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 i
5、n lowering AP and JP of the follower 1-8. Higher order polynomials are combined piecewise for constructing 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
6、ends, called knots, in such a way that the curve along with its derivatives, up to and including the derivative 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
7、 be continuous. The velocity, acceleration and jerk will be continuous but the fourth derivative, the ping, 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 vari
8、ed rather arbitrarily. Also no general method for varying the APs was proposed. An attempt was made to present a general method for manipulating the knot parameters of 6-order classical splines that yielded satisfactory. results 9. In 9, the intermediate knots were called the control points (CP) and
9、 were characterized by two parameters - AP and FD.In this work classical 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
10、 so interrelated that lowering of peak value of one causes rise to that 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 fundament
11、al principle 11 that guides the synthesis of cam displacement functions. 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 e
12、ntire interval.This means that every cam function must have third order 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 displ
13、acement function.A classical spline of order 6 conform the said fundamental 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 interme
14、diate knots are CPs 9.Following specifications are considered for the 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 velocit
15、y of the camshaft is taken as constant. Two CPs are introduced at Aps. equal 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.The
16、 displacement equations of the above four polynomial pieces are 11. here are 24 unknown coefficients like a1,. . .,a4, b1,. . . ,b4, c1,. . .,c4, d1,. . .,d4, e1,. . .,e4, f1,. . .,f4 necessitating 24 equations for solution. Successive derivations of the Eqs. (1)(4) give sets of equations for veloci
17、ty, acceleration, jerk and ping. Fifteen smoothness equations, three interpolation equations, six boundary condition equations, i.e. a total of 24 equations 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
18、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 multidimensional space. A good overview of the many practical applications of the GA is found in 13. The algorithm starts from an initial set of candidate indi
19、viduals called the initial population and, using genetic 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
20、the complexity of the problem, the algorithm finds the 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
21、second objective function f2(x) is ignored. The optimization 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
22、) and the design parameters contained in vector X1 are 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
23、 objective function to obtain f1maxx). These values denoted as /1min,/2max,/2min and fmax, respectively, are used to form the global objective function.The membership function is expressed in general terms as From 15 it is observed that the search space is concave in nature. This is obvious from the
24、 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 set of Eq. (5) the followi
25、ng membership function is formed: Then minimizing Ui(x) in Eq. (7) is equivalent to maximizing lfix. Let the global objective F(x) take the form The 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 pr
26、oblem Let Eq. (9) represents a straight line with gradient A2/A1,which passes through the point (0,h1) in thef1(x),f2(x) coordinate plane. 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 lab
27、eled 1 in Fig. 3 moves parallel towards the optimal solution. In this optimization process, the line will pass through the point/*, if the 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
28、 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 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 to Eq. (11
29、) 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 point to be found.5.Case s
30、tudyFor 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 118.5537 units for acceleration
31、 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 the KPs by manipulating
32、 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 described later.6.1. Optimizati
33、on 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 fixed y of 45, the accelera
34、tion 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) shows that for a fixed
35、 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). 6.2. Optimization of jerkFor obtaining the optimum JP the same procedure is followed, i.e. first, the FD at AP = y and 3y, is varied to fi
36、nd out for what value of the FD the JP becomes the lowest (Fig. 5(1). The value of the jerk obtained is 286.335, which is far lower than previously found value of 587.808 (Table 2). The corresponding AP is obtained as 124.633. Secondly, for a fixed FD, the AP is varied to find out for what value of
37、the AP the jerk attains the lowest mark (Fig. 5(2). The value of the jerk and the corresponding value of the acceleration are 295.026 and 128.003, respectively.6.3. Optimization of acceleration and jerk by manipulating both the CP characteristicsBoth acceleration and jerk can be optimized independen
38、tly by controlling simultaneously the values of the FD and the AP. The optimum values of the KPs thus obtained are found to be the lowest of all values foundpreviously (Tables 2 and 3). In this process, the minimum acceleration is obtained as 82.356 (/1min) with corresponding minimum jerk value of 5
39、11.333 (fmax). The minimum jerk is obtained as 255.186 (fmin) with corresponding minimum acceleration value of 115.031 (/1max).For global optimization the GA can be applied effectively with much less effort. For the examples presented in this paper, the problem can be completely defined by Eq. (11),
40、 where Xe AP, FD and X is the weighting co-efficient. Assuming objective function F(X) = 1, 1 , the minimization problem can be converted to maxi-mization problem.7. ResultAssuming values f1min and f2min lower than the previously assumed one and taking k equals to 0 and 1 the optimization of Ap and
41、Jp is done separately. The results fully tally with previous results and the searching process is much faster than the earlier one. The optimization process is performed assuming following values:Limits of the variable is AP = 0 90 and FD = 0 40; crossover probability (pcross) is 0.8; mutation proba
42、bility (pmute) is 0.1; initial population size (Ipopsize) is 20; and max number of function evaluation (maxeval) is 10,000.Optimized values of the acceleration and jerk obtained by different approaches are shown in Tables 2 and 3, respectively. Table 2 shows how both acceleration and jerk have their
43、 peak values lowered by different methods of optimization while Table 3 shows the sharp decrease of the JP. When both the parameters of the CPs are manipulated we obtain the most acceptable results.It is seen that for all cases AP is also lowered when optimization of acceleration is done (Table 2).
44、Here, the lowest value is 82.2814 in case of 7-order spline that is 87.44% of the value obtained by IC (94.1046). Similarly, Jps are gradually lowered and the lowest value is 255.186 in case of 6-order spline (Table 3), which is 36% of the value obtained by IC (707.691). Though when jerk is optimize
45、d, acceleration becomes higher than the value obtained by IC (Table 3).To minimize AP, only 6- and 7-order splines produce almost equal result. But if the corresponding JP is noticed the 8-order spline performs much better without much deviation from the least AP in the entire search space. To minim
46、ize JP, only 6-order spline produces the lowest value but we can also use 7-order spline if lowering of the corresponding Ap will be the objective.A map comprising of optimum AP and JP is drawn (Fig. 6). Along each line AP is constant but FD varies. Shifting from one line to another varies the AP. T
47、he single arrow indicates the variation of the FD while the double arrow the variation of the AP. A denotes the point of optimum AP while C denotes the point of optimum JP. B denotes a point, where both AP and JP are optimized.Analyzing the Fig. 6 it can be said that the global objective function is
48、 not a convex one but concave in nature. For that reason the desired point cannot be reached, where both of the two components of global objective function are minimum. After giving different weight on two separate objectives by changing the value of k, the different combination according to our req
49、uirement can be reached. The results are shown in Table 4.8. Synthesis of displacement function by B-splineCPs are introduced similarly in B-spline cam displacement functions. Case studies are done for 6- and 8-orders B-splines. Fundamental principle 11 of cam function design states that jerk should be finite across the entire interval of cam rotation and it is called the jerk finite function 17. The fundam
