Curvature theory for a twodegreeoffreedom planar linkage

1、available online at ,mechanismsciencedirectandmachine theorymechanism and machine theory xxx (2007) xxxxxxcurvature theory for a two-degree-of-freedom planar linkageg.r. pennockschool of mechanical engineering, purdue university, west lafayette, in 47907-2088, usa received 23 october 2006; accepted

2、26 march 2007abstractthis paper shows that the method of kinematic coefficients can be applied in a straightforward manner to the kinematic analysis of mechanisms with more than one input. for illustrative purposes, the paper presents an example of a two-input linkage or two-degree-of-freedom linkag

3、e; namely, the well-known planar five-bar linkage. the two inputs are the side links of the five-bar linkage which are assumed to be cranks. since the linkage is operated by two driving cranks independently then the linkage can produce a wide variety of motions for the two coupler links. the kinemat

4、ic coefficients are partial derivatives of the two coupler links with respect to the two input crank angles and separate the geometric effects of the mechanism from the operating speeds. as such they provide geometric insight into the kinematic analysis of a mechanism. a practical application of the

5、 five-bar linkage is to position the end-effector of an industrial robotic manipulator, for example, the general electric model p50 robotic manipulator. the paper then presents closed-form expressions for the radius of curvature and the center of curvature of an arbitrary coupler curve during the co

6、mplete operating cycle of the linkage. the analytical equations that are developed in the paper can be incorporated, in a straightforward manner, into a spreadsheet that is oriented towards the path curvature of a multi-degree-of-freedom linkage. the author hopes that, based on the results presented

7、 here, a variety of useful tools for the kinematic design of planar multi-degree-of-freedom mechanisms will be developed for planar curve generation. 2007 elsevier ltd. all rights reserved.keywords: kinematic coefficients; planar five-bar linkage; kinematics of a coupler point; geometry of a coupler

8、 curve; curvature theory; instantaneous centers of zero velocity; finite difference1. introductionthe method of kinematic coefficients provides a concise description of the geometric properties of a planar linkage 1,2. the method has been applied to the kinematic analysis of a wide variety of single

9、-degree-of-free-dom planar mechanisms; for example, a variable-stroke engine 3, a planar eight-bar linkage 4, and two cooperating robots manipulating a payload 5. the velocity problems of two robots manipulating a rigid pay-load have also been investigated using the method of kinematic coefficients

10、68. insight into the geometric properties of point trajectories in planar kinematics can also be obtained from this method. for example, the effects of the generating pin size and placement on the curvature and displacement of epitrochoidale-mail address: 0094-114x/s - see front

11、 matter 2007 elsevier ltd. all rights reserved. doi:10.1016/j.mechmachtheory.2007.03.012please cite this article in press as: g.r. pennock, curvature theory for a two-degree-of-freedom planar linkage, mech. mach. theory (2007), doi:10.1016/j.mechmachtheory.2007.03.012article in press2g.r. pennock/ m

12、echanism and machine theory xxx (2007) xxxxxxgerotors 9 and the path curvature of a single-degree-of-freedom geared seven-bar mechanism 10 were obtained from this technique. the method was also applied to the force analysis of the apex seals in the wan-kel rotary compressor 11. more recently, the me

13、thod of kinematic coefficients has also been used to provide insight into the duality between the kinematics of gear trains and the statics of beam systems 12.this paper will show that the method of kinematic coefficients can be extended in a straightforward manner to mechanisms with more than one i

14、nput (that is, multiple-degree-of-freedom mechanisms). closed-form expressions for the kinematic coefficients of the two-degree-of-freedom mechanism and the radius of curvature of the path traced by a coupler point will also be presented. these expressions are most useful in developing a systematic

15、procedure in the kinematic design of planar mechanisms. for illustrative purposes, the paper will focus on the kinematics of a two-input linkage or two-degree-of-freedom linkage; namely, the well-known planar five-bar linkage. the two inputs are the side links of the five-bar linkage which are pinne

16、d to the ground and are assumed to be cranks. the kinematic coefficients are partial derivatives of the two input crank angles and separate the geometric effects from the operating speeds of the mechanism. as such they provide geometric insight into the kinematic analysis of the mechanism. since the

17、 linkage is operated by two driving cranks independently then the linkage can produce a wide variety of motions for the two coupler links.the paper also presents two graphical techniques to check the first-order kinematic coefficients and the angular velocities of the two coupler links. the two tech

18、niques are based on locating the instantaneous centers of zero velocity (henceforth referred to as instant centers) of the linkage in the given configuration. the first technique will use the ratio of the two known input angular velocities and the second technique will use the method of superpositio

19、n. the concept of an instant center for two rigid bodies in planar motion was presented by johann bernoulli 13 and extended by chasles to include general spatial motion using the instantaneous screw axis 14. the importance of an instant center for two rigid bodies in planar motion is well known 15,1

20、6. instant centers are useful for determining both the velocity distribution in a given link and the motion transmission between links. they are also helpful in the kinematic analysis of mechanisms containing higher pairs, for example, gear trains and cam mechanisms 17. the method has proved to be v

21、ery efficient in finding the inputoutput velocity relationships of complex linkages 18. when combined with the conservation of energy, instant centers also provide an efficient method to obtain the inputoutput force or torque relationships. also, instant centers make a significant contribution to ou

22、r understanding of the kinematics of planar motion 19. for example, the loci of the instant centers fixed to the ground link and the coupler link of a planar linkage define the fixed and the moving centrodes, respectively, which are important in a study of path curvature theory 20. a graphical metho

23、d to locate the secondary (or unknown) instant centers for single-degree-of-freedom indeterminate linkages, such as the double flier and the single flier eight-bar linkages, was presented by foster and pennock 21,22. this paper also includes the finite difference method to check the first-order kine

24、matic coefficients of the two coupler links of the five-bar linkage. however, the primary reason for introducing this method is to check the second-order kinematic coefficients of the two coupler links.the paper is arranged as follows. section 2 presents a kinematic analysis of a two-input, or two-d

25、egree-of-freedom, mechanism; i.e., the well-known planar five-bar linkage. this linkage is operated by two driving cranks independently and can produce a wide variety of motions for the two coupler links. a practical application of the planar five-bar linkage is to position the end-effector of an in

26、dustrial robotic manipulator, for example, the general electric model p50 manipulator 2. section 3 presents a procedure to determine the radius of curvature and the center of curvature of the path traced by a coupler point. then section 4 presents a numerical example to illustrate the systematic pro

27、cedure and highlights several techniques to check the computation and the results. finally, section 5 presents several important conclusions and some suggestions for future research.2. kinematic analysis of the five-bar linkagea schematic drawing of the planar five-bar linkage is shown in fig. 1. th

28、e linkage is operated by driving cranks 2 and 3 independently and can produce a wide variety of motions for the coupler links 4 and 5. if the input cranks are replaced by the pitch circles of two gears which have rolling contact then the resulting mechanism is commonly referred to as a geared five-b

29、ar linkage 6. this is a single-degree-of-freedom linkage which can provide more complex motions than the well-known planar four-bar linkage. the vectors for aplease cite this article in press as: g.r. pennock, curvature theory for a two-degree-of-freedom planar linkage, mech. mach. theory (2007), do

30、i:10.1016/j.mechmachtheory.2007.03.012article in pressg.r. pennock / mechanism and machine theory xxx (2007) xxxxxx3fig. 1. the planar five-bar linkage and the corresponding vectors.kinematic analysis of the linkage are also shown in fig. 1. the first-order and the second-order kinematic coefficient

31、s of the five-bar linkage can be obtained from the loop-closure equation for the linkage which can be written asvvr2 +f4 -r50(2-1)where 62 and 63 are the input angles, denoted by the symbol i in eq. (2.1), and the coupler angles 64 and 65 are unknown angular variables. the time rate of change of the

32、 dependent variable 64, that is, the angular velocity of the coupler link 4 can be written as4 = d94/dt = (064/062)(d62/d) + (004/063) dd/dt)2.2a)where the coupler angle 64 is a function of the two independent variables 62 and 63. similarly, the time rate of change of the dependent variable 65; that

33、 is, the angular velocity of the coupler link 5 can be written as5 = d9s/dt = (065/062) (d62/dt) + (065/063) (dbi/dt)2.2b)where the coupler angle 65 is also a function of the two independent variables 62 and 63. eq. (2.2) can be written as(04 = 64 = 662 + 64363(2.3a)anda 5 = 65 = 05292 + 65363(2.3b)

34、where the prime notation denotes the first-order kinematic coefficients; that is, the derivatives of the angular velocities of the links with respect to the input angular velocities, 6-. = wi/wj, as follows:642 = 064/062, 643 = 064/063, 652 = 065/062, and 653 = 065/063(2-4)differentiating eq. (2.2)

35、with respect to time, the angular accelerations of the coupler links can be written asa4 = d264/d?2 = (064/062)(d262/d2) + (064/063)(d263/d?2) + (0264/062)(d62/d?)+ 2(0264/062063)(d62/do(d63/do(0264/062)(d63/do2(2.5a)anda5 = d 65/dt2 = (065/062)(d 92/dt2) + (065/063)(d d/dt2) 4-+ 2(0265/062063)(d62/

36、do(d63/do + (0265/062)(d63/do:(0265/062)(d62/do2(2.5b)please cite this article in press as: g.r. pennock, curvature theory for a two-degree-of-freedom planar linkage, mech. mach. theory (2007), doi:10.1016/j.mechmachtheory.2007.03.012article in press4g.r. pennock / mechanism and machine theory xxx (

37、2007) xxxxxxusing the prime and the dot notation, eq. (2.5) can be written as94292 + 04303 + 04222 + 204230203 + 333a4 = 04 = 04202 + 04303 + 042202 + 2002 #3 + 04333(2.6a)anda5 = 05 = 05202 + 05303 + 052202 + 29239293 + 053303(2.6b)where0422 = 0204/002, 0423 = 0204/002003, 0433 = 0204/002(2.7a)0522

38、 = 9205/902, 0523 = q295/d92d93, and 0533 = 0205/002(2-7b)are the second-order kinematic coefficients of the coupler links relating the angular accelerations of the these links to the two input angular velocities and accelerations.the first-order and second-order kinematic coefficients of the five-b

39、ar linkage can be obtained from the following procedure. write the scalar position equations of eq. (2.1), and solve for the unknown position variables 04 and 05; that is,r2 cos 02 + r4 cos 04 r5 cos 05 r3 cos 03 r1 = 0(2.8a)andr2 sin 02 + r4 sin 04 r5 sin 05 r3 sin 03 = 0(2.8b)then differentiate eq

40、. (2.8) partially with respect to input position 02; that is,r2 sin 02 r4 sin 04042 + r5 sin 05 052 = 0(2.9a)andr2 cos 02 + t4 cos 04042 ?5 cos 05052 = 0(2-9b)solve eq. (2.9) for the first-order kinematic coefficients 042 and 052. also, differentiate eq. (2.8) partially with respect to input positio

41、n 03; that is,r4 sin 04043 + r5 sin 05053 + r3 sin 03 = 0(2.10a)andr4 cos 04043 r5 cos 05053 r3 cos 03 = 0(2.10b)solve eqs. (2.10) for the first-order kinematic coefficients 043 and 053. next differentiate eqs. (2.9) partially with respect to the input position 02; that is,r2 cos 02 t4 sin 040422 4

42、cos 04042 + ?5 sin 050522 + r5 cos 05052 = 0(2.11a)andr2 sin 02 + t4 cos 040422 r4 sin 04042 5 cos 050522 + r5 sin 05052 = 0(2.11b)solve eqs. (2.11) for the second-order kinematic coefficients 9422 and 0522. also, differentiate eqs. (2.9) partially with respect to the input position 03; that is,t4 s

43、in 646423 r4 cos 0404243 + r5 sin 5523 + r5 cos 55253 = 0(2.12a)andr4 cos 040423 r4 sin 44243 r5 cos 5523 + r5 sin 55253 = 0(2.12b)solve eqs. (2.12) for the second-order kinematic coefficients 9423 and 9523. note that differentiating eqs. (2.10) partially with respect to the input position 02 will g

44、ive the same result. finally, differentiate eqs. (2.10) partially with respect to the input position 03; that is,t4 sin 040433 r4 cos 04043 + r5 sin 050533 + r5 cos 05053 + r3 cos 03 = 0(2.13a)please cite this article in press as: g.r. pennock, curvature theory for a two-degree-of-freedom planar lin

45、kage, mech. mach. theory (2007), doi:10.1016/j.mechmachtheory.2007.03.012article in pressg.r. pennock / mechanism and machine theory xxx (2007) xxxxxx5andr4 cos 64600 r4 sin 6460 r5 cos 95900 + r5 sin 6500 + r3 sin 63 = 0(2.13b)finally, solve eqs. (2.13) for the second-order kinematic coefficients 6

46、433 and 6533.in the interest of computational efficiency, the first-order and second-order kinematic coefficients can be determined by writing eqs. (2.9) and (2.10) in the forma1d42 + b1652 = c12, a2 642 + 2652 = c22, a1643 + b1643 = e13, and 02643 + 62643 = c23 also, eqs. (2.11)(2.13) can be written in the formfl14022 + 15022 = 12) a204022 + b29 05022 = 3(3.7a)or asvp = (xp2(02 + xp3co3)i + (yp2(o2 + yp3(o3)(3.7b)where xp2,xp3, yp2, and yp3 are referred to as the first-order kinematic c