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1、Curvature theory for a two-degree-of-freedom planar linkageG.R. PennockSchool of Mechanical Engineering, Purdue University, West Lafayette, IN 47907-2088, USAAbstractThis paper shows that the method of kinematic coefficients can be applied in a straightforward manner to the kinematic analysis of mec
2、hanisms with more than one input. For illustrative purposes, the paper presents an example 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 assumed to be cranks. Since the link
3、age is operated by two driving cranks independently then the linkage can produce a wide variety of motions for the two coupler links. The kinematic coefficients are partial derivatives of the two coupler links with respect to the two input crank angles and separate the geometric effects of the mecha
4、nism from the operating speeds. As such they provide geometric insight into the kinematic analysis of a mechanism. A practical application of the five-bar linkage is to position the end-effector of an industrial robotic manipulator, for example, the General Electric model P50 robotic manipulator. Th
5、e paper then presents closed-form expressions for the radius of curvature and the center of curvature of an arbitrary coupler curve during the complete operating cycle of the linkage. The analytical equations that are developed in the paper can be incorporated, in a straightforward manner, into a sp
6、readsheet that is oriented towards the path curvature of a multi-degree-of-freedom linkage. The author hopes that, based on the results presented here, a variety of useful tools for the kinematic design of planar multi-degree-of-freedom mechanisms will be developed for planar curve generation. Keywo
7、rds: Kinematic coefficients; Planar five-bar linkage; Kinematics of a coupler point; Geometry of a coupler curve; Curvature theory; Instantaneous centers of zero velocity; Finite difference1. IntroductionThe method of kinematic coefficients provides a concise description of the geometric properties
8、of a planar linkage 1,2. The method has been applied to the kinematic analysis of a wide variety of single-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 ro
9、bots manipulating a rigid pay-load have also been investigated using the method of kinematic coefficients 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 t
10、he curvature and displacement of epitrochoidal gerotors 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
11、method 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
12、 input (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 systemati
13、c 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 pin
14、ned 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 t
15、he 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 te
16、chniques 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 superposit
17、ion. 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
18、,16. 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
19、 very 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
20、our 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 met
21、hod 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 ki
22、nematic 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 tw
23、o-degree-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
24、 industrial 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
25、procedure 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
26、. The 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 fi
27、ve-bar linkage 6. This is a single-degree-of-freedom linkage which can provide more complex motions than the well-kinematic analysis of the linkage are also shown in Fig. 1. The first-order and the second-order kinematic coefficients of the five-bar linkage can be obtained from the loop-closure equa
28、tion for the linkage which can be written as 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 dependent variable 64, that is, the angular velocity of the coupler link 4 can be wr
29、itten as 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 is, the angular velocity of the coupler link 5 can be written as where the coupler angle 65 is also a function of the two independent va
30、riables 62 and 63. Eq. (2.2) can be written as 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:Differentiating Eq. (2.2) with respect to time
31、, the angular accelerations of the coupler links can be written as Using the prime and the dot notation, Eq. (2.5) can be written as and where are the second-order kinematic coefficients of the coupler links relating the angular accelerations of the these links to the two input angular velocities an
32、d accelerations. The first-order and second-order kinematic coefficients of the five-bar 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, Then differentiate Eq. (2.8) partially wit
33、h respect to input position 02; that is. Solve Eq. (2.9) for the first-order kinematic coefficients 042 and 052. Also, differentiate Eq. (2.8) partially with respect to input position 03; that is, Solve Eqs. (2.10) for the first-order kinematic coefficients 043 and 053. Next differentiate Eqs. (2.9)
34、 partially with respect to the input position 02; that is, 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, Solve Eqs. (2.12) for the second-order kinematic coefficients 9423 and 9523
35、. Note that differentiating Eqs. (2.10) partially with respect to the input position 02 will give the same result. Finally, differentiate Eqs. (2.10) partially with respect to the input position 03; that is Finally, solve Eqs. (2.13) for the second-order kinematic coefficients 6433 and 6533. In the
36、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 form where the coefficient matrix is From Eqs. (2.9) and (2.10), the coefficient matrix is The determinant of the coefficient matrix, see Eq. (2.17a
37、), can be written as Note that the determinant will be zero when 65 = 64 or 65 = 64 180. These values occur when the coupler links are either fully extended or folded on top of each other; i.e., when the five-bar linkage is in the configuration of a quadrangle with the coupler links aligned. Eq. (2.
38、15) can be written in matrix form as where the coefficient matrix A is the same as given by Eq. (2.17a) and the matrix. Now that the first and second-order kinematic coefficients of the linkage are known, a study of the path curvature of the path traced by a coupler point can be undertaken. 3. Path
39、curvature of a coupler pointTwo independent inputs are required for a point fixed in coupler link 4, or coupler link 5, to follow a unique path. Therefore, there is a limitless variety of curves that can be generated by a coupler point of the planar five-bar linkage. For purposes of illustration, co
40、nsider the arbitrary point P fixed in coupler link 4 as shown in Fig. 2.The radius of curvature of the path traced by coupler point P can be written as. The vectors defining the location of point P are also shown in Fig. 2. Therefore, the vector equation for this point can be written as. Differentia
41、ting this constraint equation with respect to the two independent inputs, the first-order and second-order kinematic coefficients areDifferentiating this equation with respect to time, the velocity of point P can be written asDifferentiating Eqs. (3.5) with respect to the input position 92 and using
42、 Eq. (3.4), the first-order kinematic coefficients of point P areSimilarly, differentiating Eqs. (3.5) with respect to the input position 93 and using Eq. (3.4), the first-order kinematic coefficients of point P areThe magnitude of the velocity of point P can be written asSubstituting Eq. (3.7b) int
43、o this equation, the magnitude of the velocity of point P can be written as The direction of the velocity of point P can be written as. The unit tangent vector to the path of point P is the unit vector pointing in the direction of the velocity of point P and can be written as Substituting Eqs. (3.7b
44、) and (3.9b) into this equation, the unit tangent vector can be written as counterclockwise from the unit tangent vector; i.e. Differentiating Eq. (3.7b) with respect to time, the acceleration of point P can be written as where Xp22,Xp23,Xp33, Yp22, Yp23, and Yp33 are referred to as the second-order
45、 kinematic coefficients of point P. Differentiating Eqs. (3.8a) and (3.8b) with respect to the input position 92, and using Eq. (3.4), the second-order kinematic coefficients of point P are. Similarly, differentiating Eqs. (3.8a)-(3.8d) with respect to the input position 93, and using Eq. (3.4), the
46、 second-order kinematic coefficients of point P are. The direction of the acceleration of point P can be written as. Substituting Eq. (3.13) into this equation, the direction of the acceleration of point P can be written as.For the special case where the input links 2 and 3 are rotating with constan
47、t angular velocities; i.e., the input angular accelerations a2 = 0 and a3 = 0, then Eq. (3.15b) can be written as.Substituting Eqs. (3.12) and (3.13) into Eq. (3.2b), and performing the dot product, the normal acceleration of point P can be written asSubstituting Eqs. (3.9b) and (3.17) into Eq. (3.1
48、) gives the radius of curvature of the path traced by coupler point P. For the special case where the input links 2 and 3 are rotating with constant angular velocities then Eq. (3.17) can be written asSubstituting Eqs. (3.9b) and (3.18) into Eq. (3.1), and simplifying, the radius of curvature of the
49、 path traced by coupler point P can be written as Sign convention: A negative value for the radius of curvature indicates that the unit normal vector is pointing away from the center of curvature of the path traced by coupler point P.The coordinates of the center of curvature of the path traced by point P can be written aswhere XP and YP are given by Eqs. (3.5) and the X and Y components of the unit normal vector are given by Eq. (3.12).4. Numerical exampl
