A Note on Equivalent Linkages of Direct-Contact Mechanisms

In this paper, the inequivalence of the direct-contact mechanisms and their equivalent four-bar linkages in jerk analysis is discussed. Kinematic analyses for three classical types of direct-contact mechanisms consisting of: (a) higher pairs with permanently invariant curvature centers, (b) higher pairs with suddenly changed curvature, and (c) higher pairs with continuously varying curvature are performed, respectively, through their representative case studies. The analyzed results show that the equivalent four-bar linkage cannot give a correct value of jerk for most situations in the three case studies. Subsequently, the concept of “equivalent six-bar linkage” for direct-contact mechanisms is proposed in order to discuss the infeasibility of the equivalent four-bar linkage for jerk analysis. It is found that the suddenly changed or continuously varying curvature of the higher pairs is not considered in sudden or continuous link-length variations of the equivalent four-bar linkage, which further leads to inconsistency between the angular accelerations of the coupler and the contact normal, and finally results in the infeasibility of the equivalent four-bar linkage for jerk analysis of most direct-contact mechanisms. It is also found that the concept of equivalent six-bar linkage could be applied to evaluate more higher-order time derivatives for most direct-contact mechanisms. The presented case studies and discussion can give demonstrations for understanding the inequivalence of the direct-contact mechanisms and their equivalent four-bar linkages in the aspect of jerk analysis.


Kinematic Analysis for Direct-Contact Mechanisms Consisting of Higher Pairs with Permanently Invariant Curvature Centers: Planar Gear Mechanisms with Involute Spur Gears
For a direct-contact mechanism consisting of higher pairs with permanently invariant curvature centers, its equivalent four-bar linkage can also maintain a permanently invariant configuration at all instants. The planar gear mechanism with involute spur gears is a representative one of this type of direct-contact mechanisms. Thus, kinematic analysis for such a planar gear mechanism is presented in this Section. Figure 2a illustrates a planar gear mechanism, consisting of a frame (link 1), a driving gear (link 2), and a driven gear (link 3). The driving and driven gears are pivoted to the frame on points O 2 and O 3 , respectively, while the center distance is f. The driving and driven gears are both typically involute spur gears that have involute tooth profiles generated from their base circles with radii of r b2 and r b3 , respectively. Thus, a constant angular velocity ratio between the two mating gears can be determined through the radii of their pitch circles r p2 and r p3 . Points K 2 and K 3 are the centers of curvature of a pair of involute tooth profiles in contact at point A, respectively. The common normal at the contact point A (i.e. the line of action) must always pass through points K 2 , K 3 and also the pitch point P. It is known that the line of action must always be the interior common tangent of the base circles of the two mating gears, while points K 2 and K 3 must always be coincident to the two points of tangency regardless of any positional change of the contact point A. Additionally, lines O 2 K 2 and O 3 K 3 are always parallel to each other, and the subtending acute angle between each of them and line O 2 O 3 is always equal to the pressure angle φ of the gear mechanism. As a result, the equivalent linkage of this planar gear mechanism is the four-bar linkage O 2 K 2 K 3 O 3 shown in Figure 2b, in which the coupler (link 4) of the linkage connects the centers of curvature of a pair of involute tooth profiles. The configuration of the equivalent four-bar linkage O 2 K 2 K 3 O 3 shown in Figure 2b is a permanently invariant crossed quadrilateral at all instants, according to the geometric characteristics of involute tooth profiles.

Kinematic Analysis for Direct-Contact Mechanisms Consisting of Higher Pairs with Permanently Invariant Curvature Centers: Planar Gear Mechanisms with Involute Spur Gears
For a direct-contact mechanism consisting of higher pairs with permanently invariant curvature centers, its equivalent four-bar linkage can also maintain a permanently invariant configuration at all instants. The planar gear mechanism with involute spur gears is a representative one of this type of direct-contact mechanisms. Thus, kinematic analysis for such a planar gear mechanism is presented in this Section. Figure 2a illustrates a planar gear mechanism, consisting of a frame (link 1), a driving gear (link 2), and a driven gear (link 3). The driving and driven gears are pivoted to the frame on points O and O, respectively, while the center distance is f. The driving and driven gears are both typically involute spur gears that have involute tooth profiles generated from their base circles with radii of rb and rb, respectively. Thus, a constant angular velocity ratio between the two mating gears can be determined through the radii of their pitch circles rp and rp. Points K and K are the centers of curvature of a pair of involute tooth profiles in contact at point A, respectively. The common normal at the contact point A (i.e. the line of action) must always pass through points K, K and also the pitch point P. It is known that the line of action must always be the interior common tangent of the base circles of the two mating gears, while points K and K must always be coincident to the two points of tangency regardless of any positional change of the contact point A. Additionally, lines OK and OK are always parallel to each other, and the subtending acute angle between each of them and line OO is always equal to the pressure angle  of the gear mechanism. As a result, the equivalent linkage of this planar gear mechanism is the four-bar linkage OKKO shown in Figure  2b, in which the coupler (link 4) of the linkage connects the centers of curvature of a pair of involute tooth profiles. The configuration of the equivalent four-bar linkage OKKO shown in Figure 2b is a permanently invariant crossed quadrilateral at all instants, according to the geometric characteristics of involute tooth profiles.  For the four-bar linkage that is shown in Figure 2b, its link lengths are r 1 = O 2 O 3 = f, r 2 = O 2 K 2 = r b2 = r p2 cosφ, r 3 = O 3 K 3 = r b3 = r p3 cosφ, and r 4 = K 2 K 3 = f sinφ, and the angular positions of links, θ 2 , θ 3 , and θ 4 , are measured counterclockwise from line O 2 O 3 . Since the equivalent four-bar linkage is a permanently invariant crossed quadrilateral at all instants, thus the angular positions are invariantly θ 2 = φ, θ 3 = φ + 180 • , and θ 4 = φ + 270 • . By substituting the values of θ 2 , θ 3 , and θ 4 into Equations (1) and (2), the angular velocities of links 3 and 4 of the equivalent linkage can be expressed as Equation (8) shows that the angular velocity of the coupler (link 4) is zero, which agrees with the kinematic property of the stationary common normal. Afterwards, by substituting the values of θ 2 , θ 3 , and θ 4 , and also ω 4 = 0 into Equations (3) and (4), the angular accelerations of links 3 and 4 of the equivalent linkage can be expressed as Equation (10) shows that the angular acceleration of the coupler (link 4) is always greater than zero, which does not agree with the kinematic property of the stationary common normal. Furthermore, by substituting the values of θ 2 , θ 3 , and θ 4 and also ω 4 = 0 into Equations (5) and (6), the angular jerks of links 3 and 4 of the equivalent linkage can be expressed as Equation (12) shows that the angular jerk of the coupler (link 4) is zero if and only if α 2 = α 3 = 0 when ω 2 0 and ω 3 0, otherwise it does not agree with the kinematic property of the stationary common normal. When considering that the driving gear rotates at a constant angular velocity ω 2 , its angular acceleration and jerk are α 2 = 0 and ζ 2 = 0, respectively. The driven gear will also rotate at a constant angular velocity ω 3 , and its angular acceleration and jerk should be α 3 = 0 and ζ 3 = 0, respectively. From Equation (7), it is found that Equation (13) undoubtedly agrees with the fundamental law of gearing [17], which is, the angular velocity ratio between links 2 and 3 of the equivalent linkage is equal to that between the mating driving and driven gears. Hence, the equivalent four-bar linkage of the planar gear mechanism can give a correct value of angular velocity. Subsequently, by substituting α 2 = 0 into Equation (9), the angular acceleration of link 3 of the equivalent linkage is It shows that the angular acceleration of link 3 of the equivalent linkage is zero, which is equal to that of the driven gear. Hence, the equivalent four-bar linkage of the planar gear mechanism can provide a correct value of angular acceleration. Furthermore, by substituting α 2 = 0, α 3 = 0, and ζ 2 = 0 into Equations (11) and (12), the angular jerks of links 3 and 4 of the equivalent linkage are From Equations (7) and (15), the angular jerk of link 3 of the equivalent linkage can be further expressed as From Equation (17), it can be found that It shows that the angular jerk of link 3 of the equivalent linkage is zero if and only if r 2 = r 3 . In other words, the angular jerk of link 3 of the equivalent linkage is equal to that of the driven gear if, and only if, r p2 = r p3 . Hence, only when the angular velocity ratio between the two mating gears (ω 3 /ω 2 ) is exactly negative one (−1), the equivalent four-bar linkage of the planar gear mechanism can give correct values of angular velocity, acceleration, and jerk. Otherwise, the equivalent four-bar linkage of the planar gear mechanism can only give correct values of angular velocity and acceleration.
In this case study, it is found that the equivalent four-bar linkage of a planar gear mechanism with a pair of involute spur gears is not able to give a correct value of jerk, unless the angular velocity ratio between the two mating gears is exactly negative one (−1).

Kinematic Analysis for Direct-Contact Mechanisms Consisting of Higher Pairs with Suddenly Changed Curvature: Disk Cam Mechanisms with a Circular-Arc Cam and an Oscillating Roller Follower
For a direct-contact mechanism consisting of higher pairs with suddenly changed curvature (i.e., discontinuous curvature), two equivalent four-bar linkages may simultaneously exist at a certain instant. The disk cam mechanism with a circular-arc cam and an oscillating roller follower is a representative one of this type of direct-contact mechanisms. Thus, kinematic analysis for such a disk cam mechanism is presented in this Section. Figure 3a illustrates a disk cam mechanism, consisting of a frame (link 1), a circular-arc cam (link 2), and an oscillating roller follower (link 3). The cam and the follower are pivoted to the frame on points O 2 and O 3 , respectively, while the center distance is f. The profile of the circular-arc cam is essentially composed of four circular arcs with three radii R 1 , R 2 , and R 3 , respectively. Points K 1 , K 2 , K 3 , and K 4 are the circular centers (also the centers of curvature) of the four circular arcs, respectively. Points A 12 , A 23 , A 34 , and A 14 on the cam profile are the points of tangency of each two adjacent circular arcs, respectively. The circular arc between points A 12 and A 14 has a radius of R 1 and a circular-arc angle of γ 1 , the circular arc between points A 12 and A 23 or between points A 14 and A 34 has a radius of R 2 and a circular-arc angle of γ 2 , and the circular arc between points A 23 and A 34 has a radius of R 3 and a circular-arc angle of γ 3 . In this case, points O 2 , K 1 , and K 3 are collinear, while the distance between points O 2 and K 1 is e, and that between points K 1 and K 3 is c. The follower has an arm length of l and a roller radius of r f . The cam rotation angle θ is measured counterclockwise from line O 2 O 3 to line O 2 K 1 , while the angular position of the follower ξ is measured clockwise from line O 2 O 3 to line O 3 C for point C being the roller center. When the values of R 1 , R 2 , R 3 , and c are given, then from ∆K 1 K 2 K 3 (or ∆K 1 K 3 K 4 ) and the cosine law, the circular-arc angles γ 1 , γ 2 , and γ 3 can be determined by It should be noted that the geometric relation of γ 1 + 2γ 2 + γ 3 = 360 • must be satisfied. Once the cam profile has been designed, the angular position of the follower ξ = ξ(θ) can be accordingly determined.
The contact normal can also be determined by locating instant center I 23 . By labeling instant center I 23 as Q and O 2 Q = q, as shown in Figure 3a, point Q can be located by [11,12,17] From ∆O 3 QC and the cosine law, line QC, which is collinear with the contact normal, can then be determined by Robotics 2020, 9, 38 7 of 27 distance between points O and K is e, and that between points K and K is c. The follower has an arm length of l and a roller radius of r f . The cam rotation angle  is measured counterclockwise from line OO to line OK, while the angular position of the follower  is measured clockwise from line OO to line OC for point C being the roller center. When the values of R, R, R, and c are given, then from △KKK (or △KKK) and the cosine law, the circular-arc angles , , and  can be determined by It should be noted that the geometric relation of  + 2 + = 360 must be satisfied. Once the cam profile has been designed, the angular position of the follower  = () can be accordingly determined. The contact normal can also be determined by locating instant center I. By labeling instant center I as Q and OQ = q, as shown in Figure 3a, point Q can be located by [11,12,17] From △OQC and the cosine law, line QC, which is collinear with the contact normal, can then be determined by  Thus, from ∆O 3 QC and the sine law, the angular position of the contact normal λ, measured counterclockwise from line O 2 O 3 to line QC, can be expressed as The equivalent linkage of this disk cam mechanism is a four-bar linkage whose coupler connects the circular center of one of the four circular-arc profiles and the roller center. The equivalent linkage corresponding to each circular-arc profile is an invariant one (i.e., its link lengths are invariant). Thus, the disk cam mechanism can be successively replaced by four invariant equivalent linkages (with different constant link lengths), as the circular-arc cam rotates a complete cycle. However, when considering a special situation that the contact point between the cam and the follower is point A 23 (or one of the remaining three points of tangency), the equivalent four-bar linkages at that instant are shown in Figure 3b. It can be observed that two equivalent four-bar linkages O 2 K 2 CO 3 and O 2 K 3 CO 3 simultaneously exist at that instant. For clarity of illustration, the four-bar linkage O 2 K 2 CO 3 consists of links 1, 2a, 3a, and 4a, and the four-bar linkage O 2 K 3 CO 3 consists of links 1, 2b, 3b, and 4b. The links 3a and 3b overlap with each other. The couplers 4a and 4b, also overlapping with each other, are collinear with the common normal at the contact point A 23 that passes through points Q, K 2 , K 3 , and C at that instant.
For the four-bar linkages that are shown in Figure 3b, their link lengths are The angular positions of links, θ 2a , θ 2b , θ 3 , and θ 4 , are measured counterclockwise from line O 2 O 3 . From ∆O 2 K 2 K 3 and the cosine law, the link length of r 2a can be further determined by From ∆O 2 K 3 C and the cosine law, the distance O 2 C at that instant can be determined by Hence, from ∆O 2 K 3 C, ∆O 2 O 3 C, and the cosine law, the values of the cam rotation angle θ and the angular position of the follower ξ at that instant can be determined by Subsequently, the values of θ 2a and θ 2b at that instant can be determined by Additionally, the values of θ 3 and θ 4 at that instant can be determined by It is known that when the circular-arc cam rotates a complete cycle, the follower will instantaneously undergo non-continuous accelerations and infinite jerks as the circular-arc cam is in contact with the follower at one of the four points of tangency (i.e., points A 12 , A 14 , A 34 , and A 23 ) successively, although the velocity of the follower can be continuous at those instants. A practical example is given in order to evaluate such phenomena. A circular-arc cam with R 1 = 50 mm, R 2 = 120 mm, R 3 = 32.5 mm, c = 40 mm, and e = 20 mm is designed to drive an oscillating roller follower with l = 100 mm and r f = 15 mm, while the center distance is f = 120 mm. From Equations (19) to (21), the circular-arc angles are γ 1 = 203.832 • , γ 2 = 26.570 • , and γ 3 = 103.028 • . Such a disk cam mechanism is proportionally shown in Figure 3a. The circular-arc cam is specified to rotate counterclockwise with a constant angular velocity of ω = 1 rad/s. The angular position of the follower, ξ, in a complete cam rotation cycle is shown in Figure 4a, while its corresponding time derivatives (i.e., the angular velocity are and θ 4 = 24.654 • can be obtained. For the equivalent four-bar linkage O 2 K 2 CO 3 (consisting of links 1, 2a, 3a, and 4a), its angular velocities, accelerations, and jerks at that instant, calculated by using Equations (1) to (6) with ω 2 = 1 rad/s, α 2 = 0 rad/s 2 , and ζ 2 = 0 rad/s 3 , are For the equivalent four-bar linkage O 2 K 3 CO 3 (consisting of links 1, 2b, 3b, and 4b), its angular velocities, accelerations, and jerks at that instant, calculated by using Equaions (1) to (6), are It is found that ω 3a = ω 3b = 0.484 rad/s and α 3a α 3b , which agree with the situations that are shown in Figure 4b,c that the follower undergoes a non-continuous angular acceleration, although its angular velocity can be continuous at that instant. However, it is also found that ζ 3a = −0.209 rad/s 3 and ζ 3b = −0.047 rad/s 3 are finite values, which cannot correctly reflect an instantaneously infinite follower jerk as that shown in Figure 4d. As to the results of the coupler, it can be found that ω 4a ω 4b , which agrees with the situation shown in Figure 5b that the contact normal undergoes a non-continuous angular velocity, although its angular position can be continuous at that instant. However, the remaining results of α 4a = 0.242 rad/s 2 , α 4b = 0.018 rad/s 2 , ζ 3a = −0.209 rad/s 3 , and ζ 3b = −0.047 rad/s 3 are all finite values, which cannot correctly reflect the instantaneously infinite acceleration and jerk as those shown in Figure 5c,d.    In this case study, when the circular-arc cam is in contact with its follower at a point of tangency of two adjacent circular arcs, the coupler acceleration of the equivalent linkage, α 4 , does not agree with the angular acceleration of the contact normal, .. λ, at that instant. Unreasonable values of α 4 obtained in acceleration analysis [with the use of Equation (4)] unavoidably lead to incorrect values of ζ 3 and ζ 4 when performing jerk analysis because α 4 is involved in both Equations (5) and (6).
In addition, if a special case of R 1 = R 2 = R 3 and c = 0 is given, the circular-arc cam will degenerate to an eccentric circular cam, while points K 1 , K 2 , K 3 , and K 4 will coincide with the circular center of the eccentric circular cam. It is known that an eccentric circular cam mechanism has an invariant equivalent four-bar linkage (without link-length variations) that can completely duplicate the motion transmission between the cam and the follower [3]. Such an invariant equivalent four-bar linkage can give correct values of angular velocity, acceleration, and jerk for all instants. As compared with the eccentric circular cam profile having constant curvature, the curvature of the circular-arc cam is suddenly changed at the points of tangency of each two adjacent circular arcs. As the circular-arc cam and its follower are in contact at such a point of tangency, the suddenly changed curvature of the cam profile is not considered in sudden link-length variations of the equivalent four-bar linkages. Thus, the two equivalent four-bar linkages simultaneously existing at that instant are both not able to correctly reflect an infinite jerk of the follower.

Kinematic Analysis for Direct-Contact Mechanisms Consisting of Higher Pairs with Continuously Varying Curvature: Disk Cam Mechanisms with a Double-Dwell Cam and an Oscillating Roller Follower
For a direct-contact mechanism consisting of higher pairs with continuously varying curvature, some link lengths of its equivalent four-bar linkage may continuously vary during the whole motion cycle. The disk cam mechanism with a double-dwell cam and an oscillating roller follower is a representative one of this type of direct-contact mechanisms. Thus, kinematic analysis for such a disk cam mechanism is presented in this Section. Figure 6 illustrates a disk cam mechanism, consisting of a frame (link 1), a double-dwell cam (link 2), and an oscillating roller follower (link 3). The cam and the follower are pivoted to the frame on points O 2 and O 3 , respectively, while the center distance is f. The follower has an arm length of l and a roller radius of r f . The contact normal always passes through the roller center C, the contact point A, the center of curvature of the cam profile K, and also the instant velocity center I 23 . By setting up a Cartesian coordinate system X-Y fixed on the cam and with its origin at the fixed pivot O 2 , the cam profile coordinates may be expressed in terms of the cam rotation angle θ, which is measured against the direction of cam rotation from the reference radial on cam to line O 2 O 3 . The cam is to rotate clockwise with a constant angular velocity of ω rad/s. By labeling instant center I 23 as Q and O 2 Q = q, the parametric vector equations of the theoretical cam profile coordinates can be expressed as [12,17,18] where in which, ξ(θ) is the angular displacement function of the oscillating follower [12,17,18]: where, r b is the radius of the base circle and s(θ) is the prescribed angular displacement program (or called displacement curve [17]) of the follower. Thus, v(θ) [= ds(θ)/dθ] in Equation (35) is the angular velocity curve of the follower (in unit of rad/rad or dimensionless). It can be seen that line QC is the contact normal between the cam and the roller follower. Additionally, the parametric vector equations of the pitch curve coordinates can be expressed as [18] The radius of curvature of the pitch curve, ρ C , can then be determined by [18] where, a = a(θ) [= dv(θ)/dθ] is the angular acceleration curve of the follower (in unit of rad/rad 2 or 1/rad). Therefore, the radius of curvature of the cam profile, ρ, can be obtained by [18] Robotics 2020, 9, 38 14 of 27 Figure 6. Illustration of a disk cam mechanism with a double-dwell cam and an oscillating roller follower The equivalent linkage of this disk cam mechanism is a four-bar linkage whose coupler connects the center of curvature of the cam profile, K, and the roller center, C, as shown in Figure 7. the corresponding values of ρ C and ρ are negative, and the corresponding coupler length will be r 4 = KC = −ρ C = −(ρ + r f ). From ∆O 2 QK and the cosine law, the link length of r 2 can be further determined by in which, the length QK corresponding to a convex cam profile, as shown in Figure 7a, is Subsequently, the values of θ 2 , θ 3 , and θ 4 can be determined by Because the double-dwell cam profile is with continuously varying curvature, the link lengths r 2 and r 4 of its equivalent linkage therefore continuously vary with respect to the cam rotation angle θ. A practical example is given in order to evaluate the kinematic characteristics of the equivalent four-bar linkage. A cam system requires the oscillating roller follower to oscillate 25 clockwise with cycloidal motion [17,19] while the cam rotates clockwise from 0 to 120, dwell for the next 40, return with cycloidal motion [17,19] for 120 cam rotation, and dwell for the remaining 80. The design parameters for the cam mechanism are: f = 80 mm, l = 52 mm, r b = 40 mm, and r f = 8 mm. Such a disk cam mechanism is proportionally shown in Figures 6 and 7a. The double-dwell cam is specified to rotate clockwise with a constant angular velocity of  = 1 rad/s. Figure 8   A practical example is given in order to evaluate the kinematic characteristics of the equivalent four-bar linkage. A cam system requires the oscillating roller follower to oscillate 25 • clockwise with cycloidal motion [17,19] while the cam rotates clockwise from 0 • to 120 • , dwell for the next 40 • , return with cycloidal motion [17,19] for 120 • cam rotation, and dwell for the remaining 80 • . The design parameters for the cam mechanism are: f = 80 mm, l = 52 mm, r b = 40 mm, and r f = 8 mm. Such a disk cam mechanism is proportionally shown in Figures 6 and 7a. The double-dwell cam is specified to rotate clockwise with a constant angular velocity of ω = 1 rad/s. Figure 8 shows the variations of link lengths and angular positions for the equivalent linkage of the disk cam mechanism. As seen, the link lengths of r 2 and r 4 are continuously variable. The link length of r 2 is zero when θ = 120 •~1 60 • and θ = 280 •~3 60 • , for which the follower dwells. The link length of r 4 essentially agrees with the radius of curvature of the pitch curve, ρ C . The variational curve of θ 2 is specially drawn by dashed lines when θ = 120 •~1 60 • and θ = 280 •~3 60 • , since the situations of r 2 = 0 and QK = 0 simultaneously occur to yield an undefined result of tan −1 (0/0) in Equation (44). Additionally, the variational curves of θ 3 and θ 4 essentially agree with the variational trends of (180 • − ξ) and λ, respectively. Robotics 2020, 9, 38 15 of 27 Additionally, the variational curves of  and  essentially agree with the variational trends of (180 − ) and , respectively. considerably ranges between 1.88 rad/s 3 . As a result, the equivalent four-bar linkage of the presented disk cam mechanism with a double-dwell cam and an oscillating roller follower is not able to give a correct value of jerk.
The double-dwell cam that is presented in this case study is an ordinary disk cam profile with continuously varying curvature. Thus, its equivalent four-bar linkage corresponding to two infinitesimally closed profile points in contact with the follower cannot exactly have identical link lengths. For most disk cam mechanisms, it can be deduced that their equivalent four-bar linkages should not be able to give a correct value of jerk, because the continuously varying curvature of the cam profiles is not considered in continuous link-length variations of their equivalent four-bar linkages. ξ. However, the numerical result for sum of ( ... ξ + ζ 3 ) considerably ranges between ±1.88 rad/s 3 . As a result, the equivalent four-bar linkage of the presented disk cam mechanism with a double-dwell cam and an oscillating roller follower is not able to give a correct value of jerk.

Kinematic Analysis for Direct-Contact Mechanisms Consisting of Higher Pairs with Continuously Varying Curvature via the Concept of Equivalent Six-Bar Linkage
In the above three case studies, almost all of the results indicate that the equivalent four-bar linkage for a direct contact mechanism cannot give a correct value of jerk. Especially the two case studies for disk cam mechanisms both indicate that the influence of suddenly changed or continuously varying curvature of the cam profile on sudden or continuous link-length variations of the equivalent four-bar linkage is not considered, which could be a major cause of incorrectness in jerk analysis. In order to justify this deduction, the concept of "equivalent six-bar linkage" for disk cam mechanisms considering the time rate of change of the cam profile curvature is presented The double-dwell cam that is presented in this case study is an ordinary disk cam profile with continuously varying curvature. Thus, its equivalent four-bar linkage corresponding to two infinitesimally closed profile points in contact with the follower cannot exactly have identical link lengths. For most disk cam mechanisms, it can be deduced that their equivalent four-bar linkages should not be able to give a correct value of jerk, because the continuously varying curvature of the cam profiles is not considered in continuous link-length variations of their equivalent four-bar linkages.

Kinematic Analysis for Direct-Contact Mechanisms Consisting of Higher Pairs with Continuously Varying Curvature via the Concept of Equivalent Six-Bar Linkage
In the above three case studies, almost all of the results indicate that the equivalent four-bar linkage for a direct contact mechanism cannot give a correct value of jerk. Especially the two case studies for disk cam mechanisms both indicate that the influence of suddenly changed or continuously varying curvature of the cam profile on sudden or continuous link-length variations of the equivalent four-bar linkage is not considered, which could be a major cause of incorrectness in jerk analysis. In order to justify this deduction, the concept of "equivalent six-bar linkage" for disk cam mechanisms considering the time rate of change of the cam profile curvature is presented in this Section. Kinematic analysis of the equivalent six-bar linkage for the disk cam mechanism that is presented in Section 5 is performed in order to evaluate whether the equivalence of a disk cam mechanism and its equivalent six-bar linkage in jerk analysis can exist.
For the disk cam mechanism that is shown in Figure 11a (the same as those shown in Figures 6  and 7a), its equivalent six-bar linkage shown in Figure 11b consists of a frame (link 1), a frame-pivoted linear actuator (links 2 and 5), a floating linear actuator (links 4 and 6), and a frame-pivoted arm (link 3). Each linear actuator essentially consists of two links that are connected by a prismatic joint, such as a slider in its guide, so that the two links can have a relative translation in one direction. Links 4 and 5 in the linear actuators are pinned on the center of curvature of the cam profile, K, while links 2 and 6 in the linear actuators are pinned on the fixed pivot O 2 and on the roller center C, respectively. The link lengths of the six-bar linkage are defined as r 1 = O 2 O 3 = f, r 2 = O 2 K, r 3 = O 3 C = l, and r 4 = KC = ρ C = ρ + r f , and the angular positions of links, θ 2 , θ 3 , θ 4 , θ 5 , and θ 6 , are measured counterclockwise from line O 2 O 3 . Although the configuration of the six-bar linkage is identical to that of the four-bar linkage shown in Figure 7b, its link lengths of r 2 and r 4 are both variables that can exactly reflect the time rate of change of the cam profile curvature. That is, at any instant, r 2 is a variable that describes the relative displacement in the frame-pivoted linear actuator (links 2 and 5) between points O 2 and K, and r 4 is a variable that describes the relative displacement in the floating linear actuator (links 4 and 6) between points K and C. For such a single-loop six-bar linkage with three degree-of-freedom, by specifying θ 2 , r 2 , and r 4 as the three input parameters, a constrained motion can be achieved. Since θ 2 = θ 5 and θ 4 = θ 6 , by using the vector loop method, the two unknowns θ 3 and θ 4 in displacement equations can be solved simultaneously. Alternatively, Equations (40) to (46) can also be used to determine the values of r 2 , r 4 , θ 2 , θ 3 , and θ 4 .
Robotics 2020, 9,38 18 of 27 in this Section. Kinematic analysis of the equivalent six-bar linkage for the disk cam mechanism that is presented in Section 5 is performed in order to evaluate whether the equivalence of a disk cam mechanism and its equivalent six-bar linkage in jerk analysis can exist. For the disk cam mechanism that is shown in Figure 11a (the same as those shown in Figures 6  and 7a), its equivalent six-bar linkage shown in Figure 11b consists of a frame (link 1), a frame-pivoted linear actuator (links 2 and 5), a floating linear actuator (links 4 and 6), and a frame-pivoted arm (link 3). Each linear actuator essentially consists of two links that are connected by a prismatic joint, such as a slider in its guide, so that the two links can have a relative translation in one direction. Links 4 and 5 in the linear actuators are pinned on the center of curvature of the cam profile, K, while links 2 and 6 in the linear actuators are pinned on the fixed pivot O and on the roller center C, respectively. The link lengths of the six-bar linkage are defined as r = OO = f, r = OK, r = OC = l, and r = KC =  C =  + r f , and the angular positions of links, , , , , and , are measured counterclockwise from line OO. Although the configuration of the six-bar linkage is identical to that of the four-bar linkage shown in Figure 7b, its link lengths of r and r are both variables that can exactly reflect the time rate of change of the cam profile curvature. That is, at any instant, r is a variable that describes the relative displacement in the frame-pivoted linear actuator (links 2 and 5) between points O and K, and r is a variable that describes the relative displacement in the floating linear actuator (links 4 and 6) between points K and C. For such a single-loop six-bar linkage with three degree-of-freedom, by specifying , r, and r as the three input parameters, a constrained motion can be achieved. Since  =  and  = , by using the vector loop method, the two unknowns  and  in displacement equations can be solved simultaneously. Alternatively, Equations (40) to (46) can also be used to determine the values of r, r, , , and . Figure 11. Illustration of a disk cam mechanism with a double-dwell cam and an oscillating roller follower and its equivalent six-bar linkage; (a) the disk cam mechanism and (b) the equivalent six-bar linkage.
By using the vector loop method to solve the velocity equations simultaneously, the analytical expressions of the angular velocities of link 3 () and links 4 and 6 () can be derived as Figure 11. Illustration of a disk cam mechanism with a double-dwell cam and an oscillating roller follower and its equivalent six-bar linkage; (a) the disk cam mechanism and (b) the equivalent six-bar linkage.
By using the vector loop method to solve the velocity equations simultaneously, the analytical expressions of the angular velocities of link 3 (ω 3 ) and links 4 and 6 (ω 4 ) can be derived as in which, ω 2 (= . θ 2 ), . r 2 , and . r 4 are the known input velocities, which can be determined through the first time derivatives of θ 2 , r 2 , and r 4 shown in Equations (40) to (44). Subsequently, by using the vector loop method to solve the acceleration equations simultaneously, the analytical expressions of the angular accelerations of link 3 (α 3 ) and links 4 and 6 (α 4 ) can be derived as in which, α 2 (= .. .. r 2 , and .. r 4 are the known input accelerations, which can be determined through the second time derivatives of θ 2 , r 2 , and r 4 shown in Equations (40) to (44). Furthermore, by using the vector loop method to solve the jerk equations simultaneously, the analytical expressions of the angular jerks of link 3 (ζ 3 ) and links 4 and 6 (ζ 4 ) can be derived as in which, ζ 2 (= ... θ2), ... r 2 , and ... r 4 are the known input jerks, which can be determined through the third time derivatives of θ 2 , r 2 , and r 4 shown in Equations (40) to (44). The derived analytical expressions can be used to evaluate the kinematic characteristics of the equivalent six-bar linkage obtained from a direct-contact mechanism.
The practical example that is given in Section 5 is adopted again in order to evaluate the kinematic characteristics of the equivalent six-bar linkage. The variations of link lengths and angular positions for the equivalent six-bar linkage are the same as those shown in Figure 8. It must be noted that, according to the influence of the time rate of change of the cam profile curvature on the link-length variations, the angular velocity of the frame-pivoted linear actuator (ω 2 ) and the linear velocities of both linear actuators ( . r 2 and . r 4 ) should not be constants while the cam rotates clockwise with a constant angular velocity of ω = 1 rad/s (this situation can also be directly observed in Figure 8 that the variation curves of θ 2 , r 2 , and r 4 are not horizontals). Thus, the variations of ω 2 , . r 2 , and . r 4 with respect to the cam rotation angle θ (that is proportional to time), as well as those for α 2 , ζ 2 , .. ξ, and ζ 3 = − ... ξ can be intuitively observed in the figure. Comparison results between the obtained motion curves are presented in Figure 13 in order to justify the situations. The numerical result for sum of ( . ξ + ω 3 ) merely ranges between −4.57 × 10 −7 and 2.03 × 10 −7 rad/s, and that for sum of ( .. ξ + α 3 ) also merely ranges between −7 × 10 −6 and 7.6 × 10 −6 rad/s 2 . As to the numerical result for sum of ( ... ξ + ζ 3 ), it meaningfully ranges between −1.91 × 10 −3 and 1.87 × 10 −3 rad/s 3 , although the accuracy of ζ 3 [calculated by using Equation (51) ξ, and ζ 3 = − ... ξ . In this case study, it is found that the equivalent six-bar linkage of a disk cam mechanism with a double-dwell cam and an oscillating roller follower is able to give a correct value of jerk. Likewise, the equivalence of most disk cam mechanisms and their equivalent six-bar linkages in jerk analysis can exist.
Robotics 2020, 9,38 20 of 27 mechanism with a double-dwell cam and an oscillating roller follower is able to give a correct value of jerk. Likewise, the equivalence of most disk cam mechanisms and their equivalent six-bar linkages in jerk analysis can exist.
According to the presented case study, the reasonability of the deduction mentioned above can therefore be justified because the link-length variations of the equivalent six-bar linkage can exactly reflect the time rate of change of the cam profile curvature and result in the correctness in jerk analysis.

Discussion
Some issues addressed in this study are considered and discussed in this Section.

Major Cause of Infeasibility of Equivalent Four-Bar Linkage
In Sections 5 and 6, it is found that, for a disk cam mechanism whose cam profile is with continuously varying curvature, its "equivalent six-bar linkage", rather than its equivalent four-bar linkage, is able to give a correct value of jerk. The coupler angle  is always identical to the angle of the contact normal, , as can be observed in Figures 3, 7, and 11. The time derivatives of the coupler According to the presented case study, the reasonability of the deduction mentioned above can therefore be justified because the link-length variations of the equivalent six-bar linkage can exactly reflect the time rate of change of the cam profile curvature and result in the correctness in jerk analysis.

Discussion
Some issues addressed in this study are considered and discussed in this Section.

Major Cause of Infeasibility of Equivalent Four-Bar Linkage
In Sections 5 and 6, it is found that, for a disk cam mechanism whose cam profile is with continuously varying curvature, its "equivalent six-bar linkage", rather than its equivalent four-bar linkage, is able to give a correct value of jerk. The coupler angle θ 4 is always identical to the angle of the contact normal, λ, as can be observed in Figures 3, 7 and 11. The time derivatives of the coupler angle θ 4 , as well as the link-length variations of r 2 and r 4 , are actually dominated by the positions (and their time derivatives) of curvature centers of the cam and the follower.
For the practical example that is given in Section 5, the time derivatives of the coupler angle θ 4 of the equivalent four-bar linkage, i.e., ω 4 , α 4 , and ζ 4 calculated by using Equations (2), (4), and (6), respectively, are shown in Figure 14a. Similarly, those of the equivalent six-bar linkage, i.e., ω 4 , α 4 , and ζ 4 calculated by using Equations (48), (50), and (52), respectively, are shown in Figure 14b. Additionally, the time derivatives of angle λ (i.e., . λ, .. λ, and ... λ) are shown in Figure 14c. As seen, each variation curve of ω 4 calculated by using Equation (2) or (48) can agree with that of . λ. However, the variation curves of α 4 and ζ 4 calculated by using Equations (4) and (6) (4)] will further lead to incorrect values of ζ 3 and ζ 4 when performing jerk analysis because α 4 is involved in both Equations (5) and (6). Such a situation is consistent with that observed in Section 4 (when the circular-arc cam is in contact with its follower at a point of tangency of two adjacent circular arcs, the coupler acceleration does not agree with the angular acceleration of the contact normal at that instant). Therefore, for a disk cam mechanism whose cam profile is with continuously varying curvature, its equivalent four-bar linkage can provide correct values of angular (or linear) velocity and acceleration for the driven link (i.e., the follower), but can only give correct values of angular velocity for the coupler (i.e., the contact normal).  For the practical example that is given in Section 5, the time derivatives of the coupler angle  of the equivalent four-bar linkage, i.e., , , and  calculated by using Equations (2), (4), and (6), respectively, are shown in Figure 14a. Similarly, those of the equivalent six-bar linkage, i.e. , , and  calculated by using Equations (48), (50), and (52), respectively, are shown in Figure 14b errors can justify the correctness of . ζ 3 = −ξ (4) . As a result, it is found that the equivalent six-bar linkage of a disk cam mechanism with a double-dwell cam and an oscillating roller follower is still able to give a correct value of ping. It can be deduced that the concept of equivalent six-bar linkage could also be applied to evaluate more higher-order time derivatives for most direct-contact mechanisms.   The practical example that is given in Section 5 is adopted again to perform the ping analysis of the equivalent six-bar linkage. The result for the equivalent six-bar linkage [calculated by using Equation (53)], as well as the angular ping of the follower, are both shown in Figure   15a. The situation of

Potential Future Work
The equivalent four-bar linkages that are studied in this paper are typical linkages with four revolute joints. For direct-contact mechanisms with a translating driven link (a slider), such as disk cam mechanisms with a translating roller/flat-faced follower, constant-breadth cam mechanisms, and gear-and-rack mechanisms, their equivalent four-bar linkages are slider-crank linkages or Scotch yokes. Extended work for evaluating the inequivalence of these types of direct-contact mechanisms and their equivalent four-bar linkages in jerk analysis can be undertaken in the future.
Based on the concept of equivalent linkage, some applications in the aspect of reverse engineering may be carried out, as described in Section 1. The main disadvantage of using the concept of equivalent four-bar linkage in these applications is that the kinematic analysis for the existing direct-contact mechanism is obviously limited to velocity and acceleration. According to the results shown in Sections 6 and 7.2, for an existing direct-contact mechanism with an unknown input-output relation, the kinematic analysis for the existing direct-contact mechanism can be indirectly performed through its "equivalent six-bar linkage" to obtain the velocity, acceleration, jerk, and also ping functions of the driven link, when the geometric information for the members of the existing higher pair (such as measured coordinates or curved-fitted equations of their actual profiles, and also some measured link-length related dimensions) is known. The input-output relation (or the s-v-a-j diagrams [19,20]) for the existing direct-contact mechanism would be constructed without solving cumbersome equations for the higher-pair contact analysis [7][8][9]. When the physical properties for the members of the existing higher pair (such as their masses, centers of masses, and mass moments of inertia) are also known, dynamic force analysis for the existing direct-contact mechanism can be accordingly performed in order to evaluate the shaking forces and moments [5] and their time rate of changes in the existing direct-contact mechanism. The study for demonstrating the advantages of such applications via practical examples would be significant future work.
It must be emphasized that the kinematic equivalence between a direct-contact mechanism and its equivalent linkage is based on rigid body mechanics. If so, the kinematic equivalence should not be affected by dynamic forces (and also shaking forces and moments) in the mechanism. However, dynamic response of the driven link caused by vibrations certainly will affect the kinematic equivalence because all members in a direct-contact mechanism are actually not rigid bodies. The extent of dynamic response on affecting the kinematic equivalence could also be studied with established dynamic models [19,20] in the future.

Conclusions
The inequivalence of the direct-contact mechanisms and their equivalent four-bar linkages in jerk analysis has been discussed in this paper.
For the discussed planar gear mechanism with a pair of involute spur gears, the configuration of its equivalent four-bar linkage is a permanently invariant crossed quadrilateral at all instants. It is found that the equivalent four-bar linkage for the planar gear mechanism is not able to give a correct value of jerk, unless the angular velocity ratio between the two mating gears is exactly negative one (−1). For the discussed disk cam mechanism with a circular-arc cam and an oscillating roller follower, it can be successively replaced by four invariant equivalent linkages (with different constant link lengths) as the circular-arc cam rotates a complete cycle. When the circular-arc cam is in contact with its follower at a point of tangency of two adjacent circular arcs, the two equivalent four-bar linkages simultaneously existing at that instant are both not able to correctly reflect an infinite jerk of the follower, because the suddenly changed curvature of the cam profile is not considered in sudden link-length variations of the equivalent linkages. For the discussed disk cam mechanism with a double-dwell cam and an oscillating roller follower, some link lengths of its equivalent four-bar linkage continuously vary as the double-dwell cam rotates a complete cycle. The equivalent four-bar linkage for such an ordinary disk cam mechanism is still not able to give a correct value of jerk, because the continuously varying curvature of the cam profiles is not considered in continuous link-length variations of the equivalent four-bar linkage.
Subsequently, the concept of "equivalent six-bar linkage" for direct-contact mechanisms has been proposed in order to discuss the infeasibility of the equivalent four-bar linkage for jerk analysis. It is found that the equivalent six-bar linkage, rather than the equivalent four-bar linkage, of the discussed disk cam mechanism is able to give a correct value of jerk. Because the suddenly changed or continuously varying curvature of the higher pairs is not considered in sudden or continuous link-length variations of the equivalent four-bar linkage, which further leads to the inconsistency between the angular accelerations of the coupler and the contact normal, and finally results in the infeasibility of the equivalent four-bar linkage for jerk analysis of most direct-contact mechanisms. It is also found that the equivalent six-bar linkage of the discussed disk cam mechanism is still able to provide a correct value of ping. Thus, the concept of equivalent six-bar linkage could be applied to evaluate more higher-order time derivatives for most direct-contact mechanisms.
The presented case studies and discussion can give demonstrations for understanding the inequivalence of the direct-contact mechanisms and their equivalent four-bar linkages in the aspect of jerk analysis. Therefore, the results that are presented in this paper should verify that for most three-link direct-contact mechanisms, their "representative equivalent four-bar linkages" (with a coupler connecting the centers of curvature of the driving and driven links) cannot be used to perform the jerk analysis.