To overcome these disadvantages, this paper introduces a new numerical approach for the design of a four-bar linkage。 Classifications are also presented for the possible trajectory shapes of a crank-rocker four-bar linkage: elliptical, semi-elliptical, crescent, and intersectional。 These classifications were used to figure out the first and second-order derivatives (slope and change in angle of slope) of the coupler point, which reflect the characteristics of each shape type and the size of the entire trajectory。 The root-mean-square error (RMSE) of these derivative values between the desired and obtained trajectories is used as the objective function。

This method has three advantages compared to conventional numerical approaches。 First, using the derivative of the trajectory, which is a continuous function, the desired trajectory can be set as a continuous and closed loop。 In contrast, methods that min- imize TE require points for the desired trajectory。 Second, if the four-bar linkage is designed to follow the derivative value profile according to the input angle, the method can obtain the optimal solution without the possibility of generating an unintended shape。 Finally, by adjusting the interval of two peak points of the derivative profile of the desired trajectory, the method can take account of the velocity of each section of the coupler curve with constant input velocity。

The performance of the numerical method was investigated using a new index called the goodness of traceability (GT)。 GT is defined as TE as a function of the input angle of the four-bar linkage。 This allows the shape and velocity to be considered simultaneously。  GT can be  used to compare  the performance  of  each method    objectively。

This paper is organized as follows。 The trajectory classification of the four-bar linkage is presented in Section 2。 Section 3 presents the new design method, and Section 4 presents the performance of the method in comparison to a conventional method。 The GT index is also demonstrated in this section。 Discussions and a conclusion are given in Section 5 and Section 6, respectively。

2。 Classification of trajectories of crank-rocker four-bar linkage coupler point

The trajectories of the coupler point of a crank-rocker four-bar linkage need to be classified to gain insight about them。 The shape classifications  are  explained,  and the  mathematical  properties  of  each  shape  type are described。

2。1。 Coupler point of crank-rocker four-bar linkage

To form a crank-rocker four-bar linkage, the relationships between each link length need to satisfy the Grashof conditions [17]:

T 1 ¼ l4 þ l2 — l1 — l3 N 0 ð1aÞ

T 2 ¼ l3 þ l4 — l1 — l2 N 0 ð1bÞ

Fig。 1。 Four-bar  Linkage  mechanism  in  a  global  coordinate  system。

T 3 ¼ l3 þ l2 — l1 — l4 N 0 ð1cÞ

Each variable can be found in Fig。 1。 If the input link (l1) rotates once, a coupler curve is generated that is the same as the trajectory of the coupler point of the four-bar linkage。 Fig。 1 shows the position of coupler point C on a four-bar linkage, which is described  as follows:

Cx ¼ l1 cos ðθ1 þ θ0Þ þ lcx cos ðθ2 þ θ0Þ — lcy sin ðθ2 þ θ0Þ ð2aÞ

Cy ¼ l1 sin ðθ1 þ θ0Þ þ lcx sin ðθ2 þ θ0Þ þ lcy cos ðθ2 þ θ0Þ