Introduction This paper presents a new design methodology for crank-rocker four-bar linkages based on differ- ential objective functions。 Firstly, to avoid obtaining a mechanism with an unintended trajectory shape, we classified the trajectory of the linkage into four types of shapes。 Two-step optimization for a specific shape type is proposed。 In the first step, the shape of the trajectory is obtained by minimizing the root mean square error (RMSE) between the slope (first-order derivative) of the coupler point and that of the target trajectory。 The size of the trajectory is then determined by minimizing the RMSE between the change in angle of slope (second-order derivative) of the coupler point and that of the desired trajectory。 This new approach has three advantages: i) the desired trajectory can be set as a continuous and closed loop, ii) the optimal solution can be ob- tained without the possibility of generating a mechanism with an unintended coupler curve, and iii) the method can take account of the velocity of the output for each section with a constant input velocity。 Three case studies were conducted to verify the advantages of the new approach based on a new index called the goodness of traceability。72604

Four-bar linkages are widely used in mechanical devices due to their simple structure, ease of manufacturing, and low cost。 The path synthesis of planar four-bar linkages has been studied with a variety of methods during the past 60 years [1–7]。 The methods in these studies  can be classified  as  graphical,  analytical,  or   numerical。

Hrones and Nelson used a graphical method that involves an atlas of coupler curves [1]。 They developed a four-bar linkage atlas with almost 7000 coupler curves。 Similarly, Zhang et al。 [8] proposed an atlas of five-bar geared linkage coupler curves。 Alternatively, other graphical methods involve drawing by hand [9]。 Graphical methods are quick and straightforward, but the accuracy of these approaches is limited due to drawing error, which can be critical for the design of precision mechanical devices。 Also, because of the complexity of obtaining solutions with a reasonable result, the geometric construction may have to be repeated many times。

Analytical approaches were first addressed by Sandor [2] and by other researchers thereafter [10,11]。 Methods for finding suitable four-bar linkages that can precisely trace desired precision points analytically have been developed, and the number of desired precision points has increased from four to nine。 Once a mechanism is modeled mathematically and coded for a computer simulation, parameters such as the lengths of each link are easily handled to create new solutions without further programming。 However, there is no analytical solution to the general problem of four-bar linkage synthesis for more than nine target points。

Thus, this methodology cannot be applied for the design of four-bar linkages whose coupler point can trace a large number of target  points or continuous and closed  loops。  This  problem may be solved  using a numerical     method。

There are two types of numerical method。 One is using numerical atlas database of coupler curve with Fourier series method [12,13,14]。 The other method is optimizing parameters to minimize an objective function, and obtain the solution numerically。 The most widely used objective function is the tracking error (TE), which is defined as the sum of the square of the Euclidean distance between the desired points and the obtained coupler points。 To the best of our knowledge, the first to address this objective func- tion was Han [3]。 Due to the ease of calculation, TE has been used in various studies [5,15,16]。 However, using TE for the objective function could generate unintended trajectory shapes。 Furthermore, when the input angular velocity is constant, it is not easy to take account of the velocity  of  the coupler point that traces the trajectory  of  the four-bar    linkage。