Because the four-bar linkage has one degree of freedom, the angle θ2 can be represented by the input angle θ1。 Freudestein's equation  was used to determine  the relationship  between θ2  and the input  angle    [18]:

K1 cos θ3 — K2 cos θ1 þ K3 ¼ cosðθ1 — θ3Þ ð3aÞ

K1 cos θ2 — K4 cos θ1 þ K5 ¼ cosðθ1 — θ2Þ ð3bÞ

From Eqs。 (3a) and  (3b),  the solutions of θ2  and θ3  regarding  the input  angle θ1  are obtained  as    follows:

Fig。 2。 The result of a case study for determining a four-bar linkage that can trace a human gait trajectory obtained by motion capture。

E ¼ —2 sin θ1

F ¼ K1 — ðK2 þ 1Þ cos θ1 þ K3

Substituting Eq。 (4a) into Eqs。 (2a) and (2b) allows for the derivation of the coupler point position of a four-bar linkage using a single variable  in the range  of [0  2π]。

2。2。 Classification of shape of trajectories

2。2。1。 Motivation: unintended shape

When designing a four-bar linkage that can trace a target trajectory by minimizing TE, unintended shapes of the coupler curve can sometimes be obtained, as shown in Fig。 2。 To avoid this, the coupler curves were classified into specific types of shapes and used to design a mechanism。 For example, the solution for the desired trajectory demonstrated in Fig。 2, which is part of the  “non-intersectional  shape” category,  would  not  normally  be obtained。

2。2。2。 Classification methodology

The four shape categories are shown in Fig。 3。 The ellipse-like shape is Type-I, the shape that includes linear motions is Type-II, the crescent-like shape is Type-III, and the shape that has an intersection point is Type-IV。 Two methods were used to validate this classification。 The first method is visual inspection of the coupler curves of the atlas by Hrones and Nelson [1]。 Approximately 7000 coupler curves were categorized。 This method does not need further work such as coding, so it is easy and simple to apply this method to the atlas。 However, because visual inspection is done with the human eye, the accuracy and reproducibility of  the  result  may low。

The second method is generating link lengths of a four-bar linkage randomly and checking whether or not each trajectory be- longs to one of the four types using the geometrical characteristics of each type。 Over 100,000 coupler curves were examined。 Since the shape is the only thing to be considered in the classification, the generated trajectories were rotated to have the max- imum width, as shown in Fig。 4。 We found a vector which has maximum length in the trajectory, and rotated the trajectory by the slope of the vector relative to the origin point。 After rotating the trajectory, the geometrical characteristics were considered。 The details are described in the next section。 Once the geometrical features are coded for a computer simulation, it is easy to classify a large number of coupler curves。 The simulation also enables very high accuracy and reproducibility of the results compared to visual inspection。  However,  the geometrical  characteristics  are needed, and  programing  takes  more time。

Fig。 3。 Categories of the trajectories of crank-rocker four-bar linkage: (a) Type-I (ellipse shape), (b) Type-II (semi-ellipse shape), (c) Type-III (crescent-like   shape),