Because φ1,1 and φ2,1 are assumed to be the same, Eqs。 (6a) and (6b) are the same。 This implies that the similarity ratios n1 and n2  are  the same。

ð2Þ i ¼ k

If  φ1,k =φ2,k, then  Δ O p1,k  p1,k+1  and Δ O p2,k  p2,k+1  are  similar。  The similarity  ratio  is  nkrk  ¼   nk。

Fig。 6。 Diagram  for  the  proof  of  Proposition 1。

The tangent values of angles φ1,k and φ2,k are described as  follows:

Because  φ1,k  and φ2,k  are assumed to be the same,  Eqs。  (7a)  and  (7b)  are the  same。  This  implies  that the similarity  ratios nk

and nk+1  are the  same。

By mathematical induction, the similarity ratio of step k (nk) is the same as that of step 1 (n1)。 This means that if two plane figures have the same first-order derivative in all discrete segments, they are mathematically similar。 This can also be applied to a continuous  system by decreasing the interval angle    ξk。

3。2。 Geometrical meaning of second-order derivative (change in angle of slope) of coupler point

In the previous section, the sign of the second-order derivative was also used to distinguish the coupler points。 Thus, we pro- pose  the  following  geometrical  interpretation  of  the derivative:

Proposition 2。 If two plane figures  have the same first  and second-order  derivative  profiles, they are the same    figure。

Proof Of Proposition 2。 The second-order  derivative  value  can be obtained  as  follows:

If the second-order derivative values of each plane figure are the same, Eq。 (8a) is equal to Eq。 (8b)。

2 1 ∂2 C

9Þ

∂Cx 2 big ¼ nk ∂Cx 2 small ð

Eq。 (9) shows that the similarity ratio nk  has to be one。 In other words, the two plane figures are exactly the same。

3。3。 New approach for four-bar linkage design

To obtain an optimal trajectory numerically without the possibility of generating an unintended trajectory shape, the following two-step  optimization  problem  was developed:

First  step  of optimization:

min RMSð1st deriv:G — 1st deriv:D Þ

Xlink ∈R5

Subject  to:  Desired  shape  (distinguished  by  geometrical features)。

T1 , T2 , T3 N 0 (Grashof condition)。

Second  step  of optimization:

min RMS 2nd deriv:G 2nd deriv:D

Xsize ∈R

Subject to: Xlink  ¼ Xlinkm:

where:

RMS(x): Root mean square value of  x

n  th derive。G:  n  th-order  derivative  equation  of  generated trajectory

n th derive。D:  n th-order  derivative  equation  of  desired  trajectory

Xlink  ¼ 。 l2      l3    l4    lcx     lcy 。=l1

Xlinkm:  Optimal  value  of  first  step  of  optimization

Xsize  ¼ l1

Because of Proposition 1, the shape of the trajectory is determined in the first step, in which the RMS value of the first-order derivative error between the generated trajectory and target is minimized。 After that, due to Proposition  2, the size of the  trajec- tory is determined by the second step of optimization, in which the RMS value of the second-order derivative error between the generated trajectory and target is minimized。 The reason for piding the optimization process into two steps is that the design variables can be separated into two groups: shape-dependent and size dependent variables。 The shape-dependent variable is Xlink,  which  contains the  rations  of the  link lengths  to l1。  The only  size-dependent variable  is l1,  which is  the standard  of   Xlink。