(1) Analyse the situation when the cutter circle is internally tangential to the current offset circle at point G on the corner bisector and calculate the other intersection point S between the cutter circle and the bisector curve (see Fig. 13a).

Then, calculate the other offset circle externally tangential to the cutter circle at point S and set its centre as the point M. The in-centre point M will be taken as a limit point, and the next trochoid circle will be obtained between M and N.

(2) Establish a function

Ftro_ave=Fstr * (1+t1);

Z(P)= Ftro(P)–Ftro_ave≤ε (23)

where t1 is a specified value (for example t1=15%), Ftro (P) is the average milling force when the cutter rolls within the current offset circle, and ε is a relatively small control value (e.g., ε=1).

 First, for M and N, find the midpoint P((MX+NX)/2, (MY+NY)/2) on the corner bisector. Take it as a centre and create a new circle that is tangential to UW and VW (see Fig. 13b).

 Take the circle centred by P and the circle centred by N as the present analysis objects of the trochoidal trajectory. Calculate the average milling force when the cutter rolls along the current arc EGH, and set the angle as atro(P).

 Calculate Z (P) according to formula (23). If Z (P) >0 and Z (P) ≤ε, the point P is the required trochoidal-circle centre; exit the calculation. Otherwise, if Z (P) <0, then set M=P, or if Z (P) >0, set N=P.

 Repeat the above three steps until an appropriate P is obtained. Make the circle centred by P offset inward a distance of r; then, the next required trochoid circle can be obtained.

Sometimes the maximum milling force Ftro max  is adopted as the analysed object.

Therefore, the average engagement angle in formula (23) will be replaced by the maximum milling force Ftro max.

If the engagement angle is used as a substitute for the milling force, formula (23) is converted to

Z(P)= atro(P)–atro_ave≤ε (24)

The algorithm is similar to that above.

4.2.2 Corner trochoidal machining with isometric circles

The method of trochoidal milling corners with isometric circles can facilitate calculation and improve the continuity of the trajectory. It can also effectively reduce the cutting tool load. According to whether the trochoid circle can coat the corner arc formed in previous machining, two implementation methods can be obtained, as shown in Fig. 14a and Fig. 14b.

U-V is the current trajectory in Fig. 14, PU-PV is the previous trajectory, and PSQ is the corner arc formed after the previous machining. The distance from P to UW is |PH1|. Make a circle with |PH1| as diameter; then, internally offset by a distance of r (tool radius). Then,

rh1   

PH1  / 2 r

(25)

| A1 xP  B1 yP  C1  F1 (t1  r) | / 2 r

If the radius of the selected trochoid circle rh satisfies rh>rh1, this indicates that the trochoid circle can coat the former corner arc formed by previous milling. Thus, the method in Fig. 14a can be used. To calculate its machining trochoid, a suitable centre distance (L) of neighbouring circles should be first calculated so that the trochoidal machining can satisfy the limits of the engagement angle and the milling force. Then, the curve U is offset by rh to obtain the curve Up. According to the geometric characteristics of the corner curve PU-PV, we determine the initial circle (O1) on the curve Up and then calculate the trochoid circles guided by the curve Up one by one and store the results in the path list. Assuming that the coordinate of the current trochoid