Modelling the 4 Bar Linkage

Modelling the 4 Bar Linkage

It is convenient to build the initial model in a spreadsheet. The results can be displayed as a graph:

First steps would be to start at the Axle (0,0) on the black base plate and calculate the red crank position:

X1 = L1*COS(RADIANS(A0))+X0

Y1 = L1*SIN(RADIANS(A0))+Y0

Where:

(X0,Y0) is the Axle position
A0 is the rotation angle (clockwise from the X axis)
L1 is the Thigh length
(X1,Y1) is the Crank position

Next is the find the intersection of two circles from the Crank and the Hip:

There are generally two solutions and the correct one needs to be selected. If the order of the parameters are consistent, then the selected solution will be consistent.

One method to solve this problem follows:

Source: https://math stackexchange.com/questions/256100/how-can-i-find-the-points-at-which-two-circles-intersect

Here are the Circle Intersection Calculations:

CX1 = Crank X

CY1 = Crank Y

R1 = Thigh length (L2)

CX2 = Hip X

CY2 = Hip Y

R2 = Idler Length (L3)

D = SQRT((CX1-CX2)^2+(CY1-CY2)^2)

L = (R1^2-R2^2+D^2)/2/D

H = SQRT(R1^2-L^2)

IX1 = L/D*(CX2-CX1)+H/D*(CY2-CY1)+CX1

IY1 = L/D*(CY2-CY1)-H/D*(CX2-CX1)+CY1

IX2 = L/D*(CX2-CX1)-H/D*(CY2-CY1)+CX1

IY2 = L/D*(CY2-CY1)+H/D*(CX2-CX1)+CY1

The two solutions are:

(IX1,IY1)
(IX2,IY2)

For the way I ordered my circles, the solution 2 (IX2,IY2) was chosen for the Knee position:

X2 = IX2

Y2 = IX2

Finally the Foot has to be extended (L4) from the Knee at 90 degrees toward the Floor:

Thigh Angle: A = DEGREES(ATAN2(X2-X1,Y2-Y1))
Foot X: X3 = L4*COS(RADIANS(A+90))+X2
Foot Y: Y3 = L4*SIN(RADIANS(A+90))+Y2

Note: Spreadsheets use atan2(Cos,Sin) while C code uses atan2(Sin,Cos).

Having satisfied myself that I the equations are working, I can code the model in C code for optimisation.

AlanX

Optimising the 4 Bar Linkage

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