How to formulate 2D frames if I have the origin of the frames (x, y) and the inclination angle of the frame with respect to the x-axis?

How to formulate 2D frames if I have the origin of the frames (x, y) and the inclination angle of the frame with respect to the x-axis? Then I need to compute the transformation matrix of the frames w.r.t the reference frame (which is normal x and y directions)??

The attached files are the origins and the corrosponding angles.

Answer 1

[Matt J]

T=makehgtform('translate',[x,y,0],...
              'zrotate',angleDegrees*pi/180, ...
              'translate',[-x0,-y0,0]);

T=T([1,2,4],[1,2,4]); %transformation matrix

[M]

@Matt J, I attached the origins and the corrosponding angles files, Can you please help how to pass them to your code to test it. Thanks

[M]

@Matt J, could you please help.

[Matt J]

angleDegrees=load('Angles').Angles;
xy=load('origins').origins; 
xy=vertcat(xy{:});  x=xy(:,1); y=xy(:,2);
x0=x(1); y0=y(1);

T=nan(3,3,23);
for i=1:23
    
      t=makehgtform('translate',[x(i),y(i),0],...
                   'zrotate',angleDegrees(i)*pi/180, ...
                   'translate',[-x0,-y0,0]);

       T(:,:,i)=t([1,2,4],[1,2,4]); %transformation matrix
    
    
end
whos T

  Name      Size              Bytes  Class     Attributes

  T         3x3x23             1656  double              

[M]

@Matt J, I got the following erorr:

Undefined variable origins.

Also, Can you please expline what is x0 and y0? and why did you take the negative sign here 'translate',[-x0,-y0,0]);

What does this line mean "T=T([1,2,4],[1,2,4]); %transformation matrix"?

Thanks

[Matt J]

I fixed it.

T=T([1,2,4],[1,2,4]) removes the 3rd column and row of T. Only row/column 1,2, and 4 are kept.

[M]

@Matt J Thank you. I have the following questions:

1- what are x0 and y0 represent?

2- Why did you take these values as the following: "x0=x(1); y0=y(1);"?

3- Why did you take the negative sign here: "'translate',[-x0,-y0,0]);"?

[Matt J]

1- what are x0 and y0 represent?

The origin in the first frame

2- Why did you take these values as the following: "x0=x(1); y0=y(1);"?

x(1), y(1) are the origin coordinates in the first frame after reorganizing origin data from your file. BTW, you should get out of the habit of storing matrices of equal size in cells.

3- Why did you take the negative sign here: "'translate',[-x0,-y0,0]);"?

To move the origin to (0,0) to make rotation easier. But now that you mention it, I think it should really be

T=makehgtform('translate',[x0,y0,0],...
              'zrotate',angleDegrees*pi/180, ...
              'translate',[-x,-y,0]);

You will of course test these matrices to make sure they do what you want.

[M]

@Matt J Thanks, But why you dont make the origins of the based frame x0=0 and y0=0 from the beginning?

[Matt J]

I've no way of "making" the origin whatI want. You're the one who defined the origins in your origins.mat file.

[M]

@Matt J Ok, but my question why did you choose the first origin in the .mat file to make it the reference, then you translate other origins to that origin ?

Also, Could you please tell me what does the fourth column in the transformation matrix represents?