base & another which is at the next joint. You want to describe the

position and orientation of the 2nd frame with respect to the first in

terms of position and orientation. For the position, you use a

translation matrix.

The orientation is described by a rotation matrix which rotates the 2nd

frame from it's coincident position with the first frame and brings it

to it's final position.

Let the 1st frame have axis X, Y, Z

Let 2nd frame have axis x, y, z

To describe frame 2 in terms or frame 1, you start with both frames very

fully coincident at the beginning (i.e. even in orientation).

Then you have 2 methods. (You have many methods, actually)

1) Fixed Axis angles.

Then

- you rotate frame 2 by t1 about X - Rx(t1)

- you rotate frame 2 by t2 about Y - Ry(t2)

- you rotate frame 2 by t3 about Z - Rz(t2)

So new orientation of frame 2 is given by

Rz(t3)

*** Ry(t2) ***Rx(t1)

(Obviously, you pre-multiply the 1st matrix by the 2nd. And the

premultiply the result with the 3rd matrix)

2) Euler Angles

2nd way of describing it is by Euler angles - i.e. you rotate the 2nd

frame about one of it's own axis (x or y or z), instead of (X, Y or Z)

- rotate frame 2 by t3 about x - (y becomes y' & z becomes z')

- rotate frame 2 by t2 about y' - (x becomes x' & z becomes z'')

- rotate frame 2 by t1 about z" - (x' becomes x'' & y becomes y")

Now this transform is described again by

Rz(t3)

*** Ry(t2) ***Rx(t1)

I want to know how is this 2nd transform derived?

I know how the first one is derived because I know how to find the

rotation matrix for rotating a point about about a fixed axis. And I

know that if you are working with column vectors (for the point), you

premultiply the 1st rotation matrix by the 2nd rotation matrix.

However, I am not able to grok how you write the transformation matrices

for the 2nd case.