Multiple Rotations via Matrix Multiplication

Question

so I am having an issue understanding why you right multiply for current frame rotation vs left multiply for fixed frame.

Let us say you want to: Rotate about x, then Rotate about current y, then rotate about newer z.

So we have Rx, then Ry'(prime because it is the new y, not original) and then Rz''

It makes sense that it should be like this

You take a vector P and you rotate it about x to create a new vector p'. That is p'=Rxp. Then you take this new vector (p') and rotate it about z to create p'', that is: p''=Rz''p' = Rz''(Rxp)

That means that you left multiply when you are doing successive rotation about the current frame. However it is the opposite, and I can't understand why...

Answer 1

Something to remember about rotation matrices is that they have multiple applications (that each has a different role in rigid-body motion):

1. representing an orientation relative to a reference frame,
2. changing the reference frame of a variable (e.g., a vector), or
3. rotating a vector or a frame without changing its reference frame.

The first half of your question is talking about what happens when you need to produce a chain of successive rotations, which is only applicable when you want to represent the final orientation relative to the original fixed frame. You could then use this relative orientation to change the reference frame of a variable from the fixed frame to the "current" frame or to rotate a vector or frame to align with the "current" frame without changing its frame of reference.

However, the second half of your question is talking about rotating the vector p relative to the same fixed frame twice in succession, so the fact they are both left-multiplied makes sense. You would need to change the frame of reference of the vector in order for the phrase "current" frame to make sense since you haven't defined any frame of reference (or any frame) other than the fixed frame.

Check out section 3.2.1 of the Modern Robotics textbook for a more thorough explanation or have a look at the associated video on YouTube: https://www.youtube.com/watch?v=6KIPusOv5fA.