geometric meaning of rotation matrices related to different frames

Question

Consider the following rotation matrices:

$^{a}R_{b}$, $^{a}R_{c}$ and $^{a}R_{d}$

in which

• $^{x}R_{y} \in SO(3)$ describes the orientation of a generic frame $y$ wrt the coordinates system of another frame $x$
• $a$ represents the fixed world frame
• $b$, $c$ are two different fixed
• $d$ is a rotating frame
• all frames shares the same origin $o \in R_{3}$ (i.e. no translations involved)

$$$$ What is the geometric meaning, if any, of computing the following operation: $\overline{R} \text{ } = \text{ } ^{a}R_{b} \text{ } \cdot \text{ } ^{c}R_{d}$ ?

( $^{c}R_{d} \text{ } = \text{ } ^{a}R_{c}^{T} \text{ } \cdot \text{ } ^{a}R_{d}$ )

$$$$ Can we say that $\overline{R}$ represents the orientation of frame $d$ wrt world frame $a$, computed considering that the change in orientation between $c$ and $d$ is the same change in orientation as between $b$ and $d$?

Does this make any sense from a mathematical and/or geometric point of view?

Answer 1

Using the notation you have given, the intuitive geometric meaning of rotation matrix multiplication is most clear when the subscript of the first matrix is equal to the superscript of the second (i.e. ${}^{x}R_{a} \cdot {}^{a}R_{y}$). If the column vectors of a rotation matrix ${}^{0}R_{1}$ represent the x, y, and z-axes of frame $1$ in frame $0$, then multiplying that matrix with another rotation matrix will rotate those axis vectors to represent frame $1$ in frame $x$ (whatever frame you just rotated "into").

So, what intuition do we have for ${}^{a}R_{b} \cdot {}^{c}R_{d}$? Short answer, not a lot.

We might put words to this equation as "rotating the axes vectors of frame $d$ represented in frame $c$ by the rotation of frame $b$ relative to frame $a$". Since there is no clear relationship between $b$ and $c$, it is hard to intuit anything useful from this.

Now, from the information you have given, there is enough to "connect" the rotation you described. define ${}^{b}R_{c} \triangleq {}^{a}R_{b}^{T} \cdot {}^{a}R_{c}$, then you have:

$\bar{R} = {}^{a}R_{b} \cdot {}^{b}R_{c} \cdot {}^{c}R_{d}$

for whatever that's worth!

---

As a final note, I think the last portion here conveniently summed up your question about relating the fixed frames - although that's not necessarily what I had intended. Since frame $b$ and frame $c$ are both fixed frames, the rotation matrix you gave as $\bar{R}$ is only one rotation away from expressing $d$ in the $a$ frame, That rotation being $({}^{a}R_{c} \cdot {}^{a}R_{b}^{T}) \cdot \bar{R}$ (this being the $\bar{R}$ you have defined).

Answer 2

I believe your question regarding the geometric meaning of rotation matrices to different frames I explain on Medium