Rotation Matrices

A rotation matrix $R$ describes the orientation of one coordinate frame relative to another, and it also acts as an operator that rotates a vector or a frame. This note covers how orientations are represented, how reference frames are changed, and the premultiply versus postmultiply rule.

---

Representing an Orientation

$R_{sc}$ is the representation of the orientation of frame $\{C\}$ with respect to a fixed frame $\{S\}$. When the reference frame is understood, it is also written $R_c$.

---

Changing Reference Frame

Rotations chain through a shared frame:

$$R_{ac}=R_{ab}{R}_{bc}$$

• the inner subscripts match on the shared middle frame $b$
• this lets you re-express an orientation in a different reference frame

---

Rotation About Coordinate Frame Axes

The basic rotations about each axis:

$$\text{Rot}(\hat{x},\theta )=\left(\begin{array}{ccc}1& 0& 0\\ 0& \cos \theta & -\sin \theta \\ 0& \sin \theta & \cos \theta \end{array}\right),\text{Rot}(\hat{y},\theta )=\left(\begin{array}{ccc}\cos \theta & 0& \sin \theta \\ 0& 1& 0\\ -\sin \theta & 0& \cos \theta \end{array}\right)$$ $$\text{Rot}(\hat{z},\theta )=\left(\begin{array}{ccc}\cos \theta & -\sin \theta & 0\\ \sin \theta & \cos \theta & 0\\ 0& 0& 1\end{array}\right)$$

---

General Axis-Angle Rotation

For a general unit axis $\hat{\omega }=({\hat{\omega }}_1,{\hat{\omega }}_2,{\hat{\omega }}_3)$, the rotation by angle $\theta$ is:

$$\text{Rot}(\hat{\omega },\theta )=\left(\begin{array}{ccc}{c}_{\theta }+{\hat{\omega }}_1^2(1-c_{\theta })& {\hat{\omega }}_1{\hat{\omega }}_2(1-c_{\theta })-{\hat{\omega }}_3{s}_{\theta }& {\hat{\omega }}_1{\hat{\omega }}_3(1-c_{\theta })+{\hat{\omega }}_2{s}_{\theta }\\ {\hat{\omega }}_1{\hat{\omega }}_2(1-c_{\theta })+{\hat{\omega }}_3{s}_{\theta }& c_{\theta }+{\hat{\omega }}_2^2(1-c_{\theta })& {\hat{\omega }}_2{\hat{\omega }}_3(1-c_{\theta })-{\hat{\omega }}_1{s}_{\theta }\\ {\hat{\omega }}_1{\hat{\omega }}_3(1-c_{\theta })-{\hat{\omega }}_2{s}_{\theta }& {\hat{\omega }}_2{\hat{\omega }}_3(1-c_{\theta })+{\hat{\omega }}_1{s}_{\theta }& c_{\theta }+{\hat{\omega }}_3^2(1-c_{\theta })\end{array}\right)$$

where $s_{\theta }=\sin \theta$ and $c_{\theta }=\cos \theta$, with the symmetry $\text{Rot}(\hat{\omega },\theta )=\text{Rot}(-\hat{\omega },-\theta )$. This is the matrix form of the axis-angle rotation described in SO(3).

---

Rotation as an Operator

A rotation matrix can be read two ways:

• as a description, $R_{sb}$ is the orientation of body frame $\{b\}$ in space frame $\{s\}$
• as an operator, $R=\text{Rot}(\hat{\omega },\theta )$ actively rotates a vector or a frame

---

Premultiply vs Postmultiply

The single rule: an operator's axis is interpreted in the frame adjacent to it on the left, and if nothing is on the left it is interpreted in the space frame. Always read a product right to left.

Premultiply, $R_{sb}^{\prime }=R{R}_{sb}$:

• $R$ has nothing on its left, so its axis is the space frame's axis ${\hat{z}}_s$
• read it as rotating the body about the base (space) axis
• it takes the current description of $\{b\}$, applies $R$, and gives the new description $\{b^{\prime }\}$

Postmultiply, $R_{sb}^{\prime \prime }=R_{sb}R$:

• $R$ has $R_{sb}$ on its left, so its axis is the body frame's axis ${\hat{z}}_b$
• read it as rotating the body about its own axis
• it starts with $R_{sb}$ and applies $R$, but interprets the axis in the body frame $\{b\}$ rather than the space frame

---

Worked Cases

Reading longer products right to left, the operator nearest to $R_{sb}$ acts first:

• $R_{new}=R_{sb}{R}_1{R}_2$ applies $R_1$ then $R_2$ about the body's own axes
• $R_{new}=R_1{R}_2{R}_{sb}$ rotates about the world axes, since $R_1{R}_2$ has nothing on its left
• $R_{new}=R_1{R}_{sb}{R}_2$ mixes the two, where $R_2$ acts in the body frame and $R_1$ acts in the world frame (right to left)