SO(3) & Rotation Vector

The Special Orthogonal Group $SO(N)$ is the set of all rotation matrices in $N$ dimensions. $SO(2)$ is the set of all rotations in 2D and $SO(3)$ is the set of all rotations in 3D. This note also covers the axis-angle (rotation vector) way of describing a 3D rotation, which is the form URScript uses.

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$SO(2)$, Rotations in 2D

A 2D rotation by angle $\theta$ is:

$$R(\theta )=\left(\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right)$$

Every 2D rotation lives in $SO(2)$, and it has 1 degree of freedom:

• $\theta$ is the only variable, so one parameter fully specifies the rotation
• every point on the unit circle corresponds to one rotation angle
• one parameter gives one loop, which is 1 DOF

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$SO(3)$, Rotations in 3D

$SO(3)$ is the set of all rotations in 3D space. It has 3 degrees of freedom, since we can rotate about any of the $x$, $y$, $z$ axes, and we need 3 numbers (for example roll, pitch, yaw) to fully specify a 3D rotation.

Counting the DOF straight from the matrix:

• a 3D rotation matrix has 9 entries
• orthogonality imposes 6 constraints
• $9-6=3$, so $SO(3)$ has 3 DOF

Two properties hold for any rotation matrix $R$:

$$R^{-1}=R^{T}R{R}^T=I$$

The full per-axis rotation matrices are detailed in Rotation Matrices.

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$SO(2)$ vs $SO(3)$

A key structural difference:

• $SO(2)$ is commutative, so the order of two 2D rotations does not change the result
• $SO(3)$ is non-commutative, so in general $R_1{R}_2\ne R_2{R}_1$

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Angular Velocity Vector

A short aside on how a rotating frame moves over time:

• it is written $\omega =\hat{\omega }\dot{\theta }$, a unit axis $\hat{\omega }$ times the rate of rotation $\dot{\theta }$
• if a body frame's $x$ axis traces a circle, the linear velocity of that axis is tangent to the circle, ${\dot{\hat{x}}}_b={\omega }_s\times {\hat{x}}_b$

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The Rotation Vector (Axis-Angle)

A 3D rotation can be packed into a single rotation vector $\vec{r}=\theta \hat{\omega }$:

• the direction of $\vec{r}$ is the rotation axis
• the magnitude of $\vec{r}$ is the rotation angle

Given $\vec{r}=(r_x,r_y,r_z)$, recover the axis and angle by:

$$\hat{\omega }=\frac{\vec{r}}{|\vec{r}|}(\text{extract axis})\theta =|\vec{r}|(\text{extract angle})$$

In URScript the rotation vector is the last three numbers of a pose, $[r_x,r_y,r_z]$ inside $p[x,y,z,r_x,r_y,r_z]$.

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Worked Examples

Rotate 90 degrees about the z axis:

• axis $\hat{\omega }=(0,0,1)$, angle $\theta =\pi /2$
• rotation vector $\vec{r}=(0,0,\pi /2)\approx (0,0,1.5708)$
• in URScript, p[x, y, z, 0, 0, 1.5708]

Rotate 180 degrees about the x axis:

• axis $\hat{\omega }=(1,0,0)$, angle $\theta =\pi$
• rotation vector $\vec{r}=(\pi ,0,0)\approx (3.1416,0,0)$
• in URScript, p[x, y, z, 3.1416, 0, 0]

Rotate 60 degrees about the diagonal axis $(1,1,1)/\sqrt{3}$:

• axis $\hat{\omega }\approx (0.577,0.577,0.577)$, angle $\theta =\pi /3\approx 1.047$
• rotation vector $\vec{r}\approx (0.604,0.604,0.604)$
• in URScript, p[x, y, z, 0.604, 0.604, 0.604]

A full pose from the notes is p[0.2, 0.5, 0.1, 0, 0, 3.14], where the orientation part is $\theta =\pi \approx 3.14$ about the axis $\hat{r}=[0,0,1]$, a half turn about $z$.

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Rodrigues' Formula

To convert a rotation vector $\vec{r}$ into a rotation matrix $R$, use Rodrigues' formula. This matters because URScript stores orientation as axis-angle, but the controller composes rotations as matrices, so converting between the two means going through the matrix form.

For a unit axis $\hat{\omega }$ and angle $\theta$:

$$R=I+\sin \theta [\hat{\omega }]+(1-\cos \theta )[\hat{\omega }{]}^2$$

Where:

• $I$ is the $3\times 3$ identity matrix
• $[\hat{\omega }]$ is the skew-symmetric matrix of $\hat{\omega }$
• $[\hat{\omega }{]}^2$ is that matrix multiplied by itself

$$[\hat{\omega }]=\left(\begin{array}{ccc}0& -{\hat{\omega }}_3& {\hat{\omega }}_2\\ {\hat{\omega }}_3& 0& -{\hat{\omega }}_1\\ -{\hat{\omega }}_2& {\hat{\omega }}_1& 0\end{array}\right)$$