[3D Rotation Matrix] Screw Parameter

• Which comes from Euler’s Rotation Theorem.

 

[θ, w1, w2, w3]

- θ : A rotation angle, how much to rotate around that axis

- w1, w2, w3 : A unit vector, euler axis(the rotation axis)

 

• Any rotation of a rigid body around a fixed point is equivalent to a single rotation about a fixed axis by some angle θ. In other words, Instead of thinking of a 3D rotation as a sequence of yaw-pitch-roll (Euler angles), I can always represent it as a single twist around one axis.
• It avoids gimbal lock.

---

• We can compose a 3x3 rotation matrix by Rodrigues’ rotation formula.

\[ R = I + \sin\theta \,[\hat{w}]_{\times} + (1 - \cos\theta)\,[\hat{w}]_{\times}^{2} \]

where 

\[ [\hat{w}]_{\times} \] is the skew-symmetric matrix of \[ [\hat{w}] \] , like: \[ [\hat{w}]_{\times} = \begin{bmatrix} 0 & -w_{3} & w_{2} \\ w_{3} & 0 & -w_{1} \\ -w_{2} & w_{1} & 0 \end{bmatrix} \]

Rodrigues' formula builds a full 3×3 rotation matrix using just the axis and angle.

---

For example, given :

\begin{aligned} \text{Rotation axis: } &\quad \hat{w} = [1,0,0] \;\rightarrow\; \text{rotate around the x-axis} \\ \text{Rotation angle: } &\quad \theta = 90^\circ = \frac{\pi}{2} \text{ radians} \end{aligned}

---

Step 1: Skew-symmetric matrix \[ [\hat{w}]_{\times} = \begin{bmatrix} 0 & -w_{3} & w_{2} \\ w_{3} & 0 & -w_{1} \\ -w_{2} & w_{1} & 0 \end{bmatrix} = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0 \end{bmatrix} \]

---

Step 2: Plug into Rodrigues' formula \[ R = I + \sin(\theta)[\hat{w}]_{\times} + (1 - \cos(\theta))[\hat{w}]_{\times}^{2} \] We know: \[ \sin\!\left(\frac{\pi}{2}\right) = 1 \] \[ \cos\!\left(\frac{\pi}{2}\right) = 0 \] So: \[ R = I + [\hat{w}]_{\times} + [\hat{w}]_{\times}^{2} \]

---

Step 3: Compute \[ [\hat{w}]_{\times}^{2} = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0 \end{bmatrix} \cdot \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0 \end{bmatrix} = \begin{bmatrix} 0 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{bmatrix} \]

---

Step 4: Add up the matrices \[ R = I + [\hat{w}]_{\times} + [\hat{w}]_{\times}^{2} \] \[ R = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} + \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0 \end{bmatrix} + \begin{bmatrix} 0 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0 \end{bmatrix} \]

---

Final Answer: \[ R = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0 \end{bmatrix} \]