Question

Which ONE or MORE among the following homogeneous representations in robotics involve(s) rotation about Y-axis by an angle \( \theta \)?

• A. \[ \begin{pmatrix} \cos\theta & -\sin\theta & 0 & 0 \\ \sin\theta & \cos\theta & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} \]
• B. \[ \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & \cos\theta & -\sin\theta & 0 \\ 0 & \sin\theta & \cos\theta & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} \]
• C. \[ \begin{pmatrix} \cos\theta & 0 & \sin\theta & 0 \\ 0 & 1 & 0 & 0 \\ -\sin\theta & 0 & \cos\theta & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} \]
• D. \[ \begin{pmatrix} \cos\theta & 0 & \sin\theta & 3 \\ 0 & 1 & 0 & 4 \\ -\sin\theta & 0 & \cos\theta & 2 \\ 0 & 0 & 0 & 1 \end{pmatrix} \]

Hint:

A pure Y-axis rotation matrix keeps the middle row and column as 0, 1, 0; a translation column does not change which axis the rotation happens about.

Answer 1

The Correct Option is C, D

Step 1: Understanding the Question:
A 4x4 homogeneous transformation matrix combines a 3x3 rotation block in its top-left corner with a 3x1 translation column on the right.
We only need to look at the rotation block of each option and check whether it matches the standard rotation matrix for a turn about the Y-axis.

Step 2: Standard Rotation Matrix About the Y-axis:
The rotation of a point by angle \( \theta \) about the Y-axis is given by
\[ R_y(\theta) = \begin{pmatrix} \cos\theta & 0 & \sin\theta \\ 0 & 1 & 0 \\ -\sin\theta & 0 & \cos\theta \end{pmatrix} \]
The telltale sign of a Y-axis rotation is that the middle row and middle column contain the untouched \( 1, 0, 0 \) pattern, because rotating about Y leaves the Y-coordinate unchanged.

Step 3: Checking Each Option:
Option (A) has the untouched row and column in the third (Z) position, with \( \cos\theta, -\sin\theta \) sitting in the top-left 2x2 block, so this is a rotation about the Z-axis, not Y.
Option (B) has the untouched row and column in the first (X) position, so this is a rotation about the X-axis, not Y.
Option (C) has \( \cos\theta, 0, \sin\theta \) in the first row, \( 0, 1, 0 \) in the second row, and \( -\sin\theta, 0, \cos\theta \) in the third row, which matches \( R_y(\theta) \) exactly, with zero translation.
Option (D) has the exact same rotation block as option (C), matching \( R_y(\theta) \), but this time with a nonzero translation column \( (3, 4, 2) \). Since the question asks which matrices involve a Y-axis rotation, and this one clearly contains that rotation alongside a translation, it still counts.

Final Answer:
Only options (C) and (D) contain the Y-axis rotation block, so both are correct even though (D) also carries a translation. \[ \boxed{\text{C and D involve rotation about the Y-axis}} \]