Understanding Y-Axis Rotation Matrices in Robotics

Homogeneous transformation matrices are used in robotics to represent both rotations and translations. A rotation specifically about the Y-axis by an angle $\theta$ is represented by the following 3x3 rotation matrix, often denoted as $R_y(\theta)$: $ R_y(\theta) = \begin{pmatrix} \cos\theta & 0 & \sin\theta \\ 0 & 1 & 0 \\ -\sin\theta & 0 & \cos\theta \end{pmatrix} $ In robotics, these 3x3 rotation matrices are embedded within a 4x4 homogeneous transformation matrix. For a pure rotation about the Y-axis, the homogeneous matrix takes the form:

If translation is also involved, the matrix becomes:

The question asks which representations involve rotation about the Y-axis. This means we need to check if the top-left 3x3 submatrix matches $R_y(\theta)$.

Analyzing Homogeneous Representations

Let's examine each option:

• Option 1: The top-left 3x3 submatrix is $ \begin{pmatrix} \cos\theta & -\sin\theta & 0 \\ \sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end{pmatrix} $. This represents rotation about the Z-axis ($R_z(\theta)$), not the Y-axis.
• Option 2: The top-left 3x3 submatrix is $ \begin{pmatrix} 1 & 0 & 0 \\ 0 & \cos\theta & -\sin\theta \\ 0 & \sin\theta & \cos\theta \end{pmatrix} $. This represents rotation about the X-axis ($R_x(\theta)$), not the Y-axis.
• Option 3: The top-left 3x3 submatrix is $ \begin{pmatrix} \cos\theta & 0 & \sin\theta \\ 0 & 1 & 0 \\ -\sin\theta & 0 & \cos\theta \end{pmatrix} $, which is exactly $R_y(\theta)$. This matrix represents a pure rotation about the Y-axis.
• Option 4: The top-left 3x3 submatrix is $ \begin{pmatrix} \cos\theta & 0 & \sin\theta \\ 0 & 1 & 0 \\ -\sin\theta & 0 & \cos\theta \end{pmatrix} $, which is also $R_y(\theta)$. Although this matrix includes a translation component (3, 4, 2), it fundamentally involves rotation about the Y-axis.

Therefore, the homogeneous representations that involve rotation about the Y-axis are Options 3 and 4.