# 1. Drawing: Frames {A} and {B} - **Frame {A}: Fixed reference frame. - **Frame {B}:** Oriented by a 60° counter-clockwise rotation about the X-axis of {A}, then translated by \( Y = 4 \), \( Z = 8 \) (in {A} coordinates). ``` Z_A | | | Z_B | / | / | / (Frame {B}) | / | / | / +-------- Y_A / / / Y_B Origin of {B} is at (0, 4, 8) in {A}. X_B aligns with X_A (rotation is about X). ``` - **Translation vector:** \([0, 4, 8]^T\) - **Rotation:** 60° about X-axis. # 2. Compute Transformation Matrix ## a. Rotation Matrix (about X by 60°) \[ R_x(\theta) = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos\theta & -\sin\theta \\ 0 & \sin\theta & \cos\theta \end{bmatrix} \] \[ \theta = 60^\circ = \frac{\pi}{3} \] \[ \cos\theta = 0.5;\ \sin\theta = 0.866 \] \[ R_x(60^\circ) = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0.5 & -0.866 \\ 0 & 0.866 & 0.5 \end{bmatrix} \] ## b. Translation Vector \[ t = \begin{bmatrix} 0 \\ 4 \\ 8 \end{bmatrix} \] ## c. Homogeneous Transformation Matrix \({}^A T_B\) \[ {}^A T_B = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 0.5 & -0.866 & 4 \\ 0 & 0.866 & 0.5 & 8 \\ 0 & 0 & 0 & 1 \end{bmatrix} \] # 3. Transform the Point into Frame {A} Given point \( S_B = \begin{bmatrix} x_B \\ y_B \\ z_B \\ 1 \end{bmatrix} \), its coordinates in {A} are: \[ S_A = {}^A T_B \cdot S_B \] **Example:** If \( S_B = \begin{bmatrix} x_B \\ y_B \\ z_B \\ 1 \end{bmatrix} \), then \[ \begin{bmatrix} x_A \\ y_A \\ z_A \\ 1 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 0.5 & -0.866 & 4 \\ 0 & 0.866 & 0.5 & 8 \\ 0 & 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x_B \\ y_B \\ z_B \\ 1 \end{bmatrix} \] \[ \implies \begin{cases} x_A = x_B \\ y_A = 0.5 y_B - 0.866 z_B + 4 \\ z_A = 0.866 y_B + 0.5 z_B + 8 \end{cases} \] # 4. Physical Interpretation The new position \( S_A \) expresses the location of point \( S \) (originally given in {B}) relative to the base frame {A}. It accounts for both the rotation and translation that relate {B} to {A}, providing the absolute position of \( S \) in the overall reference frame.

# Introduction Understanding the spatial relationship between different coordinate frames is fundamental in rigid body kinematics and robotics. When a frame \( \{B\} \) is positioned relative to a reference frame \( \{A\} \), it involves a combination of rotation and translation. The rotation describes the orientation difference, while the translation indicates the positional offset. In this problem, frame \( \{B\} \) is oriented by a 60° counter-clockwise rotation about the X-axis of frame \( \{A\} \) and translated by \( Y=4 \) and \( Z=8 \) in the \( \{A\} \) coordinate system. A point \( S \) is given in frame \( \{B\} \), and the goal is to determine its position in frame \( \{A\} \). This requires the use of transformation matrices, which combine rotation and translation into a single homogeneous transformation, facilitating the coordinate transformation process. **Explanation:** This introduction emphasizes the importance of coordinate transformations in spatial analysis and sets the foundation for applying rotation and translation matrices. Recognizing how frames relate via these transformations enables the accurate conversion of points between coordinate systems, which is essential in many engineering applications. --- # Presentation of Relevant Formulas Required To Solve The Question ### 1. Rotation Matrix about X-axis \[ R_x(\theta) = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos\theta & -\sin\theta \\ 0 & \sin\theta & \cos\theta \end{bmatrix} \] **Rationale:** This matrix describes the rotation of a coordinate system about its X-axis by an angle \( \theta \). It preserves the X-component and rotates the Y and Z components accordingly, which is essential when the rotation axis is aligned with the X-axis. --- ### 2. Homogeneous Transformation Matrix \[ {}^A T_B = \begin{bmatrix} R & t \\ 0_{1 \times 3} & 1 \end{bmatrix} \] where \[ R = R_x(\theta) \] and \[ t = \begin{bmatrix} t_x \\ t_y \\ t_z \end{bmatrix} \] **Relevance:** This matrix combines rotation \( R \) and translation \( t \) into a single entity, enabling straightforward transformation of points from frame \( \{B\} \) to frame \( \{A\} \). --- ### 3. Coordinate Transformation of a Point \[ S_A = {}^A T_B \cdot S_B \] or, explicitly, \[ \begin{bmatrix} x_A \\ y_A \\ z_A \\ 1 \end{bmatrix} = \begin{bmatrix} R & t \\ 0 & 1 \end{bmatrix} \begin{bmatrix} x_B \\ y_B \\ z_B \\ 1 \end{bmatrix} \] **Significance:** This operation transforms the point's coordinates from frame \( \{B\} \) to frame \( \{A\} \) by applying the combined rotation and translation. --- # A Detailed Step-by-Step Solution ### Step 1: Define the rotation matrix \( R_x(60^\circ) \) **Calculation:** \[ \cos 60^\circ = 0.5,\quad \sin 60^\circ = 0.866 \] \[ R_x(60^\circ) = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0.5 & -0.866 \\ 0 & 0.866 & 0.5 \end{bmatrix} \] **Explanation:** This matrix rotates vectors by 60° about the X-axis, affecting only the Y and Z components. --- ### Step 2: Define the translation vector \( t \) \[ t = \begin{bmatrix} 0 \\ 4 \\ 8 \end{bmatrix} \] **Explanation:** The translation is specified as \( Y=4 \) and \( Z=8 \), with no translation along \( X \), indicating the origin of \( \{B\} \) is shifted relative to \( \{A\} \). --- ### Step 3: Form the homogeneous transformation matrix \( {}^A T_B \) \[ {}^A T_B = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 0.5 & -0.866 & 4 \\ 0 & 0.866 & 0.5 & 8 \\ 0 & 0 & 0 & 1 \end{bmatrix} \] **Explanation:** This matrix combines the rotation and translation, enabling the transformation of points from \( \{B\} \) to \( \{A\} \). --- ### Step 4: Transform the point \( S_B \) into \( \{A\} \) Suppose the point in \( \{B\} \) is \( S_B = \begin{bmatrix} x_B \\ y_B \\ z_B \\ 1 \end{bmatrix} \). Then: \[ S_A = {}^A T_B \cdot S_B \] which expands to: \[ \begin{bmatrix} x_A \\ y_A \\ z_A \\ 1 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 0.5 & -0.866 & 4 \\ 0 & 0.866 & 0.5 & 8 \\ 0 & 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x_B \\ y_B \\ z_B \\ 1 \end{bmatrix} \] Breaking down the components: \[ x_A = x_B \] \[ y_A = 0.5 y_B - 0.866 z_B + 4 \] \[ z_A = 0.866 y_B + 0.5 z_B + 8 \] **Explanation:** This linear combination accounts for the rotation and translation applied to the original point in \( \{B\} \), resulting in the point's coordinates in \( \{A\} \). --- ### **Summary of the transformation:** - The position of point \( S \) in \( \{A\} \) is obtained by applying the combined rotation (60° about X) and translation (Y=4, Z=8). - The transformation matrix encapsulates these operations, enabling straightforward coordinate conversion. --- # Conclusion The transformation matrix \( {}^A T_B \) effectively combines rotation about the X-axis and translation, enabling the conversion of any point from frame \( \{B\} \) to frame \( \{A\} \). The explicit coordinate transformation equations illustrate how the original point's coordinates are modified by the rotation and translation, providing the absolute position of the point in the reference frame \( \{A\} \). This process highlights the fundamental role of homogeneous transformation matrices in spatial coordinate transformations within robotics and mechanical systems.