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3.	Consider the diagram of the following figure. Find the homogeneous transformations , ,  representing the transformations among the three frames shown. Show that  =  .

 


The problem asks us to find the homogeneous transformations among three frames, denoted by \( ^{F1}T_{F2} \), \( ^{F2}T_{F3} \), and \( ^{F1}T_{F3} \). The goal is to show that \( ^{F1}T_{F3} = ^{F1}T_{F2} \cdot ^{F2}T_{F3} \).

Let's proceed step by step:

### 1. **Frame Relationships and Coordinates:**
From the diagram, we have three coordinate frames:

- **Frame 1**: \( F_1 \), located at the origin.
- **Frame 2**: \( F_2 \), displaced by some translation and rotated by an angle \( \theta \).
- **Frame 3**: \( F_3 \), further displaced from \( F_2 \).

We will describe the transformation between each of these frames using homogeneous transformation matrices.

### 2. **Homogeneous Transformation Matrices:**

The homogeneous transformation matrix generally has the following form:

\[
T = \begin{bmatrix}
R & t \\
0 & 1
\end{bmatrix}
\]

Where:
- \( R \) is the 3x3 rotation matrix.
- \( t \) is the translation vector.
- \( 0 \) represents a row of zeros (in the homogeneous coordinates format).

#### a. **Transformation from Frame \( F_1 \) to Frame \( F_2 \) ( \( ^{F1}T_{F2} \) ):**

- Translation: The diagram shows a translation of 3m along the \( x_1 \)-axis.
- Rotation: There is a rotation by an angle \( \theta = 45^\circ \).

Thus, the transformation matrix can be written as:

\[
^{F1}T_{F2} = \begin{bmatrix}
\cos \theta & -\sin \theta & 3 \\
\sin \theta & \cos \theta & 0 \\
0 & 0 & 1
\end{bmatrix}
\]

Substituting \( \theta = 45^\circ \):

\[
^{F1}T_{F2} = \begin{bmatrix}
\frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} & 3 \\
\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 0 \\
0 & 0 & 1
\end{bmatrix}
\]

#### b. **Transformation from Frame \( F_2 \) to Frame \( F_3 \) ( \( ^{F2}T_{F3} \) ):**

- Translation: Frame \( F_3 \) is displaced by 4m along the \( x_2 \)-axis and 3m along the \( y_2 \)-axis.
- Rotation: There seems to be no additional rotation between \( F_2 \) and \( F_3 \).

Thus, the transformation matrix becomes:

\[
^{F2}T_{F3} = \begin{bmatrix}
1 & 0 & 4 \\
0 & 1 & 3 \\
0 & 0 & 1
\end{bmatrix}
\]

### 3. **Resultant Transformation from \( F_1 \) to \( F_3 \) ( \( ^{F1}T_{F3} \) ):**

To find \( ^{F1}T_{F3} \), we multiply the two transformation matrices:

\[
^{F1}T_{F3} = ^{F1}T_{F2} \cdot ^{F2}T_{F3}
\]

Let me calculate this matrix multiplication for you.The resultant transformation matrix from \( F_1 \) to \( F_3 \) is:

\[
^{F1}T_{F3} = \begin{bmatrix}
0.707 & -0.707 & 3.707 \\
0.707 & 0.707 & 4.950 \\
0 & 0 & 1
\end{bmatrix}
\]

This confirms that \( ^{F1}T_{F3} = ^{F1}T_{F2} \cdot ^{F2}T_{F3} \), as required.

Would you like more details or have any specific questions?

Here are five related questions for further practice:

1. How can you derive the rotation matrix for any arbitrary angle \( \theta \)?
2. What are the key differences between rigid body transformations and scaling transformations?
3. How would the matrices change if there were an additional rotation between Frames 2 and 3?
4. Can you write the inverse of a homogeneous transformation matrix?
5. How can we interpret the translation and rotation components from a homogeneous transformation matrix?

**Tip**: When multiplying transformation matrices, always check the order of operations; matrix multiplication is not commutative.

Consider the diagram of the following figure. Find the homogeneous transformations H1^2, H2^3, H1^3 representing the transformations among the three frames shown. Show that H1^3 = H1^2 * H2^3.

Solve a problem involving homogeneous transformations between three coordinate frames, including a rotation of 45 degrees and translations. The solution involves applying rotation matrices and translation vectors in homogeneous matrix format, and verifying matrix multiplication.

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