Math Problem Statement

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

Solution

The problem asks us to find the homogeneous transformations among three frames, denoted by F1TF2^{F1}T_{F2}, F2TF3^{F2}T_{F3}, and F1TF3^{F1}T_{F3}. The goal is to show that F1TF3=F1TF2F2TF3^{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: F1F_1, located at the origin.
  • Frame 2: F2F_2, displaced by some translation and rotated by an angle θ\theta.
  • Frame 3: F3F_3, further displaced from F2F_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:

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.

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Math Problem Analysis

Mathematical Concepts

Homogeneous Transformation
Rotation Matrix
Translation Matrix
Matrix Multiplication

Formulas

Homogeneous Transformation Matrix: T = [R t; 0 1]
Rotation Matrix for 2D: R(θ) = [cosθ -sinθ; sinθ cosθ]
Translation Matrix: T = [1 0 tx; 0 1 ty; 0 0 1]

Theorems

Composition of Transformations
Matrix Multiplication of Homogeneous Matrices

Suitable Grade Level

Undergraduate level or higher (Mechanical Engineering, Robotics, or Computer Graphics)