Math Problem Statement

What is the correct transformation matrix f1^0 for a two-degree freedom RR manipulator in the initial state?

Solution

The image depicts a two-degree-of-freedom RR (Revolute-Revolute) manipulator, and the question asks for the transformation matrix f10f_1^0, which represents the orientation of the coordinate system attached to link 1 with respect to the base (or frame 0) in the initial configuration.

In the initial state, both joints (θ1\theta_1 and θ2\theta_2) are set to 0 degrees, implying that the manipulator's first joint aligns with the base frame (frame 0) and the second joint aligns with the first link's frame.

Analyzing the frames:

  1. Frame 0 (Base frame):

    • x0x_0, y0y_0, and z0z_0 are the standard reference directions.
  2. Frame 1 (attached to link 1 after the first revolute joint):

    • In the initial configuration (θ1=0\theta_1 = 0), the x1x_1 axis aligns with x0x_0, the y1y_1 axis aligns with y0y_0, and the z1z_1 axis aligns with z0z_0.

    Therefore, the rotation matrix that transforms coordinates from frame 1 to frame 0 is the identity matrix, since the frames are perfectly aligned in the initial configuration.

The correct transformation matrix f10f_1^0 would be:

1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$$ Thus, the correct answer is the first option: $$f_1^0 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$$ ### Would you like more details on the reasoning behind this? Here are 5 related questions to further your understanding: 1. How does the transformation matrix change if $$\theta_1$$ is not zero? 2. How would the matrix $$f_2^1$$ look in the initial configuration? 3. What is the significance of each entry in a transformation matrix? 4. How does the Denavit-Hartenberg (DH) convention help in deriving these matrices? 5. How do you compute the overall transformation matrix for the end-effector in such manipulators? **Tip:** In robotics, always consider the initial configuration of joints carefully, as it simplifies many calculations.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Robotics
Transformation Matrices

Formulas

Transformation matrix formula for rotational transformations
Rotation matrix R = [cosθ -sinθ 0; sinθ cosθ 0; 0 0 1]

Theorems

Properties of Orthogonal Matrices
Transformation matrix properties in robotic kinematics

Suitable Grade Level

Undergraduate