Math Problem Statement
Solution
The image depicts a two-degree-of-freedom RR (Revolute-Revolute) manipulator, and the question asks for the transformation matrix , 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 ( and ) 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:
-
Frame 0 (Base frame):
- , , and are the standard reference directions.
-
Frame 1 (attached to link 1 after the first revolute joint):
- In the initial configuration (), the axis aligns with , the axis aligns with , and the axis aligns with .
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 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.Ask a new question for Free
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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
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