Question

For a frame F1​, we do the following transformations.

A) rotate F1​ by a rotation matrix R21​ with respect to F1​ to get frame F2​;
B) translate F2​ by a vector a1​ with respect to F2​ to get frame F3​;
C) rotate F3​ by a rotation matrix R32​ with respect to F2​ to get frame F4​; find the homogeneous transformation H21​,H32​,H43​, and H41​.

Answer 1

To find the required homogeneous transformation matrices, one should compute individual matrices for rotation and translation, and then chain them to get the final transformation from frame F1 to F4.

Step-by-step explanation:

The student is working with concepts related to homogeneous transformations in the context of frame transformations, which is common in robotics and kinematics within the field of engineering. To solve the question, you would compute the homogeneous transformation matrices H21​, H32​, H43​, and H41​ representing rotations and translations between frames F1, F2, F3, and F4.

Homogeneous transformation matrix combines rotation and translation into a single matrix, allowing for compact representation of transformations. For a rotation matrix R and translation vector a, the homogeneous transformation matrix H is:

• H = [R | a]
• [0 0 0 1]

In the given question, the student would need to calculate each individual transformation matrix as:

1. H21​ represents the rotation from F1 to F2 using R21​;
2. H32​ captures the translation from F2 to F3 by vector a1​;
3. H43​ is the rotation from F3 to F4 using R32​;
4. Lastly, H41​ is the resultant transformation from F1 to F4, which can be found by chaining the previous transformations together (H41​ = H21​ * H32​ * H43​).

To construct each transformation matrix, the student will use the given rotation matrices and translation vectors, with the final H41​ reflecting the combined transformations.