Algorithm (Workflow)

This algorithm is designed to orthogonalize a given homogeneous transformation (HT) matrix using Singular Value Decomposition (SVD). It ensures that the rotation part of the matrix is orthogonal and right-handed.

Input Parameters

• T: The original HT matrix, which is a 4x4 matrix used in kinematics to represent transformations including rotation and translation.

Output Results

• TS: The SVD-corrected HT matrix with an orthogonalized rotation matrix.

Algorithm Steps

1. Extract the rotation block from the input matrix T. This is the top-left 3x3 submatrix, denoted as R.
2. Perform Singular Value Decomposition (SVD) on the rotation matrix R. This decomposes R into three matrices: U, S, and V, where R = U * S * V'.
3. Reconstruct an orthogonal matrix R_ortho by multiplying U and the transpose of V (i.e., R_ortho = U * V').
4. Ensure the reconstructed matrix R_ortho is right-handed. This is done by checking the determinant of R_ortho. If the determinant is negative, adjust the last column of U by negating it, and then recompute R_ortho.
5. Replace the original rotation block in T with the orthogonalized matrix R_ortho.
6. Assign the modified matrix to TS, which is the output of the function.

Additional Information

If the function is called without output arguments, it calculates the error as the sum of absolute differences between the original and corrected matrices. It then prints this error and the original vector for debugging purposes.