Suppose I have four coordinate frames in 3D space: A, B, X and Y. A and B are fixed. X and Y can move, but there is a fixed transformation between X and Y. This transformation is denoted T_XY, and it is this 4-by-4 transformation matrix (homogeneous coordinates) that I want to determine.
Now, I can take a series of measurements. At each measurement step, I can record the value of T_AX, and T_YB. So, I can measure the transformation between A and X, and the transformation between Y and B. I can then take as many measurements of these two as I want (the measurements have noise), and create a least-squares solution. Other than T_AX and T_YB, all other transformations are unknown.
So, in summary, I have the following transformations:
T_XY : fixed and unknown (this is what I want to determine)
T_AB : fixed and unknown
T_AX : variable and measurable
T_YB : variable and measurable
Is this possible? Is this a linear or a non-linear problem? If T_AB was known, then perhaps I could gather a series of equations in the form:
T_YB * T_XY * T_AX = T_AB
and solve this as linear least-squares. However, T_AB is unknown, which to me suggests this is a non-linear problem...
Below is a diagram visualising these coordinate frames and transformations.
Thanks!
