Finding a 3D rotation that composed with another rotation R results in < angle than R
I have a 3D rotation R1 (which i describe in an axis-angle way) whose axis I don't know at all and has no restrictions, but has an angle close to 180°. And I need to have another rotation R2 that, when "applied" to R1 results in a composed rotation R3 that can have any axis but its angle should be lower than 180° (say, at least 10° lower).
Any guidance on how to come up with a rotation like R2 would be greatly appreciated.
Why do I need this? As a piece of an algorithm I need to find the rotation R1 that best matches some cloud of points to another, I have an algorithm for that that works for all axes and angles but for angles near 180° (in practice, between 178° and 182°). When It doesn't work, either the angle and the axis come out erroneous from it.
That's why I'm trying to solve for both R1 and R3, and keep the solution that gives me the lowest least squares error when comparing the clouds of points. (And I'll be keeping the rotation obtained by solving for R3 when R1 had an angle close to 180°)