A Displacement-Rotation Algorithm

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The attached sketch shows an object which can move in x-y plane and rotate around z axis only. The movements are small and in the order of micrometers.

Point a, b and c are the locations of measurement. Each of these points have measuring devices which can give linear displacement in micrometers.

If the object is displaced by external force, the measuring devices will give linear displacement values. However, with these 3 displacement values, I would like to develop an algorithm which can give me overall displacement of the object along x and y axes as well rotation around z axis.

rotation matrix interpretation.webp
 
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You have three vector equations of the form <br /> a&#039; = R_z(\theta)a + t which gives six equations in three unknowns; this system is overdetermined and may not have a solution. Your best option is to choose \theta and t to minimize the error <br /> \|a&#039; - R_z(\theta)a - t\|^2 + \|b&#039; - R_z(\theta)b - t\|^2 + \|c&#039; - R_z(\theta)c - t\|^2
 
Thank you for your reply.
 

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