Rigid body motion for overdetermined translational data

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Hello Everyone,
I am new here and I hope this question wasn't asked yet as I could not find a useful answer to my problem.
Let's say I got the data for the translational acceleration of a rigid body for 3-4 different defined points in the x,y and z direction.
What I am trying to do is get the rotational and translational data for another point on that rigid body. As I am overdetermined I was wondering If there is an easy way to get the values through a least squares problem for example.
I am happy for any answer and suggestion you can provide.
Best regards, Rolfe
  1 Comment
James Tursa
James Tursa on 27 Jun 2022
Please post an example of the data you are working with, and then post specifically what variables/data you would like for an outcome.

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Answers (1)

William Rose
William Rose on 27 Jun 2022
In a movement analysis lab, one sometimes puts four or more markers on a body segment and tracks the markers with motion capture cameras. The motion capture system produces a record of the x,y,z coordinates of each marker versus time. There is always some noisiness and error in the marker coordinates. The investigator want to find the best estimate of the position and orientation of the segment, which is assumed to be rigid, at each time point. This is an overdetermined system. Does this sound like your problem? If so , the answer is singular value decomposition, which is a least-squares method. Before I get into that, confirm if this sounds like what you need.
William Rose
William Rose on 28 Jun 2022
@Rolfe, I assume the translational acceleromters ar aligned with each other and that you know the x,y,z displacements between the sensors and the x,y,z coordinates of the coenter of mass.
To get started, consider the following:
You might look at the equation above and think: "I have measurements at each instant of aA, aB, aC, aD... at each instant, and I know rAB, rAC, rAD, rBC, rBD, rCD, ..., so I can use that info to find a least squares solution to omega... and its derivative alpha, since there's only one omega, and it is common to all points in the body..." But not so fast, because what you measured are the translational accelerations in body coordinates, but the equation above is for accelerations in world coordinates. So I think you need to do more research and analysis. It would not surprise me if a Kalman filter would be useful here. Kalman filters are often useful in sensor fusion situations, and this is one.
If the engine is mounted on a block or stand, then maybe the rotations are small enough that we could use the equation above as is, because the difference between world coordinates and body coordinates is never huge. But if the engine is in a vehicle that is spinning around, then world and body coordinates will be very different.

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