Rigid body motion for overdetermined translational data

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

[James Tursa]

Please post an example of the data you are working with, and then post specifically what variables/data you would like for an outcome.

Answer 1

[William Rose]

@Rolfe,

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]

@Rolfe,

A different tracking problem which I had to solve was to determine the translational and rotational acceleration of the center of the head, and the position of the head, based on data from an accelerometer securely attached to the back of the head, while the subjects were doing soccer heading. The results were published here and here. This may be more applicable to your problem. The solution involves quaternions. When I did a literature search, I found different approaches and different notations. It took me some time to compare the different approaches, identify errors and inconsistencies in the published papers, and come up with a solution. If the information in the studies cited above is not enough, contact me.

[Rolfe]

Thank you for your answers!

@William Rose

I will try to access your papers as soon as possible. This will probably be possible in a few hours. What I could see from the absract you used 1 accelerometer including a gyroscope. I think the calculation might be quite different than what i would need. But I can tell you more as soon as i could take a closer look at your paper.

@James Tursa

In regards to the data I am working with I can give you more details. Some acceleration sensors where glued on an engine at different positions. Next the translational acceleration in x,y,z-directions over time for different positions on the engine where messasured.

The data for one sensor looks something like this after filters have been applied:

Outcome, which I am looking for, are the translational and the angular acceleration over time for the Center of Gravity of that engine. These calculations should be done in Matlab if possible.

It might be useful to calculate not with all channels but 6 to get a determined system. But I don't know if important information is lost that way. I would have to find out.

[William Rose]

@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.