## Using two positions on a rigid body to calculate rotation

Hello all,

I'm currently a undergrad university student doing research and I'm anaylising some position data.

The data is a time-series' of Eastings (x) and Northings (y) for two points (P1, P2) on a rigid body in motion (T, E1, N1, E2, N2), with a position reported every 1 minute. I know that P1 and P2 are separated by a pre-determined distance of 10 m.

My goal is to use the two positions to calculate the rotation of the rigid body over time (which I need every 1 second).

The two sets of positions contain noise (not correlated with each other) which I first need to "remove" to best model the motion. My first thought was to bestfit (in a least squares sense) a different polynomial to all four time series (E1-T, N1-T, E2-T, N2-T) and use those polynomials to then interpolate and calculate the rotation. However, this has lead to me wondering the following:

1. Would it be better to propagate the noise in the positions through the rotation calculation and then bestfit the rotation data (instead of removing the noise at the position stage and then calculating a rotation)?

2. Is it valid for me to separate the E and N from each other and model them separately?

3. I've noticed since I best fit them separately, and calculated the distance between P1 and P2 using the polynomials, the distance between the two points (which was very close around 10 m before) now has a larger standard deviation.

Any help would be appreciated!

Cheers!
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 Recognitions: Science Advisor Because there are several different defintions of "best estimate" in statistics and because your problem appears complicated, I think you should write a computer simulation of the situation (if you haven't done so already). With a simulation, you can specify the true values, simulate the noise, generate simulated data, apply various analysis methods to the data and compare the statistical behavior of the estimated values to the true values. One question is whether the "true" rotation involes a stationary axis of rotation? Are yo trying to estimate only the rate of rotation and is the true rate of rotation constant? Or are you trying to estimate a possibly varying rate of rotation? Questions like whether the E and N noises can be treated independently are questions about the physics of the problem and the measurement devices. They don't have a definite mathematical answer until the physics is modeled in a definite way.
 Hi Stephen, thanks for your reponse. Thanks for the suggestions about simulations. I have been considering starting down this path. The true rotation does not involve a stationary axis, the axis is experiencing translational motion as well. The rate and even direction (CW or CCW) of rotation is varying over time as the body translates and rotates in space. The physics of the problem is basically this: picture a boat in the water with noone at the helm, floating freely. But consider that the boat can only translate and rotate in a plane (no change in roll or pitch). The coordinates of two points on opposite ends of the boat are being measured every minute or so. Cheers!

## Using two positions on a rigid body to calculate rotation

How can you measure every minute but want to draw a conclusion about successive seconds? That sounds unrealistic to me. I am not sure if I have understood your problem correctly, but wouldn't the perpendicular bisectors of two successive positions of the chord p1 p2 intersect at an instantaneous centre of rotation?
 Hi Pongo I'm having to model the motion based on the 1-minute interval data (I'm using a polynomial) and then interpolate using the functional model to get down to the second-level. It's not ideal, but it's the best data I have. Yes, the perpendicular bisectors would intersect at a pole of planer displacement. My apologies, the Physics of the problem is where I'm likely falling down. I'm wondering if you have any thoughts on how I could use this fact? So far, to calculate the rotation between two instants, I've been calculating the azimuth of the p1 p2 chord at t1 and the same at t2, then subtracting them. Is this valid?
 Sorry. I now realize I was mistaken about the perp bisectors. I mis-applied a principle I use in the collapse of building frames, and my case doesn't include translation. I 'll think more about it. Diagrams help to clarify. Whether your 1 minute readings to 1 sec predictions are sufficiently accurate would have a relationship to the period of oscillation, wouldn't it?

Recognitions: