# Covariance of orientation data

Discussion in 'Scientific Statistics Math' started by Dave, Jan 3, 2011.

1. ### DaveGuest

Hi all,

Suppose I have an accelerometer that gives me an over-all vector for
gravity, in the device's local coordinate system. So each of the x, y,
and z values from that accelerometer have independent noise (lets
assume it's gaussian) in each axis. I also have a magnetometer (i.e.,
compass) which gives me another vector that is subject to the same
sort of noise.

I can then derive vectors that describe the transform from the
device's local space to a global space. The global Y axis is the
negative of the accelerometer. Global X is the cross product of the
magnetometer output and Y. And Z is the cross between X and Y.
Assuming those vectors are normalized, then the matrix ( X, Y, Z ) is
the inverse of the transform taking me from the local space to global
space.

So what does this have to do with statistics? Well, the filtering
techniques that I have seen for these orientations require covariance
matrices. Since matrices are nonlinear, the filters require me to
convert this rotation matrix into a "rotation vector", which is
essentially the axis of rotation scaled by the angle of the rotation
(kinda like axis-angle representation). Then, a covariance matrix is
used to determine how new measurements are incorporated into previous
estimates (e.g., via Kalman filters, etc).

So, with the data that I am given for the two sensors (i.e., standard
deviations for the x,y,z of both the magnetometer and accelerometer),
how would I construct a covariance matrix for the rotation vector?

Thanks.

Dave

Dave, Jan 3, 2011

2. ### Ray KoopmanGuest

Preliminatry results, that assume only that the errors are
independent and unbiased: It looks like only X and Y are unbiased.
The bias in Z is -{ M1(u2 + u3), M2(u1 + u3), M3(u1 + u2) },
where M = {M1,M2,M3} is the "true" magnetometer vector,
and u1,u2,u3 are the error variances in the accelerometer readings.

Do you agree? Am I on the right track?

Ray Koopman, Jan 4, 2011