# Fastest interpolation method for attitude quaternions?

1. Jun 28, 2012

### vicjun

I hesitated between posting this in the Mathematics forum or here, but since it's fairly applied, I chose this place. Sorry if it should've gone somewhere else.

I posted another thread earlier (https://www.physicsforums.com/showthread.php?t=599737), about having trouble finding the quaternion derivative from the quaternion and the angular velocity. That problem is now resolved. I mentioned that I interpolate the quaternion derivatives to find a continous function.

The interpolation method used is least-squares fitting using a n-degree polynomial (n varies between 3 and 7, this is determined automatically), resulting in four n-degree polynomials, one for each quaternion component. This works fine.

I estimated the error in the interpolation by resampling (using the polynomials above) the quaternion component derivatives at the same dates used for the interpolation, and then calculating the angle between a quaternion derivative before and after interpolation. It is in the order of 10-5 radians. Is this a correct method of estimating the error or is there a better way?

I would also like the explore other interpolation methods, that are perhaps more suited for quaternions. The point of all this is to calculate the rotation of a spacecraft (i.e., a body-fixed frame) relative a fixed frame (in this case the local orbital frame). I found a document (http://www.geometrictools.com/Documentation/Quaternions.pdf) detailing three other methods: Spherical Linear Interpolation, Spherical Cubic Interpolation and Spline Interpolation. Are these better for interpolating quaternions? If yes, then why?

The goals of trying out other interpolation methods are mainly:

1. Precision (minimizing the interpolation error)
2. Minimizing computation time