Kalman Filtering for Quaternions
How do we represent motion (angular velocities) of quaternions in our state vector?
- Friedland, B.: Analysis Strapdown Navigation Using Quaternions, IEEE Transactions on Aerospace and Electronic Systems, Vol. 14, No. 5, pp.764-768, September 1978.
seems to provide the differential equation needed to propagate quaternion rotation over time
- that propagation formula is e.g. implemented in Java in the Quaternion class of the Java Astrodynamics Toolkit (JAT).
- see also T.J. Broida, S. Chandrashekhar, R. Chellappa, Recursive 3-D Motion Estimation from a Monocular Image Sequence, IEEE Transactions on Aerospace and Electronic Systems, 26 (4), pp. 639--656, 1990
- J. Goddard, Pose and motion estimation from vision using dual quaternion-based extended Kalman filtering, Ph.D. Thesis, Dec. 97, shows nicely where that formula comes from.
- another derivation of that formula (check if they are consistent!) is given by Edgar Kraft, A Quaternion-based Unscented Kalman Filter for Orientation Tracking, Proceedings of the 6th Int. Conf. on Information Fusion, July 2003, Cairns, Australia.
- Joã Luís Marins, Xiaoping Yun, Eric R. Bachmann, Robert McGhee, Michael J. Zyda,
An Extended Kalman Filter for Quaternion-Based Orientation Estimation Using MARG Sensors,
Proceedings of IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), October 29-November 3, 2001, Maui, Hawaii, USA, pp. 2003 - 2011.
How does a Quaternion with only 5DOF look like in 3D? Whats the "closest" Quaternion from that set of quaternions to a given quaternion?
- this is something like the projection of the estimated quaternion to the subspace of the measured set of quaternions.
- note that we do have only unit quaternions
How do we get from Euler angles to Quaternions?
- the required formula can be found in Paul Heckbert (editor), Graphic Gems IV, Academic Press, 1994, ISBN: 0123361559
- this of course depend heavily on the order of rotations
- Q: how is the order of rotation in the trackers we use?
How does Error Propagation in Quaternions work? How do we compute the covariances?
- Daniel Choukroun, Novel Methods for Attitude Determination Uisng Vector Observations, Ph.D. Thesis, 2003, Aerospace Engineering, Technion has a lot of stuff, including error covariance matrix of the queternion estimation error.
- cites results from M.D. Shuster, S.D. Oh., Three Axis Attitude Determination From Vector Observations. Journal of Guidance, Control and Navigation, 4(1):70--77, January-February 1981.
Other Literature