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
  • source code is available online.
• 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?

• the formula for the a priori estimate of the error covariance matrix is given in J. La Viola, A Comparison of Unscented and Extended Kalman FIltering for Estimating Quaternion Motion, Proceedings of the 2003 American Control Conference, IEEE Press, 2435-2440, June 2003.

• same formula can be found in W.H. Steyn, Full Satellite State Determination from Vector Observations, 13th International Federation for Automatic Control Symposium on Automatic Control in Aerospace. Palo Alto, California, USA, 1994: 193-198.

• Miao Ling-juan; Liu Xiao-guang; Lan Chun-yun; Dai Ya-ping, Derivation of nonlinear error equations of strapdown inertial navigation system using quaternion, Proceedings of the 41st SICE Annual Conference 2002, (1) 636- 640

• 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

• A. Ude, Filtering in a Unit Quaternion Space for Model-Based Object Tracking, Robotics and Autonomous Systems, vol. 28, no. 2-3, pp. 163-172, August 1999

• J. Goddard, Pose and motion estimation from vision using dual quaternion-based extended Kalman filtering, Ph.D. Thesis, Dec. 97