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What is the benefit of a square root (UD) factorized covariance? PDF Print E-mail

The classic form of the Kalman filter, as originally derived by Kalman and others, has the covariance matrix in the filter equations. The filter can become inaccurate or unstable in systems where the covariance can become ill-conditioned, asymmetric, or even indefinite. Factorized covariance algorithms can prevent numerical problems that affect the accuracy and stability of the filter.

Because most computers these days provide fast double-precision arithmetic, the need for factorized covariance algorithms is not so obvious. Ill conditioning of the covariance matrix depends on the system matrices, measurement geometry, the measurement error covariance, and the process noise covariance. In many attitude determination systems, only the attitude and gyro bias are estimated and the measurement geometry is favorable. The filter is "tuned" for acceptable performance by adjusting the measurement covariance or the process noise covariance, and the covariance does not become ill conditioned. When calibration parameters are added to the system model, the condition number for the covariance matrix can be large, thereby threatening the accuracy and stability of the filter. Ad hoc methods are often used to make the classical covariance equations work, at the expense of accuracy and reliability.

It has been argued that in steady-state the covariance of the calibration filter is well conditioned. In fact, the condition number of the covariance can easily be as high as 108 or more, so at least half of the significant digits are lost. (This was verified through analysis of covariance factors under typical operating conditions.) There are several operating conditions in which the condition number can be much larger than 108:

  • Upon initialization and re-initialization of the filter where some elements of the covariance matrix can be large;
  • During certain calibration maneuvers where some linear combinations of parameters converge before others;
  • During periods of poor measurement geometry.

While the classical covariance equations may perform adequately under benign conditions, stressing conditions such as these can cause the filter to lose accuracy or to fail. To ensure the highest possible numerical performance and reliability, factorized covariance algorithms are highly recommended.

Other reasons sometimes heard for not using a factorized covariance algorithms is that they require more computations than the classical covariance equations, or that the factorized covariance algorithms are too complicated. Depending on how the factorized covariance algorithms are implemented, they should not require significantly more computation than the classical equations. Today's computers, even space-qualified computers, are fast enough to perform a few extra computations. Complexity is alleviated by using a properly implemented library of functions to compute the factorized covariance prediction and update.

The foregoing applies to orbit determination filters as well as attitude determination and calibration filters.

The RADICAL Redundant IMU (RIMU) Attitude Determination/Calibration Filter is built on a set of UD factorized covariance prediction and update algorithms implemented in library functions. The covariance is never computed, except that variances (the diagonal of the covariance matrix) are computed for output to telemetry and to the user, and for threshold testing.

For further information on RADICAL, see the Specialized Software Tools and the Products pages. Also look for articles on RADICAL in the Conference Papers bibliography.