Algorithms for Multi-Sensor Integration

Multi-sensor or integrated navigation systems combine the information from multiple sensors to generate an estimate of a vehicle’s navigation state vector. The entries of the navigation state vector normally consist of the three vehicle’s position coordinates, the three velocity components and the parameters used to describe the vehicle’s attitude. Multi-sensor navigation algorithms used in commercial aviation must not only generate an accurate estimate of the navigation state vector but also meet stringent reliability requirements. One of these reliability requirements is that they must provide a precise bound on the probability with which hazardous navigation errors occur quantified by the metric of integrity.

In navigation applications integrity risk is defined to be the probability of an undetected navigation error or failure that results in hazardously misleading information.   For navigation algorithms used in applications with safety-of-life implications (e.g., UAV operations in and around populated areas, commercial aviation, etc) the required integrity risk is normally on the order of 10-6 or less.   Thus, the validation of whether a navigation algorithm has the requisite integrity must be done analytically since it is impractical (if not impossible) to conduct the large number of trails required to validate them experimentally.   This involves generating an accurate probabilistic description of the navigation system output errors.   If the statistical descriptions of the inputs to the navigation algorithm (i.e., stochastic sensor error models) are known accurately, generating a statistical description of the navigation output errors is conceptually straight forward; it is a matter of mapping the statistical descriptions of sensor errors through the mathematical model of the navigation algorithm.  Implementing real-time algorithms for this, however, presents challenges and my research deals with some of these challenges.

One of the major challenges is the fact that integrity risks on the order of 10-6 imply one has to accurately characterize the tails of the probability density functions involved. These calculations are sensitive to any uncertainties in the statistical descriptions of the input noises.  For problems that are linear (or non-linear but where errors due to linearization are small) a method known as Gaussian overbounding has been used successfully.  It involves replacing the input sensor error distributions by Gaussian distributions which guarantee that the integrity estimates generated will be conservative.  However, if the input error distributions have heavy tails, then Gaussian overbounding leads to overly conservative results.  Furthermore, if the input sensor errors are correlated in any way (e.g., time correlation) Gaussian overbounding does not apply.

Our work in this area is developing sensor fusion algorithms which possess a quanitfiable level of integrity. A related aspect to this problem is that of observability.   Some integrated navigation problems of significant interest tend to have navigation states that are “hidden” in that they cannot be estimated (or only estimated poorly) form all the available information.  For linear time invariant navigation problems, assessing systems observability is straight forward but challenging for for time-varying and non-linear problems. In the latter case, observability depends on many factors including vehicle maneuvers; in the presence (or lack) of certain maneuvers, the solution of certain unobservable (or weakly observable) states can diverge or converge to a false solution.  In this regard, we have dealt with the specific navigation problem (inertial navigation-GPS integration).  In this regard our work has led to developing a test that can be used to examine the stochastic observability of a particular class of these problems.

Relevant Publications

Y. Shao and Gebre-Egziabher, D., "Stochastic and Geometric Observability of Aided Inertial Navigators,"  Proceedings of the ION-GNSS Conference, Forth Worth, TX.  September 2006.  pp. 2723 -- 2732.

G. Phanomchoeng, D. Gebre-Egziabher, J. Rife, "A Numerical Procedure for Approximating Overbounds on Navigation Systems Error Distributions,"  Proceedings of the ION-GNSS Conference, Forth Worth, TX. Sep. 2006.  pp 1614 -- 1619.

J. Rife. and D. Gebre-Egziabher, "Symmetric Overbounding of Correlated Errors," to appear in Navigation