Gaussian-Pareto Overbounding (GPO) for Safety-critical UAS Sensor Systems (GSUSS)

Recent advances in computing, communication, and sensing have created great potential for Uncrewed Aircraft Systems (UAS) to be furthered developed as a platform for applications. However, these applications create more safety-critical dependence on the complex and nondeterministic sensor systems, including navigation systems, air data systems (ADS), power monitoring systems (PMS), and collision avoidance systems (CAS). Verification and Validation (V&V) of the error characteristics of these safety-critical sensor systems requires determining the risk of critical failure of the entire vehicle due to hazardously misleading information. For such safety-critical operations, i.e. where failure could result in loss of life, significant property damage, and/or damage to the vehicle.

 The current approach to statistically analyzing these operations in the aircraft community is a technique known as overbounding, which has been notably demonstrated for navigation systems based on the Global Navigation Satellite System (GNSS). The overbounding technique was developed in the early 2000s because estimated error models were too imprecise to adequately model the extreme probabilities associated with system V&V required for certification. Hence techniques for rigorously bounding these probabilities were developed that ensured that the probability of extreme errors for the system, modeled by an overbound, was never less than the true probability for the system. In addition, overbounds on the inputs to a linear data fusion algorithm produce overbounds on the output of that algorithm. This method has led to the widespread use of the Gaussian distribution for overbounding since their convolutions through the Kalman Filter algorithm remain Gaussian. For nonlinear data fusion algorithms this has led to the practice of linearizing the system and performing Gaussian overbounding for V&V.

However, Gaussian overbounding methods may be inadequate for modern V&V for several reasons. Firstly, linearizable approximations may not be sufficient for all data fusion algorithms used onboard UAS. Secondly, Gaussian models introduce conservatism into the overbounding process because of their inadequacy to truly overbound the entire error distribution due to a faster decay rate, i.e. heavy-tails, than is seen in many sensor systems, such as navigation and air data system. Thirdly, the methods for calculating a safety margin are based on using many data sets for providing some level of confidence in the overbound.
​
The proposed Gaussian-Pareto Overbounding approach uses rigorous statistical results from Extreme Value Theory (EVT) for improved overbounding models, advanced numerical algorithms for analyzing non-Gaussian and nonlinear stochastic systems. The foundational Pickands-Balkema-de Haan theorem states that the tail distribution of many* heavy-tailed random variables converge to a Pareto Distribution. Recent work by has shown that this Pareto model is not only an advantageous model for estimating the extreme portions of distributions, i.e. the tails, but also for overbounding them. This has led to a novel GPO model which is essential for data-efficient V&V algorithms.

A webinar hosted by the Institute of Navigation (ION) is available for an overview of some of the preliminary results for the Gaussian-Pareto Overbound.

Relevant Papers:

  • J. D. Larson, D. Gebre-Egziabher, and J. H. Rife, “Improving Navigation System Availability using Gaussian-Pareto Overbounding,” in 2019 ION GNSS+ Conference, pp. 3153-3161, September 2019
  • J. D. Larson, D. Gebre-Egziabher, and J. H. Rife, “Gaussian-Pareto Overbounding of DGNSS Pseudoranges from CORS,” in NAVIGATION, the Journal of the Institute of Navigation, vol. 66, no. 1, pp. 139-150, March 2019
  • J. D. Larson, D. Gebre-Egziabher, and J. H. Rife, “Multivariate Overbounding with a Gaussian-Pareto Model,” in AIAA Journal of Aerospace Information Systems, vol. 16, no. 4, pp. 148-161, January 2019
  • J. Larson and D. Gebre-Egziabher, “Conservatism Assessment of Extreme Value Theory Overbounds,” in IEEE Transactions on Aerospace and Electronic Systems, vol. 53, no. 3, pp. 1295-1307, June 2017
  • J. Larson and D. Gebre-Egziabher, “Constructing EVT-Based Confidence Bounds Using Bootstrapping,” 2017 IEEE Aerospace Conference, pp. 3363-3368, March 2017
  • J. Larson and D. Gebre-Egziabher, “Analysis and Utilization of Extreme Value Theory for Conservative Overbounding,” 2016 IEEE/ION Position, Location, And Navigation Symposium, pp. 462-471, April 2016