Application of the Theory of Formal Languages to the Modeling of Trajectory Uncertainty and the Analysis of its Impact in Future TBOs

1. Introduction

The main objective of this PhD research is to develop a formal framework for the modeling and analysis of trajectory prediction uncertainty in the context of future Trajectory-Based Operations (TBO). Such a framework will allow evaluating the impact of trajectory prediction errors on the performance of the advanced airborne and ground-based automation tools required to enable TBO. These automation tools will rely on trajectory prediction to support decision-making in a variety of functions within TBO, such as flight planning and optimization (design of Business Trajectories), flight management and guidance (flying Business Trajectories) and trajectory de-confliction (separation assurance between Business Trajectories, both ground-based and airborne-based). The ability to assess the impact of trajectory prediction errors and taking them on board in the decision-making process is paramount to the design of efficient and robust automation tools for future TBO.

2. Research Objectives

The main objectives of the proposed PhD Program are the following:

  • Develop a formal theoretical framework for the analysis of trajectory prediction uncertainty in the context future trajectory management applications. This framework will provide the means to model the different stochastic factors affecting the aircraft trajectory (considering their mutual correlations) based on the Theory of Formal Languages and leveraging the trajectory modeling paradigm defined by the Aircraft Intent Description Language (AIDL) [1][2].
  • Detailed study and classification of the stochastic factors affecting trajectory prediction in ATM.
  • Definition of robust strategies for managing uncertainty effectively accordingly to the confidence required by the future DSTs.
  • Provide practical methods and tools to evaluate the impact of the different stochastic factors on trajectory prediction accuracy, with a view to facilitating the development of requirements for advanced automation tools regarding uncertainty management in future TBO.
  • Identification of high level requirements for trajectory prediction tools capable of taking into account uncertainty in a rigorous manner and of generating stochastic trajectory prediction outputs.

3. Methodology

The proposed formal language model of trajectory prediction uncertainty will include an alphabet, which will contain a finite set of descriptors of the different types of stochastic factors affecting trajectory prediction, together with a grammar, which will govern how to combine such factors in a meaningful way, e.g. which factors can simultaneously affect the aircraft motion. Each of the descriptors will be a symbol or letter of the alphabet. The feasible combinations of stochastic factors determined by the grammar, i.e. the grammar productions, will be the strings or words of the language. These words will be characterized by a statistical description of the results of the specific combination of stochastic factors on the trajectory prediction uncertainty. These statistical descriptions will provide the semantics of the language. The development of such a grammar will involve the application of concepts and tools from the field of Stochastic Processes [3]. As an example, consider the combination of two possible stochastic factors: wind modeling error and lateral guidance error. They could be modeled by a word of the language formed by two letters (two types of stochastic factors). The combined result of these two sources of uncertainty will produce a given error distribution on the estimated time over a specific waypoint for a given look-ahead time. Such error distribution will be the meaning of the word in question.
The development of the proposed formal language model will involve the following steps:

  • Detailed study and classification of the key stochastic factors affecting trajectory prediction.
  • Statistical modeling and characterization of the effects of each of those stochastic factors and their combinations on the aircraft trajectory.

As part of the PhD work, an experimental stochastic trajectory prediction tool (based on Monte Carlo simulation [4]) will be prototyped to support the characterization of the meaning of the words of the language (i.e. to support the assessment of the impact on the trajectory predictions of the different stochastic factors and their combinations) and validate the theoretical results.
The validation of the results will be based on comparisons with operational recorded flight data as well as with predictions obtained from automations tools such as a Flight Management System (FMS). To that aim, a Boeing 737 FMS located at the BR&TE ATM lab will be at the disposal of the PhD candidate.

4. Implication for Research/ Expected Results

This PhD research will involve developing a formal model of the different types of information involved in a trajectory prediction process, focusing on the stochastic factors affecting such process. A stochastic factor is defined as a source of trajectory prediction uncertainty, i.e. of random deviations between a nominal predicted trajectory and the actual observed trajectory flown by the aircraft, usually referred to as trajectory prediction errors. In this context, a trajectory is understood in its classical sense to be a time-ordered sequence (time series) of aircraft states, with the aircraft state considered as multi-dimensional (e.g. including velocity and mass besides position).

The innovative aspects of the proposed research lie mainly in the application of concepts from other areas of knowledge (mainly the Theory of Formal Languages in this case) to an ATM problem.

5. References

  1. Vilaplana, Miguel A., E. Gallo, F. Navarro, S. Swierstra, 2005, Towards a Formal Language for the Common Description of Aircraft Intent, 24th Digital Avionics Systems Conference, Washington D.C., USA.
  2. López-Leonés, Javier, 2007, Definition of an Aircraft Intent Description Language for ATM, PhD Thesis. University of Glasgow, UK (to be published in 2011).
  3. Papoulis, A., Unnikrishna Pillai, S. Probability, random Variables and Stochastic Processes, Fourth Edition, 2002. McGraw Hill.
  4. Rubinstein R.Y. and Kroese D.P. Simulation and the Monte Carlo Method – 2nd Editon, 2008. Wiley Series in Probabilistic and Statistics.