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Modeling and Optimization of Aircraft Trajectories: a review. Maria Pia Fanti, Giovanni Pedroncelli , Gabriella Stecco, Walter Ukovich. contact: Outlines. Aim of the paper.

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Modeling and optimization of aircraft trajectories a review

Modeling and Optimization of Aircraft Trajectories: a review

Maria Pia Fanti, Giovanni Pedroncelli, Gabriella Stecco,

Walter Ukovich


Aim of the paper

Airspace is more and more crowded because the air transportation flow is constantly increasing. The current Air Traffic Management (ATM) is unsuitable for the future expansion of the air traffic.

A review of the main contributions about the modeling and optimization of the aircraft trajectories with two objectives:

i) resolving conflicts with other aircrafts (search of the optimal path from origin point to destination point which is conflict-free) ii) avoiding hazardous weather (search of the optimal route of deviation from weather encounters).

CD&R Systems

The Free Flight concept has been conceived as a new and better ATM system in which every pilot can select the optimal route for the aircraft in respect of safety constraints, avoiding conflicts with the other aircrafts.

In this environment, we consider each aircraft surrounded by two virtual cylindrical shapes, the Protected Zone and the Alert Zone. A conflict or loss of separation between two aircrafts occurs whenever the protected zones of the aircrafts overlap.

The goal for the Conflict Detection and Resolution (CD&R) system is to predict that a conflict is going to occur in the future, communicate the detected conflict to a human operator and, in some cases, assist in the resolution of the conflict situation.

CD&R Systems

If the Alert Zone of an aircraft is violated, air traffic controllers will intervene to assist in conflict avoidance. To ensure safety, the Protected Zone should never be penetrated.

The probability of conflict avoidance success depends on the size of the Alert Zone.

J. K. Kuchar and L.C. Yang, “Prototype Conflict Alerting System for Free Flight”. Journal of Guidance, Control and Dynamics; Volume 20; No. 4; 768-773, 1997.

CD&R Systems: multi-agent environments

Some CD&R approaches consider the airspace as a multi-agent environmentin which aircrafts are considered as agents occupying zones of the space.

Some of these methods (e.gSislak et al.) use the principles of game theory, generally assuming cooperation between aircrafts (considered as agents in negotiation). The zones of the airspace are considered as resources contended by the agents.

Other methods (e.g. Resmerita et al.) make use of the graph theory. The airspace is discretized as a grid with each node of the graph representing a zone of the space. Usually there is no assumed cooperation: agents are greedy and there is no inter-agent communication during the conflict resolution.

D. Sislak, P. Volf and M. Pechoucek, “Agent-Based Cooperative Decentralized Airplane-Collision Avoidance,” IEEE Transactions on Intelligent Transportation Systems, Vol. 2, No. 1, pp. 36-46, 2011.

S. Resmerita, M. Heymann and G. Meyer, “A framework for conflict resolution in Air Traffic

Management”. IEEE Conference on Decision and Control, pp.2035-40, 2003.

CD&R Methods

Other CD&R methods (e.g. Pallottino et al.) are based on the optimization of a cost function. This function is usually associated with a set of cost metrics (e.g. time of flight, fuel, projected separation, workload).

Some other methods are based on a path planningapproach. For example, Schouwenaars et al. present a cooperative, decentralized path planning method which makes use of Mixed Integer Linear Programming (MILP) problems resolution methods and receding horizon strategy.

L. Pallottino and E. Feron, “Conflict Resolution Problems for Air Traffic Management Systems Solved with Mixed Integer Programming,” IEEE Trans. on Intelligent Transportation Systems, Vol.3 No.1, pp.3-11, 2002

T. Schouwenaars, J. How and E. Feron, “Decentralized Cooperative Trajectory Planning of Multiple Aircraft with Hard Safety Guarantees,” AIAA Guidance, Navigation, and Control Conference and Exhibit, Providence, Aug. 2004.

Weather is one of the leading causes of delays for aircrafts and these delays increase as the volume of air traffic increases.

Weather events cause an additional workload for the controller who has to perform tactical decisions about the weather impact reduction.

Weather phenomena can also cause damages to the electronic and navigational equipment of an aircraft or danger to the passengers’ lives.

In a Free Flight environment, weather avoidance will be more complicated as pilots will have autonomous control and efficient and robust weather avoidance algorithms will be needed to ensure safety.

A weather modelis the way of representing weather events in order to be avoided by the pilots: it’s the first step in developing a fully automated system for decisional support integrating also flight information, trajectory modeling to help the pilot performing better decisions and reducing workload.

CWAM is a model proposed by DeLaura et al. It generates estimates of pilot deviation probabilities in the case of weather encounters as a function of Vertical Integrated Liquid(VIL, that is the amount of liquid water detected in a vertical column of the atmosphere) and echo top(i.e. the radar observed height of a convective system).

R. DeLaura, M. Robinson, M. Pawlak and J. Evans, “Modeling Convective Weather Avoidance in En Route Airspace,” 13th ARAM Conference , New Orleans, LA, Jan. 2008

The products of CWAM are Weather Avoidance Fields(WAFs), 3D grids which represent the estimates of pilot deviation probability in the encounter of weather. These products are used in some other weather avoidance systems: e.g. in Windhorst et al. CWAM forecasts define the contours of airspace containing convective weather, each with the probability that it will be avoided by the pilot.

R. Windhorst, M. Refai and S. Karahan, “Convective Weather Avoidance with Uncertain Weather Forecasts,”

28th Digital Avionics Systems Conf., Oct. 2009.

In Kangapouret al. VIL measurements are quantizedinto 6 levelswithlevel 3 and higherindicating a recommendedno-fly zone. Thenweathermaps are created and eachstorm in the airspaceisenclosed in a minimum volume boundingellipsewhichisanover-approximationof the weather.

M. Kangapour, V. Dadok and C. Tomlin, “Trajectory Generation for Aircraft Subject to Dynamic Weather Uncertainty,” 49th IEEE CDC, Atlanta, GA, USA, pp. 2063-2068, 2010.

The weathermodelproposedbyNilimet al. associates to eachstorm a 2-states Markovchain: state “0” correspods to having no storm in a particularregion and timeframe; “1” corresponds to having a storm. Ifthere are m stormstherewillbe a 2nMarkovchain and a 2nx2ntransitionmatrixasan input of the optimizationalgorithm.

A. Nilim, L. El Ghaoui, M. Hansen and V. Duong, “Trajectory-based Air Traffic Management (TB-ATM) under Weather Uncertainty,” ATM 2001, Santa Fe, New Mexico, 2001.

Ng et al. make use of a DP algorithm to find the shortest path from a starting point to an ending point. The airspace is modeled as a graph with each node representing a point and edges representing the possible trajectories from one point to another. It uses CWAM forecasts.

  • The objective cost functioncan be constructed by adding different components to represent obstacles or constraints:

  • components relative to the estimated fuel cost,

  • the cost of deviation due to weather,

  • the cost associated with crossing a congested region.

H.K. Ng, S. Grabbe and A. Mukherjee, “Design and Evaluation of a Dynamic Programming Flight Routing Algorithm Using the Convective Weather Avoidance Model,” AIAA Guidance, Navigation, and Control Conference, Chicago, Illinois, Aug. 2009.

Kangapour et al. propose a non-cooperative algorithm of trajectory optimization which makes use of a reference trajectory (planned in advance) and of a cost function that penalizes deviations from this reference trajectory.

The optimization problem is solved by means of the receding horizon technique: the planning horizon is chosen as the discrete number of steps for which the trajectory planning is performed. After solving the problem in the defined horizon, it is shifted by a certain number of time units (in this case that is 5 minutes, the time for which new weather forecasts are available). Every aircraft which reaches its destination is removed from the problem and the algorithm terminates when all the aircrafts have reached their destination.

M. Kangapour, V. Dadok and C. Tomlin, “Trajectory Generation for Aircraft Subject to Dynamic Weather Uncertainty,” 49th IEEE CDC, Atlanta, GA, USA, pp. 2063-2068, 2010.

Nilim et al. address trajectory optimization for a single aircraft, minimizing the expected delay facing bad weather by a DP algorithm.Even if the paper deals with the trajectory optimization of a single aircraft, the novelty is considering the weather constraints in the trajectory optimization. First, a cost is assigned to each zone of the discretized airspace; then the solution is the point which will be occupied by the aircraft for the next 15 minutes.

A. Nilim, L. El Ghaoui, M. Hansen and V. Duong, “Trajectory-based Air Traffic Management (TB-ATM) under Weather Uncertainty,” ATM 2001, Santa Fe, New Mexico, 2001.

Pannequin et al. aim to solve both the weather and conflict avoidance problems, considering also the effects of strong winds.

They make use of the receding horizon technique and the cost function is obtained by solving the Hamilton-Jacobi partial differential equation (HJ PDE).

The solution to HJ PDE provides a value function which contains the minimum travel time from any point of the airspace to the destination.

Each destination point has a corresponding value function and the set of these value functions will serve as a look-up table for the cost function.

Each deviation of the aircraft from its optimal trajectory is penalized in the objective function definition.

J. Pannequin, A.M.Bayen, I.M Mitchell, H. Chung and S. Sastry, “Multiple Aircraft Deconflicted Path Planning with Weather Avoidance Constraints,” AIAA Conf. on Guidance, Control and Dynamics, South Carolina, 2007.


We summarize the proposed methods and approaches to model and optimize the flight paths in the following table, according to their most important features.


  • Discretization: the airspace is subject to a discretization in the process of resolution of the conflict or weather encounter. The optimization is performed by solving MILP problems (like in Schouwenaars) or applying DP algorithms (like in Ng).

  • Aircrafts: some models are conceived for a single aircraft (Single) in the airspace: in this case the possibility of a conflict with other aircrafts is not considered. Other contributions consider the presence of multiple aircrafts (Multiple) and have to find solutions for the avoidance of conflicts.

  • Centralization: the weather or conflict problem can be solved by a centralized approach (e.g. the airspace controller) or by distributing the task among the different aircrafts in the airspace. This feature is considered only for multi-aircraft systems. In Schouwenaars the resolution of the problem is distributed among aircrafts which act in a cooperative manner. This approach leads to the optimization of the available computation resources and is one of the central ideas of the Free Flight concept.


We presented how the modeling techniques and the optimization techniques are applied in order to optimize the trajectories in air traffic management.

The problem involves potentially conflicting objectives such as minimizing deviations, weather avoidance, reducing emissions, minimizing distance travelled and hard constraints like aircraft performance.

The problem is still open and future research should address, in particular, sustainability objectives such as reduction of emissions and pollution.

Future work

This paper is a starting point for the developing of a real-time flight path optimization system.

The path will be described by a set of 4D waypoints: 3D points in the airspace (determined by latitude, longitude and altitude) and the time of flight over them.

The objective is to compute this flight path minimizing pollutant emissions (CO2, NOx and noise) in a non-conflict airspace (i.e. for a single aircraft).

In particular, CO2 depends on the phase of flight and NOx on weather conditions such as air pressure, temperature and relative humidity.

Modeling and optimization of aircraft trajectories a review1

Modeling and Optimization of Aircraft Trajectories: a review