Why do we need trajectory optimization?

Today’s economy is facing tough competition at increased scarcity of resources in a more and more globalized world. In order to improve efficiency and to reduce cost it is of utmost importance to exploit the full potential of a dynamic system, be it a car, a single aircraft or even the whole traffic at an airport. And when trying to achieve peak performance with a given dynamic system, numerical optimization is inevitable to generate verifiable results, enabling countless new applications.

To gain a deep understanding of optimization theory and to improve real world applications, the Institute of Flight System Dynamics at TUM (FSD) has been conducting research in this field for several decades now. With its vast experience in numerical optimization, the trajectory optimization research group at FSD develops and applies optimal control methods especially tailored to flying systems and air traffic related problems.

The following video shows one of the applications which is the calculation of noise minimal approach routes for a generic airport surrounded by populated areas. (The video is only available in German.)


The methods developed at the institute enable a vast number of applications some of which are listed below.

Calculation of noise minimal approach and departure routes

Today, major airports are facing more than a thousand aircraft movements a day leading to considerable impact on the surrounding communities in terms of noise and air pollution. Hence, plans for the extension of existing airports or the construction of new airports are strongly opposed by the local population. However, due to the rise of modern means of navigation (GBAS/SBAS, RNAV) it is nowadays possible to define curved approach and departure procedures instead of the conventional straight flight tracks. This allows the definition of environmentally friendly procedures that minimize e.g. the noise impact on ground.

To this end, a bi-level optimal control problem is formulated that can be solved with the optimal control framework developed at FSD. Within this bi-level optimal control problem, multiple lower level optimal control problems (OCP) representing approaching or departing aircraft are embedded in an upper level optimization framework, which determines the optimal position of procedure waypoints to minimize the noise impact on ground. At the same time, the aircraft in the lower level optimal control problems have to follow the route prescribed by the waypoints while minimizing their direct operational cost.

Optimized approach route for a given airport with population distribution.

Optimization of air race tracks

Half Cuban eights, chicanes, high-speed turns and accelerations up to 12g! Air races are truly thrilling for the spectators and utmost challenging for the pilots but safety has to be the number one priority for the organizers. Thus, next to the possibility to calculate the fastest trajectory through a given race course, the FSD optimization framework allows to optimize the race course itself with respect to safety and other criteria. This way, it is possible to have a great and spectacular track layout while taking risk factors such as distance to crowd, directed energy, pilot blinding, etc. into account at the same time.

To achieve this, FSD built a bi-level optimal control problem, where in the lower optimal control problem the optimal race courses for different aircraft types are calculated. In the upper level optimization problem the positions of the air race gates are altered such that either the safest race course or the fairest one results. The race track is regarded as fair if the basic layout does not favor any of the participating types of aircraft. The limits for the race gate positions can in this algorithm be defined by the user and have to be chosen such that the course still remains interesting for the spectators in any case. (Website of Red Bull Air Race)

Optimized trajectory for a given race course.

Calculation of fuel minimal trajectories

Especially airlines have a great interest in fuel optimal operations of their fleet for cost reduction. Such fuel minimal trajectories can be calculated using the optimal control framework developed at FSD. In this application it is of primary importance to take care for all aircraft related envelope constraints as well as all regulatory constraints. These constraints may be waypoints, maximum or minimum altitude limits or any other restrictions published in the approach or departure charts or any other respective documents.

Optimization of ATM scenarios

In 2005 the European Commission set the following high-level goals for the “Single European Sky” to be met by 2020:

  • Enable a 3-fold increase in air traffic capacity
  • Improve safety by a factor of 10
  • Enable a 10% reduction in the effects flights have on the environment
  • Provide ATM services to the airspace users at a cost of at least 50% less

To achieve this goal, huge research programs (e.g. SESAR and NextGEN) have been initiated to shape the future of Air Traffic Management. One of the general trends is the increasing use of automation in this field, where the main question to be solved is how to safely and efficiently bring as many aircraft as possible through a given airspace. At FSD, optimization theory is used to guide multiple aircraft through a given sector and minimize the environmental impacts or the operational costs for the airlines.

Of course, safety has the highest priority and separation constraints have to be guaranteed at any time.

Tunnel in the Sky

The optimization methods developed at the institute have been used to calculate optimized approach and departure routes that were used in the market offered tunnel in the sky system 3D-Pilot developed by Aviontek.

Screenshot of the tunnel in the sky system used in a simulator.


The aircraft trajectory optimization group at the Institute of Flight System Dynamics developed and implemented an optimal control toolbox that can be used to solve trajectory optimization problems from several branches. This includes aircraft trajectory optimization as well as other areas like optimal robot control or optimal trajectories for ground based vehicles. The framework has been developed in MATLAB© and allows the use of model dynamics implemented in Simulink© as well.

Direct discretization methods

For the solution of optimal control problems, the framework described before allows the use of different direct discretization schemes like direct single shooting, direct multiple shooting and direct collocation with different integration schemes. For the solution of the resulting numerical optimization problems currently SNOPT, IPOPT, WORHP or the MATLAB internal solver fmincon can be used.

Methods used to solve optimal control problems.

Bi-Level optimal control

Besides the possibility to solve regular optimal control problems in the optimal control framework developed at FSD, the code is capable of solving so called Bi-level optimal control problems. In this class of problems, the solution of an upper level parameter optimization problem depends on the optimal solution of one or multiple lower level optimal control problems. Where the dependence of the optimal solution of the lower level problem on the upper level optimization parameters is efficiently computed using the post-optimal sensitivity analysis.

Optimal control problems including discrete controls and decisions

A special class of optimal control problems are so called MINLP problems (Mixed Integer Non-Linear Programming). In these problems not only continuous values like the control and state histories are sought but additional discrete controls or decisions are to be determined. For flight systems this is especially relevant when discrete controls are involved in the optimization process like the flaps settings or the gear positions that can only take predefined positions with no intermediate values allowed. Next to standard approaches like multiple-phase formulation and approximation using the hyperbolic tangent, the optimal control framework directly supports sophisticated methods such as outer and inner convexification to deal with MINLP, which are at the user’s disposal.

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Contact any one of us: Tuğba Akman, Felix Schweighofer, Alexander Zwenig