This paper studies the trajectory planning problem for multiple aircraft with logical constraints in disjunctive form which arise in modeling passage through waypoints, distance-based and time-based separation constraints, decision-making processes, conflict resolution policies, no-fly zones, or obstacle or storm avoidance. Enforcing separation between aircraft, passage through waypoints, and obstacle avoidance is especially demanding in terms of modeling efforts. Indeed, in general, separation constraints require the introduction of auxiliary integer variables in the model; for passage constraints, a multiphase optimal control approach is used, and for obstacle avoidance constraints, geometric approximations of the obstacles are introduced. Multiple phases increase model complexity, and the presence of integer variables in the model has the drawback of combinatorial complexity of the corresponding mixed-integer optimal control problem. In this paper, an embedding approach is employed to transform logical constraints in disjunctive form into inequality and equality constraints which involve only continuous auxiliary variables. In this way, the optimal control problem with logical constraints is converted into a smooth optimal control problem which is solved using traditional techniques, thereby reducing the computational complexity of finding the solution. The effectiveness of the approach is demonstrated through several numerical experiments by computing the optimal trajectories of multiple aircraft in converging and intersecting arrival routes with time-based separation constraints, distance-based separation constraints, and operational constraints.
In air traffic management (ATM), the flight of several aircraft can be modeled as a hybrid dynamical system, which can be regarded as a set of interacting continuous dynamical systems. A number of frameworks have been proposed to model hybrid dynamical systems, in which, in general, differential equations describe the dynamics of each system, whereas logical constraints describe the behavior of the systems during the interactions among them and the interaction with the environment in which they operate. In the ATM context, logical constraints describe, for instance, policies to apply in conflict detection and resolution and operational constraints to be fulfilled during flight. The main operational constraints to be fulfilled during flight are separation constraints, keep-out constraints to avoid no-fly zones, and passage constraints through or by waypoints [
Given a set of aircraft, separation constraints between them can be expressed as follows: pairwise, they must keep a vertical distance greater than a minimum vertical safety distance or a horizontal distance greater than a minimum horizontal safety distance. The minimum horizontal separation distance can be fixed or variable. In the latter case, it can be established based on the turbulence generated by the preceding aircraft and the ability of the following aircraft to resist turbulence [
It is easy to see that all the constraints mentioned above are expressed in disjunctive form. Standard modeling techniques are able to tackle constraints in disjunctive form using binary variables. The four-dimensional (4D) trajectory planning problem for multiple aircraft with logical constraints in disjunctive form can be solved as an optimal control problem (OCP) for a hybrid dynamical system and a common approach for solving this class of problems is to formulate them as a mixed-integer programming problem. In [
In this paper, the embedding technique proposed in [
This study can be classified into the category of continuous descent operations (CDO) [
Most of the previous research on CDOs based on optimal control theory focused on the trajectory optimization of a single aircraft. In [
In [
Less research efforts have been devoted to combined optimization of trajectories of multiple aircraft and sequencing for approaching a terminal manoeuvring area (TMA) in which all aircraft follow CDAs, while satisfying the operational requirements. This fact motivated the study presented in this paper.
Two types of CDA exist depending on the lateral path followed, generally referred to as CDA under vectoring and advanced CDA. In the first case, the lateral path followed by the aircraft is assumed to be specified through instructions provided by the air traffic control (ATC). In the second case, the lateral path of the aircraft is predefined and is based on the STAR.
Each of the problems that has been solved to validate the method proposed in this paper can be stated as follows. Given the dynamic models of a set of aircraft, their initial and final states, a set of operational constraints, find the optimal trajectories that steer the aircraft from the initial to the final states, fulfilling all the constraints and optimizing an objective functional.
In particular, the optimal trajectories of multiple aircraft in converging arrival routes are computed taking into account time separation constraints, distance separation constraints, and their optimized profile descent along a STAR lateral profile. The problem has been solved using optimal control techniques. In particular, the OCP is transcribed using a Hermite-Simpson collocation method [
The paper is structured as follows. In Section
The multiaircraft flight planning problem considered in this paper can be regarded as a multitrajectory optimization problem in which the motion of each aircraft has been modeled as a differential algebraic dynamic system
Since this multiaircraft flight planning problem also involves operative performances and flight envelope conditions for multiple aircraft, as well as the optimization of a specified performance index, the multitrajectory optimization problem can be formulated as an OCP of a set of dynamic systems in which the goal is to find the trajectories and the corresponding control inputs that steer the states of the systems between two configurations, satisfying a set of constraints on the state and/or control variables while minimizing an objective functional.
Therefore, the optimal control problem considered in this work can be stated as follows:
The objective function
Functions
In the objective function (4), the Lagrange term represents a running cost, whereas the Mayer terms represent a terminal cost. A usual Lagrange objective function is to minimize the total amount of fuel consumed during the flight. A typical Mayer objective function is to minimize the duration of the flight. Equations (
Hence, the optimal control problem (
A direct numerical method has been employed to transcribe the OCP into a NLP problem. More specifically, a Hermite-Simpson direct collocation method [
The set of constraints of the resulting NLP problem includes the Hermite-Simpson system constraints that correspond to the differential constraint (
To solve the resulting NLP problem, the open source IPOPT solver [
Following [
The following equations of motion of the aircraft have been considered:
The three dynamic equations in (
Aircraft state and forces.
Top view
Front view
Lateral view
The states of the system (
The parameter
Note that differential equations in (
Flight envelope constraints are derived from the geometry of the aircraft, structural limitations, engine power, and aerodynamic characteristics. The performance limitation model and the parameters have been obtained from the Base of Aircraft Data (BADA), version 3.6 [
In (
Note that inequality constraints in (
In this section, the approach proposed in [
It was shown in [
In order to include the logical constraint (
Transcribing the conjunction in (
Thus, taking into account (
To transcribe these disjunctions into a set of inequality constraints, a continuous variable
It is immediate to check that if
Note that, as expected, equality and inequality constraints in (
One of the problems of most interest in ATM is the conflict detection and resolution problem [
Let
Following the technique described above to tackle constraints in disjunctive form, if we define new variables
Equation (
The last constraint in (
In the second instance, to model collision avoidance among different aircraft, at each endpoint of the subintervals of the discretization (
Let
As before, condition
Again, following the technique described above, new variables
The last constraint in (
In the third instance, an obstacle avoidance problem has been considered. Following [
Two opposite corners of a cuboid in the 3D space.
Let
Note that constraints (
Following the same technique used in the previous sections, new variables
Again, as mentioned above, the last constraint in (
Finally, the modeling of an aircraft flying through a waypoint has been considered. The model has been based on the use of cuboids similar to those considered in Section
Let
Note that, for one hand, constraints (
To overcome this drawback, a second auxiliary cuboid is considered to modelled free flight mode of the aircraft. Let
Then, the transcription into a logical disjunction which allows to select along the whole trajectory between flying by the waypoint mode (WM) or free flight mode (FM), namely,
Once again, following the technique described above, if we define new variables
Equation (
The last constraint in (
On the one hand, the optimal solution could provide a trajectory in which the aircraft flies in FM for each discretization instant
On the other hand, the optimal solution could provide a trajectory in which the aircraft flies by the waypoint more than once. This situation can be easily avoid introducing in the model the following simple constraint:
Note that this waypoint constraint setting can be straightforward extended to multiwaypoint and multiaircraft settings. Moreover, once again, since this embedded logical constraint approach is quite general, any other waypoint constraint described in terms of Equation (
To show the effectiveness of the methodology describe in Section
All of these numerical experiments involve Airbus A-320 BADA 3.6 aircraft models in which the performance index is the sum of the duration of the flights of the three aircraft. In some of them, constraints derived from current flight regulation have been introduced, namely, time-based separation and distance-based separation operational constraints have been imposed. In particular, a minimum horizontal distance separation between aircraft of 5000 m and vertical separation between aircraft of 1000 m has been considered, whereas the considered minimum time separation between aircraft has been 200 s. These specific values have been chosen taking as a reference the aviation regulation [
The numerical experiments have been conducted on an Intel Core i7 2.8 GHz CPU with 16 GB RAM.
In this experiment, a CDA under vectoring has been considered, that is, the lateral path followed by the aircraft has been assumed to be specified through instructions provided by the ATC. In particular, the boundary conditions of the state variables have been selected from the chart of the Adolfo Suárez Madrid-Barajas (LEMD/MAD) TMA shown in Figure
Chart of the Adolfo Suárez Madrid-Barajas (LEMD/MAD) STAR 10-2A1.
Boundary conditions for Experiment 1 and Experiment 3.
Symbol | Unit | Aircraft 1 | Aircraft 2 | Aircraft 3 |
---|---|---|---|---|
m | 7400 | 7000 | 7200 | |
m | 3350 | 3350 | 3350 | |
deg | 39.526 | 39.116 | 39.000 | |
deg | 40.575 | 40.575 | 40.575 | |
deg | -5.327 | -4.448 | -3.325 | |
deg | -3.422 | -3.422 | -3.422 | |
m/s | 130 | 130 | 130 | |
m/s | 110 | 110 | 110 | |
deg | 0 | 0 | 0 | |
deg | 0 | 0 | 0 | |
deg | 356 | 294 | 24 | |
kg | 65000 | 65000 | 65000 |
Results of Experiment 1.
# aircraft | Final time (s) | Final mass (kg) | |
---|---|---|---|
Without time constraint | 1 | 1539 | 64023 |
2 | 1419 | 64142 | |
3 | 1350 | 64151 | |
With time constraint | 1 | 1750 | 64109 |
2 | 1550 | 64216 | |
3 | 1350 | 64151 |
In Figure
Experiment 1. 3D view of the paths with (thin lines) and without (thick lines) time-based separation constraints.
Experiment 1. Horizontal profiles with (TC, dashed lines) and without (FF, solid lines) time-based separation constraints.
Experiment 1. Vertical profiles with (TC, dashed lines) and without (FF, solid lines) time-based separation constraints.
The computation time to find the solution has been
In Figure
In this case, the computation time to find the solution has been
Experiment 1. Mass consumption with (TC, dashed lines) and without (FF, solid lines) time-based separation constraints.
This experiment has been designed to test the effectiveness of the logical constraint formalism described in Section
Neither time-based nor distance-based separation constraints have been included in the model. The initial mass of the three aircraft has been assumed equal to the maximum landing weight of the aircraft. The specific boundary conditions of the state variables are given in Table
Boundary conditions for Experiment 2.
Symbol | Unit | Aircraft 1 | Aircraft 2 | Aircraft 3 |
---|---|---|---|---|
m | 7400 | 7400 | 7400 | |
m | 3350 | 3350 | 3350 | |
deg | 39.000 | 39.000 | 38.701 | |
deg | 40.575 | 40.575 | 40.848 | |
deg | -5.327 | -3.325 | -4.320 | |
deg | -3.430 | -5.215 | -4.320 | |
m/s | 130 | 130 | 130 | |
m/s | 110 | 110 | 110 | |
deg | 0 | 0 | 0 | |
deg | 0 | 0 | 0 | |
deg | 356 | 294 | 294 | |
kg | 65000 | 65000 | 65000 |
Positions of the aircraft at conflict region (at time 980 s) without distance constraint for Experiment 2.
Symbol | Unit | Aircraft 1 | Aircraft 2 | Aircraft 3 |
---|---|---|---|---|
m | 4176 | 4170 | 4194 | |
deg | 39.8414 | 39.8449 | 39.8458 | |
deg | -4.3243 | -4.3165 | -4.3184 |
Results of Experiment 2.
# aircraft | Final time (s) | Final mass (kg) | |
---|---|---|---|
Without distance constraint | 1 | 1838 | 63787 |
2 | 1838 | 63824 | |
3 | 1838 | 63751 | |
With distance constraint | 1 | 1841 | 63772 |
2 | 1839 | 63900 | |
3 | 1843 | 63772 |
In Figure
Experiment 2. 3D view of the paths with (thin lines) and without (thick lines) distance-based separation constraints.
Experiment 2. Horizontal profiles with (DC, dashed lines) and without (FF, solid lines) distance-based separation constraints.
Experiment 2. Vertical profiles with (DC, dashed lines) and without (FF, solid lines) distance-based separation constraints.
To avoid the conflict, a new experiment has been conducted, which includes distance-based separation logical constraints of at least 5000 m (
It can be seen from Figures
Experiment 2. Distance among aircraft with distance-based separation constraints (DC).
Experiment 2. Mass consumption with (DC, dashed lines) and without (FF, solid lines) distance-based separation constraints.
In this experiment, a STAR-based CDA has been considered, that is, the lateral path followed by the aircraft has been assumed to be specified in a navigation chart. In particular, as in Experiment 1, the boundary conditions of the state variables have been selected from the chart of the Adolfo Suárez Madrid-Barajas (LEMD/MAD) TMA shown in Figure
The initial mass of the three aircraft has been assumed equal to the maximum landing weight of the aircraft. The specific boundary conditions of the state variables are the same as in Experiment 1 and are given in Table
Results of Experiment 3.
# aircraft | Final time (s) | Final mass (kg) | |
---|---|---|---|
Without logical constraints | 1 | 1539 | 64023 |
2 | 1419 | 64142 | |
3 | 1350 | 64151 | |
With waypoint constraints | 1 | 1971 | 63813 |
2 | 1626 | 64071 | |
3 | 1965 | 63863 | |
With waypoint and time constraints | 1 | 2109 | 63908 |
2 | 1619 | 64127 | |
3 | 1909 | 63956 |
Therefore, a new experiment has been conducted, which, besides waypoint constraints, also includes time-based separation logical constraint of at least 200 s between aircraft as described in Section
Experiment 3. 3D view of the paths with (thin lines) and without (thick lines) waypoints and time-based separation constraints.
Experiment 3. Horizontal profiles with (WT, dashed lines) and without (FF, solid lines) waypoints and time-based separation constraints.
Experiment 3. Vertical profiles with (WT, dashed lines) and without (FF, solid lines) waypoints and time-based separation constraints.
It can be seen that, as expected, there are significant changes in both the horizontal and vertical profiles. The final time of the three aircraft is 2109 s, 1619 s, and 1909 s, respectively. The difference between the final time of aircraft 1 and 3 is 200 s, whereas the difference between the final time of aircraft 3 and 2 is 290 s. Notice that, unlike Experiment 1, aircraft 3 and 2 do not maintain the minimum required time separation of 200 s. This is due to the fact that the performance index includes in this case, besides the minimization of the duration of the flights, the penalty term associated with the waypoints constraints. Therefore, the saturation of the time-based constraints may not happen in this setting. In this case, the computation time to find the solution has been
The mass consumption is depicted in Figure
Experiment 3. Mass consumption with (WT, dashed lines) and without (FF, solid lines) waypoints and time-based separation constraints.
In this paper, the trajectory planning problem for multiple aircraft has been studied in which logical constraints in disjunctive form are included in the model. The logical constraints in disjunctive form have been transformed into inequality and equality constraints which involves only continuous auxiliary variables. In this way, the optimal control problem with logical constraints has been converted into a smooth optimal control problem which has been solved using standard techniques. This approach has been applied to the computation of the optimized profile descent of multiple aircraft in converging and intersecting arrival routes within the Adolfo Suárez Madrid-Barajas (LEMD/MAD) TMA. The results show the effectiveness of the proposed technique.
No data were used to support this study.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This work has been supported by the Spanish grants TRA2013-47619-C2-2-R and TRA2017-91203-EXP.