International Journal of Aerospace Engineering

Volume 2019, Article ID 5158967, 15 pages

https://doi.org/10.1155/2019/5158967

## Multiaircraft Optimal 4D Trajectory Planning Using Logical Constraints

Correspondence should be addressed to Dinesh B. Seenivasan; se.cjru@hsenid.nasavinees

Received 15 January 2019; Accepted 19 September 2019; Published 3 December 2019

Academic Editor: Jose Carlos Páscoa

Copyright © 2019 Dinesh B. Seenivasan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. Introduction

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 [1].

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 [2]. Obstacles and no-fly zones are in general polyhedral regions of airspace. However, the corresponding keep-out constraints are usually introduced by bounding ellipsoids around obstacles. In some cases, this is a coarse approximation. Keep-out constraints from a polyhedral region of airspace can be expressed as follows: each aircraft must stay outside one of the half-spaces defined by the planes that supports the faces of the polyhedron. This method for modeling keep-out constraints from obstacles can be extended to model passage constraints through windows or waypoints in the airspace. In this manner, the multiphase modeling of the problem is avoided. This is an interesting possibility, since multiphase optimal control models imply the introduction of additional variables to the problem, such as the duration of the phases and additional constraints such as the linkage constraints between them to enforce continuity of the state variables between contiguous phases [3].

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 [4], the optimal cooperative three-dimensional (3D) conflict resolution problem among multiple aircraft has been solved in which separation constraints among aircraft expressed in disjunctive form have been included in the model using continuous auxiliary variables. In [5], the optimal path planning problem for multiple unmanned aerial vehicles in the horizontal plane with collision avoidance has been studied, in which constraints for collision avoidance with rectangular obstacles expressed in disjunctive form are included in the model using continuous auxiliary variables. In [6], the trajectory optimization problem for multiple aircraft landing on a single runway in the presence of constraints on the airspace has been treated. The constraints considered are passage constraints through windows in the airspace, and optimal trajectories have been determined by solving a nonsequential constrained multiple-phase optimal control problem.

In this paper, the embedding technique proposed in [5] to model rectangular obstacle avoidance in the horizontal plane has been extended to model time-based and distance-based separation constraints and passage through waypoints constraints in trajectory optimization for multiple aircraft. The modeling of passage constraints through waypoints has been done by defining vertical walls in the airspace with a cuboidal window around the waypoint. In this way, introducing multiple phases in the model to enforce passage through waypoints is avoided. Moreover, the dimensions of the windows can be easily calibrated to induce a fly-by or a fly-through the waypoint.

This study can be classified into the category of continuous descent operations (CDO) [7]. During CDO, aircraft descend from the cruise altitude to the final approach fix at or near idle thrust without level segments at low altitude minimizing the need for high thrust levels to remain at a constant altitude and reducing the environmental impact. Actually, the term CDO makes reference to the different techniques to maximize operational efficiency and, at the same time, fulfilling local airspace requirements and constraints. These operations are known as continuous descent arrivals, optimized profile descents (OPDs), tailored arrivals, 3D path arrival management, and continuous descent approaches (CDA). In particular, an OPD is a descent profile normally associated with a standard terminal arrival route (STAR) and designed to allow maximum use of a CDO. Planning CDOs is one of the functions of the so-called arrival managers (AMANs) whose purpose is to ensure an optimal sequencing and spacing of arrival traffic [8].

Most of the previous research on CDOs based on optimal control theory focused on the trajectory optimization of a single aircraft. In [9], a multiphase optimal control method based on the pseudospectral technique has been employed to optimize vertical trajectories for individual aircraft in CDAs. Since the lateral path is assumed to be given by a STAR procedure, this work focused on optimizing vertical profile only using time and fuel consumption as performance indices. All the phases are formulated based on operational constraints and flap/gear schedules. The initial along track distance is free. Hence, it is possible to calculate both the optimal top-of-descent (TOD) and CDA trajectory. The optimal trajectories have been computed for two aircraft types: a Boeing 737-500 and a Boeing 767-400.

In [10], the vertical trajectory optimization for the en route descent phase of an aircraft has been studied in the presence of both along-track and cross winds, which are both modeled as functions of altitude. Flight idle thrust was assumed during the entire descent phase. The problem is formulated as an optimal control problem. The flight range was specified from a point during the latter stages of the cruise to the meter fix. Calibrated air speed (CAS) and Mach constraints, which are the state path constraints, are considered, along with flight path angle constraints, and a maximum descent rate limit, which is a mixed input and state path constraint. The descent trajectory is optimized with respect to two cost functionals: fuel and emissions. The effects of wind speed, windshear, and cross-wind on the optimal trajectory have been analyzed using the models of two types of aircraft, Boeing 737-500 and Boeing 767-400.

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 [11]. The resulting nonlinear programming (NLP) problem has been solved using the NLP solver IPOPT [12].

The paper is structured as follows. In Section 2, the general optimal control problem for multiple dynamical systems is stated and the direct collocation approach for its resolution is described. In Section 3, the aircraft equation of motion and the flight envelope constraints are stated. In Section 4, the general approach to model logical constraints is presented, which is then particularized to model time-based and distance-based separation constraints between aircraft, obstacle avoidance constraints, and waypoint constraints. In Section 5, the results of the application of the proposed method to solve several trajectory optimization problems for multiple aircraft with logical constraints in disjunctive form are reported and discussed. Finally, in Section 6, some conclusions are drawn.

#### 2. Optimal Control Approach

##### 2.1. Statement of the Optimal Control Problem

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
for , where describes the right-hand side of the differential equation
and describes the algebraic constraints
where and are the state and control sets, respectively, is a -dimensional state variable, is a -dimensional control input, is a vector of parameters, and represents time, in which and denote the initial time and final time for aircraft *p*, with .

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: where

The objective function is given in Bolza form. It is expressed as a combination of a Mayer term and a Lagrange term

Functions are assumed to be twice differentiable. Function is assumed to be piecewise Lipschitz continuous within the time interval , and the derivative of the algebraic right-hand side function with respect to , that is, is assumed to be regular within the time interval , where denotes the -dimensional algebraic variable, that is, the state variables without time derivative. Vector represents the initial conditions given at the initial time and function provides the terminal conditions at the final time , and it is assumed to be twice differentiable. The system must also satisfy algebraic path constraints within the time interval given by the vector function with lower bound and upper bound . Function is assumed to be twice differentiable.

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 (4b) and (4c) represent the differential-algebraic equation system that governs the motion of the dynamical system, e.g., the aircraft. Equation (4d) models the physical performance limitations of the dynamical system, typically expressed as upper and lower bounds on both states and control variables. Equations (4e) and (4f) denote the boundary (initial and final, respectively) conditions of the aircraft. Note that Equations (4c) and (4d) will also include the logical constraints that model conflict detection and resolution, and operational constraints as described in Section 4, which are of special interest for the problem studied in this paper.

Hence, the optimal control problem (4a), (4b), (4c), (4d), (4e) and (4f) consists in finding an admissible control such that the set of aircraft follow an admissible trajectory between the initial and final state that minimizes the performance index . The final time, , may be fixed or free.

##### 2.2. Direct Collocation Transcription of 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 [11] has been used. The time interval has been subdivided into subintervals of equal length, whose endpoints are with and . In each subinterval , , the Hermite-Simpson numerical integration scheme has been used.

The set of constraints of the resulting NLP problem includes the Hermite-Simpson system constraints that correspond to the differential constraint (4b) and the discretized versions of the other constraints of the optimal control problem. They include the algebraic constraints (4c), the state and control envelope constraints (4d), and the boundary conditions (4e) and (4f). The unknowns of the NLP problem are the values of the state and the control variables at the endpoints of each subinterval , .

To solve the resulting NLP problem, the open source IPOPT solver [12] has been employed. It implements an interior point line search filter method and it is able to handle properly large-scale sparse nonconvex NLP problems, with a large number of equality and inequality constraints. Source and binary files are available at the Computational Infrastructure for Operations Research (COIN-OR) website (https://www.coin-or.org/).

#### 3. Aircraft Model Description

Following [13], a common three-degree-of-freedom dynamic model has been used which describes the point variable-mass motion of the aircraft over a spherical Earth model. In particular, a symmetric flight has been considered. Thus, it has been assumed that there is no sideslip and all forces lie in the plane of symmetry of aircraft.

##### 3.1. Equations of Motion

The following equations of motion of the aircraft have been considered:

The three dynamic equations in (14) are expressed in an aircraft-attached reference frame , and the three kinematic equations are expressed in a ground-based reference frame as shown in Figure 1.