Hybridization of Nonlinear and Mixed-Integer Linear Programming for Aircraft Separation With Trajectory Recovery

International audienceThe approach presented in this article aims at finding a solution to the problem of conflict-free motion planning for multiple aircraft on the same flight level with trajectory recovery. One contribution of this work is to develop three consistent models, from a continuous-time representation to a discrete-time linear approximation. Each of these models guarantees separation at all times as well as trajectory recovery, but they are not equally difficult to solve. A new hybrid algorithm is thus developed in order to use the optimal solution of a mixed integer linear program as a starting point when solving a nonlinear formulation of the problem. The significance of this process is that it always finds a solution when the linear model is feasible while still taking into account the nonlinear nature of the problem. A test bed containing numerous data sets is then generated from three virtual scenarios. A comparative analysis with three different initialisations of the nonlinear optimisation validates the efficiency of the hybrid method

Comparison of Mixed-Integer Linear Models for Fuel-Optimal Air Conflict Resolution With Recovery

International audienceAny significant increase in current levels of air traffic will need the support of efficient decision-aid tools. One of the tasks of air traffic management is to modify trajectories when necessary to maintain a sufficient separation between pairs of aircraft. Several algorithms have been developed to solve this problem, but the diversity in the underlying assumptions makes it difficult to compare their performance. In this article, separation is maintained through changes of heading and velocity while minimizing a combination of fuel consumption and delay. For realistic trajectories, the speed is continuous with respect to time, the acceleration and turning rate are bounded, and the planned trajectories are recovered after the maneuvers. After describing the major modifications to existing models that are necessary to satisfy this definition of the problem, we compare three mixed integer linear programs. The first model is based on a discretization of the airspace, and the second relies on a discretization of the time horizon. The third model implements a time decomposition of the problem; it allows only one initial maneuver, and it is solved periodically with a receding horizon to build a complete trajectory. The computational tests are conducted on a benchmark of artificial instances specifically built to include complex situations. Our analysis of the results highlights the strengths and limits of each model. The time decomposition proves to be an excellent compromise

A space-discretized mixed-integer linear model for air-conflict resolution with speed and heading maneuvers

International audienceAir-conflict resolution is a bottleneck of air traffic management that will soon require powerful decision-aid systems to avoid the proliferation of delays. Since reactivity is critical for this application, we develop a mixed-integer linear model based on space discretization so that complex situations can be solved in near real-time. The discretization allows us to model the problem with finite and potentially small sets of variables and constraints by focusing on important points of the planned trajectories, including the points where trajectories intersect. A major goal of this work is to use space discretization while allowing velocity and heading maneuvers. Realistic trajectories are also ensured by considering speed vectors that are continuous with respect to time, and limits on the velocity, acceleration, and yaw rate. A classical indicator of economic efficiency is then optimized by minimizing a weighted sum of fuel consumption and delay. The experimental tests confirm that the model can solve complex situations within a few seconds without incurring more than a few kilograms of extra fuel consumption per aircraft

Two decomposition algorithms for solving a minimum weight maximum clique model for the air conflict resolution problem

International audienceIn this article, we tackle the conflict resolution problem using a new variant of the minimum-weight maximum-clique model. The problem involves identifying maneuvers that maintain the required separation distance between all pairs of a set of aircraft while minimizing fuel costs. We design a graph in which the vertices correspond to a finite set of maneuvers and the edges connect conflict-free maneuvers. A maximum clique of minimal weight yields a conflict-free situation that involves all the aircraft and minimizes the costs induced. The innovation of the model is its cost structure: the costs of the vertices cannot be determined a priori, since they depend on the vertices in the clique. We formulate the problem as a mixed integer linear program. Since the modeling of the aircraft dynamics and the computation of trajectories is separated from the solution process, the model is flexible. As a consequence, our mathematical framework is valid for any hypotheses. In particular, the aircraft can perform dynamic velocity, heading, and flight-level changes. To solve instances involving a large number of aircraft spread over several flight levels, we introduce two decomposition algorithms. The first is a sequential mixed integer linear optimization procedure that iteratively refines the discretization of the maneuvers to yield a trade-off between computational time and cost. The second is a large neighborhood search heuristic that uses the first procedure as a subroutine. The best solutions for the available set of maneuvers are obtained in less than 10 seconds for instances with up to 250 aircraft randomly allocated to 20 flight levels

Mathematical optimization methods for aircraft conflict resolution in air traffic control

Air traffic control is a very dynamic and heavy constrained environment where many decisions need to be taken over short periods of time and in the context of uncertainty. Adopting automation under such circumstances can be a crucial initiative to reduce controller workload and improve airspace usage and capacity. Traditional methods for air traffic control have been exhaustively used in the last decades and are reaching their limits, therefore automated approaches are receiving a significant and growing attention. In this thesis, the focus is to obtain optimal aircraft trajectories to ensure flight safety in the short-term by solving optimization problems. During cruise stage, separation conditions require a minimum of 5 Nautical Miles (NM) horizontally or 1000 feet (ft) vertically between any pair of aircraft. A conflict between two or more aircraft is a loss of separation among these aircraft. Air traffic networks are organized in flight levels which are separated by at least 1000 ft, hence during cruise stage, most conflicts occur among aircraft flying at the same flight level. This thesis presents several mathematical formulations to address the aircraft conflict resolution problem and its variants. The core contribution of this research is the development of novel mixed integer programming models for the aircraft conflict resolution problem. New mathematical optimization formulations for the deterministic aircraft conflict resolution problem are analyzed and exact methods are developed. Building on this framework, richer formulations capable of accounting for aircraft trajectory prediction uncertainty and trajectory recovery are proposed. Results suggest that the formulations presented in thesis are efficient and competitive enough with the state-of-art models and they can provide an alternative solution to possibly fill some of the gaps currently present in the literature. Furthermore, the results obtained demonstrate the impact of these models in solving very denser air space scenarios and their competitiveness with state-of-the-art formulations without regarding variable discretization or non-linear components

A Two-Stage Optimization-based Motion Planner for Safe Urban Driving

Recent road trials have shown that guaranteeing the safety of driving decisions is essential for the wider adoption of autonomous vehicle technology. One promising direction is to pose safety requirements as planning constraints in nonlinear, non-convex optimization problems of motion synthesis. However, many implementations of this approach are limited by uncertain convergence and local optimality of the solutions achieved, affecting overall robustness. To improve upon these issues, we propose a novel two-stage optimization framework: in the first stage, we find a solution to a Mixed-Integer Linear Programming (MILP) formulation of the motion synthesis problem, the output of which initializes a second Nonlinear Programming (NLP) stage. The MILP stage enforces hard constraints of safety and road rule compliance generating a solution in the right subspace, while the NLP stage refines the solution within the safety bounds for feasibility and smoothness. We demonstrate the effectiveness of our framework via simulated experiments of complex urban driving scenarios, outperforming a state-of-the-art baseline in metrics of convergence, comfort and progress.Comment: IEEE Transactions on Robotics (T-RO), 202

Safety and Convergence Analysis of Intersecting Aircraft Flows under Decentralized Collision Avoidance

Safety is an essential requirement for air traffic management and control systems. Aircraft are not allowed to get closer to each other than a specified safety distance, to avoid any conflicts and collisions between aircraft. Forecast analysis predicts a tremendous increase in the number of flights. Subsequently, automated tools are needed to help air traffic controllers resolve air born conflicts. In this dissertation, we consider the problem of conflict resolution of aircraft flows with the assumption that aircraft are flowing through a fixed specified control volume at a constant speed. In this regard, several centralized and decentralized resolution rules have been proposed for path planning and conflict avoidance. For the case of two intersecting flows, we introduce the concept of conflict touches, and a collaborative decentralized conflict resolution rule is then proposed and analyzed for two intersecting flows. The proposed rule is also able to resolved airborn conflicts that resulted from resolving another conflict via the domino effect. We study the safety conditions under the proposed conflict resolution and collision avoidance rule. Then, we use Lyapunov analysis to analytically prove the convergence of conflict resolution dynamics under the proposed rule. The analysis show that, under the proposed conflict resolution rule, the system of intersecting aircraft flows is guaranteed to converge to safe, conflict free, trajectories within a bounded time. Simulations are provided to verify the analytically derived conclusions and study the convergence of the conflict resolution dynamics at different encounter angles. Simulation results show that lateral deviations taken by aircraft in each flow, to resolve conflicts, are bounded, and aircraft converged to safe and conflict free trajectories, within a finite time. The proposed rule is powerful when the pilots of the collaborating aircraft, resolving a potential conflict, are either humans or robots. However, when a human pilot is collaborating with a robot pilot for the first time, the robot control should optimize its control criteria to collaborate well with the human. Basically, the robot must adapt its output as it learns from the human. We study the situation of the Human-in-the-Loop, assuming that human will follow lateral maneuvers for conflict resolution. We model the human as an optimal controller that minimizes the risk of collision. Based on that model, we use differential game analysis to get an accurate estimate for the human control criteria. We propose a new algorithm, based on least square minimization, to estimate the Kalman gain of the human's model, and therefore accurately estimate his optimal control criteria. Simulations of this learned rule show that robot pilot can successfully learn from the human pilot actions, and both can cooperate successfully to resolve any conflicts between their aircraft

A Comprehensive Survey on Particle Swarm Optimization Algorithm and Its Applications

Particle swarm optimization (PSO) is a heuristic global optimization method, proposed originally by Kennedy and Eberhart in 1995. It is now one of the most commonly used optimization techniques. This survey presented a comprehensive investigation of PSO. On one hand, we provided advances with PSO, including its modifications (including quantum-behaved PSO, bare-bones PSO, chaotic PSO, and fuzzy PSO), population topology (as fully connected, von Neumann, ring, star, random, etc.), hybridization (with genetic algorithm, simulated annealing, Tabu search, artificial immune system, ant colony algorithm, artificial bee colony, differential evolution, harmonic search, and biogeography-based optimization), extensions (to multiobjective, constrained, discrete, and binary optimization), theoretical analysis (parameter selection and tuning, and convergence analysis), and parallel implementation (in multicore, multiprocessor, GPU, and cloud computing forms). On the other hand, we offered a survey on applications of PSO to the following eight fields: electrical and electronic engineering, automation control systems, communication theory, operations research, mechanical engineering, fuel and energy, medicine, chemistry, and biology. It is hoped that this survey would be beneficial for the researchers studying PSO algorithms

A Hybrid Tabu/Scatter Search Algorithm for Simulation-Based Optimization of Multi-Objective Runway Operations Scheduling

As air traffic continues to increase, air traffic flow management is becoming more challenging to effectively and efficiently utilize airport capacity without compromising safety, environmental and economic requirements. Since runways are often the primary limiting factor in airport capacity, runway operations scheduling emerge as an important problem to be solved to alleviate flight delays and air traffic congestion while reducing unnecessary fuel consumption and negative environmental impacts. However, even a moderately sized real-life runway operations scheduling problem tends to be too complex to be solved by analytical methods, where all mathematical models for this problem belong to the complexity class of NP-Hard in a strong sense due to combinatorial nature of the problem. Therefore, it is only possible to solve practical runway operations scheduling problem by making a large number of simplifications and assumptions in a deterministic context. As a result, most analytical models proposed in the literature suffer from too much abstraction, avoid uncertainties and, in turn, have little applicability in practice. On the other hand, simulation-based methods have the capability to characterize complex and stochastic real-life runway operations in detail, and to cope with several constraints and stakeholders’ preferences, which are commonly considered as important factors in practice. This dissertation proposes a simulation-based optimization (SbO) approach for multi-objective runway operations scheduling problem. The SbO approach utilizes a discrete-event simulation model for accounting for uncertain conditions, and an optimization component for finding the best known Pareto set of solutions. This approach explicitly considers uncertainty to decrease the real operational cost of the runway operations as well as fairness among aircraft as part of the optimization process. Due to the problem’s large, complex and unstructured search space, a hybrid Tabu/Scatter Search algorithm is developed to find solutions by using an elitist strategy to preserve non-dominated solutions, a dynamic update mechanism to produce high-quality solutions and a rebuilding strategy to promote solution diversity. The proposed algorithm is applied to bi-objective (i.e., maximizing runway utilization and fairness) runway operations schedule optimization as the optimization component of the SbO framework, where the developed simulation model acts as an external function evaluator. To the best of our knowledge, this is the first SbO approach that explicitly considers uncertainties in the development of schedules for runway operations as well as considers fairness as a secondary objective. In addition, computational experiments are conducted using real-life datasets for a major US airport to demonstrate that the proposed approach is effective and computationally tractable in a practical sense. In the experimental design, statistical design of experiments method is employed to analyze the impacts of parameters on the simulation as well as on the optimization component’s performance, and to identify the appropriate parameter levels. The results show that the implementation of the proposed SbO approach provides operational benefits when compared to First-Come-First-Served (FCFS) and deterministic approaches without compromising schedule fairness. It is also shown that proposed algorithm is capable of generating a set of solutions that represent the inherent trade-offs between the objectives that are considered. The proposed decision-making algorithm might be used as part of decision support tools to aid air traffic controllers in solving the real-life runway operations scheduling problem