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Research Papers

Efficient Computation of Separation-Compliant Speed Advisories for Air Traffic Arriving in Terminal Airspace

[+] Author and Article Information
Alexander V. Sadovsky

NASA Ames Research Center,
Moffett Field, CA 94035
e-mail: alexander.v.sadovsky@nasa.gov

Damek Davis

UCLA,
Department of Mathematics,
Los Angeles, CA 90095-1555

Douglas R. Isaacson

NASA Ames Research Center,
Moffett Field, CA 94035

1Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received February 2, 2012; final manuscript received February 21, 2014; published online May 8, 2014. Editor: J. Karl Hedrick.

J. Dyn. Sys., Meas., Control 136(4), 041027 (May 08, 2014) (10 pages) Paper No: DS-12-1043; doi: 10.1115/1.4026957 History: Received February 02, 2012; Revised February 21, 2014

A class of problems in air traffic management (ATM) asks for a scheduling algorithm that supplies the air traffic services authority not only with a schedule of arrivals and departures but also with speed advisories. Since advisories must be finite, a scheduling algorithm must ultimately produce a finite data set, hence must either start with a purely discrete model or involve a discretization of a continuous one. The former choice, often preferred for intuitive clarity, naturally leads to mixed-integer programs (MIPs), hindering proofs of correctness and computational cost bounds (crucial for real-time operations). In this paper, a hybrid control system is used to model air traffic scheduling, capturing both the discrete and continuous aspects. This framework is applied to a class of problems, called the fully routed nominal problem. We prove a number of geometric results on feasible schedules and use these results to formulate an algorithm that attempts to compute a collective speed advisory, effectively piecewise linear with finitely many vertices, and has computational cost polynomial in the number of aircraft. This work is a first step toward optimization and models refined with more realistic detail.

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Figures

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Fig. 1

A portion of the LAX terminal airspace for arriving traffic. The thin arrows indicate the traffic directions.

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Fig. 2

Examples of three discrete modes with edge set E = {e1, e2,…, e6} and moving agent set A = {1,2,3} (A = 3). The agents are shown as numbered gray squares. The discrete modes shown are: (a) μ1:μ1(1) = e1,μ1(2) = e1,μ1(3) = e4; (b) μ2:μ2(1) = e1,μ2(2) = e2,μ2(3) = e4; (c) μ3:μ3(1) = e1,μ3(2) = e3,μ3(3) = e4.

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Fig. 3

(a)–(c) The continuous state spaces Xμ1,Xμ2,Xμ3 corresponding to discrete modes μ1, μ2, μ3 from Fig. 2. Among the discrete modes the system may enter from μ1 are μ2 and μ3, accordingly as agent 2 enters e2 or e3. Conflicting states not shown. (d) A gluing of Xμ1 to Xμ2 and to Xμ3 (see Fig. 3). Glued, all three state spaces share a face, but no two have any other points in common. Conflicting states not shown.

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Fig. 4

Agents 1, 2 on their respective rectilinear edges e1,e2, which share a common vertex, taken as the origin 0 in R2. The orientation of the edges is not specified. (a) The unit vectors a1,a2 are collinear with the respective edges, but their directions do not necessary agree with the edges' orientations. (b) With suitably chosen scalars coefficients c1,c2, the vectors c1a1 and c2a2 are the respective position vectors of the two agents.

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Fig. 5

(a) An example of two elliptical sectors in the c1c2-plane corresponding to conflicting states. (b) An example of two stripes in the c1c2-plane corresponding to conflicting states of two agents on the same edge.

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Fig. 6

An example of two agents whose paths overlap. The black star shows the beginning of the overlap in (a) and the corresponding state (both agents being at that point) in (b); the white star, the end of the overlap in (a) and the corresponding state (both agents being at that point) in (b). The system, shown in (a), has seven discrete modes with both agents in the transportation network. Each mode's set of separation-violating states, shown in (b) as a connected [31] gray region, is “glued” to some of the others. The result of the gluing is the connected region shown in (c).

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Fig. 7

The case of A = 3 aircraft: (a) the distal edges and the target set and (b) an illustration of the distal boundary

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Fig. 8

The shaded regions are the orthogonal projections of the half-spaces (a) Hα1,α2;-1, (b) Hα1,α2;+1, and (c) Fα1,α2 onto the yα1yα2-plane. Their intersection is the polygonal approximation, shown in Fig. 8(d), of the conflict zone for α1,α2. (d) An approximation of a pairwise conflict zone (Fig. 6(c)) by a polygonal region equal to the intersection of three half-planes. (e) The geometry of the safe wedges (shaded) and DICPs.

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Fig. 9

An example with two aircraft. The conflict zone is as in Fig. 8(d). (a) A collective state trajectory y(t), with initial state y(0) = y0. (b) The cone of attainability C(y0) (shown in the lighter shade) at y0. (c) The cone is positioned so that there are no feasible solutions.

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Fig. 10

An example with A = 3 aircraft (compare with Fig. 7). There are (A2) = 3 conflict zones. The orthogonal projection of zone Zα1,α2 onto the yα1yα1-coordinate plane (onto the appropriate face of the parallelotope) has the form shown, as a shaded region, in Fig. 9(a).

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Fig. 11

An example illustrating Lemma 6.5. There are A = 3 aircraft, and the initial state y0 lies in a safe wedge. By going from y0 in the direction d, one reaches the distal boundary (in state p) without running into a conflict zone.

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Fig. 12

An example, with A = 3 aircraft, illustrating the algorithm. In steps 4 and 5, one finds a DICP p closest to y0 and proceeds from p along the direction of −d until reaching a state y1 on the boundary of C(y0). The segment y0y1 may or may not be feasible. The latter case is illustrated in (a) (the infeasible part of y0y1 is shown dashed) and is handled by step 7, which is illustrated in (b). If y0y1 had turned out feasible, the piecewise linear trajectory y0y1p would have constituted a first portion of a solution, and this would have been detected and handled in step 6.

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Fig. 13

The initial state of a traffic scenario for 23 aircraft. To capture airspace adequately, an aspect ratio different from 1 is used. The thin arrows indicate the directions of the traffic.

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