American Institute of Aeronautics and Astronautics

1

Sensor Resource Management to Support UAS Integration

into the National Airspace System

Nicholas Hanlon1, Dr. Elad Kivelevitch

2, and Dr. Kelly Cohen

3

University of Cincinnati, Cincinnati, OH 45221

The National Air Space (NAS) is often described as a complex aviation system-of-systems

that seamlessly works in harmony to provide safe transit for all aircraft within its domain.

The number of aircraft within the NAS is growing and according the FAA1, “[o]n any given

day, more than 85,000 flights are in the skies in the United States...this translates into

roughly 5,000 planes in the skies above the United States at any given moment. More than

15,000 federal air traffic controllers in airport traffic control towers, terminal radar

approach control facilities and air route traffic control centers guide pilots through the

system". The expected increase in projected density of aircraft, including integration of

UAVs, requires innovative resource allocation methodologies to task and re-task sensors

(radar systems) to effectively monitor airborne vehicles and ensure aircraft adhere to the

minimum separation requirement. Our approach is to use an evolutionary algorithm to

evolve the task assignment. In addition, we utilize the Kalman Filter’s covariance matrices

to determine positional uncertainty in our estimates to predict if a separation requirement is

violated.

I. Introduction

The National Air Space (NAS) is often described as a complex aviation system-of-systems that seamlessly works

in harmony to provide safe transit for all aircraft within its domain. The number of aircraft within the NAS is

growing and according the FAA1, “[o]n any given day, more than 85,000 flights are in the skies in the United

States...this translates into roughly 5,000 planes in the skies above the United States at any given moment. More

than 15,000 federal air traffic controllers in airport traffic control

towers, terminal radar approach control facilities and air route

traffic control centers guide pilots through the system". Figure 1

provides a glimpse of the air traffic within the NAS. The advent of

World War II introduced RADAR (RAdio Detection and Ranging)

systems, leading to today's current and primary Air Traffic Control

System2,3

. ATC Systems rely on two radar systems, Primary

Surveillance Radar (PSR) and Secondary Surveillance Radar

(SSR), to manage air traffic. PSR systems detect and track objects

by reflecting electromagnetic waves off the object; SSR systems

supplement PSR systems by communicating with the object to

gather aircraft information and flight characteristics but relies on an

onboard transponder installed on the aircraft. The FAA does

provide Radar Traffic Information Service4 (according to the FAA

Aeronautical Information Manual (AIM), Chapter 4, Section 4-1-15) for the purpose of "assisting and advising a

pilot that a particular radar target's position and track indicates it may intersect or pass in such proximity to that

pilot's intended flight path that it warrants attention." The caveat to this service lies in the human interface, as

explicitly stated in the provision section that traffic volume, workload, and communications frequency congestion

may prevent the controller from providing timely instruction. The goal of this research is to begin the foundational

1 Graduate Student, Department of Aerospace Engineering and Engineering Mechanics, AIAA Student Member.

2 Assistant Professor, Department of Aerospace Engineering and Engineering Mechanics, AIAA Senior Member.

3 Professor, Department of Aerospace Engineering and Engineering Mechanics, AIAA Associate Fellow.

Figure 1. Aircraft Distribution within NAS

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AIAA Infotech @ Aerospace

5-9 January 2015, Kissimmee, Florida

AIAA 2015-0360

Copyright © 2015 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

AIAA SciTech

American Institute of Aeronautics and Astronautics

2

steps towards an innovative resource allocation algorithm based on evolutionary algorithms based on a unique

approach of evaluating aircraft separation uncertainty.

The rest of the paper is organized as follows: in Section II we present the problem formulation, including our

aircraft model, Kalman Filter, constraints, and assumptions. Section III describes the structure of the evolutionary

algorithm and its application. Section IV gives the results achieved in this work and Section V draws some

conclusions and recommendations for future work.

II. Problem Formulation

A. Aircraft Model

A commercial aircraft flight path is generally described by the series of events such as takeoff, climb, cruise,

descent, and landing. The majority of the flight time is spent in cruise mode travelling between waypoints with

standard rate turns at the waypoint to correct course direction towards the next waypoint. The two modes are

summarized as a Uniform Motion (UM) Model and Coordinated Turn (CT) Model. For purposes of this work, we

are considering only the UM Model. The state of the aircraft is defined within the Cartesian coordinate system as:

𝐱 = [𝑥 �̇� 𝑦 �̇�]𝑇 (1)

The UM Model is defined as a linear dynamic model:

𝐱k = 𝐅𝐱k−1 + 𝐯 (2)

where the transition matrix 𝐅 is defined as

𝐅 = [

𝐅x ⋮ 𝟎⋯ ⋮ ⋯𝟎 ⋮ 𝐅y

] (3)

and 𝐅𝐱,𝐲 are

𝐅𝐱 = [1 ∆t0 1

] 𝐅𝐲 = [1 ∆t0 1

] (4)

The acceleration noise vector 𝐯 is modeled as white, zero-mean Gaussian.

B. Kalman Filter

Kalman5 originally proposed his filter and prediction in 1960 and, along with variants for non-linear systems, it

has been used extensively for target tracking applications. The algorithm is a two-step process that predicts the next

state and error covariance, and then corrects its estimates based on input measurements. The linear system

prediction steps are:

Predicted State 𝐱k|k−1 = 𝐅𝐱k−1|k−1 + 𝐆uk (5)

Predicted Covariance 𝐏k|k−1 = 𝐅𝐏k−1|k−1𝐅𝐓 + 𝐐 (6)

Given that a measurement 𝑧𝑘 is made at time step k, the state estimate and covariance matrix are updated:

Measurement Residual 𝐲k = 𝐳k − 𝐇𝐱k|k−1 (7)

Innovation 𝐒k = 𝐇𝐏k|k−1𝐇𝐓 + 𝐑 (8)

Kalman Gain 𝐊k = 𝐏k|k−1𝐇𝐓𝐒k

−1 (9)

Updated State 𝐱k|k = 𝐱k|k−1 + 𝐊k𝐲k (10)

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Updated Covariance 𝐏k|k = (𝐈 − 𝐊k𝐇k)𝐏k|k−1 (11)

where 𝐇 is the observation matrix, 𝐐 is process noise, 𝐑 is measurement noise, and 𝐆 is the control matrix.

The key lesson is that projecting the covariance matrix does not rely on taking a measurement. If one does not

intend on taking a measurement, the covariance is predicted by Eq. (6). If one plans to take a measurement, the

covariance matrix is calculated using Eq. (8), (9), and (11). Therefore, future covariance matrices can be projected

out a specified time frame.

C. Formulation and Constraints

FAA radar systems rotate 360 degrees scanning the environment and collecting target measurements. The

accurate performance of the sensors are entailed, but a simplified model can be characterized by a few parameters,

such as, rotations per minute (RPM), horizontal beam width (𝜃𝑎𝑧), pulse repetition frequency (PRF), and pulse

repetition time (PRT = 1/PRF). In a single target case, the number of echo signals returned to the radar is termed

“Hits per Scan" (HPS)6 as depicted in Figure 2.

For example, the HPS is calculated in Eq. (12) using notational ASR-9 parameters: 𝜃𝑎𝑧 = 1.4°, RPM = 12.5, and

PRF = 325.

HPS =

𝜃𝑎𝑧 ∗ 60

360° ∗ 𝑅𝑃𝑀 ∗ 𝑃𝑅𝑇=

1.4° ∗ 60

360° ∗ 12.5 ∗ 1/325= 6.60667 ≈ 6 (12)

Theoretically, the radar would collect 6 target measurements per sweep or 75 per minute given that the

Probability of Detection (𝑃𝐷) = 100% In addition, a single pulse may return multiple echoes to indicate multiple

targets, assuming that the Probability of False Alarm (𝑃𝐹𝐴) = 0%. However, the processing demand stresses the

sensor's capabilities limiting the number of targets that can be tracked simultaneously. This concept is abstracted for

this research. For a time horizon of 𝑥 minutes, a radar will collect up to

Number of Data Measurements = x ∗ RPM ∗ HPS (13)

target measurements. Using the numbers above for the ASR-9, in 1-minute, a total of 1 ∗ 12.5 ∗ 6.06667 =75.8333 ≈ 76 data measurements are made on a single target. The radar is also limited in the number of pulses sent

out based on the pulse repetition frequency. Using the same data, the number of pulses is

Number of Transmitted Pulses = x ∗ 60 ∗ PRF (14)

or for the 1-minute horizon, 1 ∗ 60 ∗ 325 = 19,500 pulses. Based on the formulation, the resource allocation

problem becomes evident: how to allocate the pulse distribution among targets under two constraints: (a) each

sensor is limited to the number of pulses described in Eq. (14), and (b) each target is limited to the hits per scan

described in Eq. (13).

D. Assumptions

Since the goal of this work is to focus on sensor allocation for aircraft separation validation, a few assumptions

have been made. First, the aircraft are travelling with the same velocity vector so their position relative to one

another remains the same. Second, the actual sensor measurement is removed since the optimization algorithm

relies only on the state covariance matrix from the Kalman Filter.

Figure 2. Hits Per Second

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III. Approach

A. Evolutionary Algorithm

1. Structure

Unlike a genetic algorithm where the chromosome length is fixed with binary values, this solution must handle

dynamic chromosomes with integer values as each genome represents the tracked targets and their task assignment.

It is not necessary that each sensor contain each aircraft in its list as some aircraft might be outside the range of that

particular sensor. The following is a simplified example of a candidate solution to illustrate the chromosome

structure. Sensors A and B performance characteristics limits the number of pulses transmitted to 30 pulses per time

horizon where each target has a HPS of 20 pulses. Sensor C has a higher PRF, limiting the number of pulses

transmitted to 45 pulses per time horizon where each target has a HPS of 25 pulses.

Note that it is possible for a target to not have any assignment (e.g., Sensor B – Aircraft 3). In this case, the

target is left in the sensor's genome as a placeholder to indicate that it is still an active track but no sensor resources

are assigned to the target for the next time horizon.

2. Elitism

Elitism is an exploitative mechanism7 to leverage the knowledge gained by maintaining a fixed percentage of

solutions as benchmark solutions for performance comparison as the algorithm continuously evolves. The design in

this work maintains the top 5% of solutions and is added to the next generation without any evolutionary operations.

3. Selection

There are a few different mechanisms developed over time to prune the population7. Roulette Selection selects

individuals proportional to their fitness value compared to the overall fitness of the population. This was extended to

the Stochastic Universal Sampling that ensures fit individuals are always selected. The issue with the above

methodologies requires that the value of the fitness score has to be meaningful enough during the comparison, and

not just the ranking between the solutions. The Tournament Selection overcomes this issue by relying on the rank

order of the solutions. This work employs the Tournament Selection technique given its simple design and

implementation, computationally quick, and tunable.

4. Mutation

The standard mutation in genetic algorithms is a bit-flip, that is, replacing a 0 (zero) with a 1 (one), and vice-

versa. Since the genomes are not binary in design, a different mutation design is required to achieve the same affect.

Therefore, the bit-flip is replaced with a mutation vector [𝑚 𝑑], that provides a magnitude of change, m, and

direction (positive or negative), d. In other words, a positive magnitude increases the sensor's processing on a target.

The magnitude of change is drawn from a normal distribution where the variance is based on the relative

performance of the solution compared to the elite solution. This permits good solutions to further exploit their

knowledge and fine-tune its assignments. On the other hand, poor solutions may have larger magnitude changes due

to the larger variance in the distribution, resulting in a more explorative change.

Sensor A Aircraft 1 Aircraft 2 Aircraft 3

17 10 3

Sensor B Aircraft 1 Aircraft 3 Aircraft 4

18 0 12

Sensor C Aircraft 4

25 Table 1. Distribution of Aircraft Assignment

Sensor A Sensor B Sensor C

Aircraft 1 Aircraft 2 Aircraft 3 Aircraft 1 Aircraft 3 Aircraft 4 Aircraft 4

17 10 3 18 0 12 25 Table 2. Chromosome Structure

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A challenge in evolutionary algorithms is ensuring that mutation operations still produce feasible solutions. It is

likely that the mutation operation for the FAA problem will violate the maximum number of targets tracked

constraint or the maximum number of data point associations. In these cases, the value is capped at the maximum

value.

5. Crossover

Similar to the mutation operation, crossover should also produce feasible solutions. To ensure only feasible

solutions result, the crossover is limited to only occur between sensors rather than at the sensor-task assignment,

guaranteeing that offspring are still feasible solutions as the constraint on task assignment is at the sensor level.

B. Objective Function

The goal of this work is to provide optimal task assignment for maximum situational awareness for ATC

controllers in terms of aircraft separation. To do so, we must reduce the uncertainty in the aircraft state to provide

reliable assessments on the proximity of aircraft from one another. First, assume a single airspace such that the

requirement of separation is uniform throughout. The objective function that aggregates all the aircraft positional

uncertainty is formulated as

𝐽 = ∑𝑔(𝑥𝑖)

𝑁

𝑖=1

(15)

where 𝑁 is the total number of aircraft in the airspace and 𝑔(𝑥𝑖) is the positional uncertainty for aircraft 𝑖. This represents a single objective optimization problem and attempts to find the task allocation that minimizes

the uncertainty for the entire system. However, this approach has inherent flaws as all aircraft are treated equally. An

aircraft traveling at 37,000 feet with no other aircraft within 50 miles is evaluated equally to an aircraft at 5,000 feet

with ten other aircraft within close proximity. To resolve this issue, a weighted sum design is included to show

preference of one aircraft over others:

𝐽 = ∑𝑤𝑖𝑔(𝑥𝑖)

𝑁

𝑖=1

(16)

where 𝑤𝑖 > 0. Arbitrary weight coefficients will likely lead to arbitrary results unless meaning is provided to the

coefficients. Since we are interested in reducing the positional uncertainty of aircraft, the weight represents the

density of aircraft in a volume of space where the higher the density, the larger the weight. However, Coello8 notes

that “the weighting coefficients do not reflect proportionally the relative importance of the objectives, but are only

factors which, when varied, locate points in the Pareto set." In the event that the weights are optimized, the

formulation still does not fully achieve the stated goal. For instance, suppose multiple aircraft are within a volume of

space and an increased level of positional certainty is required. We want to increase the positional certainty to a

specified level but based on the `Law of Diminishing Returns`, any additional time spent on the targets provides

minimal reduction, while simultaneously, reducing the ability to track other targets. In a worst case scenario, the

allocation of pulses drives the positional uncertainty of a target below the threshold required on a target, and loses

track of other targets. It is not necessary to reduce the positional uncertainty beyond what is required at the risk of

losing other targets but rather, strive to achieve a positional uncertainty goal. Charnes and Cooper9 and Ijiri

10

originally designed the concept of goal programming, which, when applied to this problem yields:

𝐽 = ∑|𝑔(𝑥𝑖) − �̂�𝑖|

𝑁

𝑖=1

(17)

where �̂�𝑖 represents the desired positional uncertainty for target 𝑖. The challenge is deciding on a proper �̂�𝑖 per

target when evaluating the separation of multiple aircraft. A different approach is to indirectly reduce the

covariance matrix by focusing on the relationship between targets. This alleviates the problem of formulating a

desired positional uncertainty �̂�𝑖 , but rather minimizes the uncertainty in the distance between targets based on their

state estimate and covariance matrices. Therefore, the objective function is formulated as:

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𝐽 = ∑ 𝑔(𝑥𝑖 , 𝑥𝑗)

𝑁

𝑖,𝑗=1

𝑖≠𝑗

(18)

Similarly, the outcome may still produce suboptimal results as it may drive the uncertainty lower then needed to

validate separation requirements. Once a separation is confidently achieved, there is not a desire to expend valuable

sensor resources on that target any longer. Therefore, the objective function leverages goal programming to where

the desired goal is more explicit, that is, the required separation 𝑆𝑖𝑗 between the targets 𝑖 and 𝑗. Beyond the desired

goal, no award is provided. Thus, 𝑔(𝑥𝑖 , 𝑥𝑗) in Eq. (18) is evaluated as:

𝑔(𝑥𝑖, 𝑥𝑗) = {

|𝑓(𝑥𝑖, 𝑥𝑗) − 𝑆𝑖𝑗|, 𝑓(𝑥𝑖, 𝑥𝑗) − 𝑆𝑖𝑗 < 0

0, 𝑓(𝑥𝑖, 𝑥𝑗) − 𝑆𝑖𝑗 ≥ 0

(19)

where 𝑓(𝑥𝑖 , 𝑥𝑖) represents the minimum distance between the target with 95% confidence, and is calculated as

follows.

The distance and variance between the two target positional estimates are:

𝑑𝑖𝑗 = √(𝑥𝑗 − 𝑥𝑖)

2+ (𝑦𝑗 − 𝑦𝑖)

2 (20)

𝜎𝑖𝑗

2 = 𝐸[(𝑑𝑖𝑗 − 𝑑𝜇)(𝑑𝑖𝑗 − 𝑑𝜇)] (21)

𝜎𝑖𝑗

2 = 𝐸[∆𝑑𝑖𝑗 ∙ ∆𝑑𝑖𝑗𝑇 ] (22)

The distance equation is non-linear so it is linearized:

𝑓(𝑥, 𝑦) ≈ 𝑓(𝑎, 𝑏) +

𝜕𝑓(𝑎, 𝑏)

𝜕𝑥|𝑎,𝑏

(𝑥 − 𝑎) +𝜕𝑓(𝑎, 𝑏)

𝜕𝑦|𝑎,𝑏

(𝑦 − 𝑏) (23)

Rearrange the terms

𝑓(𝑥, 𝑦) − 𝑓(𝑎, 𝑏) ≈

𝜕𝑓(𝑎, 𝑏)

𝜕𝑥|𝑎,𝑏

(𝑥 − 𝑎) +𝜕𝑓(𝑎, 𝑏)

𝜕𝑦|𝑎,𝑏

(𝑦 − 𝑏) (24)

Substitute 𝑓(𝑥, 𝑦) for 𝑑 and f(a, b) for 𝑑𝜇

𝑑𝑖𝑗 − 𝑑𝜇 = ∆𝑑𝑖𝑗 ≈

𝜕𝑑

𝜕𝑥𝑖|𝑥𝑖=𝑥𝑖,𝜇

∆𝑥𝑖 +𝜕𝑑

𝜕𝑦𝑖|𝑦𝑖=𝑦𝑖,𝜇

∆𝑦𝑖 +𝜕𝑑

𝜕𝑥𝑗|𝑥𝑗=𝑥𝑗,𝜇

∆𝑥𝑗 +𝜕𝑑

𝜕𝑦𝑗|𝑦𝑗=𝑦𝑗,𝜇

∆𝑦𝑗 (25)

Rearrange again into matrix form

∆𝑑𝑖𝑗 ≈ [𝜕𝑑

𝜕𝑥𝑖|𝑥𝑖=𝑥𝑖,𝜇

𝜕𝑑

𝜕𝑦𝑖|𝑦𝑖=𝑦𝑖,𝜇

𝜕𝑑

𝜕𝑥𝑗|𝑥𝑗=𝑥𝑗,𝜇

𝜕𝑑

𝜕𝑦𝑗|𝑦𝑗=𝑦𝑗,𝜇

]

[ ∆𝑥𝑖

∆𝑦𝑖

∆𝑥𝑗

∆𝑦𝑗] (26)

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Let

𝐷 = [

𝜕𝑑

𝜕𝑥𝑖|𝑥𝑖=𝑥𝑖,𝜇

𝜕𝑑

𝜕𝑦𝑖|𝑦𝑖=𝑦𝑖,𝜇

𝜕𝑑

𝜕𝑥𝑗|𝑥𝑗=𝑥𝑗,𝜇

𝜕𝑑

𝜕𝑦𝑗|𝑦𝑗=𝑦𝑗,𝜇

] (27)

and

𝐶 =

[ ∆𝑥𝑖

∆𝑦𝑖

∆𝑥𝑗

∆𝑦𝑗] (28)

Place Eq. (26) into Eq.(22).

σij2 = E[(𝐷𝐶)(𝐷𝐶)𝑇] = E[𝐷𝐶𝐶𝑇𝐷𝑇] (29)

Assume that there is no correlation between target covariance matrices (i.e., ∆𝑥𝑖∆𝑦𝑗 = 0), then 𝐶𝐶𝑇 is computed

as:

𝐶𝐶𝑇 =

[

∆𝑥𝑖2 ∆𝑥𝑖∆𝑦𝑖 0 0

∆𝑦𝑖∆𝑥𝑖 ∆𝑦𝑖2 0 0

0 0 ∆𝑥𝑗2 ∆𝑥𝑗∆𝑦𝑗

0 0 ∆𝑦𝑗∆𝑥𝑗 ∆𝑦𝑗2

]

(30)

𝐷 is calculated by taking the partial derivatives, respective to the appropriate variable

𝐷 = [

−(𝑥𝑗 − 𝑥𝑖)

𝑑𝑖𝑗

(𝑥𝑗 − 𝑥𝑖)

𝑑𝑖𝑗

−(𝑦𝑗 − 𝑦𝑖)

𝑑𝑖𝑗

(𝑦𝑗 − 𝑦𝑖)

𝑑𝑖𝑗

] (31)

The variance is then calculated by placing Eq. (30) and Eq. (31) into Eq. (29). To check for separation between

targets, the lower bound of distance within 95% confidence is calculated. Therefore, 𝑓(𝑥𝑖 , 𝑥𝑗) in Eq. (19) becomes:

𝑓(𝑥𝑖, 𝑥𝑗) = 𝑑𝑖𝑗 − 1.96√𝜎𝑖𝑗

2 (32)

This assumes that the aircraft separation based on the covariance matrix is Gaussian. A simple Monte Carlo

execution is performed to confirm this assumption. Two aircraft are assigned the following position vector (the

velocity component is ignored in the state vector for this analysis)

𝐱𝟏 = [10

10] 𝐱𝟐 = [

30

25] (33)

and covariance matrix values

𝐏𝟏 = [1.4 0.20.2 0.8

] 𝐏𝟐 = [0.5 0.50.5 1.0

] (34)

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A Monte Carlo simulation of 500,000 trials were performed where a random sample was drawn as dictated by

the covariance matrix for each aircraft and the distance was computed (each sample is shown in the right graph of

Figure 3. The distances of each trial run was collected and are shown in the histogram, shown in the left graph of

Figure 3. The data has a kurtosis of 2.9991 and skewness of 6.5x10-4

, verifying the Gaussian distribution

assumption.

IV. Results

A select few cases are shown below with varying number of aircraft or aircraft pairing relationships. Each case

illustrates the scenario and displays the resulting optimal task allocation based on the evolutionary algorithm.

A. Case 1: Three Aircraft

Figure 4 illustrates three targets in the environment with one sensor. The asterisk represents the Kalman Filter's

state estimate and the ellipses' represent the uncertainty in the position via the state covariance matrix.

The FAA mandates a 3-naticual mile safety separation of the aircraft assuming the targets are within airspace

class B. This distance is sufficient given the Euclidean distance between the two aircraft position estimates, shown

as the dotted line. Since the estimate has uncertainty, the aircraft could theoretically be at any position within the

error ellipses. In this particular example, the constraints on each target are 60 tasking assignments per aircraft and

the summation of all assignments cannot exceed 60. All aircraft are assumed to have the same velocity vector so

Figure 3. Gaussian Distribution Test. (Left) Histogram of Distance Values (Right) Sampled Data Points from Aircraft Covariance Matrix about its Mean

Figure 4. Three Aircraft before Optimization

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their relative positions during the planned horizon do not change. The estimated position and covariance matrices

are:

𝐀𝟏 = [4

4] 𝐀𝟐 = [

0

0] 𝐀𝟑 = [

8

3] (35)

𝐏𝟏 = [0.7 0.30.3 1.0

] 𝐏𝟐 = [0.2 0.150.15 0.4

] 𝐏𝟑 = [0.15 0.050.05 0.3

] (36)

At its current state, the distance between the state estimates and the minimum separation with 95% confidence

are:

Although the state (positional) estimates show all aircraft adhere to the separation requirement, the uncertainty

via the covariance matrices indicates that 𝐴1 and 𝐴3 may be violating that requirement. Since the covariance matrix

will grow (based on the noise in system model), focusing the sensor only on 𝐴1 and 𝐴3 may cause an issue between

the 𝐴1 ↔ 𝐴2 pairing. Thus, the problem is how to allocate the tasks among the three targets efficiently and

effectively for better situational awareness. Figure 5 illustrates the result of the selected solution based on the

evolutionary algorithm execution.

The distance between the state estimates remain the same but covariance matrices were updated and the

minimum distance with 95% confidence are listed below:

𝐴1 ↔ 𝐴2 𝐴1 ↔ 𝐴3 𝐴2 ↔ 𝐴3 Distance Between Estimates - 𝑑𝑖𝑗 5.6569 4.1231 8.5440

Minimum Distance with 95%

Confidence - �̆�𝑖𝑗 3.1776 2.4695 7.1243

Table 3. Calculated Distance and Minimum Distance with 95% Confidence

Figure 5. Three Aircraft after Optimization

𝐴1 ↔ 𝐴2 𝐴1 ↔ 𝐴3 𝐴2 ↔ 𝐴3 Distance Between Estimates - 𝑑𝑖𝑗 5.6569 4.1231 8.5440

Minimum Distance with 95%

Confidence - �̆�𝑖𝑗 3.9442 2.8865 6.8566

Table 4. Updated Distance Calculations Post Execution

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The corresponding task assignment was:

In this case, the algorithm increased the confidence in separation for the 𝐴1 ↔ 𝐴2 pairing and the 𝐴1 ↔ 𝐴3

pairing. This can be seen by the larger task assignment to 𝐴1 as target had the most influence in separation between

the other targets. Although the 𝐴1 ↔ 𝐴3 pairing still appears to potentially violate the separation, we have increased

the confidence that is it closer to 3-nm than originally thought. The corresponding updated covariance matrices are

𝐏𝟏 = [0.2878 0.16390.1639 0.1784

] 𝐏𝟐 = [0.2906 0.13870.1387 0.1651

] 𝐏𝟑 = [0.2766 0.16540.1654 0.1913

] (37)

B. Case 2: Three Aircraft Modified

An interesting observation is the fact that the minimum distance between 𝐴2 and 𝐴3 exceeds the minimum

separation by a good margin. Thus, it may be worthwhile to explore disregarding target pairings given a proper

metric is satisfied, such as a distance greater than 6-nm. It is still possible to have covariance matrices that would

indicate a potential separation violation, but the volume of uncertainty of the aircraft would the aircraft lost. The

problem is reformulated (Figure 6) to only evaluate the distance certainty between the 𝐴1 ↔ 𝐴2 pairing and

the 𝐴1 ↔ 𝐴3 pairing, thus resulting in a different in task allocation.

The corresponding task assignment was:

The resulting covariance matrices are:

𝐏𝟏 = [0.2324 0.13140.1314 0.1602

] 𝐏𝟐 = [0.3169 0.15050.1505 0.1718

] 𝐏𝟑 = [0.3281 0.19700.1970 0.2078

] (38)

and the distance calculations are:

𝑨𝟏 𝑨𝟐 𝑨𝟑 26 17 17

Table 5. Three Aircraft Optimal Allocation

Figure 6. Three Aircraft (Modified) before Optimization

𝑨𝟏 𝑨𝟐 𝑨𝟑 28 16 16

Table 6. Three Aircraft Optimal Allocation

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C. Three Aircraft Analysis

Case 1 and case 2 demonstrated the approach but varied the aircrafts it utilized in the objective function. Table ()

summarizes the results to compare the effectiveness.

By excluding the 𝐴2 ↔ 𝐴3pairing from the objective function, the algorithm provided better certainty

measurements of the distances between those targets close to the separation requirements. Another method to

evaluate the performance is by measuring the volume of the covariance matrices, a topic that Kalandros11

discusses

and which details the advantages and disadvantages each, such as, matrix determinant, trace, eigenvalue, etc. Each

of these were evaluated and resulted in the same conclusion, thus only the determinant results are displayed

It is intuitive that driving down the uncertainty in 𝐴1 would be most beneficial since it affects the distance in

both the 𝐴1 ↔ 𝐴2 pairing and the 𝐴1 ↔ 𝐴3 pairing. By focusing on only those target relationships of utmost

importance, the determinant of 𝐴1 is lower in the modified case as opposed to evaluating all pairings.

D. Case 3 – Four Aircraft

The work is extended to four aircraft as depicted Figure 7. The three target scenario remains the same and a

fourth target is added at 𝐴4 = [6 10]𝑇 with an initial covariance matrix of 𝑃4 = [0.03 0.02; 0.02 0.5]. The

concept of excluding aircraft pairing where the distance 𝑑𝑖𝑗 exceeds some metric is used so only 𝐴1 is paired with

the other aircraft. Given that the sensor does not exceed the pulses that can be sent, the sensor can track a fourth

target up to the maximum number hits per scan within the planned horizon (60).

𝐴1 ↔ 𝐴2 𝐴1 ↔ 𝐴3 𝐴2 ↔ 𝐴3 Distance Between Estimates - 𝑑𝑖𝑗 5.6569 4.1231 8.5440

Minimum Distance with 95%

Confidence - �̆�𝑖𝑗 3.9907 2.8918 6.7868

Table 7. Updated Distance Calculations Post Execution

𝐴1 ↔ 𝐴2 𝐴1 ↔ 𝐴3 𝐴2 ↔ 𝐴3

Distance - 𝑑𝑖𝑗 5.6569 4.1231 8.5440

Case 1 - �̆�𝑖𝑗 3.9442 2.8865 6.8566

Case 2 - �̆�𝑖𝑗 3.9907 2.8918 6.7868 Table 8. Three Aircraft Analysis in Separation Confidence

𝐴1 𝐴2 𝐴3

Case 1 – det (𝑃𝑖) 0.0245 0.0287 0.0256

Case 2 - det (𝑃𝑖) 0.0200 0.0318 0.0294 Table 9. Three Aircraft Analysis in Separation Confidence

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The optimal allocation assignment was:

and the resulting graph is shown in Figure 8.

We find that the algorithm focused more tasks on 𝐴1 given its pairing with all the other targets. However,

intuitively, it cannot just focus on that target as the uncertainty in the other targets would continue to grow

producing a less than desirable result. Even though the 𝐴1 ↔ 𝐴3 pairing appears to be violating the separation

minimum requirement, it increased its confidence in the separation and a longer planning horizon may or may not

gain complete confidence.

Figure 7. Four Aircraft before Optimization

𝑨𝟏 𝑨𝟐 𝑨𝟑 𝑨𝟒 19 15 16 10

Table 10. Four Aircraft Optimal Allocation

Figure 8. Four Aircraft after Optimization

𝐴1 ↔ 𝐴2 𝐴1 ↔ 𝐴3 𝐴1 ↔ 𝐴4

Distance - 𝑑𝑖𝑗 5.6569 4.1231 6.3246

Before Optimization - �̆�𝑖𝑗 3.1776 2.4695 3.8130

After Optimization - �̆�𝑖𝑗 3.9106 2.7719 4.7974 Table 11. Four Aircraft Analysis in Separation Confidence

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V. Conclusions and Future Work

The FAA regulates aircraft separation requirements to ensure safety for commercial aircraft. Given the projected

growth of commercial flight and integration of unmanned aerial systems, air traffic controllers and sensor systems

will be stressed to maintain optimal situational awareness. We present an approach based on evolutionary

algorithms that optimally task sensor systems for aircraft tracking to ensure aircraft separation requirements are met.

The initial work indicates that the algorithm performs well in small scenarios and showed promise for further

expansion. The algorithm converged on a solution that provided the more useful situational awareness for an air

traffic controller. In the three cases, the aircraft that had the most pairings that result with uncertainty in separation

received the most resources as minimizing that aircraft covariance matrix provided additional benefit to the

objective function. We did not explore adjusting the initial covariance matrix of the aircraft with the most pairings.

Another aircraft with less pairings may receive more resources if its initial covariance is larger and the process noise

is configured such that state variance increase is reduced due to confidence in the aircraft models. In addition,

results are affected by the selected pairings. Thus it is desired to find a metric that disregards aircraft pairings that

are not necessary to evaluate for aircraft separation requirements.

Finally, there are several aspects that we want to expand on while this project progresses. First, we want to

extend the work from two-dimensions to three-dimensions to capture for more realistic tracking and aircraft

separation. Second, we want to implement more aviation flight characteristics beyond Uniform Motion. This

includes acceleration models and coordinated turn models. This will cause the optimization algorithm to make

decisions not only on the covariance matrix but also on the predicted state of the aircraft. In other words, two

aircraft that are projected to naturally diverge during their intended flight path will require fewer resources to ensure

separation standards are met. Conversely, aircraft that are within close proximity such as near an airport, or two

aircraft converging require additional sensor resources during the planned time horizon. In addition, we want to

incorporate unmanned aerial systems where their motion is less predictable than the standard commercial aircraft.

Third, sensor measurements are required to properly predict and update in the Kalman filter. FAA radar systems

provide range and bearing measurements which furthermore require modifying the Kalman filter to a nonlinear

option such as an Extended Kalman Filter. Fourth, since additional flight models will be implemented, we want to

construct an Interacting Multiple Model (IMM) for an improved filtering to better handle the multiple flight

trajectories.

References 1Federal Aviation Administration. Next Generation Air Transportation System (NextGen). http://www.faa.gov/nextgen. 2National Air Traffic Controllers Association. A History of Air Traffic Control.

http://www.natca.org/ULWSiteResources/natcaweb/Resources/file/Media\%20Center/ATCHistory.pdf. 3 National Air Traffic Controllers Association. Navigation Programs - History.

https://www.faa.gov/about/office_org/headquarters_offices/ato/service_units/techops/navservices/history/. 4Federal Aviation Administration. Aeronautical Information Manual.

https://www.faa.gov/air_traffic/publications/atpubs/aim/index.htm. 5Rudolph Emil Kalman. A new approach to linear filtering and prediction problems. Journal of Fluids Engineering, 82(1):35-

45, 1960. 6Christian Wol_. Radar Basics. http://www.radartutorial.eu/. 7Sean Luke. Essentials of Metaheuristics. Lulu, second edition, 2013. http://cs.gmu.edu/_sean/book/metaheuristics/. 8Carlos A Coello Coello. A comprehensive survey of evolutionary-based multiobjective optimization techniques. Knowledge

and Information systems, 1(3):269-308, 1999. 9Abraham Charnes and William W Cooper. Management models and industrial applications of linear programming.

Management Science, 4(1):38-91, 1957. 10Yuji Ijiri. Management goals and accounting for control, volume 3. North Holland Pub. Co., 1965. 11Michael Kalandros. Covariance control for multisensor systems. Aerospace and Electronic Systems, IEEE Transactions on,

38(4):1138-1157, 2002. 12Federal Aviation Association. Air Traffic. http://www.faa.gov/air_traffic/

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