Determining Follower Aircrafts Optimal Trajectory in Relation to a Dynamic Formation Ring Carl A. Gotwald Michael D. Zollars

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Determining Follower Aircraft’s Optimal Trajectory in

Relation to a Dynamic Formation Ring

Carl A. Gotwald & Michael D. Zollars

Department of Aeronautics and Astronautics

Air Force Institute of Technology

Wright-Patterson AFB, OH 45433

carl.gotwald@aﬁt.edu; michael.zollars@aﬁt.edu

Isaac E. Weintraub

Controls Science System Center

Wright-Patterson AFB, OH 45433

isaac.weintraub.1@us.af.mil

Abstract—The speciﬁc objective of this paper is to develop a tool

that calculates the optimal trajectory of the follower aircraft as it

completes a formation rejoin, and then maintains the formation

position, deﬁned as a ring of points, until a ﬁxed ﬁnal time. The

tool is designed to produce optimal trajectories for a variety

of initial conditions and leader trajectories. Triple integrator

dynamics are used to model the follower aircraft in three di-

mensions. Control is applied directly to the rate of acceleration.

Both the follower’s and leader’s velocities and accelerations are

bounded, as dictated by the aircraft’s performance envelope.

Lastly, a path constraint is used to ensure the follower avoids

the leader’s jet wash region. This optimal control problem is

solved through numerical analysis using the direct orthogonal

collocation solver GPOPS-II. Two leader trajectories are inves-

tigated, including a descending spiral and continuous vertical

loops. Additionally, a study of the effect of various initial guesses

is performed. All trajectories displayed a direct capture of the

formation position, however changes in solver initial conditions

demonstrate various behaviors in how the follower maintains

the formation position. The developed tool has proven adequate

to support future research in crafting real-time controllers ca-

pable of determining near-optimal trajectories.

TABLE OF CONTENTS

1. INTRODUCTION................................... 1

2. PROBLEM FORMULATION ........................ 2

3. SOLUTION METHODOLOGY ...................... 5

4. RESULTS .......................................... 6

5. CONCLUSIONS .................................... 10

ACKNOWLEDGMENTS ............................... 11

REFERENCES ........................................ 11

BIOGRAPHY ......................................... 11

1. INTRODUCTION

Autonomous formation control is a growing area of interest

for future Air Force operations. Autonomous aircraft are

increasingly used across the battle space to avoid putting

personnel at risk. Thus, the need for autonomous aircraft to

handle formation tasks normally requiring a pilot has grown.

One model for this interaction consists of a leader aircraft

who is accompanied by an autonomous follower aircraft.

The ability for the follower aircraft to autonomously rejoin

and maintain a designated formation position would free the

leader aircraft to focus on other duties, leading to increased

mission effectiveness. The leader aircraft often has primary

responsibility for interactions outside of the formation, so

having an autonomous wingman which can execute without

additional oversight from the lead aircraft in an efﬁcient

manner would decrease the leader workload. The scenario

U.S. Government work not protected by U.S. copyright

investigated in this paper focused on the response of a wing-

man rejoining from an arbitrary starting location to a deﬁned

formation position and maintaining that position until a ﬁxed

ﬁnal time. This same behavior could readily be applied to

the scenario of an autonomous aircraft attempting to rapidly

attain an attack position on an enemy aircraft, along with

many other combat and training applications.

Figure 1.Fighting Wing Position [1]

Extensive research in maintaining a desired formation posi-

tion, deﬁned as a single location relative to a leader aircraft,

has been conducted in articles such as [2], [3], and [4].

The author of [5] presented a model predictive controller

which added the ability to arrive at a target waypoint with

a desired velocity vector orientation. To extend this concept

beyond single-point formation positions, further research was

conducted which crafted a control architecture allowing a

follower aircraft to rejoin to a ring of points deﬁned relative

to a leader aircraft [6]. This ring of formation points was

inspired by the traditional formation position ”ﬁghting wing”

as seen in Figure 1 referenced from [1]. Future research aims

to dictate a trajectory in real-time for which an autonomous

wingman would attain and maintain a formation position de-

ﬁned in reference to a maneuvering leader while minimizing

a desired cost function. In order to attain an approximate

minimum in real-time may require the use of different control

laws depending on the initial states of the leader and follower

aircraft. Thus, a state space of boundaries could be deﬁned to

arXiv:2210.01665v1 [math.OC] 4 Oct 2022

dictate when certain control laws would be optimal. Further,

future research aims to create a strategy which will optimally

select from these predeﬁned control laws and implement them

in real-time. Since it is unlikely a fully optimal solution can

be attained in real-time for this problem, the control strategy

will aim to approximate an optimal trajectory. Therefore, the

ability to calculate the degree of the error of this approxi-

mation is needed. This paper speciﬁcally aimed to create a

validation tool that will determine the optimal path for future

comparison to the approximated trajectory generated by the

real-time control strategy.

One speciﬁc scenario was considered. The follower aircraft

being tasked to rejoin to a deﬁned formation ring in min-

imum time, and then maintaining that formation position

with minimum deviation over a ﬁxed time duration. During

this task, the leader aircraft maneuvered in all three axes,

similar to the behavior dictated by typical formation roles.

The leader’s maneuvers were determined a priori for these

simulations, and were treated as a given set of parameters

for each test case. To simulate the follower aircraft’s ﬂight

envelope, constraints were applied to the magnitude of both

the follower’s velocity and acceleration. The leader aircraft

maintained its own constraints along the predetermined path.

Two cases were analyzed and evaluated to assess the tools

effectiveness at determining the optimal path. The analysis of

this problem is organized in the following sections: Section

2 details the problem formulation, Section 3 outlines the

solution methodology, Section 4 includes discussion of the

results, and Section 5 includes the conclusions and future

recommendations.

2. PROBLEM FORMULATION

Assumptions

To aid in the initial creation of the validation tool, a num-

ber of simplifying assumptions were used. The follower

aircraft was modeled with triple integrator dynamics, with

control applied as a three-dimensional jerk vector. This

model assumed perfect knowledge of the aerodynamic forces

and thrust interactions. Full knowledge of the leader’s path

was provided to the solver for both cases, including three-

dimensional position, velocity, acceleration, and jerk vectors

deﬁned in the inertial reference frame. No exogenous inputs

were considered within the tool. Therefore, deterministic

knowledge of all the follower states was assumed, which in-

cluded three-dimensional position, velocity, and acceleration.

Both the leader and follower aircraft were modeled as point

masses about their center of gravity. For any images created

throughout this report, the velocity vector was used as a

reference to determine the pitch and the yaw of the aircraft.

However, without modeling the full Euler angles of the

aircraft the ability to determine the roll angle was limited.

To aid in graphics development, roll was visually estimated.

In formulating the optimal control problem, the existence

of an optimal solution was assumed. A scenario without

a solution could be easily crafted, such as the simple case

where a trailing follower cannot accelerate fast enough to

catch a higher performance leader aircraft in the two-minute

simulation time. This assumption was satisﬁed within this

study by choosing appropriate initial conditions and using

matching aircraft performance envelopes. This assumption

could easily be relaxed while still providing optimal solutions

in future work. Additionally, the test cases presented leader

trajectories such that the opportunity existed for an optimal

rejoin and formation position maintenance.

Lastly, the aircraft performance envelope was modeled with

a minimum velocity and acceleration to prevent aerodynamic

stall, a maximum velocity to limit maximum dynamic pres-

sure forces, and a maximum acceleration to limit maximum

structural forces.

Reference Frames

This problem required two primary reference frames: the

inertial reference frame and the leader reference frame. Both

are shown in Fig 2, reproduced from [6]. The inertial

reference frame was deﬁned as a traditional north, east, down

right-handed coordinated system with an origin ﬁxed to the

starting north and east location of the follower aircraft at zero

feet mean sea level (MSL). The body ﬁxed reference frame of

interest was ﬁxed to the center of gravity of the leader aircraft.

This leader reference frame was deﬁned with a longitudinal x

axis for which positive extended out the nose of the aircraft,

and a lateral y axis with positive deﬁned as pointing out the

right wing. The third axis of the leader frame was deﬁned

by the cross product of the x and y axis, using the right-hand

rule to determine the sign. The rotation matrix between the

two reference frames was deﬁned by the leader’s ﬂight path

angle, γ, and the course angle χ. The applicable direction

cosine matrix to rotate a vector from the leader frame to the

inertial frame was given by the following equation referenced

in [7].

L="cos χ−sin χ0

sin χcos χ0

0 0 1#" cos γ0 sin γ

0 1 0

−sin γ0 cos γ#(1)

Additionally, the rotation matrix in (1) satisﬁes the relations

(RL

i)−1= [Ri

L]>=Ri

Land det(RL

i) = 1.RL

ideﬁnes

the rotation from the inertial frame to the leader frame. A

superscript L on any vector denotes a vector deﬁned in the

leader reference frame, while all other vectors are deﬁned

in the inertial frame. The subscript land frepresent the

leader and follower vectors respectively. Lastly, the vector

pdrepresent the vector LF in the inertial frame as shown in

Figure 2. This formulation was chosen to match the research

in [6]. The follower and leader states and control jerk vector

are deﬁned in relation to the inertial coordinate system as

follows:

pi= [pix, piy, piz]>;LF =pd=pf−pl. . . . . . . . . . Position

vi= [vix, viy, viz]>..............................Velocity

ai= [aix, aiy, aiz]>..........................Acceleration

ui= [ux, uy, uz]>..................... ...... ....Control

Mathematical Formulation

As stated previously, the goal of this validation tool is to

determine the optimal path for a follower aircraft to rejoin to

a formation position, deﬁned by the set of points contained

on the formation ring in minimum time, and subsequently

maintain the formation position with least deviation until a

ﬁxed ﬁnal time. This problem was crafted into a two-phase

dynamic optimal control problem based on the GPOPS-II

standard problem formulation [8].

The cost function for phase 1 dictates a minimum time to

execute the initial rejoin from a predeﬁned starting location

to a point contained on the formation ring. The Mayer

form of a minimum time cost function was used as the ﬁnal

Figure 2.The formation ring is deﬁned in the leader frame as shown by the red circle, in trail of the leader. The

follower strives to rejoin by reaching the ring and then sustaining that position thereafter. [6]

time of phase 1. The follower’s dynamics were modeled by

three-dimensional triple integrator dynamics, with the three-

element control applied directly to the follower’s jerk vector.

The scenario begins at time zero with a known initial state of

both the leader and follower. A path constraint was enforced

to prevent the follower from entering the leader’s jet wash,

a safety hazard to the follower aircraft. The jet wash was

modeled as an inﬁnite cylinder centered about the xL-axis

centered at (yL, zL) = (0,0) with a radius of Rjw. Note

that t(i)

findicates the ﬁnal time of phase i. Thus, phase

1 included the terminal condition requiring the follower to

attain a formation position at t(1)

fwhich was enforced by the

functions:

f1(pd) = (pdx

−[Ri

LcL]x)2

f2(pd) = (pdy

−[Ri

LcL]y)2+ (pdz

−[Ri

LcL]z)2

−R2

ring

(2)

The equations in (2) provided a way to characterize the

distance from the follower aircraft to the formation ring.

f1(pd(t(1)

f)) ensured the follower aircraft attained the same

xL-coordinate in the leader reference frame as the tip of the

vector cL, which was used to deﬁne the center point of the

formation ring. f2(pd(t(1)

f)) was used to ensure that the yL

and zLcoordinates of the follower in relation to the leader

were at a distance from the ring center equal to the radius of

the ring. Simply put, when f1=f2= 0 the follower was

established in a valid formation position deﬁned by the ring

of points of radius Rring with center deﬁned by the vector cL.

In order to permit small numerical errors, a tolerance of ±10

meters was allowed as the maximum value of each function

at the terminal condition. Lastly, constraints were enforced

on the magnitude of the follower’s velocity and acceleration

vectors for the duration of the phase. The magnitude of the

follower’s velocity vector was constrained by a minimum and

maximum velocity, as dictated by the aircraft’s stall speed and

maximum speed. The magnitude of the acceleration vector

was constrained by a minimum acceleration to avoid zero g

ﬂight, and a maximum acceleration to prevent overloading

the aircraft structure. Both of these constraints were enforced

as path constraints throughout the duration of the phase.

Minimizing the cost function of phase 2 required the follower

aircraft to minimize the deviation from the formation ring for

the duration of the phase. This deviation was characterized

by squaring and summing f1and f2as deﬁned during phase

1, and integrating the resulting values over the duration of

the phase. Phase 2 used the same follower dynamics along

with the same path, state, and control constraints. The second

phase also contained the requirement for the position and

velocity to match across phases with a tolerance of ±0.1

m, ensuring a continuous trajectory throughout the entire

scenario. Additionally, the condition for the free ﬁnal time

of phase 1 to match the initial time of phase 2 was enforced.

The ﬁnal time of phase 2 was ﬁxed, and was chosen to ensure

that the follower could attain the formation ring during phase

1 before the ﬁxed ﬁnal time of phase 2.

Since the optimization software required one cost function for

the overall problem, the cost functions from both phase 1 and

phase 2 were summed to produce a cumulative cost function.

In order to account for a difference in the magnitudes between

the cost functions of each phase, a weighting parameter βwas

included in the summation.

A summary of the problem’s mathematical formulation is

presented on the following page.

Discrete Hamiltonian Derivation

A derivation of the discrete Hamiltonian is provided and was

used to ensure the validity of the optimal solution. The

continuous Hamiltonian was deﬁned as H=L+¯

λTF. The

Lagrangian of the cost function was deﬁned as L= (1 −

β){f2

1+f2

2}.¯

λcontained a column vector of the 9 costates.

Finally, Fwas a column vector of the right-hand side of the

9 dynamics equations. Since the tool produces a solution at

discrete points in time, the discrete form of the Hamiltonian

equation was used, deﬁned as Hd=Lk+¯

λT

kFk. Here the

subscript kdenotes the value of each function at the point

tk. For a problem in which the Hamiltonian is not an explicit

function of time, the value of the Hamiltonian will be constant

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DeterminingFollowerAircraft'sOptimalTrajectoryinRelationtoaDynamicFormationRingCarlA.Gotwald&MichaelD.ZollarsDepartmentofAeronauticsandAstronauticsAirForceInstituteofTechnologyWright-PattersonAFB,OH45433carl.gotwald@at.edu;michael.zollars@at.eduIsaacE.WeintraubControlsScienceSystemCenterWright-Pat...

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Determining Follower Aircrafts Optimal Trajectory in Relation to a Dynamic Formation Ring Carl A. Gotwald Michael D. Zollars

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