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AAS 20-540

OPTIMAL INTERCEPT OF EVASIVE SPACECRAFT

Luke Schoenwetter∗

, Rohan Sood†

, and Brent Barbee‡

A solution method to the orbital intercept differential game is explored. Indirect

optimization techniques are combined with the dynamics of two continuous thrust

spacecraft acted on by a point mass gravitational ﬁeld, with these dynamics be-

ing represented by a collocation transcription. Capabilities for ﬁnding optimal

solutions to the position intercept game are demonstrated, and a novel process for

obtaining an initial guess is presented. The initial control histories found using this

process are very similar to those of the ﬁnal solution, which signiﬁcantly improves

convergence. The combined methods are evaluated for soundness using a simple

linear environment, and then used to solve multiple orbital intercept games.

INTRODUCTION

Optimization problems are most often posed as having a single objective function to be minimized

or maximized. Such an optimization problem is said to be one-sided. Past research has rendered

one-sided optimization a well studied topic, and created many robust numerical methods. Spacecraft

trajectory optimization offers a broad target for these numerical methods, as small improvements

to a mission trajectory can yield large resource savings. Spacecraft rendezvous problems often ﬁt

neatly into the one-sided optimization box; however, a more recently posed and less deﬁned problem

is that of simultaneous pursuit and evasion, where the target agent acts in ways that are detrimental

to achieving rendezvous. This problem constitutes a two-sided trajectory optimization. Speciﬁcally,

it is a special class of optimization problem known as a differential game.

Differential games are two-sided optimization problems where the two objectives are mutually

exclusive, meaning that as one objective is realized, the other is lost. The mechanics at play can

be thought of as two opposing players. As one player wins the differential game, the other conse-

quentially loses it. Because of this tug-of-war nature, differential games cannot be solved using the

traditional optimization methods that the scientiﬁc community has for decades applied to one-sided

optimization problems. Rufus Isaacs pioneered the study of differential games in the mid-1900s.1, 2

Out of his work came one of the best and most well known examples of a differential game: the

homicidal chauffeur problem.3In this problem, a fast but not very maneuverable player (the dis-

agreeable chauffeur) attempts to end the game as quickly as possible by “hitting” the other player,

∗M.S. Student, Astrodynamics and Space Research Laboratory, Department of Aerospace Engineering and Mechanics,

The University of Alabama, Tuscaloosa, AL, 35487, USA

†Assistant Professor, Astrodynamics and Space Research Laboratory, Department of Aerospace Engineering and Mechan-

ics, The University of Alabama, Tuscaloosa, AL, 35487, USA

‡Aerospace Engineer, Navigation and Mission Design Branch, NASA, Goddard Space Flight Center, Greenbelt, MD,

20771, USA

1

who is highly maneuverable but slow (presumably on foot). This second player attempts to prolong

the game.

Pursuit evasion games are an intersection point between differential games and spacecraft trajec-

tory optimization. One player is termed the pursuer, and the other player is termed the evader. A

simulation is allowed to evolve, ending only when the pursuer is said to have intercepted or ren-

dezvoused with the evader. For the purposes of this discussion, intercept will be understood to be

matching position with the target, while the term rendezvous will be reserved for matching both

position and velocity. The ﬁnal time is deﬁned as the time of capture, and the difference between

the initial time and the ﬁnal time is the objective that the pursuer seeks to minimize. Conversely, the

evader seeks the maximization of the same objective.

Orbital pursuit evasion has been explored in recent times most notably by Conway and Horie

2002,4and by Pontani and Conway 2009.5These works introduce a method called the Semi-Direct

Collocation with Non-Linear Programming (semi-DCNLP), which converts the differential game

to a one-sided optimization problem by applying indirect optimization conditions to one of the

players, and allowing the other player to freely search for an extreme point. A genetic algorithm

was proposed for obtaining initial guesses. Sun et al. 20156presents the Semi-Direct Control

Parameterization (SDCP) method, which also uses a genetic algorithm to determine a starting point.

SDCP solves two one sided optimization problems, one for each player. Sufﬁcient conditions are

applied to each player in turn, while the non-constrained player attempts to minimize an objective

using a piece-wise constant control formulation. Given that the two games ﬁnd objective values

that are nearby, a saddle solution is said to exist. They also outline a way to combine SDCP with

a multiple shooting approach in order to locate nearby saddle points. Hafer 20147as well as Hafer

and Reed 20158give a method for dealing with pursuit evasion games on a meta level, proposing

that a switching function based on the value of the objective be used to turn on/off a differential

games based control method. This results in the spacecraft having the ability to autonomously

manage some primary mission alongside the secondary objective of either pursuit or evasion. More

recently, Shen and Casalino 20189offer a more simplistic solution method to the pursuit evasion

game using mostly pure indirect optimization techniques along with a few simpliﬁcations. Initial

guesses are generated via a straightforward and effective method: allowing the pursuer to minimize

the time to intercept an inert (non-thrusting) evader. Multi-agent situations (where there exists more

than two players in the game) have also been explored. Spendel 201810 shows applications of these

multi-agent games to collision avoidance situations, and Zhou et al. 201911 applies differential

games directly to a pursuer-evader-defender scenario.

The methods presented here use low complexity one-sided optimization problems in order to

obtain initial guesses, without the need for genetic algorithms. Optimal differential game solutions

are solved to a level of ﬁdelity such that the control histories are continuous and smooth. These

solutions are obtained using a collocation transcription, conditions derived from the generic indirect

optimization technique outlined by Arthur Bryson, Jr. & Yu-Chi Ho 1975,12 and a non-linear solver.

BACKGROUND

The orbital pursuit-evasion problem can be envisioned as a game played between two spacecraft,

termed pursuer and evader. The pursuing spacecraft’s objective is to intercept the evasive space-

craft as quickly as possible, while the evasive spacecraft’s objective is to prevent interception from

occurring for as long as it can. At the solution of this differential game, there exists a saddle point.

2

Figure 1: Saddle Point Concept Visualization

In the case of pursuit evasion games, the mutually exclusive objective is the time until capture. As

the pursuer acts to minimize the time span, its actions necessarily harm the objective of the evader,

as the evader tries to maximize the same time span. Solutions where both players perform optimally

are of particular interest, and represent the saddle point solutions. For example, refer to Figure 1

and assume controls for both players are zero. If the pursuer attempts to deviate, the objective (time

until capture) increases. Conversely, if the evader attempts to deviate, the objective decreases. In

either scenario, the player that deviates harms its own objective, and thus the point is deemed a

saddle point and can be declared an equilibrium for the game.

METHODS

Optimal Control of a Differential Game

The differential game at hand includes a pursuing spacecraft and an evading spacecraft. Each

player possesses a state vector of position and velocity (xP,xE), identiﬁed by the subscripts “P”

and “E” for pursuer and evader, respectively. Players inﬂuence the system through their control

vectors: uPand uE. The time derivatives of the states are functions of the states and the controls

as follows:

˙

xP=f(xP,uP)(1)

˙

xE=f(xE,uE)(2)

˙

x=˙

xP

˙

xE(3)

A vector function of terminal boundary conditions Ψand an objective function Jare also deﬁned:

Ψ(x(tf)) = 0(4)

3

J=φ(x(tf)) + Ztf

t0

L(x,u, t)dt (5)

where Jis the objective that is to be minimized by the pursuer and maximized by the evader, and

Ψcontains any prescribed equality constraints on the position and velocity components of the state

vector xat the ﬁnal time tf. Furthermore, φis the portion of the objective that can be taken purely

as a function of the state vector at the ﬁnal time, and Lis the portion of the objective that must be

integrated over time. Costate vectors will be used in conﬁrming the existence of an optimal saddle

point, and are introduced as λP,λE, and ν. These costate vectors are conjugate to xP,xE, and

Ψ, meaning that λPcontains one costate variable for each element of xP,λEcontains one costate

variable for each element of xE, and likewise for νand Ψ. The Hamiltonian of the system, H, and

the terminal condition function, Φ, are now constructed as:

H=λT˙

x+L=λT

P˙

xP+λT

E˙

xE+L(6)

Φ = φ+νTΨ(7)

The time derivative of the costate can be found as:

˙

λ=−∂H

∂x(8)

and the ﬁnal time boundary conditions for the costate can be derived based on the following equa-

tion:

λ(tf) = ∂Φ

∂xtf

(9)

where λis the concatenation of the costate of the pursuer and the evader. If a variable ﬁnal time

is desired, then the ﬁnal time of the trajectory is unconstrained, and a transversality constraint Ωis

applied at the ﬁnal time:

Ω = dΦ

dt +L=∂Φ

∂t +∂Φ

∂x

T

˙

x+L= 0 (10)

In order to ascertain whether the optimal controls have been found, path constraints are given as:

∂H

∂uP

=∂H

∂uE

= 0 (11)

∂2H

∂u2

P

≥0(12)

∂2H

∂u2

E

≤0(13)

These path constraints ensure that the Hamiltonian is stationary with respect to the control inputs,

that the pursuer is ﬁnding a minimum, and that the evader is ﬁnding a maximum. Equations 12 and

13 ensure the min-max nature of the differential game is preserved by constraining the Hessian of the

4

Hamiltonian with respect to the pursuer’s controls to be positive deﬁnite (or at least positive semi-

deﬁnite), and the Hessian of the Hamiltonian with respect to the evader’s controls to be negative

deﬁnite (or at least negative semi-deﬁnite).

The constraints of Equations 3, 4, 8, 9, 11, 12 and 13 (possibly 10, depending on whether the ﬁnal

time is constrained), along with initial conditions for the state x, constitute a two point boundary

value problem (TPBVP), and can be solved using a variety of non-linear programming approaches.

System dynamics will be enforced using a 3rd Order Legendre Gauss Lobatto (LGL) collocation

method.13, 14 An in-depth investigation of the equations in this section can be found in a work by

Arthur Bryson, Jr. & Yu-Chi Ho 1975.12

Non-Dimensional Equations of Motion

In order to facilitate numerical accuracy in the optimization process, a non-dimensional system

is constructed. The central body will be given Earth-like properties and a point mass gravity model.

The characteristic length of the system is taken as the radius of the Earth (lchar =RE). The non-

dimensional gravitational parameter µis chosen to be unity, which leads to the equation for the

characteristic time of the system:

tchar =sµ l3

char

µdim

(14)

Given that µdim =µEarth (the dimensional gravitational parameter), the characteristic time, tchar,

of the system evaluates to a little less than 15 minutes. The governing equations of motion are:

¨x=−µx

(x2+y2+z2)3

2

¨y=−µy

(x2+y2+z2)3

2

¨z=−µz

(x2+y2+z2)3

2

(15)

where x,y, and zrepresent non-dimensional scalar components of the spacecraft’s position relative

to the central body. Control is given in the form of two angles (αand β) which deﬁne the thrust

vector in inertial coordinates:

u=α

β(16)

The angle αrepresents the angle from the inertial X axis to the projection of the thrust vector into

the XY plane, and βrepresents the angle from the same projection to the actual thrust vector, as

shown in Figure 2. The sign for αfollows the standard right hand convention. The sign for βis

positive above the XY plane, and negative below the XY plane. Thrust magnitude is not given as a

control parameter. Neither of the players can choose to use less than their full allotment of thrust

and still ﬂy an optimal path, since doing so would harm their own objective. I.e., if there is an

instant in time where the pursuer is using less than its full thrust, then doing so and shifting the rest

of its trajectory/controls thereafter would result in a lower time to intercept, regardless of whether

5

Figure 2: Control Conﬁguration for Spacecraft in Three Dimensional Space

the evader’s strategy is optimal. A similar argument can be made for the evader. Thus, no control is

given to the players over their thrust level, and the controls are only used to construct the direction of

thrust. The Cartesian representation of the acceleration vector due to the continuous thrust control

can be calculated as:

a=amax

cos(α)cos(β)

sin(α)cos(β)

sin(β)

(17)

where amax is the chosen to be the maximum acceleration that the spacecraft is able to produce.

Although a point-mass Earth gravitational model was used in the development of this research, the

reader should note that the methods used to obtain the results shown in this text are applicable to any

generic system of differential equations, and consequently to any other central body or gravitational

model.

CONDITIONS FOR THE OPTIMAL INTERCEPT GAME

The optimal intercept game represents a scenario where a pursuing spacecraft attempts to align

its position vector with an evading spacecraft’s position vector in as little time as possible, while the

evading spacecraft attempts to prevent it for as long as possible. This means that the objective of

the differential game is the time until capture. The terminal constraints and objective are formulated

from:

6

Ψ=rE(tf)−rP(tf)(18)

φ= 0 (19)

L= 1 (20)

where riis the position vector of the ith player for i= [E, P ]. Equation 18 represents the position

intercept nature of the game, while Equation 19 represents the fact that the objective is not a function

of the ﬁnal state. Finally, Equation 20 represents that time until capture is the objective (note that the

integral portion of Equation 5 becomes tf−t0). Because the ﬁnal time is unconstrained, Equation

10 must also be included as a constraint.

FINDINGS

Testing Environment Using Simple Dynamics

In order to facilitate validation and testing, a set of linear dynamics were implemented. The

TPBVP outlined in the previous section is rather temperamental and difﬁcult to solve, thus any

advantage, such as linearity, is welcome both in ﬁnding and understanding the solutions. These

dynamics are chosen to ensure that at least one closed-form solution exists, and thus the numerical

methods can be tested against this closed-form solution. The simple dynamics used are as follows:

˙

rx

ry

vx

vy

=

vx

vy

0

0

(21)

where riis the position component in the ith direction and viis the velocity component in the ith

direction. The thrust acceleration vector is:

a=amax cos(α)

sin(α)(22)

where amax is the arbitrary maximum acceleration, and αis the single control variable. The angle

αspeciﬁes the direction of applied acceleration, and is referenced counter-clockwise from the X

axis, as shown in Figure 2. This single control variable carries an extra advantage for the solution

of a differential game: Equation 12 and Equation 13 will be scalar constraints instead of matrix

positive/negative deﬁnite constraints, which will further simplify the problem.

The ﬁrst step is to solve the trivial case where the initial velocities of the players are co-linear,

and the maximum accelerations are chosen so that amaxP> amaxE. This should result in a solution

where both players apply their controls along their velocity vectors (i.e., straight forward), since

a derivation of the indirect method for this problem dictates this outcome. Indeed, the solution

obtained (shown in Figure 3) for these initial conditions is as such, and both players choose to apply

their acceleration control along their velocity vectors.

After showing that the method is capable of ﬁnding the trivial solution, a non-trivial case is solved

and evaluated. Using a scenario where the the initial velocity vectors are no longer aligned, but the

7

(a) Trajectory (b) Control History

Figure 3: Solution to the Trivial Intercept Game

pursuer still possesses superior control authority, a new solution is found (Figure 4). Note that the

acceleration directions for the two players are co-linear and constant throughout the solution of this

game, holding at about −23◦.

An interesting outcome is observed when the relative acceleration of the two players is lowered.

The solution in Figure 5 begins with the exact same initial conditions as Figure 4, except that the

evader’s maximum acceleration has been increased to lessen the control authority gap between the

evader and the pursuer. Not only does this have the obvious effect of increasing the time until

capture, but it also causes the trajectories of the players to approach the alignment of the trivial

solution in an asymptotic fashion. The co-linearity of the controls can again be seen in this scenario,

and the behavior holds for all initial conditions in this linear dynamic environment.

Three-Dimensional Orbital Dynamics

Obtaining an Initial Guess: Compared to the solution of differential games in the linear dynam-

ical environment from the last section, the three-dimensional orbital case is much more difﬁcult

problem to solve. Even using a collocation transcription, the convergence radius of the solution

(as well as the speed with which it is found) is extremely dependent on the initial guess. As such,

blindly integrating the initial conditions using a static control law yields convergent initial guesses

only for very simplistic cases where the solution controls vary more or less linearly with time. A

more robust method for getting the initial guess is through the use of a traditional optimal control

problem, where the the evader ﬂies a static control law, and the pursuer minimizes the objective for

the game. The costate of the indirect optimization method is then integrated backwards in time over

the trajectory, using Equation 9 to ﬁnd what the value at the ﬁnal time. At this point, a guess must

be made in assigning the values of the terminal constraint costate vector, ν. Unity will usually prove

to be acceptable for initializing the process.

8

(a) Trajectory (b) Control History

Figure 4: Solution to a Non-Trivial Intercept Game

(a) Trajectory (b) Control History

Figure 5: A Solution Displaying Asymptotic Behavior

9

The static evader initial guess produces much better results when passed on to the non-linear

solver for the attempted solution of the full differential game. However, one extra enhancement

is still to be added. Recall from the previous section that the control directions were co-linear.

Solutions to the non-linear differential equation will not have the same property. However, it can be

found, via inspection, that solutions to the orbital intercept game often contain control histories that

are somewhat co-linear. As such, feeding in an initial guess that contains co-linear control histories

will often converge very rapidly. Thus, multiple steps are taken in the formulation of the initial

guess. The following method is used to obtain the initial guess:

1. Solve a traditional optimal control problem where the pursuer minimizes the objective and

the pursuer is constrained to some arbitrary control law. All conditions in Ψ(Equation 4)

must be enforced at this stage.

2. Re-solve the optimal control problem in step 1 using the solution from step 1 as an initial

guess, but constrain the control directions of the two players to be co-linear. This necessitates

that the evader’s arbitrary control law be removed.

3. Back-propagate the trajectory of the costate using Equations 8 and 9. This forms the initial

guess for the costate.

4. Use the results from steps 2 and 3 as the initial guess for the full differential game.

In some cases, it may be necessary to perform continuation in order to obtain solutions. This

is usually not an issue, as the method presented behaves well in a continuation scheme (slowly

stepping from an initial solution to a desired solution). Most initial conditions can be solved simply

by using the pre-solve process above. Of the cases that do not converge using the pre-solve, it is rare

that continuation will not be able to ﬁnd the solution. This can often be done easily by increasing

the pursuer’s control authority until the solution to the game becomes very obvious to the non-linear

solver, at which point, continuation is able to obtain a solution for the desired control authority.

Intercept Game: The methods outlined in previous sections will now be combined to form a

solution method for an intercept game. A TPBVP is given to the non-linear solver in the form of

Equations 3 and 8 (represented via a 3rd Order LGL collocation transcription) along with Equations

4, 9, 10, 11, 12, and 13 as constraints. The initial guess is generated according to the previous

subsection: by ﬁrst solving a one-sided optimal control problem using a static evader, then subse-

quently solving a second one-sided optimal control problem where the controls of the two players

are constrained to be co-linear.

In order to demonstrate the effectiveness of the presented methods an example problem is shown.

The initial conditions and acceleration magnitudes have been selected in order to present a visibly

understandable solution, and also to highlight some trends that have been identiﬁed in the solution

space. The two spacecraft start in circular orbits, with the evader’s orbit being slightly larger in

radius, and slightly inclined compared to the pursuer’s orbit. The native orbits of the two players

are shown as faint lines in Figure 6. The pursuer is granted signiﬁcant advantage over the evader

by way of control authority. The exact initial conditions for position and velocity along with the

maximum accelerations are given in Table 4 of Appendix A. The solution trajectory and its control

history, along with the control history of the second presolve phase, can be viewed in Figures 6,

7, and 8. The TPBVP has been solved so that both the pursuer and the evader face the worst

10

case opponent and perform optimally against that worst case opponent. As expected, the evader

immediately attempts to maneuver away from the approaching pursuer, and the pursuer directs its

thrust generally in the direction of the evader. The ﬁrst presolve phase is not shown since there is

little useful information that can be garnered from a visual inspection. The second presolve phase

(Figure 8), however, can be seen to have a control history strikingly similar to that of the actual

solution. It is this similarity that is in large part responsible for the quick convergence when solving

this particular example.

(a) Top View (b) Side View

Figure 6: Solution Trajectory for an Example Intercept Game

In the true solution, the players’ control strategies are initially offset. As the scenario unfolds, the

strategies eventually align and become co-linear. This is a trend that has been observed in nearly all

of the pursuit evasion cases that have been considered during the course of this research. However,

it remains to be seen whether this trend has discernible mathematical basis, and/or can be exploited

to form a robust closed loop controller. Regardless, the presence of this trend toward the end of the

trajectory is almost certainly due to the dynamics between the two players becoming nearly linear

(recall from the previous section, where a linear dynamic environment was considered, that the con-

trols for the two players in a linear environment are always co-linear). This conclusion is supported

by an intuitive inspection. As the pursuer closes the gap between the evader and itself, there will

come a point in time where the evader’s most effective strategy must be to accelerate away from the

pursuer’s approach direction, since the urgency caused by the pursuer’s proximity outweighs any

advantage to be gained from exploiting the gravitational dynamics. The same intuition suggests that

a similar point in time exists for the pursuer. Once both of these conditions have occurred, the thrust

directions will necessarily be nearly co-linear, as seen at the end of the solution control history in

Figure 7.

The maximum accelerations allowed in the previous example are somewhat unrealistic. Consider

two spacecraft in co-planar geosynchronous orbit (GEO), with 45◦of separation between them.

11

Assuming that both spacecraft weigh 200 kg, the pursuer’s thruster is capable of producing 1Nof

thrust, and the evader’s thruster is capable of producing 0.25 Nof thrust, that leaves the players’

maximum accelerations at 5mm/s2and 1.25 mm/s2, respectively. (Current experimental electric

propulsion engines are capable of producing just over 5Nof thrust.15) These low thrust-to-mass

ratios will elongate the trajectory and increase the time until capture. This scenario will be solved

using a few continuation steps, starting with a mock scenario where the two spacecraft have higher

maximum accelerations, and gradually stepping to the desired accelerations. The solution obtained

can be viewed in Figures 9 and 10. Both players’ orbit directions are counter clockwise. The

pursuing player completes a phasing arc to intercept the evader not far from the pursuer’s original

position, using a little less than a full day to do so. Once again, the aligning of the control vectors is

seen at the very end of the trajectory in Figure 10.

Figure 9: GEO Intercept Solution

There is a limitation to the proposed methods that becomes evident when scenarios with long

thrust arcs are at play: costate growth. Due to the dynamics at play (namely Equation 8), the costate

will shrink in magnitude along the entirety of the trajectory. From a time-reversed perspective, the

costate grows without bound as the time to intercept increases, as can be seen from the costate in

Figure 11 for the GEO example game. As the costate magnitudes increase, the numeric conditioning

of the problem deteriorates. As this deterioration progresses, there comes a point where the non-

linear solver does not converge. Thus it is desirable, in subsequent research efforts, to ﬁnd a method

for dealing with the costate growth in such games so that they may be solved more efﬁciently.

13

CONCLUSION AND FUTURE WORK

The presented solution method has proven effective when solving generic orbital intercept games.

A novel approach to forming an initial guess for the solution method was taken, and striking simi-

larity between the generated guess and the true solution was demonstrated. Although they are used

here to solve a very particular case and formulation, differential games have many applications to

astrodynamics beyond the intercept game. These could include uncertainty mitigation in scenarios

like targeted re-entry, and rendezvous situations such as those involving deep space asteroid tours.

An area of necessary improvement to the methods given in this paper is the costate initial guess.

While the control history initial guesses obtained here are very similar to that of the solution, the

costate is not similar, and new methods which correct this are desirable.

Furthermore, applications to spacecraft autonomy exist, but the computation time currently re-

quired for the indirect optimization approach has not yet proven to be capable of supporting real-

time control updates. Even using a reasonable initial guess, the number of iterations required to

solve an orbital differential game can reach into the hundreds. Recomputing a new solution at small

intervals is less than ideal, although resolving a trajectory using a perturbed previous solution as the

starting point has shown promise, and may provide a basis for a real-time controller after further

analysis.

It is also hypothesized that the tendency toward co-linearity at the end of intercept solutions is

due to the dynamics between the two players becoming very nearly linear as the pursuer’s control

strategy overtakes that of the evader. If this hypothesis holds, there is a possibility that it can be

leveraged to compute update solutions in nearly real-time. This would be especially impactful,

since as the end of the solution trajectory approaches, the two spacecraft have less and less time to

react to one another, necessitating that control updates be readily available.

REFERENCES

[1] R. Isaacs, Games of Pursuit. Santa Monica, CA: Rand Corp, 1951.

[2] R. Isaacs, Differential Games: A Mathematical Theory with Applications to Warfare and Pursuit, Con-

trol and Optimization. New York, NY: Wiley, 1965.

[3] A. Merz, “The Homicidal Chauffeur - A Differential Game,” tech. rep., Stanford University, Center for

Systems Research, 1971.

[4] B. A. Conway and K. Horie, “A new collocation-based method for solving pursuit/evasion (differential

games) problems,” Advances in the Astronautical Sciences, Vol. 109, Jan 2002.

[5] M. Pontani and B. A. Conway, “Numerical Solution of the Three-Dimensional Orbital Pursuit–Evasion

Game,” Journal of Guidance, Control, and Dynamics, Vol. 32, Apr 2009.

[6] R. L. Songtao Sun, Qiuhua Zhang and B. Li, “Numerical Solution of a Pursuit-Evasion Differential

Game Involving Two Spacecraft In Low Earth Orbit,” Journal of Industrial and Management Optimiza-

tion, Vol. 11, Oct 2015.

[7] W. Hafer, Sensitivity Methods Applied to Orbital Pursuit-Evasion. PhD thesis, Texas AM University,

Aug 2014.

[8] W. Hafer and H. Reed, “Orbital Pursuit-Evasion Hybrid Spacecraft Controllers,” AIAA Guidance, Nav-

igation, and Control Conference, Jan 2015.

[9] H.-X. Shen and L. Casalino, “Revisit of the Three-Dimensional Orbital Pursuit-Evasion Game,” Journal

of Guidance, Control, and Dynamics, Vol. 41, Aug 2018.

[10] D. Spendel, “Parameter Study of an Orbital Debris Defender using Two Team, Three Player Differential

Game Theory,” Master’s thesis, Air Force Institute of Technology, Mar 2018.

[11] J. C. S. W. Junfeng Zhou, Lin Zhao and Y. Wang, “Pursuer’s Control Strategy for Orbital Pursuit-

Evasion-Defense Game with Continuous Low Thrust Propulsion,” Applied Sciences, Aug 2019.

[12] J. Arthur Bryson and Y.-C. Ho, Applied Optimal Control. Washington DC: Hemisphere Publishing

Corporation, 1975.

15

[13] F. Topputo and C. Zhang, “Survey of Direct Transcription for Low-Thrust Space Trajectory Optimiza-

tion with Applications,” Abstract and Applied Analysis, Vol. 2014, Aug 2014.

[14] A. Herman and B. Conway, “Direct Optimization Using Collocation Based on High-Order Gauss-

Lobatto Quadrature Rules,” Journal of Guidance, Control, and Dynamics, Vol. 19, May 1996.

[15] S. Hall et al., “High-Power Performance of a 100-kW Class Nested Hall Thruster,” 35th International

Electric Propulsion Conference, October 2017.

16

APPENDIX A: INITIAL CONDITIONS

Table 1: Initial Conditions for Figure 3

Player rxryvxvyamax

P 0 0 1 0 0.04

E 3 0 1 0 0.017

Table 2: Initial Conditions for Figure 4

Player rxryvxvyamax

P 0 0 0 0.8 0.04

E 3 3 0.5 0.5 0.017

Table 3: Initial Conditions for Figure 5

Player rxryvxvyamax

P 0 0 0 0.8 0.04

E 3 3 0.5 0.5 0.024

Table 4: Non-Dimensional Initial Conditions for Figure 6

Player rxryrzvxvyvzamax

P 2 0 0 0 p1/20 0.06

E 2.2 0 0.2 0 p1/2.20 0.02

Table 5: Non-Dimensional Initial Conditions for Figure 9

Player rxryrzvxvyvzamax

P 6.6181 0 0 0 0.3887 0 0.0005

E 4.6797 4.6797 0 -0.2749 0.2749 0 0.000125

17