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OPTIMAL INTERCEPT OF EVASIVE SPACECRAFT
, 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.
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,
who is highly maneuverable but slow (presumably on foot). This second player attempts to prolong
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.
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.
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.
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
A vector function of terminal boundary conditions Ψand an objective function Jare also deﬁned:
Ψ(x(tf)) = 0(4)
J=φ(x(tf)) + Ztf
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:
Φ = φ+νTΨ(7)
The time derivative of the costate can be found as:
and the ﬁnal time boundary conditions for the costate can be derived based on the following equa-
λ(tf) = ∂Φ
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
x+L= 0 (10)
In order to ascertain whether the optimal controls have been found, path constraints are given as:
= 0 (11)
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
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
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:
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:
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
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:
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
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
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.
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:
where riis the position component in the ith direction and viis the velocity component in the ith
direction. The thrust acceleration vector is:
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
(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.
(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
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
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
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.
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.
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
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.
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APPENDIX A: INITIAL CONDITIONS
Table 1: Initial Conditions for Figure 3
P 0 0 1 0 0.04
E 3 0 1 0 0.017
Table 2: Initial Conditions for Figure 4
P 0 0 0 0.8 0.04
E 3 3 0.5 0.5 0.017
Table 3: Initial Conditions for Figure 5
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
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
P 6.6181 0 0 0 0.3887 0 0.0005
E 4.6797 4.6797 0 -0.2749 0.2749 0 0.000125