# Target Defense against Sequentially Arriving Intruders

Arman Pourghorban, Michael Dorothy, Daigo Shishika, Alexander Von Moll, and
Dipankar Maity A. Pourghorban and D. Maity are with the Department of
Electrical and Computer Engineering, at The University of North Carolina at
Charlotte, NC, 28223, USA. Emails<EMAIL_ADDRESS><EMAIL_ADDRESS>M.
Dorothy is with the Computational and Information Sciences Directorate, Army
Research Laboratory, APG, MD, 20783, USA. Email:
<EMAIL_ADDRESS>D. Shishika is with the Mechanical Engineering
Department, at George Mason University, Fairfax, VA, 22030, USA. Email:
<EMAIL_ADDRESS>A. Von Moll is with Control Science Center, Air Force
Research Laboratory, WPAFB, OH, 45433, USA. Email:
<EMAIL_ADDRESS>This research is supported by the ARL grant ARL
DCIST CRA W911NF-17-2-0181. The views expressed in this paper are those of the
authors and do not reflect the official policy or position of the United
States Air Force, United States Army, Department of Defense, United States
Government, or its components.

###### Abstract

We consider a variant of the target defense problem where a single defender is
tasked to capture a sequence of incoming intruders. The intruders’ objective
is to breach the target boundary without being captured by the defender. As
soon as the current intruder breaches the target or gets captured by the
defender, the next intruder appears at a random location on a fixed circle
surrounding the target. Therefore, the defender’s final location at the end of
the current game becomes its initial location for the next game. Thus, the
players pick strategies that are advantageous for the current as well as for
the future games. Depending on the information available to the players, each
game is divided into two phases: partial information and full information
phase. Under some assumptions on the sensing and speed capabilities, we
analyze the agents’ strategies in both phases. We derive equilibrium
strategies for both the players to optimize the capture percentage using the
notions of engagement surface and capture circle. We quantify the percentage
of capture for both finite and infinite sequences of incoming intruders.

Keywords:  Target-defense, Pursuit-evasion, Reach-avoid, Partial information
games, Target-Attacker-Defender games.

##  I Introduction

Considering the increase of usage and applications of robots in securing and
supervising regions, considerable research has been focused on the problem of
guarding a target. Isaacs [1] first analyzed a two-player differential game in
which the pursuer is tasked to chase an evader. Since his pioneering work,
several extensions to pursuit-evasion games (PEGs) have been proposed, which
incorporate real-world constraints such as obstacles [2], sensing limitations
[3] and visibility constraints [4]. Similarly, different variations of PEGs
have also been investigated in the literature. Among these, the Target-
Attacker-Defender (TAD) games [5, 6] and the reach-avoid games [7, 8, 9] are
of particular interest for our work. In the TAD game framework, the target and
the defender form a team to guard the former from being intercepted by an
attacker/intruder; whereas the attacker tries to intercept the target before
getting intercepted/captured by the defender. In reach-avoid games, the
attackers attempt to reach certain regions while avoiding the defenders that
try to prevent them from reaching their designated areas.

Perimeter defence games are a variant of the reach-avoid and TAD games. In
this variation, a group of intruders is tasked to breach a target and the
defenders’ job is to capture these intruders before they reach the target
perimeter. In some cases, the defenders’ motion is constrained on the target
perimeter [10, 11] and in other cases they move freely in the environment
[12]. The percentage of captured intruders is of particular interest in this
type of games, whereas in other variations of PEGs, the capture distance
and/or the fuel consumption is of primary interest [13].

In this paper, we consider the problem of guarding a circular target region
from incoming intruders. All the agents (target, intruders, and defender) have
limited sensing regions. Therefore, they may not have information about their
opponents all the time. This sensing limitation enforces them to consider both
positional and sensing vantage points while choosing their strategies. The
intruders appear randomly at a fixed distance from the target center (see Fig.
1). We consider a sequential arrival of the intruders in the sense that, at
any time, exactly one intruder tries to breach the target boundary. Once the
current intruder succeeds or gets captured, the next intruder appears. This is
a natural extension to the work in [12] where the target had to be defended
from a single intruder instead of a sequence of intruders. Similar problems
have been studied in [14, 15, 16, 17] where multiple intruders may attempt to
breach the target at the same time. However, in these works, the strategies
for the intruders are simple and fixed a priori. More specifically, these
works assume that the intruders come directly at the target and they do not
try to evade while being captured. Furthermore, these works do not consider
sensing capability for the intruders. In contrast to these works, we show that
the sensing capability results in a particular type of evasive maneuver where
the intruders can force the defender to pursue them and thus the defender ends
up capturing the intruders at locations which are advantageous for the next
intruders to increase their target breaching probability.

Due to the sequential arrival of the intruders, our problem becomes a sequence
of one-vs-one games where the next game starts as soon as the current game
ends. We study this problem by decomposing each one-vs-one game into two
phases: partial and full information phases. Each one-vs-one game starts with
the partial information phase and then either it proceeds to the full
information phase or the game is terminated in the breach of the target. The
objective of the defender (intruders) is to maximize (minimize) the percentage
captured intruders.

The main contributions of this paper are: (i) Formulating and solving a
sensing limited target defense game against sequentially incoming intruders
where the intruders are equipped with a limited range sensor through which
they can detect the defender from a certain distance and take evasive
maneuver, (ii) Analyzing the equilibrium strategies for the agents by defining
and using the concepts of engagement surface and capture circle where the
former provides the favourable configurations to engage with an intruder and
the latter is the set of all possible capture locations. Under the equilibrium
strategies, the set of all possible capture locations become a a circle with
fixed radius and center, (iii) Analytically computing the capture percentage
under both finite and infinite number of intruder arrivals. (iv) Numerically
validating (using Monte-Carlo type random trials of experiments) the
theoretically found capture probability, and finally, (v) Characterizing the
relationship between the capture probability and the problem parameters (e.g.,
sensing radius, speed ratio etc.) through simulations.

The rest of the paper is organized as follows: In Section II, we formulate our
problem, provide the assumptions for this work, and discuss some useful
definitions and necessary background materials. The two phases (partial and
full information) of the one-vs-one games are discussed in Sections III and
IV, respectively. We analyze the whole game for the entire sequence of
arrivals in Section V and describe the progression of the game in Algorithm 1.
In this section we also derive the capture percentage. Numerical results are
discussed in Section VI. We conclude the paper in Section VII.

Notation: All vectors are denoted with lowercase bold symbols, e.g.,
$\mathbf{x}$. We use $\hat{\mathbf{u}}(\theta)$ to denote the unit vector
$[\cos\theta,~{}\sin\theta]^{\intercal}$.

##  II Problem Formulation

$\rho_{{}_{T}}$

Figure 1: gh: Target region with radius $r_{{}_{T}}$, gh: intruders sensing
region with radius $\rho_{{}_{A}}$, gh: Target’s sensing anulus with radius
$\rho_{{}_{T}}$.

We consider a target guarding problem in ${\mathbb{R}^{2}}$ where a defender
is tasked to protect a circular target region
$\mathcal{R}_{T}=\\{\textbf{x}\in{\mathbb{R}^{2}}\ |\ \|\textbf{x}\|\leq
r_{{}_{T}}\\}$ from an incoming sequence of intruders as schematically shown
in Fig. 1. The target has a sensing annulus of radius $\rho_{{}_{T}}$ through
which it can detect the presence of an intruder within this annulus region. We
will refer to this annulus as Target Sensing Region (TSR). The intruders
appear on the TSR boundary sequentially. That is, the next intruder appears on
the TSR boundary as soon as the current one gets captured by the defender or
breaches the target. Typically in these types of games, the intruders are able
to escape from the TSR and the defender is unable to pursue those intruders
afterwards [12]. To prevent this scenario, we will consider a large enough TSR
(see our parametric assumptions in (2)) so that an intruder is not able to
escape when it does not have a strategy to breach the target. Therefore, each
one-vs-one game terminates either in breach of the target or in capture of the
intruder. Each intruder appears at the TSR boundary randomly with a uniform
probability that is independent of the previous arrivals. Due to the random
arrivals, the number of capture (equivalently, the percentage of capture) is a
random variable. The objective of this work is to find a strategy for the
defender that maximizes the expected percentage of capture. We will consider
both the cases of finite and infinite sequences of arrivals and analyze the
expected percentage of capture for both cases.

Let $\mathbf{x}_{A}(t),\mathbf{x}_{D}(t)\in\mathbb{R}^{2}$ denote the
positions of the intruder and the defender at time $t$. Both the defender and
intruder are assumed to have first-order dynamics, i.e.,

$\displaystyle\dot{\mathbf{x}}_{A}=v_{A}\hat{\mathbf{u}}(\psi_{A}),\qquad\dot{\mathbf{x}}_{D}=v_{D}\hat{\mathbf{u}}(\psi_{D}),$
(1)

where the defender (intruder) directly controls its speed and heading angle by
selecting $v_{D}$ and $\psi_{D}$ ($v_{A}$ and $\psi_{A}$), respectively. In
this problem, we assume that the defender and the intruder have the speed
limit of 1 and $\nu$, respectively, i.e., $|v_{D}(t)|\leq 1$ and
$|v_{A}(t)|\leq\nu$ for all $t$. Furthermore, we assume that $\nu<1$, i.e.,
the defender is faster. Without the constraint $\nu<1$, it is impossible for
the defender to prevent any breaching and the percentage of capture will be
zero. Thus the outcome of the game is trivial for $\nu\geq 1$.

After arriving at the TSR boundary, the intruder moves radially toward the
target center with maximum speed until it senses the defender. The target and
the defender work as a team and whenever an intruder is within the TSR, the
defender has access to the intruder’s instantaneous position
$\mathbf{x}_{A}(t)$. The intruder is also equipped with its own sensor and is
able to sense the defender only if they are within a distance of
$\rho_{{}_{A}}$ or less. Using this sensing capability, the intruder is able
to find the right breaching point on the target, or able to get out of the TSR
uncaptured, or is able to evade for some time before getting captured by the
defender. This evasive maneuver is an important capability for the intruder
because this forces the defender to pursue and capture the intruder at a
location that is unfavorable for the defender to start the next game. Thus,
while getting captured, each intruder can maximize the chances of winning for
the next intruder, which would not have been possible if $\rho_{{}_{A}}=0$.

###  II-A Parametric Assumptions

Agent’s strategies and the overall outcome of this game depend on the game
parameters $r_{{}_{T}},\rho_{A}$, $\rho_{T}$ and $\nu$. In this paper we focus
on a specific parameter regime given by the following condition

$\displaystyle\max\\{(1+\nicefrac{{2\nu}}{{(1-\nu^{2})}})\rho_{{}_{A}},~{}~{}~{}\nu
r_{{}_{T}}+\nicefrac{{2\rho_{{}_{A}}\nu^{2}}}{{(1-\nu^{2})}}\\}\leq\rho_{{}_{T}}$
(2)

The first condition (i.e.,
$(1+\nicefrac{{2\nu}}{{(1-\nu^{2})}})\rho_{{}_{A}}\leq\rho_{{}_{T}}$) is
necessary to ensure that there exists a strategy for the defender to capture
the intruder inside the TSR. Otherwise, the intruder can evade outside the TSR
where the defender cannot detect it. Thus, there can be a deadlock situation
where every time the intruder cannot breach it gets out of the TSR and keeps
trying indefinitely (and thus, not allowing any other intruders to appear).
The second condition (i.e., $\nu
r_{{}_{T}}+\nicefrac{{2\rho_{{}_{A}}\nu^{2}}}{{(1-\nu^{2})}}\leq\rho_{{}_{T}})$
is to ensure that if the defender does not have a strategy to capture the
intruder and loses the current game, then it has enough time to reach the
target center from the capture circle (to be defined later) before this
intruder breaches the target boundary and the next game starts. The need for
these assumptions will be apparent from Remark 1.

###  II-B Apollonius Circle and its Importance

Given the locations $\mathbf{x}_{A}(t)$ and $\mathbf{x}_{D}(t)$ of the agents
at time $t$, one can construct the locus of the points x such that the ratio
of the distances from x to $\mathbf{x}_{A}(t)$ and to $\mathbf{x}_{D}(t)$ is
$\nu$, i.e.,

$\displaystyle\|\textbf{x}-\mathbf{x}_{A}(t)\|=\nu\|\textbf{x}-\mathbf{x}_{D}(t)\|.$

For $\nu<1$, the set of such points lie on a circle with center
$\mathbf{x}_{C}(t)$ and radius $r_{C}(t)$ given by

$\displaystyle\mathbf{x}_{C}(t)=\alpha\mathbf{x}_{A}(t)-\beta\mathbf{x}_{D}(t),\quad
r_{C}(t)=\gamma\|\mathbf{x}_{A}(t)-\mathbf{x}_{D}(t)\|,$ (3)

where

$\displaystyle\alpha=(1-\nu^{2})^{-1},\quad\gamma=\nu\alpha,\quad\beta=\nu\gamma.$

The interior of the Apollonius circle is called the intruder’s dominance
region. That is, the intruder can reach any point within this region before
getting captured. Both agents can reach any point on the perimeter of the
Apollonius circle at the same time. The following result from [18, Theorem 1]
shows that the intruder cannot exit the Apollonius circle without getting
captured. That is, any point outside of the Apollonius circle is the dominance
region of the defender.

######  Lemma 1 ([18])

Regardless of the strategy of the intruder, the defender has a strategy to
capture the intruder arbitrarily close to the Apollonius circle. $\triangle$

Therefore, the Apollonius circle is a crucial component in the analysis of our
game as it uniquely determines the reachable set of the intruder before it
gets captured.

###  II-C Game Phases and Guarded Arc

Full information phase: In this phase, both agents can sense each other, i.e,
$\|\mathbf{x}_{A}(t)\|<r_{{}_{T}}+\rho_{{}_{T}}$ and
$\|\mathbf{x}_{A}(t)-\mathbf{x}_{D}(t)\|\leq\rho_{{}_{A}}$.

Partial/Asymmetric information phase: In this phase only the defender sees the
intruder, i.e., $\|\mathbf{x}_{A}(t)\|<r_{{}_{T}}+\rho_{{}_{T}}$ and
$\|\mathbf{x}_{A}(t)-\mathbf{x}_{D}(t)\|>\rho_{{}_{A}}$. In this phase, the
intruder comes radially toward the target until it senses the defender- at
which point the full information phase starts.

###### Lemma 2

Let the defender is located at $r\hat{\mathbf{u}}(\theta_{D})$ with
$r\leq\tilde{\rho_{{}_{T}}}\triangleq\rho_{{}_{T}}+r_{{}_{T}}$. This defender
can capture any intruder incoming at an angle of $\theta_{A}$ such that

$\displaystyle|\theta_{A}-\theta_{D}|\leq\begin{cases}\Theta_{\rm
g}\qquad&\text{ if }r>\frac{\rho_{{}_{T}}}{\nu}-r_{{}_{T}},\\\
\pi\qquad&\text{ otherwise,}\end{cases}$

where

$\Theta_{\rm g}=2\cos^{-1}\\!\Bigg{(}\\!\\!\sqrt{\frac{\Big{(}(r_{{}_{T}}+\nu
r)^{2}-(\tilde{\rho_{{}_{T}}}-\nu
r_{{}_{T}})^{2}\Big{)}\Big{(}(\tilde{\rho}_{{}_{T}}+\nu r_{{}_{T}})^{2}-(\nu
r-r_{{}_{T}})^{2}\Big{)}}{16\nu^{2}r_{{}_{T}}^{2}r\tilde{\rho}_{{}_{T}}}}\Bigg{)}.$

$\triangle$

###### Proof:

A proof of this lemma is omitted due to space limitation. ∎

The above lemma shows that if the defender is located within
$\nicefrac{{\rho_{{}_{T}}}}{{\nu}}-r_{{}_{T}}$ distance from the target center
then it can capture any incoming intruder. Otherwise, it can capture only the
intruders that are coming with an (absolute) angular separation of no more
than $\Theta_{\rm g}$. We further notice that $\Theta_{\rm g}$ depends on the
radial position of the defender (i.e., $r$), and the nature of this dependence
is presented in the following corollary.

###### Corollary 1

$\Theta_{\rm g}$ is a decreasing function of $r$. $\hfill\triangle$

###### Proof:

A proof of this corollary is omitted due to space limitation. ∎

Therefore, it is desirable for the defender to stay as close to the target
center as possible. This observation will be an instrumental component in
determining the strategies for the defender and the intruders.

Next we analyze each of the game phases in detail.

##  III Partial Information Phase

In this phase, the intruder moves radially toward the center of the target
with full speed until it senses the defender and starts the full information
phase.111 In this phase, the defender has information about the intruder’s
location and can strategically position itself to capture this intruder
whenever capture is possible. On the other hand, if the intruder does not come
radially toward the target center, it spends a longer time within the TSR
without knowing which direction is beneficial to move for avoiding the
defender, which only benefits the defender. A formal proof of the last
statement is beyond the scope of this paper and we simply assume that the
intruders are prescribed to move radially in this phase.  The defender’s
objective in this phase is to leverage the sensing (information) advantage it
has and use it to start the next phase in a favorable configuration.
Therefore, to discuss the strategy of the defender in this phase, we need to
understand what configurations are favorable for the next phase and whether
such a configuration is achievable by the defender. This is discussed in the
following.

###  III-A Engagement Surface

Suppose the full information phase starts after a $\tau$ amount of time from
the appearance of the intruder at the TSR boundary. Without any loss of
generality, we assume that each intruder appears on the TSR boundary at time
$0$ at the point $[r_{{}_{T}}+\rho_{{}_{T}},~{}0]$. Since the intruder moves
radially toward the target center with maximum velocity during the partial
information phase, the intruder shall be at a location
$\mathbf{x}_{A}(\tau)=[r_{{}_{T}}+\rho_{T}-\tau\nu,\ 0]^{\intercal}$ at time
$\tau$ when the full information phase starts, i.e., the intruder first senses
the defender. Let the defender be located at a $\theta$ angle on intruder’s
sensing boundary at that time, i.e.,

$\mathbf{x}_{D}(\tau)=\mathbf{x}_{A}(\tau)+\rho_{A}\hat{\mathbf{u}}(\theta).$
(4)

Therefore, from (3), the center of the Apollonius circle is at
$\mathbf{x}_{C}(\tau)=\mathbf{x}_{A}(\tau)-\beta\rho_{{}_{A}}\hat{\mathbf{u}}(\theta)$
and the radius is $\gamma\rho_{{}_{A}}$. To guarantee capture of the intruder,
the Apollonius circle must be contained within the TSR without intersecting
with interior of the target, i.e.,

$\displaystyle\|\mathbf{x}_{C}(\tau)\|\geq r_{{}_{T}}+\gamma\rho_{{}_{A}},$
(5a) $\displaystyle\|\mathbf{x}_{C}(\tau)\|\leq
r_{{}_{T}}+\rho_{{}_{T}}-\gamma\rho_{A}.$ (5b)

Notice that, due to properties of the Apollonius circle and Lemma 1, equation
(5) is necessary and sufficient to ensure capture. See Fig. 2 for a graphical
representation.

breachcapture

Figure 2:  The blue and red dots represent the positions of the defender and
the intruder, respectively. The green region is a part of the circular target
and the brown arc represents a part the TSR boundary. The black circles
represent the Apollonius circles. In the left figure the intruder can breach
the target. Such a breaching location is shown by the dashed red vector. The
right figure shows a configuration that leads to capture.

Satisfiability of the two conditions in (5) depends on the choice of $\tau$
and $\theta$ in (4), and there could be multiple such choices for $\tau$ and
$\theta$. We next analyze which of these choices are going to help the
defender in the current game and maximize the probability of winning the
subsequent games.

Since the next intruder appears at the TSR boundary as soon as the current one
is captured (or breaches the target), the defender needs to ensure that the
current one is captured at a location that maximizes the chance of capturing
the next one. Recall from Corollary 1 that the capture angle is maximized if
the defender is able to capture the current one as close to the target center
as possible. Or in other words, the Apollonius circle should be as close to
the target center as possible while satisfying (5). Therefore, $\theta$ and
$\tau$ should be chosen such that the Apollonius circle is tangent to the
target boundary, i.e.,

$\displaystyle\|\mathbf{x}_{C}(\tau)\|=r_{{}_{T}}+\gamma\rho_{A}.$ (6)

Notice that, due to Assumption (2), equation (6) ensures satisfaction of both
the conditions in (5). Equation (6) simplifies to

$\sin^{2}(\theta/2)=\frac{(r_{{}_{T}}+\gamma\rho_{{}_{A}})^{2}-(r_{{}_{T}}+\rho_{{}_{T}}-\tau\nu-\beta\rho_{{}_{A}})^{2}}{4\beta\rho_{A}(r_{{}_{T}}+\rho_{{}_{T}}-\tau\nu)},$
(7)

which provides a feasible engagement time and location pair $(\tau,\theta)$.
The locus of these engagement points, i.e., $\mathbf{x}_{A}(\tau_{\rm
eng})+\rho_{{}_{A}}\hat{\mathbf{u}}(\theta_{\rm eng})$ is plotted in Fig. 3.

######  Definition 1 (Engagement Surface)

This is the collection of the pairs $(\tau_{\rm eng},\theta_{\rm eng})$ that
satisfy (7). That is, $S_{\rm engage}=\\{(\theta_{\rm eng},\tau_{\rm
eng})~{}:~{}\theta_{\rm eng},\tau_{\rm eng}\text{ satisfy
}\eqref{eq:engagementSurface}\\}.$ $\triangle$

If the full information phase starts at a point on the engagement surface,
then (6) is satisfied and due to Assumption (2) and Lemma 1 the intruder
cannot breach or get out of the TSR before getting captured. Therefore, the
intruder’s objective in this case is to get captured the farthest from the
target center. This will increase the breach probability for the next intruder
due to Corollary 1. The farthest capture point will be at a radial location of
$r_{{}_{T}}+2\gamma\rho_{{}_{A}}$ since the radius of Apollonius circle is
$\gamma\rho_{{}_{A}}$ and the center is at a distance
$r_{{}_{T}}+\gamma\rho_{{}_{A}}$ from the target center (see (6)). The circle
with radius $r_{{}_{T}}+2\gamma\rho_{{}_{A}}$ and concentric with the target
is referred to as the capture circle, since all the captures of the intruders
will happen on this circle.

$\mathbf{x}_{A}(\tau_{\rm eng})$$\theta_{\rm eng}$$x_{1}$$D_{1}$$D_{2}$$x_{2}$

Figure 3:  (Left) The cyan oval-shaped curve represents the locus of the
engagement points $\mathbf{x}_{A}(\tau_{\rm
eng})+\rho_{{}_{A}}\hat{\mathbf{u}}(\theta_{\rm eng})$. We demonstrate two
configurations where a defender $D_{1}$ (or $D_{2}$) starts the full
information phase by engaging at the point $x_{1}$ (or $x_{2}$) when the
intruder is at $\mathbf{x}_{A}(\tau_{\rm eng})$. (Right) The zoomed out
version of the left figure. The outer most (brown) arc represents a part of
the TSR boundary. The gray arc is the capture circle. The orange part of the
capture circle represents all the possible locations of the defender such that
it can reach the magenta star before time $\tau_{\rm eng}$. The magenta star
represents one possible engagement point. The locations of the intruder at the
time of engagement and its initial location on TSR boundary are shown using
solid red dots and a hollow red circle, respectively. The trajectories taken
by the agents are shown using (dotted-)dashed lines.

###  III-B Reachability to the Engagement Surface

Let the defender be located at a point $r\hat{\mathbf{u}}({\theta_{D}})$ at
the time the intruder enters the TSR boundary at
$[r_{{}_{T}}+\rho_{{}_{T}},0]^{\intercal}$. The following theorem provides the
conditions on $r$ and $\theta_{D}$ such that the defender is able to reach the
engagement surface at $(\tau_{\rm eng},\theta_{\rm eng})\in S_{\rm engage}$.

######  Theorem 1 (Necessary)

A necessary condition for a defender located at
$r\hat{\mathbf{u}}({\theta_{D}})$ to reach $(\tau_{\rm eng},\theta_{\rm
eng})\in S_{\rm engage}$ before getting detected by the intruder is
$\theta_{D}\leq\theta_{\max}(\tau_{\rm eng},\theta_{\rm eng})$, where

$\displaystyle\theta_{\max}(\tau_{\rm eng},\theta_{\rm
eng},r)=\cos^{-1}\bigg{(}\frac{r_{\rm eng}^{2}+r^{2}-\tau_{\rm
eng}^{2}}{2r_{\rm eng}r}\bigg{)}+\phi_{\rm eng}$ (8) $\displaystyle\phi_{\rm
eng}=\sin^{-1}\bigg{(}\frac{\rho_{{}_{A}}\sin(\theta_{\rm eng})}{r_{\rm
eng}}\bigg{)}$ $\displaystyle r_{\rm
eng}=\Big{(}(r_{{}_{T}}+\rho_{{}_{T}}-\tau_{\rm eng}\nu-\rho_{{}_{A}})^{2}$
$\displaystyle\qquad\quad+4(r_{{}_{T}}+\rho_{{}_{T}}-\tau_{\rm
eng}\nu)\rho_{{}_{A}}\cos^{2}(\theta_{\rm
eng}/2)\Big{)}^{1/2}.\qquad\hfill\triangle$

###### Proof:

The proof is omitted due to space limitation. ∎

Theorem 1 provides the maximum allowed initial angular separation between the
intruder and the defender to ensure capture. Since there are many possible
engagement points $(\tau_{\rm eng},\theta_{\rm eng})$, we shall optimize over
$(\tau_{\rm eng},\theta_{\rm eng})$ to maximize $\theta_{\max}(\tau_{\rm
eng},\theta_{\rm eng})$. Without loss of generality, henceforth we will denote
$(\tau_{\rm eng},\theta_{\rm eng})$ to be the one that maximizes
$\theta_{\max}(\tau_{\rm eng},\theta_{\rm eng})$.

Theorem 1 gives the necessary condition on the defender’s initial location to
ensure reachability to the engagement surface. However, the theorem does not
provide any guarantee whether the defender will be detected by the intruder
along its way to $\textbf{x}_{\rm eng}$. For example, a scenario like the one
presented in Fig. 4 may occur. In the following theorem we formally prove that
a scenario like Fig. 4 shall not occur under a certain condition on $r$. In
other words, the necessary condition in Theorem 1 is also sufficient.

$\mathbf{x}_{A}(\tau_{\rm
eng})$$\mathbf{x}_{A}(\tau_{1})$$\rho_{{}_{A}}$$\underset{{r_{{}_{T}}+\rho_{{}_{T}}-r}}{\underbrace{~{}~{}}}$$\theta$

Figure 4:  $\medbullet$ and  $\medbullet$ denote the location of the intruder
and defender at times $\tau_{1}$ and $\tau_{\rm eng}$. The hollow red and blue
circles denote their initial locations. The brown arc denotes a part of the
TSR boundary. The gray arc has a radius of $r$ and it is concentric to the
target center (denoted by the green point $O$). The defender moves along the
dotted line. The angle between the defender’s path and the intruder’s path is
denoted by $\theta$.

######  Theorem 2 (Sufficient)

If $r\leq r_{{}_{T}}+\rho_{{}_{T}}-\rho_{{}_{A}}$, then
$\theta_{D}\leq\theta_{\max}(\tau_{\rm eng},\theta_{\rm eng})$ is a sufficient
condition for a defender located at $r\hat{\mathbf{u}}({\theta_{D}})$ to reach
$(\tau_{\rm eng},\theta_{\rm eng})\in S_{\rm engage}$ before getting detected
by the intruder. $\hfill\triangle$

###### Proof:

The proof is omitted due to space limitation. ∎

The following remark is due to Theorem 2 and Assumption (2). This remark is
important for analyzing the entire game with multiple arrivals.

###### Remark 1

If a defender is located on the capture circle at the time an intruder appears
on the TSR boundary, then
$r=r_{{}_{T}}+2\gamma\rho_{{}_{A}}<r_{{}_{T}}+\rho_{{}_{T}}-\rho_{{}_{A}}$ due
to Assumption (2). Therefore, if $\theta_{D}\leq\theta_{\max}(\tau_{\rm
eng},\theta_{\rm eng})$ then due to Theorem 2, the defender is able to reach
the engagement surface and the intruder will then be captured on the capture
circle. As a consequence, the condition (on $r$) for Theorem 2 will be
satisfied for the next game too. On the other hand, if
$\theta_{D}>\theta_{\max}(\tau_{\rm eng},\theta_{\rm eng})$, then the defender
will lose the current game. Therefore, in that case, the defender goes back to
the target center to ensure that it wins the next game with probability 1. Due
to the Assumption that $\rho_{{}_{T}}/\nu>r_{{}_{T}}+2\gamma\rho_{{}_{A}}$ in
(2), the defender can reach the target center before the next game starts.
$\triangle$

Based on the above remark, we conclude this section with the following
proposition on the defender’s strategy during the partial information phase.

######  Proposition 1 (Defender Strategy)

If $\theta_{D}\\!>\\!\theta_{\max}(\tau_{\rm eng},\theta_{\rm eng})$, then the
optimal strategy for the defender is to go to the center of the target,
otherwise, it should go to the point $(\tau_{\rm eng},\theta_{\rm eng})\in
S_{\rm engage}$ to start the full information phase. $\hfill\triangle$

###### Proof:

When $\theta_{D}>\theta_{\max}(\tau_{\rm eng},$ $\theta_{\rm eng})$, the
defender is not able to reach to engagement surface, and therefore, at the
full information phase the Apollonius circle will have an intersection with
the target. This will lead to a breach. Therefore, the defender’s attempt to
engage with the intruder is futile and at the end of this engagement the
defender might be at a non-zero radial location which will decrease its
probability of winning the next game (Corollary 1). Thus, an optimal strategy
for the defender is to maximize the probability of the next game, which is
achieved by being closest to the target center.

On the other hand, when $\theta_{D}\leq\theta_{\max}(\tau_{\rm eng},$
$\theta_{\rm eng})$, then the defender can reach the engagement surface or it
may choose not to engage with this intruder and rather position itself better
for the next intruder. Given that the intruders are arriving on the TSR
boundary with uniform probability, it can be verified (by considering the
expected number of captures) that the optimal choice is to engage with the
current intruder since it is going to be a win with probability 1 (and some
strictly positive probability for capturing the next). ∎

##  IV Full Information Phase

In the previous section we discussed the favourable configurations for
starting the full information game and derived the conditions on the
defender’s location to ensure that such a favorable configuration is
achievable. In this section we discuss the progression of the game once the
engagement starts.

As soon as the intruder detects the defender, it can compute the instantaneous
Apollonius circle and verify whether the configuration leads to a breach or
capture. Recall from Lemma 1 that the intruder cannot get out of this
Apollonius circle without being captured. Therefore, if breach is possible
(i.e., the Apollonius circle has a nonempty intersection with the interior of
the target), then the intruder moves toward a breaching point (see left image
in Fig. 2). Otherwise, the intruder picks a strategy to draw the defender
farthest away from the target center. In this way, the intruder ensures the
radial distance of the defender is maximized before the next game starts and
the defender has a lesser guarding angle (see Corollary 1). The following
lemma provides the strategy for an intruder that gets captured.

###### Lemma 3

The intruder goes to the point $\textbf{x}_{p}$ in a straight line with
maximum velocity, where

$\displaystyle\textbf{x}_{p}=\mathbf{x}_{A}(\tau_{\rm
eng})-\beta\rho_{{}_{A}}\hat{\mathbf{u}}({\theta_{\rm
eng}})+\gamma\rho_{{}_{A}}\hat{\mathbf{u}}(\varphi)$

Where $\varphi$ is equal to:

$\displaystyle\varphi=\tan^{-1}\bigg{(}\frac{\beta\rho_{{}_{A}}\sin({\theta_{\rm
eng}})}{\beta\rho_{{}_{A}}\cos({\theta_{\rm
eng}})-(r_{{}_{T}}+\rho_{{}_{T}}-\tau_{\rm eng}\nu)}\bigg{)}$ (9)

$\hfill\triangle$

###### Proof:

The full information phase starts with intruder being at
$\mathbf{x}_{A}(\tau_{\rm eng})=[r_{{}_{T}}+\rho_{{}_{T}}-\tau_{\rm
eng}\nu,0]^{\intercal}$ and the defender being at $\mathbf{x}_{D}(\tau_{\rm
eng})=\mathbf{x}_{A}(\tau_{\rm
eng})+\rho_{{}_{A}}\hat{\mathbf{u}}({\theta_{\rm eng}})$. Therefore, from (3)

$\displaystyle\mathbf{x}_{C}(\tau_{\rm eng})=\mathbf{x}_{A}(\tau_{\rm
eng})-\beta\rho_{{}_{A}}\hat{\mathbf{u}}({\theta_{\rm
eng}})=(r_{{}_{T}}+\gamma\rho_{{}_{A}})\hat{\mathbf{u}}({\varphi}),$

where $\varphi$ is given (9). Thus, the farthest point on the Apollonius
circle from the target center is at

$\displaystyle\textbf{x}_{p}=\mathbf{x}_{C}(\tau_{\rm
eng})+\gamma\rho_{{}_{A}}\hat{\mathbf{u}}(\varphi).$

By substituting the expression of $\mathbf{x}_{C}(\tau_{\rm eng})$ in the
above equation, the lemma is proved. ∎

##  V Analysis of the Game

The first game starts with defender being at the target center. Due to
Assumption (2) and Lemma 2, this leads to a capture of the first intruder with
probability 1. The defender is on the capture circle at the end of the first
game. Afterwards, at the end of each game it will either be on the capture
circle or on the target center depending on the arrival angles of the
intruders. The defender strategy (and the progression of the game) is
described by the following algorithm.

Algorithm 1 Defender’s Strategy

1:Initialize $\mathbf{x}_{D}\leftarrow[0,0]^{\intercal}$, $N_{\rm
capture}\leftarrow 0$, and $N$

2:for $n=1:N$ do

3: $\theta_{A}\sim{\mathcal{U}}(-\pi,\pi)$ $\triangleright$ Uniform random
arrival of intruder

4: if $\mathbf{x}_{D}$ is at target center then $\triangleright$ Capture
happens

5: ${\textbf{x}}_{\rm
eng}\leftarrow(r_{{}_{T}}-\nicefrac{{\rho_{{}_{A}}}}{{(1+\nu)}})\hat{\mathbf{u}}(\theta_{A})$

6: Defender goes to ${\textbf{x}}_{\rm eng}$

7: $N_{\rm capture}\leftarrow N_{\rm capture}+1$

8:
$\mathbf{x}_{D}\leftarrow(r_{{}_{T}}+2\gamma\rho_{{}_{A}})\hat{\mathbf{u}}(\theta_{A})$
$\triangleright$ $\mathbf{x}_{D}$ after capture

9: else if
$\mathbf{x}_{D}=(r_{{}_{T}}+2\gamma\rho_{{}_{A}})\hat{\mathbf{u}}(\theta_{D})$
then

10: if $|\theta_{D}-\theta_{A}|\leq\theta_{\max}$ then $\triangleright$
Capture happens

11: ${\textbf{x}}_{\rm eng}\leftarrow\mathbf{x}_{A}(\tau_{\rm
eng})+\rho_{{}_{A}}\hat{\mathbf{u}}(\theta_{\rm eng})$

12: Defender goes to ${\textbf{x}}_{\rm eng}$

13: $N_{\rm capture}\leftarrow N_{\rm capture}+1$

14: $\mathbf{x}_{D}\leftarrow\textbf{x}_{p}$ from Lemma 3 $\triangleright$
$\mathbf{x}_{D}$ after capture

15: else$\triangleright$ Breach happens

16: Defender goes to the target center

17: $\mathbf{x}_{D}\leftarrow[0,0]^{\intercal}$

18: end if

19: end if

20:end for

###  V-A Percentage of capture

If the defender is located at the target center, then capture is guaranteed
regardless of the angle $\theta_{A}$ the intruder appears at the TSR boundary.
Otherwise (when it is on capture circle), it can only capture the intruders
that appear on the TSR boundary with an angular separation of $\theta_{\max}$
(Theorems 1, 2). Given that the intruders appear on the TSR boundary
independently with uniform probability, the capture probability is

$\displaystyle p^{*}\triangleq\frac{\theta_{\max}}{\pi}.$ (10)

###  V-B Percentage of Capture with Finite Arrival of Intruders

As discussed before, the first intruder gets captured with probability 1.
After the first capture, the defender is on the capture circle and can capture
the next randomly appearing intruder with probability $p^{*}$. If the next
intruder is captured, then the defender will remain on the capture circle;
otherwise, it will be at the target center at the time the third game starts.
This process is repeated as the intruders keep coming.

Every time the defender goes back to the center of the target (i.e., an
intruder is able to breach) we call this a reset of the game. The (random)
sequence of capture in-between two resets is called a travel. The length of a
travel is the number of captures in that travel. Let the random variable
$L_{i}$ denote the length of the $i$-th travel. Since each intruder appears
independently with uniform probability at the TSR boundary, one may verify
that $L_{i}$ is a geometric random variable with

$\displaystyle P(L_{i}=k)=(p^{*})^{k-1}(1-p^{*}),$ (11)

for all positive integer $k$, where $p^{*}$ is the quantity defined in (10).
The probability of capturing a total of $n$ intruders in $m$($\leq n$) travels
is therefore

$\displaystyle
P\bigg{(}\sum_{i=1}^{m}L_{i}=n\bigg{)}=\binom{n-1}{m-1}(p^{*})^{n-m}(1-p^{*})^{m},$

which follows a negative binomial distribution.

Let $N$ denote the total number of intruders that have arrived at the TSR
boundary so far and $S_{N}$ denote the number of resets at the end of the
$N$-th game. Therefore, we have

$\displaystyle N\geq\sum_{i=1}^{S_{N}}L_{i}+S_{N}$

with probability almost surely. Notice that the event $\\{S_{N}\leq m\\}$ is
equivalent to the event $\\{\sum_{i=1}^{m+1}L_{i}+m\geq N\\}$. Therefore,

$\displaystyle P(S_{N}>m)=1-P(S_{N}\leq m)$
$\displaystyle=1-P\bigg{(}\sum_{i=1}^{m+1}L_{i}+m\geq N\bigg{)}$
$\displaystyle=P\bigg{(}\sum_{i=1}^{m+1}L_{i}+m\leq
N-1\bigg{)}=\sum_{j=m+1}^{N-m-1}P\bigg{(}\sum_{i=1}^{m+1}L_{i}=j\bigg{)}$
$\displaystyle=\sum_{j=m+1}^{N-m-1}\binom{j-1}{m}(p^{*})^{j-m-1}(1-p^{*})^{m+1}.$
(12)

At the end of the $N$-th game, a total of $S_{N}$ intruders have been able to
breach, and therefore, the capture fraction is $\nicefrac{{(N-S_{N})}}{{N}}$,
which is a random variable due to the randomness of $S_{N}$. From the
(cumulative) distribution function of $S_{N}$ in (12), we compute the
expectation $\operatorname{\mathbb{E}}[S_{N}]$ at the end of the $N$-th game,
from which we find the expected percentage of capture at the end of the $N$-th
game to be

$\displaystyle{\rm
percentage}(N)=\frac{N-\operatorname{\mathbb{E}}[S_{N}]}{N}\times 100,$ (13)

for any finite $N$.

###  V-C Percentage of Capture with Infinite Arrival of Intruders

After the $k$-th travel, the total number of captures is $\sum_{i=1}^{k}L_{i}$
and the total number of breach is $k$. Therefore, the capture fraction $c(k)$
after the $k$-th travel is

$\displaystyle c(k)$
$\displaystyle=\frac{\sum_{i=1}^{k}L_{i}}{\sum_{i=1}^{k}L_{i}\ +k}.$

Notice that, $c(k)$ is a random variable for each $k$ since $L_{i}$’s are
random variables. For the case of infinite arrival of intruders the number of
resets (i.e., $k$) also approaches infinity. We take the limit $k\to\infty$ to
obtain the asymptotic capture fraction $c_{\infty}$. Therefore,

$\displaystyle
c_{\infty}=\lim_{k\to\infty}c(k)=\frac{\lim_{k\to\infty}\frac{1}{k}\sum_{i=1}^{k}L_{i}}{\lim_{k\to\infty}\frac{1}{k}\sum_{i=1}^{k}L_{i}\
+1}.$

Although $c(k)$ is a random variable, due to the (strong) law of large numbers
we obtain that
$\frac{1}{k}\sum_{i=1}^{k}L_{i}\to\operatorname{\mathbb{E}}[L_{i}]$ almost
surely as $k\to\infty$. Recall that $L_{i}$ is a geometric random variable and
from (11), we obtain that
$\operatorname{\mathbb{E}}[L_{i}]=\nicefrac{{1}}{{1-p^{*}}}$. After
substituting
$\lim_{k\to\infty}\frac{1}{k}\sum_{i=1}^{k}L_{i}=\nicefrac{{1}}{{1-p^{*}}}$,
we obtain $c_{\infty}=\nicefrac{{1}}{{(2-p^{*})}}.$ Therefore, the asymptotic
percentage of capture is

$\displaystyle\mathrm{percentage}(\infty)=\frac{100}{~{}2-p^{*}}.$ (14)

Using the expression of $p^{*}$ from (10), we may also write
$\mathrm{percentage}(\infty)=\nicefrac{{100\pi}}{{(2\pi-\theta_{\max})}}$.

##  VI Simulation Results

We simulate the game with the following parameters $r_{{}_{T}}=5$,
$\rho_{{}_{A}}=1,~{}\rho_{{}_{T}}=10$, and $\nu=0.8$. Under this parametric
choice we conducted $100$ random trials of the game. In each trial we
considered a sequence of $200$ incoming intruders that are uniform randomly
generated on the TSR boundary. The percentage of capture for each trial is
plotted in Fig. 5.

Figure 5: Percentage of Capture versus number of arrivals from 100 trials.
Each trial is represented with a gray line. Their empirical average is
represented by the cyan line. The magenta line represents the theoretically
predicted capture percentage in (13).

The abscissa in this figure denotes the number of arrivals ($N$) and the
ordinate denotes the percentage of capture (i.e.,
$100(1-\nicefrac{{S_{N}}}{{N}})$) for that number of arrivals. We compute the
empirical mean of the percentage of capture from these 100 random trials, and
that is shown by the cyan line in Fig. 5. To compare this simulation result
with our theoretical analysis, we plot the expected percentage (i.e.,
${\rm{percentage}}(N)$ from (13)) in the same figure using the magenta line.
We observe that the empirical mean is very close to the theoretically
predicted quantity. In this plot we also report the value of the asymptotic
capture percentage (i.e., $100c_{\infty}$), and we notice that the random
trials and $\mathrm{percentage}(N)$ converge ($\sim$ exponentially) to
$100c_{\infty}$ as $N$ increases. A short simulation video can be accessed at
[19].

###  VI-A Parameter variations

In the next simulation, we vary the parameters $\nu,\rho_{{}_{A}}$ and
$\rho_{{}_{T}}$ and plot the asymptotic and some of the finite time expected
capture percentages in Figs. 6-7.

Figure 6: Expected percentage of capture when $\rho_{{}_{A}}$ and
$\rho_{{}_{T}}$ are varied for a fixed $\nu=0.75$. From left to right, we plot
the percentage capture for $N=20,50,100$ and $\infty$, respectively.

Figure 7: Expected percentage of capture when $\rho_{{}_{A}}$ and $\nu$ are
varied for a fixed $\rho_{{}_{T}}=12$. From left to right, we plot the
percentage capture for $N=5,20,50$ and $\infty$, respectively.

In Fig. 6, we fix $\nu=0.75$ and vary the parameters $\rho_{{}_{A}}$ and
$\rho_{{}_{T}}$. We use (12)-(13) to compute $\textrm{percentage}(N)$ for
$N=20,50$ and $100$. These are the first three subplots in Fig. 6. Then we use
(14) to compute the asymptotic percentage, which is shown in the right-most
plot in Fig. 6. One of the unexpected observations is to note that many of the
level sets (i.e., curves with constant percentages) appear to be linear in the
$\rho_{{}_{A}}-\rho_{{}_{T}}$ plane, which is not apparent from the nonlinear
relationship between these parameters and $\theta_{\max}$ (or equivalently
$p^{*}$) in Theorem 1. Another immediate observation is that the separation
between these level sets is not proportional to the difference between the
percentages. In fact, the separation among these lines monotonically decreases
even though the percentage is increased at a regular interval.

For this particular experiment, the slope of these lines appear to be
approximately $2.5$. An immediate conclusion from this linear behavior is that
for each unit of change in $\rho_{{}_{A}}$, we need $2.5$ units of change in
$\rho_{{}_{T}}$ to maintain the same expected capture percentage (since the
slope is 2.5). An interesting future direction would be to investigate these
lines and find the relationship between their properties (e.g., slope and
intercept) and the problem parameters ($\nu,r_{{}_{T}}$ etc.).

Another observation from Fig. 6 is that the plots do not change much as we
move from $N=20$ to $N=\infty$. This is consistent with our observation in
Fig. 5 where the expected capture percentage (the magenta line) quickly
converges to the asymptotic capture percentage. While Fig. 5 showcases this
behavior for one particular parameter setting, Fig. 6 demonstrates the same
over a range of parameters.

Next we fixed $\rho_{{}_{T}}$ while varying $\nu$ and $\rho_{{}_{A}}$ to
compute the expected capture percentages. This is plotted in Fig. 7. When the
condition in (2) is not satisfied, we do not compute the capture percentages
and hence some parts (top-right) of the plots in Fig. 7 are empty. Some level
sets of the capture percentages are also plotted in these plots. Similar to
the plots in Fig. 6, these plots also do not change a lot as we vary $N$.

##  VII Conclusion

In this paper, we formulated a target defense game against a sequence of
incoming intruders. At any time only one intruder attempts to breach the
target, which decomposes the entire game into a sequence of 1-vs-1 games. The
terminal configuration of the current game becomes the initial configuration
for the next game. Based on the available sensor information, each game is
divided into two phases and the strategies for both agents are derived for
these two phases. We define the concept of engagement surface and capture
circle to construct the strategies for the agents. We analyzed the entire game
for both finite and infinite sequences of intruder arrivals and analytically
computed the expected percentage of capture for both the cases. Numerical
experiments demonstrate further interesting and unexpected characteristics in
the levels sets (see discussion on Fig. 6).

A natural extension of this work would be to consider different arrival
patterns for the intruders (e.g., periodic arrival, non-uniform probability of
arrival locations, multiple arrivals). Furthermore, one may also consider a
heterogeneous team for the intruders where different intruders may have
different speed and sensing capabilities. Another possible extension is to
consider a defender with its own sensing region.

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