## I Introduction

### I-1 Motivation

Various cooperative techniques are used in [isaacs1999differential, venkatesan2014target, pachter2017differential, harris2020abnormal, mohanan2020target, huang2011differential, pierson2016intercepting, chen2017multiplayer, shishika2020cooperative, coon2017control, zheng2020time] for a team of defenders to effectively capture multiple attackers in order to defend a safety-critical area. However, none of these studies considers active collision avoidance among the defenders. In this paper, we develop a collaborative defense strategy for a team of defenders to intercept as many of the attackers as possible before the attackers reach the protected area, while ensuring no collisions between the defenders.

### I-2 Related work

Due to curse of dimensionality, approaches for 1-vs-1 area defense games such as

[isaacs1999differential, venkatesan2014target, pachter2017differential, harris2020abnormal, mohanan2020target, huang2011differential] are extended to multi-player scenarios using a “divide and conquer” approach. That is, the 1-vs-1 problems, or problems with small number of agents for all such combinations of the agents are solved first, and then the defenders are assigned to the attackers based on these 1-vs-1 games to solve the multiplayer case.The authors in [pierson2016intercepting] develop a distributed algorithm for the cooperative pursuit of multiple evaders by multiple pursuers using area-minimization strategy based on a Voronoi tessellation in a bounded convex environment. In [chen2017multiplayer], the authors solve the reach-avoid game for each pair of defender and attacker operating in a compact domain with obstacles using a Hamilton-Jacobi-Issacs (HJI) reachability approach. The solution is then used to assign defenders against the attackers in multiplayer case using graph-theoretic maximum matching. In [shishika2020cooperative] defenders are restricted to move on the perimeter of protected area. Local games between small teams of defenders and attackers are solved and then assignments are done using a polynomial time algorithm.

The aforementioned studies provide useful insights to the area or target defense problem, however, are limited in application due to the use of single-integrator motion model, and lack of consideration of inter-defender collisions. In [coon2017control], authors consider Target-Attacker-Defender (TAD) game with agents moving under double-integrator dynamics and use the isochrones method to design time-optimal control strategies for the players. In [zheng2020time], authors develop a time-optimal strategy for a Dubins vehicle to intercept a target moving with constant velocity, an assumption that could be limiting in practice.

### I-3 Overview and summary of our contributions

In this paper, we resort to the common “divide and conquer” paradigm. We build on the time-optimal guidance problem for isotropic rocket [bakolas2014optimal], that uses a damped double-integrator motion model, and formulate a non-zero sum game between each pair of defender and attacker. We then use the time of interception by each defender against each attacker, and the times of possible collisions on the defenders’ optimal trajectories, to assign the defenders to the attackers. We call this assignment as collision-aware defender-to-attacker assignment (CADAA). Furthermore, we use exponential control barrier functions (ECBF) [nguyen2016exponential, wang2017safe] in a quadratically constrained quadratic program (QCQP) to augment defenders’ optimal control actions in order to avoid collision with other fellow defenders when such collisions are unavoidable solely by CADAA.

In summary, the contributions of this paper are: (1) a non-zero-sum game between a defender and an attacker to obtain a time-optimal defense strategy, and (2) a mixed-integer quadratic program (MIQP) to find CADAA to capture as many attackers as possible and as quickly as possible, while preventing or delaying the collisions among the defenders.

### I-4 Organization

Section II provides the problem statement. The interception strategy for the 1-defender-vs-1-attacker area defense game is given in Section LABEL:sec:one-vs-one_interception_game and that for multiple defenders vs multiple attackers is discussed in Section LABEL:sec:many-vs-many_interception_game. Simulation results and conclusions are given in Section LABEL:sec:simulations and Section LABEL:sec:conclusions, respectively.

## Ii Modeling and Problem Statement

Notation: denotes the Euclidean norm of its argument. denotes absolute value of a scalar argument and cardinality if the argument is a set. A ball of radius centered at the origin is defined as and that centered at is defined . denotes all the elements of the set that are not in the set . We define , .

We consider attackers denoted as , , and defenders denoted as , , operating in a 2D environment that contains a protected area , defined as . The number of defenders is no less than that of attackers, i.e., . The agents and are modeled as discs of radii and , respectively.

Let and

be the position vectors of

and , respectively; , be the velocity vectors, respectively, and , be the accelerations, which serve also as the control inputs, respectively, all resolved in a global inertial frame (see Fig.1). The agents move under double integrator (DI) dynamics with linear drag (damped double integrator), similar to isotropic rocket [bakolas2014optimal]:(1) |

where , is the known, constant drag coefficient. The accelerations and are bounded by , as: The defenders are assumed to be at least as fast as the attackers, i.e., . The dynamics in (1) take into account the drag experienced by the agents modeled as a linear function of the velocity. Note also that the damped double integrator model inherently poses a speed bound on each agent under a limited acceleration control, i.e., and , and does not require an explicit constraint on the velocity of the agents while designing bounded controllers, as in earlier literature. So we have , for all , where and , for all , where . We denote by the configuration space of all the agents. We also make the following assumption:

###### Assumption 1

Each player (either defender or attacker) knows the states of the all the other players.

Each defender is endowed with an interception radius , i.e., the defender is able to physically damage an attacker when .

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