## I Introduction

Swarms of low-cost agents such as small aerial robots may pose risk to safety-critical infrastructure such as government facilities, airports, and military bases. Interception strategies [chen2017multiplayer, coon2017control] against these threats may not be feasible or desirable in an urban environment due to posing greater risks to humans and the surrounding infrastructure. Under the assumption of risk-averse and self-interested adversarial agents (attackers) that tend to move away from the defending agents (defenders) and from other dynamic objects, herding can be used as an indirect way of guiding the attackers to a safe area.

In our recent work [chipade2019swarmherding, chipade2020swarmherding], we developed a herding algorithm, called ‘StringNet Herding’, to herd a swarm of adversarial attackers away from a safety-critical (protected) area. A closed formation (‘StringNet’) of defending agents connected by string barriers is formed around a swarm of attackers staying together to confine their motion within given bounds, and guide them to a safe area. However, the assumption that the attackers will stay together in a connectivity region, and they will react to the defenders collectively as a single swarm while attacking the protected area, can be quite conservative in practice.

In this paper, we build upon our earlier work on ‘StringNet Herding’ [chipade2020swarmherding] and study the problem of defending a safety-critical (protected) area from adversarial agents that may or may not stay together. We propose a ‘Multi-Swarm StringNet Herding’ approach that uses clustering-based defender assignment, and the ‘StringNet Herding’ method to herd the adversarial attackers to known safe areas.

### I-1 Related work

Several approaches have been proposed to solve the problem of herding. Some examples are: the -wavefront algorithm [gade2015herding, paranjape2018robotic], where the motion of the birds on the boundary of the flock is influenced based on the locations of the airport and the safe area; herding via formation control based on a potential-field approach [pierson2018controlling]; biologically-inspired "wall" and "encirclement" methods that dolphins use to capture a school of fish [haque2011biologically]; an RRT approach that finds a motion plan for the agents while maintaining a cage of potentials around the sheep [varava2017herding]; sequential switching among the chased targets [licitra2017single]. In general, the above approaches suffer from one or more of the following: 1) dependence on knowing the analytical modeling of the attackers’ motion, 2) lack of modeling of the adversarial agents’ intent to reach or attack a certain protected area, 3) simplified motion and environment models. The proposed ‘StringNet Herding’ approach relaxes the first and the third issue above, and takes into account the second one for control design.

Clustering of data points is a popular machine learning technique

[xu2015comprehensive]. There are various categories of clustering algorithms: 1) partition based (K-means

[macqueen1967some]), 2) hierarachy based (BIRCH [zhang1996birch]), 3) density based (DBSCAN [ester1996density]), 4) stream based (STREAM [o2002streaming]), 6) graph theory based (CLICK [sharan2000click]). Spatial proximity of the agents is crucial for the problem at hand so our focus will be mostly on the density based approaches in this paper.Assignment problems have also been studied extensively [burkard2012assignment]. In this paper, we are interested in a generalized assignment problem (GAP) [oncan2007survey], in which there are more number of objects than knapsacks to be filled. GAP is known to be NP-hard but there are approximation algorithms to solve an arbitrary instance of GAP [oncan2007survey].

### I-2 Overview of the proposed approach

The proposed approach involves: 1) identification of the clusters (swarms) of the attackers that stay together, 2) distribution and assignment of the defenders to each of the identified swarms of the attackers, 3) use of ‘StringNet Herding’ approach by the defenders to herd each identified swarm of attackers to the closest safe area.

More specifically, we use the “Density based Spatial Clustering of Application with Noise (DBSCAN)" algorithm [ester1996density] to identify the swarms of the attackers in which the attackers stay in a close proximity of the other attackers in the same swarm. We then formulate a generalized assignment problem with additional constraints on the connectivity of the defenders to find which defender should go against which swarm of attackers and herd it to one of the safe areas. This connectivity constrained generalized assignment problem (C2GAP) is modeled as a mixed integer quadratically constrained program (MIQCP) to obtain an optimal assignment solution. We also provide a hierarchical algorithm to find the assignment quickly, which along with the MIQCP formulation is the major contribution of this paper.

### I-3 Structure of the paper

Section II describes the mathematical modeling and problem statement. The StringNet herding approach is briefly discussed in Section LABEL:sec:herding. The approach on clustering and the defenders-to-attackers assignment for multiple-swarm herding is discussed in Section LABEL:sec:multi_swarm_herding. Simulations and conclusions are provided in Section LABEL:sec:simulations and LABEL:sec:conclusions, respectively.

## Ii Modeling and Problem Statement

Notation: The set of integers greater than 0 is denoted by

. Vectors and matrices are denoted by small and capital bold letters, respectively (e.g.,

r, P). denotes the Euclidean norm of its argument. denotes the absolute value of a scalar, and cardinality if the argument is a set. is a factorial of .
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