## I Introduction

Increased attention has been paid recently to multi-agent networks which rely on distributed control algorithms to distribute roles and workload among themselves. In this work, a class of distributed coverage control problems is considered in which the agents of the multi-agent network are required to deploy over a given region while avoiding obstacles at all times such that their spatial distribution matches very closely a desired time-varying reference distribution.

Literature Review: The problem addressed herein lies under the category of deployment problems. Some examples include target tracking [10] and area coverage [2, 6, 7, 1]. The objective of the deployment problem is to distribute the agents over a region in which a density function which is either static [2] or time-varying [6, 7] describes the relative importance of each subset of the region of interest. A key problem in the deployment of multi-agent networks is how the agents maneuver while avoiding collisions among themselves and with obstacles in a cluttered domain of interest. Methods used for collision avoidance include leader-follower control strategies [13], potential field based methods [12] and velocity obstacle methods [4]. While extensive research has been made on the obstacle avoidance of agents in several applications, most algorithms depend on the robots’ dynamics [8]. Therefore, if the robot is subject to uncertainty or its dynamics are highly nonlinear, these algorithms may result in collisions with obstacles due to controller’s poor performance. In addition, some of the proposed solution algorithms are not suitable for multi-agent systems as they do not guarantee safety among the agents.

Another method for collision avoidance guarantees is the Voronoi-based coverage control; yet, this method guarantees collision among agents but not obstacle avoidance [2]. If the latter method is modified, then effective obstacle collision can be guaranteed [10]. Hence, in this work we utilize the Obstacle-Aware Voronoi Cell (OAVC) method [10] that incorporates collision avoidance into path planning by generating a safe area around each agent. This method is based on the Voronoi-coverage control; however, to guarantee collision avoidance with obstacles, each generated Voronoi cell needs to be modified. One approach is to create buffered cells such that any agent located at the boundary of its cell does not collide with neighbours [14, 11]. However, [14] use static weights to create the buffered cells which is useful if the offset between agents is known and the region of interest has no large obstacles. In contrast, in this paper we use the approach in [10] which uses an algorithm that puts dynamic weights between the obstacles and agents such that the boundaries of the Voronoi cells are always tangent to the obstacles and never intersect or collide.

Our Contributions: In this paper we propose a two-level approach to the distributed control deployment problem in an environment with obstacles. The proposed approach aims to transport the multi-agent network from an initial Gaussian mixture (GM) distribution to a desired terminal GM distribution while guaranteeing collision safety and obstacle avoidance as the agents maneuver to achieve the goal spatial disposition. In particular, we show that there exists a path of GM connecting the initial and goal GMs which can be characterized in closed form in terms of the evolution of its components (mean, covariance and mixing proportions) as functions of time. The GM solution acts as a reference density to the low-level problem, which corresponds to an obstacle avoidance coverage problem with time-varying density in which collision avoidance is supposed to be guaranteed at all times. To solve the latter problem, we propose a new distributed control algorithm which is a variation of Lloyd’s algorithm in which OAVCs are utilized to guarantee obstacle avoidance as the agents move toward the locations that conform to the desired terminal distribution. In addition, our work considers a third problem that combines the low-level and high-level problems to indirectly steer the density of the team towards the desired reference density as the distributed control problem does not guarantee that.

Outline: In Section II, we formulate the three control problems, high-level, low-level and combined coverage control problems. In Section III, we present the solutions to the first two control problems. In Section IV, we present the algorithm that solves the combined coverage control problem based on the solutions to the high-level and the low-level control problems. In Section V, we present simulation results. Finally, in Section VI we conclude the paper with a summary of remarks.

## Ii Problem Formulation

### Ii-a Preliminaries

Given two integers and with , the discrete interval to is denoted by . The space of

-dimensional vectors in

with non-negative components is denoted by . Moreover, the standard simplex is denoted by where, consists of all vectors with and . To denote that the (symmetric) matrix is positive definite, we write. We denote the probability density function (pdf) of a multi-variate (

-dimensional) Gaussian distribution

with mean and covariance with , as where(1) |

where the determinant of is denoted as .

Given a collection of Gaussian density functions ), where and , for , and a vector of mixing proportions , we define the corresponding Gaussian mixture symbolized by as follows:

(2) |

Given a compact set , and a point-set , we refer the collection of sets where

(3) |

, as the Voronoi Tessellation (VT) or Voronoi Partition (VP) [9] of the set generated by the point-set , and the set as the -th Voronoi cell of the VP. The modified Voronoi cells (OAVC) for the -th agents will be denoted as .

We denote by the set of static obstacles represented as circles centred at with radius . We write to denote the subset each obstacle occupies in the region defined as :

(4) |

while denotes the total area occupied by all the obstacles, that is, .

Finally, given a continuous function , where is assumed to be a compact set, its spatial norm is denoted by and defined as:

(5) |

### Ii-B Problem Statement

In this section, we will formulate the two obstacle avoidance coverage problems for a multi-agent network.

###### Problem 1 (High-level coverage control problem)

Given: A 2-dimensional compact domain over which agents are dispersed. Let the point-set represent the agents’ locations at time , which is assumed to be a known initial Gaussian mixture distribution defined as where , where and for all and . Moreover,
we are given a terminal Gaussian mixture density distribution with components and corresponding mixing proportions, defined as , where , where and for all and .

Goal: Find a reference coverage density function such that: (i) is over , (ii) is continuous for all , (iii) , and (iv) the following boundary conditions are satisfied at and as :

(6) | ||||

(7) |

Thus, Problem 1 seeks for a density-path that will connect the initial density with the terminal density .

Next, the low-level control problem is introduced. The objective of the latter problem is to find the individual inputs of the agents that will make the team of agents evolve such that its density will eventually converge (in the sense) to the goal terminal density while avoiding obstacles in a cluttered region. For this purpose, a dynamic coverage optimization problem which utilizes the solution to (Problem 1) as its time-varying reference density is proposed. We consider a group of homogeneous mobile robots distributed over , having a set of static obstacles centered at for all . The motion of the mobile robots is described as follows:

(8) |

where = is the position of th agent in at time and is its velocity vector (control input).

###### Problem 2 (Low-level coverage control problem)

Given: The set of static obstacles , the set of initial locations of the agents such that for all and a reference density function that solves Problem 1. In addition, a configuration cost function that measures how well the agent is positioned which is defined as:

(9) |

Goal: Design distributed control algorithms to control the motion of agents in order to minimize the locational cost defined in (9). In other words,

(10) | |||||

subject to |

where .

Due to the fact that neither the high-level coverage control problem nor the low-level problem result independently in a satisfactory solution to the multi-agent control problem that we are trying to solve comprehensively, we will propose a combined version of these two problems, which will be solved practically. To be specific, Problem 1 seeks a density path that will connect the initial GM density to a desired terminal GM density but overlooks the agents’ dynamics, the obstacles and the individual control inputs that these agents need to apply for this density path to be realized. On the other hand, Problem 2 seeks for the individual control inputs that will account for collision avoidance and solve a locational optimization to help the network obtain the goal spatial distribution specified in the high-level problem, yet with no assurance that the solution to the latter problem will allow the density of the agents to approach the goal density. Thus, the previous reasons prompt us to consider a third problem that seeks for the computation of the individual inputs of the agents (similar to Problem 2) that will steer the density of the agents towards the desired terminal density (similar to Problem 1).

To formulate the third problem, we consider an approximation of the team’s probability density by a Gaussian mixture (density of the actual distribution of the network’s agents), which we refer to as the

fine approximation and denote as . For the computation of the fine approximation at each instant of time , the set of positions of the agents is fitted into a Gaussian mixture density:(11) |

with the use of, for instance, the Gaussian Mixture Model Likelihood Optimization (GMMLO) algorithm. Having the point-set

and a positive integer as input to the GMMLO algorithm, the output of the latter algorithm will be a Gaussian mixture consisting of Gaussian mixtures, which form the collection , with corresponding mixing proportion vector .###### Problem 3 (Combined coverage control problem)

Given:
Let be the fine approximation of the Gaussian mixture density distribution of the population based on the dispersion of the agents at time .

Goal: Find the individual control inputs , for , that will steer the agents emerging from the point-set , whose motion is described by (8), to the terminal destinations forming a point-set

, that corresponds to (approximately) a sample deduced from a probability distribution with density

. In other words, we seek to enforce the following limiting behaviors:(12) | ||||

(13) |

## Iii Proposed Solution and Analysis

### Iii-a Solution of the High-Level Coverage Control Problem

In this section, we will introduce a solution to Problem 1. For this purpose, consider a function which corresponds to a time-varying Gaussian mixture reference coverage density defined as:

(14) |

for all , where and are given by

(15) | ||||

(16) |

where and for denote the means and covariances at and and , for , symbolize the (time-varying) means and covariances, respectively, of the Gaussian densities that determine, together with the vector of mixing proportions , the Gaussian mixture density path at each time .

Our goal is to find a time-varying Gaussian mixture , which satisfies the following boundary (limiting) conditions:

(17) | |||

(18) |

where , , and are defined as in Problem 1. Next, we provide a closed-form solution to Problem 1.

###### Proposition 1

Let us consider the collection of Gaussians , where

(19) | ||||

(20) |

for , and , where , . In addition, let for with

(21) | ||||

(22) |

for all , where and is the vector of mixing proportions corresponding to . Also, . Then, the function , where satisfies the boundary conditions (6) and (7) and thus solves Problem 1.

###### Proof:

From (19) and (20), it follows readily that

(23) | ||||||

(24) |

for . In addition, (21) and (22) imply

(25) | ||||||

(26) |

Equations (21) and (22) also imply that . In addition, we will show that , for all , where

and we will conclude that , for all . Indeed, we have

(27) |

for all , where in the last equality, we have used the fact that . Therefore, , for all . We conclude that the collection of Gaussians with corresponding mixing proportions determine a Gaussian mixture for all , whose density satisfies equations (III-A). Next, we show that the density , which is defined in (III-A), satisfies the boundary conditions (6) and (7). From (23)-(24) and (25)-(26), we conclude

(28) | ||||

(29) |

where in the last derivation we have used the fact that and .This completes the proof.

### Iii-B Solution of the Low-Level Coverage Control Problem

To solve Problem (2) we design a distributed control algorithm that will make the agents track the time-varying centroids of their Voronoi cells and asymptotically converge to them while avoiding collisions with obstacles and among themselves. The solution approach proposed to address the low-level coverage control problem depends on a variation of Lloyd’s algorithm for the case of a time-varying density. The approach will include 1) computation of the modified Voronoi tessellations of the spatial domain comprised of the Obstacle Aware Voronoi Cells ([10]) generated by the current locations of the agents to avoid any obstacles in the domain, 2) characterization of the agent’s individual control inputs based on the information obtained from their own Voronoi cells. The Lloyd’s approach includes generating the modified Voronoi tessellations and computing the mass and the centroids of the th OAVC and iteratively modifying the agents’ positions to the centroids .

#### Iii-B1 Obstacle Aware Voronoi Cell

To create a modified Voronoi cell for obstacle collision avoidance, we adopt the approach proposed in [10]. When the agents calculate the Voronoi boundaries between other agents and themselves, they use the standard Voronoi tessellation method [3] which allows collision avoidance among the agents. However, to account for the presence of bigger obstacles, the agents will dynamically assign weights for each obstacle such that the boundaries of the modified Voronoi cells are tangent to the obstacle boundaries. The modified Voronoi cell is referred to as the “Obstacle-Aware Voronoi Cell” (OAVC), defined by:

(30) |

where is the dynamic weight that forces the boundary lines of the Voronoi cell to be tangent to the obstacles defined as

(31) |

for all and for all . Therefore, with the dynamic weight in the OAVC, the largest possible convex cell for the agent around static obstacles will be created. In addition, by maintaining a convex cell, the agents will be able to utilize move-to-centroid distributed controllers and satisfy asymptotic convergence to the centroids while they guarantee collision avoidance with other agents or obstacles.

Given a density function , the mass and centroid of the th OAVC are defined as follows:

(32a) | |||

(32b) |

where their derivatives with respect to time are computed as follows:

(33a) | ||||

(33b) | ||||

(33c) |

Thus, in order for the agents to achieve asymptotic tracking of the time-varying centroids of their Voronoi cells, a feedback controller that will ensure that will decrease along the agents’ trajectories must be designed. In particular we have,

(34) |

and by expanding the above expression and using (32a) and (32b) we obtain

(35) |

where 0 and when . We can compute the derivative of as

(36) |

Thus, to ensure that is negative semi-definite, we propose the following feedback control law:

(37) |

### Iii-C Analysis of the Low-Level Solution and Results

Following [5], it can be shown that when is bounded from above, the existing upper bound can be used for the design of a controller that will solve Problem 2.

###### Proposition 2

Let us assume that there exists such that

(38) |

Moreover, let us assume that the proportional gain of the controller given in (37) satisfies the following inequality

(39) |

Then, the controller given in (37) makes the time derivative of the locational cost which is defined in (9), negative-semi definite along the trajectories of the agents of the network, that is,

###### Proof:

###### Proposition 3

The system of mobile agents located at at time , for , driven by the feedback control law (37) will converge to the trajectories of the time-varying Voronoi centroids which evolve based on the reference time-varying density .

Next, we will prove that the mobile robots will track and eventually converge to the (moving) centroids of their Voronoi cells (the latter correspond to critical points of the locational cost for the given spatial domain of interest ). The proof of convergence we will provide is based on the following lemma.

###### Lemma 1 (Barbalat)

Consider a function which satisfies the following properties:

(1) V(p,t) is lower bounded,

(2) is negative semi-definite,

(3) is uniformly continuous in time

Then,

###### Proof:

To prove the statement, we will show that the following candidate Lyapunov function satisfies all the three properties of Lemma 1.

(1) Since the time-varying Gaussian mixture density is assumed to be strictly positive, then it is clear that the locational cost (9) is positive definite. Hence and thus is lower bounded.

(2) In light of (38) and (III-C), (III-C) would be negative semi-definite. ().

(3) To prove that is uniformly continuous in time, it suffices to prove that the second time derivative of the Lyapunov function is bounded.
We have,

(42) |

it can be observed that all the terms on the right hand side of (42) are finite and bounded, which suggests that is finite and bounded. Also in view of the fact that all the hypotheses of Barbalat’s Lemma are satisfied, it follows that , thus concluding that the agents asymptotically converge to their respective time-varying Voronoi centroids.

###### Proposition 4

###### Proof:

By the definition of (III-B1), each agent’s cell is convex. The centroid of any convex set lies inside the convex hull of the vertices of . Therefore, since the agents move to the centroids of their cells, they will only move within the collision free safe-area avoiding any collisions. In addition, because does not intersect with any obstacle or other agents, then all subsequent calculation of will not intersect .

## Iv Solution to the Combined Coverage Control Problem

This section presents the solution to Problem 3 by iteratively combining the solutions to Problems 1 and 2. The proposed approach is based on estimating the fine approximation of the team’s density at different instants of time which form a finite increasing sequence , where and is a positive integer; in particular, for , where is a given time-step. At each time instant , we compute the fine approximation which is determined by the mean and covariance of the locations of the team of agents at that time, which are computed by the GMMLO at . Therefore,

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