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The paper presents a novel methodology for the control management of a swarm of autonomous vehicles. The vehicles, or agents, may have different skills, and be employed for different missions. The methodology is based on the definition of descriptor functions that model the capabilities of the single agent and each task or mission. The swarm motion is controlled by minimizing a suitable norm of the error between agents’ descriptor functions and other descriptor functions which models the entire mission. The validity of the proposed technique is tested via numerical simulation, using different task assignment scenarios.

Control of swarms of vehicles has received a lot of attention from the scientific community for the theoretical challenges and potential use. The concept of swarm has a number of advantages in many aerospace applications due to the decentralized nature and capability of performing missions not possible for large UAVs and single platforms. One of the most difficult problems in swarm motion is the design of a control management structure general enough to accommodate vehicles with different properties and with limited information exchange, which move in an adverse environment to achieve the same objective. The problem under consideration here consists of a number of agents, which may be heterogeneous in terms of size, autonomy of decision, payload capability, and task assignment. A high level controller should (loosely speaking) optimize the dynamics of the swarm (

There are several methods and strategies that can be used in order to address the above problem; one of the most promising is the set of knowledge tools derived from biology (especially bird flocking, fish schooling, insect foraging techniques, predator hunting, boids,

The problem of control management of a swarm of vehicles or a distributed allocation of resources over some environment has been addressed in the past using different tools, depending on the driving application and the mathematical background. The following paragraphs give a brief summary of the current state of the art.

The understanding and development of tools derived from biology (especially bird flocking, fish schooling, foraging techniques, predator hunting, boids,

The consensus protocol is a technique that formalizes what we call the “agreement” on a set of shared variables defining some properties common to the entire swarm. This asymptotic convergence is reached via local communications, which means that the algorithm is decentralized in nature. Consensus problems have a long history in computer science and statistical analysis forming the core of the so called distributed computing [

Coverage control deals, in a general sense, with the optimal distribution of a large amount of resources over some environment. These resources can be robots, sensor networks, and diversified assets, all characterized by a decentralized and distributed evolution. In coverage approaches, the focus is on the use of tools such as proximity graphs (Voronoi diagrams, Delaunay tessellations,

The use of mathematical and physical abstractions is an appealing concept in swarm control management. One useful application of mathematical abstractions is the identification of a limited number of variables as a way of reducing the swarm shape to a “small” manageable set from the computation standpoint, as well as from the point of maintaining full capability of management. The literature presents several approaches that use this idea. In a paper by Belta and Kumar [

The objective of this paper is to present a new technique for the high level control management of swarms of vehicles which is capable of modeling a large class of coordination problems. The environment and mission(s) under consideration are as general as possible, as well as the characteristics of the agents which may be all identical or heterogeneous. This condition is seldom found in the literature regarding high level control, and it comes into consideration usually only at the task assignment level. The number of agents is assumed to be “large”, and therefore the size of a single vehicle is considered negligible. Secondly, due to that, each agent may be limited in its capability as an individual, but could help optimize the overall behavior in terms of performance, robustness, probability of success. Thirdly, we will assume that the swarm may be subjected to losses (or planned expendabilities); therefore, the performances are better evaluated as a whole, rather than at the individual level. With the above assumptions, the main idea is to associate a function to each agent, and one or more functions to the mission (see [

The Descriptor Functions method was recently introduced in [

Consider a system composed by _{j}^{n}_{i}

In a general multi-tasking framework (although not considered in the paper), the task that agent _{i}^{k}(t)_{i}: k_{i}(t)

For each task

The Current Task Descriptor Function (Current TDF) is defined as the sum of the descriptor functions of the agents that are executing that task: ^{k}^{N}^{+},

Although the above framework is set for multitasking, the present paper limits itself to a single task scenario for clarity’s sake.

Example of Descriptor Function Framework: Gaussian (

The aim of the swarm is to reduce the difference between their combined DFs and the desired one. To this end, we introduce the Task Error Function (TEF), which is defined as the difference between the Desired TDF and the Current TDF:

For the current mission and for each point of the environment, the TEF determines if there is a lack of resources (^{k}^{k}

In this context, the objective is assumed to be completed if its TEF is equal or less than 0 at each point of the environment of interest for that task. The main advantage in using descriptor functions is the potential generality of the approach. Agent DFs could for instance describe a sensor capability, a sensor type, a rotary wing vehicle as opposed to a fixed wing one,

This section presents three control laws for the motion of the agents. The first control law was originally introduced in [

Since the aim of the agents that belong to the same team is to satisfy the need of resources, a control law was designed to minimize a global measure of the TEF, which can be represented by the cost function:
^{k}^{+} and

The control law described above requires the knowledge of TEF by each agent, at each point of the environment. This may be unrealistic from a practical as well as an analytical point of view, since it would require full swarm connectivity and unlimited distribution of the descriptor function of each agent (although the sigmoidal distribution makes agents far away almost irrelevant). There are several ways to decentralize the above control law. A formal approach would require a process of computing the current task descriptor function, from the estimation of the state of neighbor agents only. This could be achieved, for instance, by a consensus estimation algorithm with associated connectivity graph, or by discretization of the environment with a finite grid [^{k}^{k}

The control law in Equation (8) can also be written in terms of attractive forces generated in the environment by the Task Error Function. If the Agent DFs are symmetric, with a peak centered at the agent position _{i}_{i}_{i}

Note that if the agents’ DFs are limited in range, _{i}_{i}_{i}

In order to solve the convergence issues of the previous controller with DFs limited in range, a modified control law is proposed, which is based on the attractive-repulsive properties of field potentials, already known in the literature. The philosophy is similar to the framework of reactive robotics (known also as behavior-based robotics) introduced by Rodney Brooks [

The control problem can be formulated in terms of Potential Field, with the objective of moving the agents such that the Current TDF equals the Desired TDF. Recalling the definition of DF error in Equation (6), we can place an attractive potential field Y at each point of the environment, where there is a need for resources, that is at

This potential field (Error Potential Field, EPF) maps the TEF at point

The total velocity of the agent is then computed as the integral of the EPFs generated by all the points

The presence of an agent reduces the current TEF at the points where the agent DF is not 0. For this reason, the EPFs generated in the environment change as the agents move. The function g(·) in Equation (10) can be seen as a measure of the importance that an agent assigns to the error at each point of the environment, and can be chosen according to the derivative of the Agent DF. In particular, it can be selected to weight the error in the whole environment even if the Agent DFs are limited.

Since we associate an attractive potential to multiple points of the environment

The resulting control law is then given by:

The control law tends to drive the agents towards regions of the space where the “density of the error” is larger. The choice of the potential gain _{PF}_{D}

The two gains _{D}_{PF}

The main drawback of this control law is that it does not guarantee that the agents will eventually stop and does not minimize any measure of the TEF. On the other hand, this control law does not suffer from the problems described in

In the previous paragraphs, two controllers were described, which have advantages and disadvantages when applied singularly. In particular, a gradient based controller is not suited in cases where the agent DF is limited in space and does not intersect, neither in a small amount, the descriptor function which describes the desired value. The controller derived using potential fields, on the other hand, has no guarantee of minimizing the computed error function. It is possible, however, to take advantage of positive aspects of both approaches, and to derive a combined controller. The Potential Field control law should be employed by the agents that are stuck in configurations for which there is no contribution to the reduction of the TEF such as when they are far from the areas where TEF > 0. The Gradient based control law should be used instead by the agents to locally optimize the cost function of Equation (7) when they are sufficiently near the goal.

Let us assume that each vehicle is capable of measuring its contribution to the reduction of the TEF as:

The agent evaluates the amount of TEF that it is covering in a neighborhood of radius ^{2}, and σ was set equal to 1.

(

Cost function comparison.

The proposed concept of descriptor function was successfully applied to a coverage scenario using the controller in

Consider a mission where _{w}_{*}_{w}

To formalize the problem, a single task is considered and the corresponding superscript _{i}_{i}

The DF of the agents and targets are spatially limited, thus:

Furthermore, a target is said to be “observed” if the TEF in its neighborhood

The simulations presented in this section are performed using agents and targets modeled with descriptor functions with Gaussian shape:
_{a}_{a}

The first set of simulations uses the switching controller and a comparison with the Potential Field controller is shown in the second set of simulations. In both cases, the cost function is given by Equation (7) with: ^{2}.

The above selection is extensively motivated in [

In this first example, we consider five agents and a changing number of targets (5, 3, and 10, respectively). The agents have all the same capability (same DF) as shown by their equal footprints in the numerical simulations results below. The selected gains for the potential field based controller were selected to be:

The initial positions of the agents are:

(

The plateau in the cost function between 12 and 24 s is due to the use of the Potential Field control by one agent (the one in the bottom left corner) while the other four agents are already covering the corresponding targets. After 24 s, the DF of this agent intersects the DF of the target, and the cost function decreases again.

Similar results are obtained if the number of the targets is less than the number of agents as shown in _{w}

This example highlights the improvement made by the switching control law over the Potential Field based controller. The latter controller weights the error over the whole environment, and it may force the agents towards unexpected or undesirable equilibrium configurations, due to asymmetries in the descriptor functions of agents and targets.

Consider a scenario composed of five agents and seven targets. All the agents and six of the targets are modeled with the descriptor function of Equation (17). The seventh target (bottom left corner) has a larger DF, indicating possibly a higher priority or a request of more resources, and given by Equation (20), with _{i}^{T}

Full use of the potential field based controller produces the trajectories shown in

Potential field controller performance (Example 2).

The agents get close but not sufficiently (for the requirements in the example) to the targets. The final shape of the task error function indicates a large remaining error not compensated by the assignment, and shown by the five multicolored peaks. The larger error peaks belong obviously to the two targets not covered by the agents.

The use of switching controller improves the overall scenario performance as shown in

Switching controller performance (Example 2).

Cost function comparison (Example 2).

The real time implementation of the concepts presented in the paper, and described by the above examples, was partially performed with RC cars, and can be found in [

The paper presents the comparison of different controllers for the management of a swarm of vehicles using the descriptor function approach. The performance of the approach are evaluated in a simple task assignment scenario, and advantages and disadvantages of the proposed controllers evaluated. Future work will be directed towards formal decentralization and multi-tasking.

Mario Innocenti thanks the National Academies and the Air Force Research Laboratory, Munitions Directorate, Eglin AFB, Florida for their partial support.

The authors declare no conflict of interest.