# Cooperative Control for Multiple Autonomous Vehicles Using Descriptor Functions

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

#### 2.1. Behavioral Approaches

#### 2.2. Consensus Protocols

#### 2.3. Coverage and Connectivity

#### 2.4. Abstractions and Models

## 3. The Descriptor Function Framework

_{j}, i = 1, … N and some mission composed of M tasks. At each time instant, the set of agents executing the same task forms a team. The number of teams may be up to M, and each agent may be part of one or more teams simultaneously. The domain where agents operate is Q ⊂ R

^{n}, defining a spatial environment, as well as non-spatial related variables, and the position of agent i at time t is defined as p

_{i}(t). Without loss of generality, we make the following assumptions: Q is closed and bounded; each vehicle is capable of multitasking, but its behavior is optimized for only one task; the agents’ motions are limited to a single integrator kinematics with unity gain (no unicycle-type motion or dynamics are considered here):

_{i}(t) and the team performing the same task is defined as:

^{k}(t) = {V

_{i}: k ∈ T

_{i}(t)}, k = 1, …, M

^{k}(p,q): P

^{N}×Q → R

^{+}, i.e.,

^{k}(p,q,t) >0) or an excess of resources (E

^{k}(p,q,t) <0), and therefore if there is a need for action by the swarm.

## 4. The Swarm Control Law

#### 4.1. Gradient-Based Control

^{k}for which f( ) ≥ 0, ∀ ∈ R

^{+}and f(t) = 0, “t < 0”; σ(q) ≥ 0, ∀q ∈ Q, is an appropriate weighting function. The assumptions on f(.) are necessary in order to penalize only the lack of resources. In fact, the excess of resources does not foreclose task completion. The control problem is then formulated as an optimization and the resulting control law for each agent i derives directly from the gradient of the cost function, via steepest descent:

^{k}(p,q)) precludes an analytical solution of the global minimum for J

^{k}. In fact, if an agent has a symmetric and limited DF, and if the TEF in its neighborhood is constant, the resulting control would be 0 yielding a local minimum. The control law in Equation (8) works well only locally and it is especially suited for high density environments, when the agents are near to each other, and their relative motion keeps the TEF non-stationary.

_{i}and decreasing as the distance d from its position increases, the integrand of Equation (8) is a vector pointing from p

_{i}towards the point in space q, whose modulus depends on the TEF at that point weighted by the derivative of the Agent DF with respect to distance. This yields:

_{i},q) is the distance between agent i and the point q, and

_{i},q) from the agent position p

_{i}, the integrand in Equation (9) becomes zero, and the total control signal u

_{i}may become zero as well when the i-th agent is far enough from all tasks.

#### 4.2. Potential Field-Based Control

_{PF}determines the strength of the force induced by the points of the environment. If the gain is large, areas of large TEF, which are far away from the agent may induce small velocities and this characterizes the tendency of the agents to prefer areas of positive TEF nearer to them. Moreover, this may cause slower convergence rates, i.e., smaller velocity commands for the agents, thus longer mission time. In order to have control both on the convergence rate and of the tendency to prefer areas of TEF > 0 in the agent’s neighborhood, an additional degree of freedom g

_{D}> 0 is introduced yielding:

_{D}and g

_{PF}constitute the tradeoff between the two objectives.

#### 4.3. Combined (Switching) Control Law

^{2}, and σ was set equal to 1.

**Figure 2.**(

**a**) Gradient controller performance. (

**b**) Gradient controller with a local minimum. (

**c**) Potential field controller performance. (

**d**) Complete controller performance.

## 5. Case Study: Target Assignment

_{w}targets. The desired D

_{*}is selected as given by N

_{w}“small” disjoint areas, possibly far from each other and from the initial positions of the agents.

_{i},q). Given the position of the targets w

_{i}∈ Q, the desired TDF is constructed as:

_{a}and the dispersion s

_{a}were selected as 1. The shape of the DF is very similar to the model of sensors that have a peak at the agent position and a decreasing performance with the distance.

^{2}.

#### 5.1. Example 1

**Figure 4.**(

**a**) Five agents and five target assignments. (

**b**) Five agents and three target assignments. (

**c**) Five agents and 10 target assignments.

_{w}agents in excess may also stop without covering any target.

#### 5.2. Example 2

_{i}= [-5 -6]

^{T}

## 6. Conclusions

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Niccolini, M.; Pollini, L.; Innocenti, M.
Cooperative Control for Multiple Autonomous Vehicles Using Descriptor Functions. *J. Sens. Actuator Netw.* **2014**, *3*, 26-43.
https://doi.org/10.3390/jsan3010026

**AMA Style**

Niccolini M, Pollini L, Innocenti M.
Cooperative Control for Multiple Autonomous Vehicles Using Descriptor Functions. *Journal of Sensor and Actuator Networks*. 2014; 3(1):26-43.
https://doi.org/10.3390/jsan3010026

**Chicago/Turabian Style**

Niccolini, Marta, Lorenzo Pollini, and Mario Innocenti.
2014. "Cooperative Control for Multiple Autonomous Vehicles Using Descriptor Functions" *Journal of Sensor and Actuator Networks* 3, no. 1: 26-43.
https://doi.org/10.3390/jsan3010026