Decentralized MPC for UAVs Formation Deployment and Reconfiguration with Multiple Outgoing Agents

Abstract

This paper presents a new decentralized algorithm for the deployment and reconfiguration of a multi-agent formation in a convex bounded polygonal area when considering several outgoing agents. The system is deployed over a two-dimensional convex bounded area, each agent being driven by its own linear model predictive controller. At each time instant, the area is partitioned into Voronoi cells associated with each agent. Due to the movement of the agents, this partition is time-varying. The objective of the proposed algorithm is to drive the agents into a static configuration based on the Chebyshev center of each Voronoi cell. When some agents present a non-cooperating behavior (e.g. agents required for a different mission, faulty agents, etc.), they have to leave the formation by tracking a reference outside the system’s workspace. The outgoing agents and their objective positions partition the convex bounded polygonal area into working regions. Each remaining agent will track a new objective point allowing it to avoid the trajectory of the outgoing agents. The computation of this objective position is based on the agent’s safety region (i.e. the intersection of the contracted Voronoi cell and the contracted working region). When the outgoing agents have left the workspace, the remaining agents resume their deployment objective. Simulation results on a formation of a team of unmanned aerial vehicles are finally presented to validate the algorithm proposed in this paper when several agents leave the formation.

Appendix: UAV Model

The state-space model of the UAV is written by considering the notations from Section 2.2. The state-space model itself is derived from Lagrangian mechanics [5]:

$$ \begin{array}{@{}rcl@{}} \dot{x} & =& v_{x} \end{array} $$

(16)

$$ \begin{array}{@{}rcl@{}} \dot{y} & =& v_{y} \end{array} $$

(17)

$$ \begin{array}{@{}rcl@{}} \dot{z} & =& v_{z} \end{array} $$

(18)

$$ \begin{array}{@{}rcl@{}} \dot{\phi} & =& \omega_{x} + \left( \omega_{y}\sin\phi + \omega_{z}\cos\phi\right)\tan\theta \end{array} $$

(19)

$$ \begin{array}{@{}rcl@{}} \dot{\theta} & =& \omega_{y}\cos\phi - \omega_{z}\sin\phi \end{array} $$

(20)

$$ \begin{array}{@{}rcl@{}} \dot{\psi} & =& \omega_{y}\frac{\sin\phi}{\cos\theta} + \omega_{z}\frac{\cos\phi}{\cos\theta} \end{array} $$

(21)

$$ \begin{array}{@{}rcl@{}} \dot{v}_{x} & =& \frac{f_{t}}{m}\left( \cos\phi\sin\theta\cos\psi + \sin\phi\sin\psi\right) \end{array} $$

(22)

$$ \begin{array}{@{}rcl@{}} \dot{v}_{y} & =& \frac{f_{t}}{m}\left( \cos\phi\sin\theta\sin\psi - \sin\phi\cos\psi\right) \end{array} $$

(23)

$$ \begin{array}{@{}rcl@{}} \dot{v}_{z} & =& \frac{f_{t}}{m}\cos\phi\cos\theta - g \end{array} $$

(24)

$$ \begin{array}{@{}rcl@{}} \dot{\omega}_{x} & =& \frac{I_{y} - I_{z}}{I_{x}}\omega_{y}\omega_{z} + \frac{\tau_{x}}{I_{x}} \end{array} $$

(25)

$$ \begin{array}{@{}rcl@{}} \dot{\omega}_{y} & =& \frac{I_{z} - I_{x}}{I_{y}}\omega_{x}\omega_{z} + \frac{\tau_{y}}{I_{y}} \end{array} $$

(26)

$$ \begin{array}{@{}rcl@{}} \dot{\omega}_{z} & =& \frac{I_{x} - I_{y}}{I_{z}}\omega_{x}\omega_{y} + \frac{\tau_{z}}{I_{z}} \end{array} $$

(27)

where m is the UAV’s mass, I_x, I_y and I_z are the moments of inertia with respect to the axes x_UAV, y_UAV and z_UAV illustrated in Fig. 1 and g is the gravitational acceleration. The numerical values of all the model parameters are presented in Table 1.

Table 1 Values of UAV model’s parameters