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Motivated by the idea of multiplexed model predictive control (MMPC), this paper introduces a new framework for unmanned aerial vehicles (UAVs) formation flight and coordination. Formulated using MMPC approach, the whole centralized formation flight system is considered as a linear periodic system with control inputs of each UAV subsystem as its periodic inputs. Divided into decentralized subsystems, the whole formation flight system is guaranteed stable if proper terminal cost and terminal constraints are added to each decentralized MPC formulation of the UAV subsystem. The decentralized robust MPC formulation for each UAV subsystem with bounded input disturbances and model uncertainties is also presented. Furthermore, an obstacle avoidance control scheme for any shape and size of obstacles, including the nonapriorily known ones, is integrated under the unified MPC framework. The results from simulations demonstrate that the proposed framework can successfully achieve robust collision-free formation flights.

Formation flight control of multiple Unmanned Aerial Vehicles (UAVs) has been a hot research topic for the last few years. Formation flight of UAVs can potentially be useful for increased surveillance coverage, better target acquisition, and increased security measures [

Different approaches have been developed for formation flight control. Linear formation flight controller has been discussed [

Model Predictive Control (MPC) is one of the frequently applied advanced optimal control methods in industry. Recently, with the advent of powerful microcontroller and small-size sensors, it has become possible to extend the MPC method to UAV systems which exhibit relatively fast dynamics. Many works have addressed the “decentralized MPC” in UAV cooperative control [

Updates all the UAVs’ control inputs simultaneously. For example, in the schemes proposed in [

Similar to Scheme

In this paper, motivated by the MMPC proposed in [

Another contribution of this paper is the extension of the obstacle avoidance algorithm in the formation flight MPC framework. Most of the existing collision avoidance scheme under the MPC formation flight treats the shape of the obstacles to be rectangle and does not take into account the small or nonapriorily known obstacles. The framework proposed will try to eliminate these limitations.

In summary, the contributions of this paper are in the development of a robust collision-free formation flight control system using the MMPC idea. Specifically, the paper addresses the following.

In the overall formation flight control scheme, extend the MMPC algorithm for formation flight application and provide the conditions (terminal cost and stabilizing feedback gain) for the whole formation group to be stable by applying the periodic linear system theory.

In the obstacle avoidance scheme, extend the current collision avoidance algorithm to include various shapes of obstacles and prevent the collision with small or nonapriorily known obstacles by using the spatial detection horizon idea, which is then translated into an additional convex position constraint.

The paper is organized as follows. Section

Each UAV is equipped with autopilot system capable of waypoint tracking.

With this assumption, the dynamics of each UAV subsystem in the formation flight group can be represented by their controlled dynamics (with autopilot), which can be modeled as linear system or piecewise linear system.

The couplings between the UAVs, if any, come only from the cost function and constraint formulations in the MPC optimization problem. The dynamical couplings, like induced vortices, are neglected.

With this assumption, the whole formation flight system can be formulated as a linear periodic system under the MMPC as detailed in Section

The overall formation flight control scheme is represented in Figure

The overall formation flight control system under MMPC.

The key idea of the proposed framework is to break the big centralized MPC formation flight problem into decentralized problems of smaller size while the stability of the whole formation flight system is guaranteed. In order to achieve this, the framework utilizes two steps: first, to apply the MMPC update scheme by formulating the whole formation flight as a linear periodic system; and second, to break the centralized periodic system into decentralized systems consisting of individual UAV, which is possible under Assumption

The use of robust decentralized model predictive control (RDMPC) scheme to update only one UAV subsystem at a time is proposed. Assume that there are

Each RDMPC controller is associated with a different vehicle and computes the local control inputs based only on its states and that of it neighbors (shown in Figure

Decentralized UAV formation flight control.

In this paper, the inputs, outputs, and states constraints are all regarded as polytopes. Some pertinent definitions are given in the Appendix

Under Assumption

In general, (

Additionally, as indicated in (

Let

Under Assumption

As we will be referring to the expression

According to our UAV input update scheme, at each fixed time step only one of UAV’s inputs,

According to [

Terminal cost term:

Terminal states constraints:

The terminal cost and the linear terminal stabilization controller can be derived from the unconstrained infinite-time linear quadratic (LQ) control theory of periodic system (See Appendix

The stability requirements for the MMPC formation flight framework from (

A graph topology [

In this research, a graph topology is used to represent the couplings and we will assume that for every wingman UAV, there is at least one path connecting it with the leader UAV, this assumption guarantees the controllability of the formation flight group [

Using the graph structure defined previously, the optimization problem can be formulated by letting

Depending on the group tasks, the interconnection function

For interagent collision avoidance, this constraints define nonconvex requirements in the following way:

Since the group of vehicles is equipped with a communication network, and that the connectivity of the network depends upon the relative distance between neighboring vehicles, maintaining network connectivity constrains the maximum allowable distance between vehicles. These constraints define convex requirements in the following way:

In general, guaranteed collision avoidance and network connectivity between any vehicle pair in a formation flight problem would necessitate the use of a complete graph for describing inter-vehicle constraints. However, for practical decentralization purposes, it is usually sufficient for each vehicle to consider only a neighboring subset of all vehicles to accomplish the formation flight. Furthermore, the allowed number of vehicles in these neighboring subsets might be limited. For example, the maximum number of similar agents whose detection cylinder is in contact with the detection cylinder of the computing agent is 6 (see Figure

Agent in the center of the cluster exchange information with most 6 neighbors.

Consider the following cost function for the

The leader in the formation fight group is denoted as the first UAV, that is,

This cost term can also be used in formation reconfiguration purposes.

Given a certain graph interconnection structure

system dynamics (without disturbance and model uncertainty).

State and control constraints

Stabilizing controller after the prediction horizon

Interconnection constraints for

(i) For the stabilizing controller around the equilibrium point, the close-loop system equation can be expressed as

The set of periodic stabilizing control gains (

(ii) The constraint 2 comes from the constraint tightening technique [

(iii) It can be shown [

Combining this scheme with the stability results in Section

The multivehicle system with the Multiplexed Robust Decentralized MPC scheme is closed-loop stable.

According to [

state constraints satisfy terminal constraint set;

control constraints satisfy terminal constraint set;

the terminal constraint set is positively invariant under a local controller;

the terminal cost is a local Lyapunov function.

Then the closed-loop stability is obtained.

In our setup, conditions (

According to [

Hence

The proof above depends on the feasibility of the constrained optimization at each step, and the feasibility is guaranteed by the constraints tightening (

Mixed Integer Linear Programming (MILP) combined with MPC [

UAVs penetrating the small size obstacle.

Receding horizons can be divided into two families, that is, temporal and spatial ones. The temporal horizons manage the time frames, while the spatial horizons manage the physical areas. A temporal horizon, as called in most of the MPC literatures [

A number of spatial horizons can also be defined. One example can be the sensor detection horizon [

In this work, a polytope is used to present various shapes of the obstacles. In order to ensure that every trajectory point of a UAV does not lie inside the boundaries of an obstacle, binary variables are introduced in the constraint formulation. For the hexagon shape obstacle for example, each edge can be regarded as a hyperplane which divides the fly zone into two half-spaces, one is safe while the other is not. In the collision avoidance constraint formulation, the predicted future point on the trajectory of each UAV is attached with an array of binary variables. In the case of a hexagon-shaped obstacle, an array of 6 binary variables is attached to each predicted point. The mathematic formulation is

The collision avoidance formulation here is an extension of that in [

The whole formation flight control optimization problem becomes nonconvex after adding the collision avoidance constraints. In order to solve the nonconvex optimization problem more efficiently, bounds on the variables are added explicitly

In this part, the small obstacle’s shape is specified as a circle or cylinder (note that if the shape of the obstacle is not a circle or cylinder, the collision avoidance algorithm can still be applied after some modifications, which will not be detailed in this paper).

In the proposed approach, position constraints are computed using the intersection points between the vehicle spatial horizon

Collision avoidance with a small obstacle.

Note that this position constraint actually prevents the next vehicle predictive position to be located outside of

Collision avoidance flow chart.

This section presents some simulation results of the multiplexed collision-free decentralized control scheme. The vehicle model used in the simulations is a small-size quadcopter with autopilot installed. Most of the simulations are done in a 2D space; however, it can be extended to a 3D space in a straightforward manner. The 2D representation is selected since it facilitates the visualization of the vehicle trajectories in a clearer manner. Throughout this section, the interconnection is described using a directed graph, which is determined by the two closest visible neighbors to each vehicle.

It is important to note that the method proposed in this paper can easily accommodate any other particular UAV dynamics described by higher fidelity, heterogeneous, higher order, more complex linear, or piecewise-linear models. Within the multiplexed robust decentralized MPC framework, each vehicle is modeled as a low-order linear system that represents the controlled dynamics of the vehicle. The details of the simulation setup are elaborated below.

The predictive horizon

The following simulations are performed using multiplexed robust decentralized scheme, and the update cycle is sequential, that is, (1, 2, 3, 1, 2, 3,

Terminal cost for (

Given the system dynamics and a “two-closest-neighbors” interconnection policy, each vehicle solves the decentralized optimization problem with the cost function defined in (

The vehicles in the formation have identical dynamics.

Linear constraints on single vehicle’ velocity is

The nonconvex interconnection constraints (collision avoidance) are represented by

Network connectivity constraints:

The weights in the cost function are chosen to be

In Figure

Triangle obstacle collision avoidance.

The scenario in Figure

Formation split and obstacle avoidance.

In this simulation, the leader (blue) is commanded to visit two target points (green circles), and the wingman is trying to maintain the formation (3 m from the leader in

Decentralized formation flight with robustness and without robustness. (a) Formation flight with robustness. (b) Formation flight without robustness.

In this simulation, we show the benefit of the small-obstacle avoidance algorithm. In Figure

Small-obstacle-collision free formation flight. (a) Formation without position constraint. (b) Formation with position constraint.

In this simulation, the performances of the multiplexed decentralized formation flight control and the nominal decentralized formation flight control are compared and presented in Figure

Multiplexed decentralized versus nominal decentralized scheme. (a) Nominal decentralized formation. (b) Multiplexed decentralized formation.

This simulation is used to demonstrate that the scheme can be easily extended to 3D case as can be seen from Figure

3D Formation Flight.

In this paper, the MMPC approach is employed to systematically handle the multiple UAV formation flight control. In its implementation, careful consideration should be given to the following.

If the number of UAV in the formation is large, updating one UAV’s inputs at each time step will be not acceptable since the update interval will be too long. In this case, update can be applied to a small group of UAVs at each time step to reduce the total update interval.

Since in the MMPC scheme, updating time of each UAV subsystem is fixed, an additional synchronization clock is required in the formation system, which may add complexities to the formation flight system.

The 3D multiplexed decentralized vehicle formation flight control problem has been formulated using decomposition in a hierarchical fashion. Under the multiplexed decentralized scheme, the leader UAV can response faster to the GCS compared to the nominal decentralized scheme, that is, when the whole group update their inputs simultaneously. Moreover, the closed-loop stability of the whole formation flight system is always guaranteed even if a different updating sequence is used which makes the scheme flexible and the capability of each UAV can be fully exploited. The stable collision-free control and the robustness have been proposed under the unified MPC framework. By integrating the spacial horizon, the collision avoidance of the small pop-up obstacles is transformed into a convex position constraint in the MPC problem formulation. Simulation results demonstrate that the proposed framework is able to maintain the desired formation geometry within the constraints posted and guarantees collision-free flight.

A convex set

A bounded polyhedron

One of the fundamental properties of polytope is that it can also be described by its vertices

Consider the system (

Using dynamic programming technique [

The set of periodic control gains

It is a standard fact that this is a necessary and sufficient condition for closed-loop stability of a linear periodic system, and that the eigenvalues of the monodromy matrix are invariant under cyclic permutations of the