BELKACEM Kada，MUNAWAR Khalid，and MUHAMMAD Shafique Shaikh

1. Department of Aerospace Engineering，King Abdulaziz University，Jeddah 21589，Saudi Arabia;2. Department of Electrical and Computer Engineering，King Abdulaziz University，Jeddah 21589，Saudi Arabia

Abstract: This paper addresses the cooperative control problem of multiple unmanned aerial vehicles (multi-UAV) systems.First，a new distributed consensus algorithm for second-order nonlinear multi-agent systems (MAS) is formulated under the leader-following approach. The algorithm provides smooth input signals to the agents’ control channels，which avoids the chattering effect generated by the conventional sliding mode-based control protocols. Second，a new formation control scheme is developed by integrating smooth distributed consensus control protocols into the geometric pattern model to achieve three-dimensional formation tracking. The Lyapunov theory is used to prove the stability and convergence of both distributed consensus and formation controllers. The effectiveness of the proposed algorithms is demonstrated through simulation results.

Keywords: cooperative control，distributed consensus，three-dimensional formation control，multiple-UAV system.

1. Introduction

The concept of cooperative control for multiple aerial vehicles has been developed in response to the increasing need in performing complex missions. Aerial multi-agent systems (MAS)，as well as many other MAS，can perform cooperative tasks with numerous advantages such as efficiency，accuracy，flexibility，robustness，and cost-effectiveness. Multiple aircraft [1-3]，helicopters [4]，spacecraft [5，6]，satellites [7，8]，missiles [9，10]，and unmanned aerial vehicles (UAV) [11-19]are among aerospace MAS.

Multi-UAV cooperative control is one of the very interesting and challenging areal MAS applications since it allows the achievement of complex missions with more flexibility，reconfiguration，adaptation to dynamic environments，and distribution. In recent years，numerous studies have tackled the problem of multi-UAV cooperative control by using different approaches and theories.

Using the contract net protocol，a task assignment model of manned/unmanned aerial vehicle formation was built in [11]to assign tasks in dynamic environments. In[12]，authors formulated the formation control problem of multi-UAV systems as a differential game problem where an open-loop Nash strategy was for each agent to build a fully distributed formation control. The virtual structure approach was used in [13]to build a nonsmooth distributed cohesive motion control for the formation of autonomous quadrotor aircraft. In [14]，a distributed formation flying control algorithm was developed by using a nonsmooth backstepping design approach. Adaptive formation control was developed in [15]by using a differential evolution algorithm to design the optimized formation control among a group of UAVs. The problem of formation-containment control for multiple multirotor UAV was solved in [16]by using a Riccati equationbased algorithm. In [17]，a distributed self-organized mission planning algorithm was proposed for the formation control of multi-UAV systems. Voronoi diagram or partition was used in [18]to design a distributed formation control without collision. The work in [19]studied the problem of path planning within the formation control strategy using the fast particle swarm optimization where chaos-based initialization，parameter optimization，and topology update were considered to reach formation with constraints and without collisions. In [20]，sliding-PID control was proposed for linear multi-agent that can be adapted for multi-UAV systems.

Although the aforementioned protocols and techniques have been shown to be effective，important issues remain to be addressed with regard to the cooperative control of MAS in general and multi-UAV systems in particular.Among these issues，one can site，control input smooth-ness，switching communication topology，partial loss of communication within MAS，and formation keeping. In this paper，we address the aforementioned issues and propose solutions within a smooth distributed consensus and formation control framework. To this end，a smooth distributed consensus control is developed for multi-agent with inherent nonlinear dynamics，which is our first contribution. The control inputs to the agent closed-loop dynamics are designed by using continuous PI-like (proportional-integral) control instead of signum-based control used in the conventional discontinuous control，which results in the chattering of controllers. The second contribution of this paper is to build an aerospace formation control model. The formation model is designed by combining translational and rotational distributed protocols with a six-degree-of-freedom dynamic model in one framework.

The rest of the paper is organized as follows. The problem of distributed consensus for the second-order MAS is formulated in Section 2. In Section 3，a new distributed consensus control algorithm is developed and its stability is analyzed by using the Lyapunov-like function. In Section 4，a three-dimension formation control of autonomous aerial flying vehicles is developed via expansion of planar formation presented in [21]. Section 5 presents simulation scenarios including aerobatic consensus following and three-dimensional helicoidal formation keeping. Concluding remarks are presented in Section 6.

2. Preliminaries and problem formulation

2.1 Graph theory

2.2 Distributed consensus problem for nonlinear second-order MAS

3. Distributed consensus control algorithm

In this section，we study the problem of smooth distributed consensus control for the second-order nonlinear MAS where we consider the case of time-varying velocities. The control objective is to design distributed individual protocolsuito achieve the following consensus agreement:

To solve this consensus problem，we propose the fol-lowing smooth distributed control protocols:

Fig. 2 Fixed-time switching topology connected interaction graph

Fig. 3 shows the three-dimensional consensus of the group of four agents from different angles of view，and Fig. 4 depicts the time history of the convergence of agents’ orientations to those corresponding to the virtual leader.

Fig. 3 Cyclic path following from different angles of view

Fig. 4 Time history of agents’ orientations

5.2 Tracking of helicoidal formation pattern

Fig. 5 Desired flying formation

Fig. 6 Fixed-time switching topology interaction graph

The effectiveness of the proposed control law is validated and compared to the protocol integral sliding mode(ISM) obtained by [24]. In this scenario，the control law(42) is run with γ =0.75，α1=5，β1=1.5，α2=β2=0，and the ISM protocol in [24]is run with α1=0.5，α2=0.67，andk1=k2=0.25. Fig. 7 shows the formation tracking using the protocol (42)，while Fig. 8 depicts the centroid formation trajectory along the motion axes for both controllers (42) and ISM. The corresponding control inputs are illustrated in Fig. 9.

Fig. 7 Formation tracking with the control law (42)

Fig. 8 Geometric pattern centroid tracking

Fig. 9 Control inputs (protocols)

5.3 Formation control under undirected topology graph

Consider the case of formation flying of three agents where the communication topology among these agents is described by the undirected graph shown in Fig. 10 [25].

Fig. 10 Triangular formation under undirected communication topology

Fig. 11 Formation tracking of triangular pattern

Fig. 12 Control effort

6. Conclusions

In this paper，a smooth distributed cooperative control for multiple flying vehicles such as UAVs is developed based on the concept of the leader-following consensus of MAS. First，in contrast to the conventional sliding-mode based consensus，a smooth distributed consensus algorithm is developed by using proportional and integral continuous functions instead of the discontinuous signum function. Second，a formation control model for three-dimensional geometric pattern tracking is designed. The flying formation scheme employs transitional and rotational smooth control inputs into a three-dimensional motion formation dynamic model. Using the Lyapunov function approach，sufficient conditions are established to ensure the convergence of the consensus and formation models. The effectiveness of the proposed models is demonstrated through complex simulation scenarios such as aerobatic maneuvering，switching communication topology，loss of communication with the leader，and formation control through the helicoidal path.

In the future，the focus will be on multi-UAV formation control with hardware-in-the-loop，formation tracking in presence of external disturbance，and obstacle avoidance among flying agents.