Three-dimensional distributed tracking control for multiple quadrotor helicopters

Abstract

This paper deals with the cooperative tracking control of a team of underactuated quadrotors in three-dimensional space when the reference signal is not available to all the vehicles. Using the Backstepping and filtering design technique and results from graph theory, a distributed cooperative controller consisting of two interconnected blocks is proposed. The first block, which is based on local distributed controllers, guarantees that the centroid of the three-dimensional formation asymptotically follows a desired path corresponding to the desired formation trajectory. The second block ensures that the attitude dynamics of the quadrotor will converge asymptotically to their actual values, which are constrained within a predefined set, such that singularities are avoided in the quadrotor׳s dynamics. Due to the coupling property between the translational and rotational dynamics, the equilibrium point of the complete closed-loop system is shown to be asymptotically stable. Numerical simulations are illustrated to show that the theoretical conclusions are effective.

Introduction

Recent years witnessed quick advances in the field of unmanned aerial vehicles (UAVs), with increasingly affordable devices endowing these types of vehicles with versatile capabilities to perform many types of tasks in several applications. Most of these applications involve search and rescue, security, civil engineering inspection and weather observation.

In this paper, we will be interested in controlling rotary-wings aircrafts. These types of aerial robotic vehicles are dynamic systems of 6-degree-of-freedom (DOF), underactuated, multi-input multi-output (MIMO), strong coupled and nonlinear. It is then highly desirable to design a reliable and robust controller that accommodates the nature of the mechanical system while achieving complex flight maneuvers. Various control algorithms for a stabilization and trajectory tracking control problem have been addressed in [4], [28], [5] and [44], [24], [9], [11], [12], [27], [39], [42], respectively.

Cooperative control of multiple quadrotors is more challenging than controlling a single one since it additionally requires that the quadrotors perform cooperative tasks to imply a faster and efficient process like load transportation [29], [25] or surveillance and area exploration [33], [38], [31]. These kinds of applications have also received extensive interest in recent years leading to significant theoretical developments. Relevant research works on flight formation tracking problems have been conducted in the framework of the leader–follower [10], [13], [17] and the virtual structure approach [36], [37]. These two approaches, although simple to implement, have trouble keeping the desired formation when the virtual or physical leader or even the followers are subject to disturbance. Another interesting approach to coordination problem is the graph theory. In [20], [19], the graph is utilized to model information exchange where every node in a graph is considered as a quadrotor which can share information with some neighboring connected quadrotors. The flight formation control problem is then addressed based on a forced consensus algorithm. This means that all quadrotors henceforth called agents must agree on and reach a certain variable of interest, so that the formation control problem is fulfilled. However in most of these works, it is assumed that a desired trajectory is available to each system. In practice, if the number of agents is large, it is more realistic to assume that a desired trajectory is available only to a portion of the system.

Although much progress has been made on consensus control problem with reference system, much work remains to be done to overcome nonlinearities on the multi-agent systems. As a matter of fact, research on distributed control of multiple nonlinear system has just recently spawned new consensus control algorithms. Interesting work on cooperative control of nonlinear systems has been proposed in [6] where a neural network-based adaptive leader–follower controller is proposed for multiple agents with uncertain dynamics. The author in [15] proposed an adaptive distributed controller for uncertain mechanical systems, where a desired trajectory is made available to only a portion of the group. Distributed neural network cooperative tracking control of multiple first-order and multiple second-order nonlinear systems has been established in [7], [8], while a general approach for distributed cooperative tracking control of higher-order integrator nonlinear systems has been addressed in [43]. The authors proposed a control design which falls in the leader–follower framework, where the leader is modeled as a higher-order dynamic that acts as a reference generator which gives command to a small portion of the networked group. All the aforementioned results were obtained considering the consensus tracking problem of homogeneous multi-agent systems, i.e., all the agents whether they are modeled as a nonlinear single or double integrators have the same dynamic behaviors. However, due to some mechanical restrictions, the dynamics of the agents are different from each other. Furthermore in these works, it is assumed that local linear velocity information of the agent is available for feedback. However, this assumption from a practical point of view cannot be realistic. For quadrotor aircraft applications, an estimate of the linear velocity information is generally obtained from approximate derivation of the successive measurements from a GPS, then fused with available measurements from accelerometers [2]. Nevertheless, the GPS signal might be attenuated in indoor and urban applications, which may deteriorate the positioning accuracy. Furthermore, implementing a redundant velocity-free controller in an aircraft equipped with high-precision GPS will improve the reliability of the system to sensor failure.

To overcome the limitations mentioned above, this paper investigates a three dimensional distributed formation control problem for a team of quadrotor UAVs using a robust consensus-based approach. In this case, the team of quadrotors come into desired formation provided that the desired trajectory is only available to a subset of the group of quadrotors. Compared with the existing coordination algorithms in many publications, the main contributions of our proposed coordinated controllers are as follows: (i) A transformation is given to convert the individual quadrotor׳s dynamic into a cascade of nonlinear subsystems between the translational dynamics and the rotational dynamics. (ii) The formation control objective is formulated for the translational subsystem dynamics into a state consensus problem. A constructive distributed control design that removes the requirement of the linear-velocity measurement in the position feedback for a group of n quadrotors under environmental disturbances is established based on graph theory and filter technique [40], [41]. This ensures that the centroid of the three dimensional desired formation shape asymptotically follows a desired path corresponding to the desired formation trajectory. In this paper, the trajectory is represented by the state of a virtual leader which generates only information on its position that is made available to a subset of a group of quadrotors. This idea basically differs from conventional result in [8], [1] which generally requires velocity measurement of the dynamical agent for feedback. (iii) The attitude control is designed with newly developed technique based on Barrier backstepping technique and continuous robust RISE feedback (Robust Integral of the Sign of the Error) [32]. It is proved that, with the proposed attitude controller, the singularities caused by the Euler angle representation of the quadrotor׳s attitude dynamics are avoided. (iv) The resulting closed-loop system basically consists of two interconnected subsystems: the coordinated translational subsystem and the closed-loop rotational subsystem. The interconnected structure motivates the use of the small-gain theorem, particularly the input-to-state-stability (ISS type) small-gain theorem [22] to conclude about the overall stability of the system.

The paper is organized as follows: in the following section, the problem formulation to be solved is stated. In Section 3, new results on distributed coordination control problem for a group of quadrotor UAVs are established based on the backstepping technique. Section 4 provides stability analysis results for the whole cooperative quadrotor systems. Simulation results illustrating the effectiveness of the proposed control design are given in Section 5.