Robust formation control for unmanned helicopters with collision avoidance

https://doi.org/10.1016/j.jfranklin.2020.01.011Get rights and content

Abstract

A systematic framework is proposed which addresses fully distributed formation control of under actuated unmanned helicopters (UHs) with resisting the external disturbances and model uncertainty capability. In this paper, a hierarchical control strategy for the whole closed-loop under-actuated UHs system is applied. The adaptive terminal sliding mode (TSM) control technique is employed to design an continuous fast robust controller in each loop and an integral filter is introduced between the position loop and the attitude loop to ensure that the attitude controller can track the desired attitude command. The stability of the whole closed-loop system is guaranteed through Lyapunov theory and input-to-state stability (ISS). Compared to the previous related works, our main contribution is that the proposed adaptive control strategy is fully distributed and scalable, which is independent any global information of the network graph and is independent of the UHs’s scale and we proved the stability of the closed-loop. Finally, the effectiveness and superiority of the designed strategy are demonstrated by numerical simulations.

Introduction

The UHs formation control problem has received a lot of interest due to its broad applications in cooperative surveillance, cooperative attack, cooperative location and observation and so on. Considering a large number of UHs, the distributed formation control approach is more suitable than centralized formation control. The objective of distributed cooperative control lies in guaranteeing a family of UH using local information from adjacent UHs to accomplish dangerous and challenging tasks which cannot be achieved by a single UH [1], [2], [3], [4].

Fully distributed formation control for UHs is that the controller gain is computed and implemented by each UH only using the relative states of agents in each other’s proximity graph. Therefore, the requirement of the smallest nonzero eigenvalue of the Laplacian matrix associated with the communication graph can be relaxed for multi-agent systems in [5], [6], [7], [8]. Recent results on fully distributed coordination mainly concerning linear multi-agent systems (MASs) [5], [6], [7] and networked Lagrange systems [8] focus on the consensus with asymptotic convergence rate. For the consensus control of linear system [5], the reason why the controller gain depends on the minimum non-zero eigenvalue of the communication topology is system state matrix A. While for distributed leader-follower flocking control with a moving leader for networked Lagrange systems [8], the stability condition relied on the controller gain lager than the smallest nonzero eigenvalue of the Laplacian matrix and the upper bound of the leader’s acceleration, that is α>σl/λmin[H(0)]. Therefore, the controller gain of leaderless MASs consensus control with state matrix or leader-follower MASs consensus tracking control dependent on the whole communication graph.

Many adaptive protocols were designed [5], [6], [7], [8], [9], [10], [11] to settle the aforementioned problem. In [5], based on relative states of agents, an adaptive law was proposed to adjust the coupling weights and to eliminate the global information caused by the control protocol for general linear and Lipschitz nonlinear systems. The fully distributed consensus problem of linear multi-agent systems was investigated for both the cases without and with a leader whose control input might be nonzero and time varying [12]. Regarding second-order nonlinear systems, Ghapani [8] studied the leader-follower flocking of a group of agents where the followers move cohesively with the moving leader. Moreover, the connectivity maintaining problem and collision avoidance problem were solved also by proposing a distributed discontinuous adaptive control algorithm and state estimator. However, most of the existing adaptive control laws [5], [8] will result in the non-stop increase of the gains in the presence of disturbances or measurement errors. Recently, several methods are designed to deal with this problem. In [8] and [13], the time-varying coupling gain was forced to stop increasing when the right hand sides of the relative states was within some given constant. By introducing a damping term [14], or applying σ-modification technique [15] to the adaptive law, the controller gain will not increase very fast at the initially moment and will eventually converge to a fixed constant. In [16], filtered coupling weights were designed to reduce the initial learn rate of the adaptive coupling gain, which can also be considered as a damping term. Among these approaches, adding a damping term to the controller may more suitable for practical application. However, it is well known that finite-time stability has better performance and robustness for disturbance rejection [17] than asymptotical stability. In the formation process of UHs, the collision avoidance and obstacle avoidance security problems must be considered. The potential energy function method is the most popular collision avoidance method [8], [19], [20]. In this paper, the potential energy function is applied to change the acceleration of UH, so as to avoid potential collision.

The finite-time fully distributed consensus [18] and formation control [19], [20] have been studied. In [18], a finite-time adaptive consensus protocol for MASs with nonlinear dynamics was proposed and the Lipschitz constants and the eigenvalues of Laplacian matrix were no need to know beforehand. In [19] and [20], disturbance observer-based fully distributed control strategies were developed based on TSM and adaptive backstepping methods. However, the existing algorithms in [19], [20] were too complex, and the algorithm in [18] can not be robust to disturbances with unknown bounds.

Motivated by the above analyses, in this paper, we aim to address a fully distributed formation control strategy for UH system with external disturbances and model uncertainty. The main contributions of this paper can be summarized as follows:

  • Compared with the asymptotic stability results in [19], [20], a new yet simple formation controller with collision avoidance is proposed to achieve robust and fully distributed formation using TSM and adaptive techniques. Specially, our fully distributed strategy can realize velocity consensus and desired formation simultaneously independent on any global informations compared to [21], [22].

  • The upper bounds of unknown disturbances, model uncertainties and the global informations are estimated and compensated to controller simultaneously, while in [23] and [24] these problems were not considered.

  • No longer rely on well-known multiple time principle [17], [19], [20], a rigorous proof of the ISS stability of the whole closed loop system is derived utilizing Lyapunov theory and ISS.

The rest of the paper is organized as follows. In Section 2, notations and graph theory, some useful lemmas and problem formulation are presented. In Section 3, a fully distributed, adaptive formation controller with collision avoidance is designed for outer-loop subsystem, and a continuous finite-time attitude tracking controller is proposed for inner-loop subsystem. Simulation results and Conclusions are given in Sections 4 and 5, respectively.

Section snippets

Notations and graph theory

Throughout this paper, the following notations will be used. λmin(L) and λmax(L) are, respectively, the minimum and maximum eigenvalues of the matrix L. For a vector x=[x1,...,xn]T, |x|=[|x1|,...,|xn|]T,x‖ is the Euclidean norm of x. diag(x) denotes the diagonal matrix of vector x. For γ ∈ R, define sigγ(x)=[sgn(x1)|x1|γ,...,sgn(xn)|xn|γ]T,where sgn(x) represents sign function. 1n=[1,...,1]TRn.

A directed graph G is pair (V,ε,A), where V=(υ1,υ2,...,υn) is the node set and εV×V is the edge

Main results

The architecture of the fully distributed control for multiple UHs is presented in Fig. 3. In the formation control of multiple UHs, a hierarchical control structure is adopted for each UH. In the outer loop, a fully distributed safety formation controller is designed to achieve formation. Then, through attitude resolution, we get the desired attitude angle command. In the inner loop, an adaptive attitude tracking controller is proposed to guarantee that the attitude can track the desired

Simulation

In the simulation, we consider a multiple UHs formation consisting of a leader UH and four follower UHs, assuming that the two UHs can communicate within the communication range. The communication graph is presented in Fig. 4. UHs formation takes off from the ground to the air to form and maintain formation. There are two obstacles in the flight process, one of which is stationary and the other is mobile. The simulation will verify that the algorithm can keep the formation flying in the case of

Conclusions

In this paper, the UHs’s fully distributed formation control with obstacle/collision avoidance was designed in a robust and safety control framework. The TSM and adaptive techniques were applied to design a novel fully distributed formation control controller and a continuous attitude tracking controller. The ISS of the whole closed loop system have been demonstrated for UHs system. The simulation results show that the proposed algorithm is effective in controlling the formation flying in

Declaration of Competing Interest

All authors declare that there is no conflict of interest.

Acknowledgements

This work is supported by Key Program of National Natural Science Foundation of China (No. 61933002), China Postdoctoral Science Foundation 2019TQ0033, National Natural Science Foundation of China (Nos. 61873031, 61803032).

References (35)

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