† Corresponding author. E-mail:
Project supported by the National Natural Science Foundation of China (Grant Nos. 61673217 and 61673214), the National Defense Basic Scientific Research Program of China (Grant No. JCKY2019606D001), and the China Scholarship Council.
Bearing-based hunting protocols commonly adopt a leaderless consensus method, which requests an entire state of the target for each agent and ignores the necessity of collision avoidance. We investigate a hunting problem of multi-quadrotor systems with hybrid bearing protocols, where the quadrotor systems are divided into master and slave groups for reducing the onboard loads and collision avoidance. The masters obtain the entire state of the target, whose hybrid protocols are based on the displacement and bearing constraints to maintain formation and to avoid the collision in the hunting process. However, the slaves’ protocols merely depend on the part state of the masters to reduce loads of data transmission. We also investigate the feasibility of receiving the bearing state from machine vision. The simulation results are given to illustrate the effectiveness of the proposed hybrid bearing protocols.
Remote-controlled rotorcrafts and their formation tactics have received much attention in recent years due to their low cost, convenient operation and excellent maneuverability. The multi quadrotor systems play a preferred role in material transportation, forest fire fighting,[1] target tracking,[2] battlefield reconnaissance and strike,[3] mapping[4] and so forth. This paper focuses on performing the hunting task efficiently and reliably.
The hunting problem is a particular case of formation control containing a definite moving target. At present, the standard formation methods mainly classify as position-based control, displacement-based control and distance-based control, as shown in Figs.
![]() | Fig. 1. Four kinds of the formation methods. (a) The position-based formation, (b) the displacement-based formation, (c) the distance-based formation, (d) the bearing-based formation. |
The position-based formation senses their positions in the global coordinate system without error feedback. It has derived a variety of straightforward methods in applications such as leader-follower[5] and virtual structure methods.[6] In the displacement-based protocol, each agent can sense the relative positions of its neighboring agents in the global coordinate system. Many displacement-based consensus protocols have been proposed to achieve consensus of a multi-agent system. In Ref. [7] a novel hybrid consensus protocol with dynamically changing interaction topologies was designed to take the time-delay into account. In Ref. [8] a new distributed protocol was presented to solve the mean square leader-following consensus problem for the nonlinear multi-agent systems, which contains a designed signal to dominate the effects of unmodeled dynamics. An adaptive finite-time control for the stochastic nonlinear systems driven by the noise of covariance was proposed in Ref. [9]. The distance-based control strategies requires less information than the others. The orientations of local coordinate systems in distance-based formation are not necessarily aligned with each other. Thus the trajectories may be different even if the initial values are equal (see e.g., Refs. [10,11]).
These three methods apply to most consensus application scenarios associated with graph theory. However, they have difficulties to adjust the formation in the hunting process. The agents persist in reaching the desired position far fast, and the formation is chaotic and unorganized before convergence. If the target rush at the gap of the formation suddenly, the quadrotor systems would generate a high overshoot, which decreases the success rate of hunting.[12] To resolve the conflict between dynamic formation and the reliability of hunting, researchers presented a new bearing-based formation method for the first time in 2011.[13] This method drives the agents to circle the target with the same angular speed, which can prevent the target from escaping from the encirclement before convergence.
In Ref. [14] the bearing-based maneuver control of multi-agent formation to arbitrary dimensions was presented. In Ref. [15] two types of position estimation laws were combined with bearing-based protocols for improving the position accuracy. The modeling and controller design methods of first-order multi-agent systems are discussed systematically in Ref. [16]. In Ref. [17] an adaptive control law was designed based on the reference acceleration for networked thrust-propelled vehicles with parametric uncertainties. Even though the bearing-based formation method has made significant progress recently, dealing with the hunting scenarios based on the bearing state is still an open problem. The current achievements mainly focus on the single-integrator kinematic models, and the acquisition of bearing information is not mentioned. Meanwhile, the common bearing-based hunting protocols ignore the necessity of collision avoidance, and the formation converges to a point. Much more importantly, the hunting protocols request the entire state of the target for each agent, which aggravates loads of data transmission.
In this paper, we study the model of multi-quadrotor systems and propose a practical controller input of each quadrotor. The simulation approximates to the real situation, which improves the feasibility of engineering applications. Meanwhile, according to the research achievements in Ref. [18] we investigate the feasibility of receiving the bearing state by machine vision. In order to reduce the onboard loads and to ensure the safe distance of the quadrotor systems, we introduce a hierarchical network approach and design protocols for masters and slaves respectively. The masters are driven by the hybrid protocols based on both displacement error and bearing constraint, which act as skeleton and avoid the collision in the hunting process. Simultaneously, the slaves merely depend on the bearing states of adjacent masters to reduce loads of data transmission to maintain the formation. We illustrate that the bearing-based hybrid formation protocol is one of the most effective solutions to the hunting problem.
The outline of this paper is given as follows. In Section
Consider a multi-agent system with at least 2 masters. One can use a graph to describe the communication topology. Let
The incidence matrix E is defined as follows:
The adjacency matrix A is defined as follows:
Donate
Define a finite collection
We define the framework between masters and slaves as
Similarly, considering the target position, we can denote the framework between target and agents as
![](cpb_29_5_050701/cpb_29_5_050701_ieqn12.gif)
Here
Considering ζk = (E ⊗ Id)p and
The quadrotor is an under-actuated system, and the coupling relationship between attitude angles and position is the core of modeling. According to the Euler–Lagrange modeling principle and the assumptions,[19] the dynamic model of the quadrotor is established as
![](cpb_29_5_050701/cpb_29_5_050701_ieqn24.gif)
![](cpb_29_5_050701/cpb_29_5_050701_ieqn25.gif)
![](cpb_29_5_050701/cpb_29_5_050701_ieqn26.gif)
![](cpb_29_5_050701/cpb_29_5_050701_ieqn27.gif)
In this dynamic model, the outer position loop drives the quadrotor toward the desired position with position controller U1i, while the attitude inner loop tracks the desired angles. The attitude controller in the tracking problem was well established in Ref. [19], so we mainly focus on the design of the position controller in this paper.
Considering Eq. (
![](cpb_29_5_050701/cpb_29_5_050701_ieqn28.gif)
In this section, we briefly describe the way to obtain the bearing information in the engineering application.
According to Ref. [21], we can obtain the absolute position estimated by a combination of the monocular visual simultaneous localization and mapping (MVSLAM), and the air pressure sensor. MVSLAM provides the bearing information and the air pressure sensor provides an absolute scale respectively. However, considering the capability of data transmission, this approach only applies to a single quadrocopter. Another similar approach is to utilize artificial markers for simplifying position estimation. In Ref. [22], Scaramuzza and Fraundorfer proposed a system based on a low-cost commercial quadrotor with a monocular front-looking camera for performing autonomous hovering and waypoints following in an unknown environment.
The slaves aim to get the bearing information of the neighbor masters. In this process, the attitude angle φ = 0 for each quadrotor. Considering this simple requirement, we can utilize Engel’s model and establish a three-dimensional coordinate system for the target in the field of camera view. The principle is shown in Fig.
![]() | Fig. 2. The schematic of camera view. (a) The three-dimensional schematic, (b) the projection of schematic. |
As shown in Fig.
The spherical mapping
It is noteworthy that the artificial markers should be easily distinguished and anti-disturbed, so conspicuous graphics or infrared markers are good choices. In this paper, we select eight highlighted yellow lights in two rows as an example, and the process of identification is shown in Fig.
![]() | Fig. 3. The identification of markers: (a) the original image, (b) the identification of markers, (c) the filtering and corrosion, (d) the connected domain. |
This method has a high demand for image quality and data transmission of a quadrotor. After the feature extraction and filtering of the image, we can construct the connected domain, which contains all the artificial markers. Somasundaram proposed remarkable spatiotemporal regions in Ref. [23] to extract the spatiotemporal features. Only a small percentage of the most salient (least self-similar) regions was considered and found using its algorithm, over which Spatio-temporal descriptors were computed.
Figures
![]() | Fig. 4. The topology constructions between target and masters (blue lines), and between masters and slaves (black lines). |
To achieve obstacle avoidance in the hunting process, the master quadrotors have to collect the displacement information D = [d1,d2,…,dnm]T from the target and the following quadrotors only receive (or collect by cameras), where di is the displacement vector from the target to the i-th quadrotor. The formation spirally shrinks the hunting radius with rotated velocity ωk(t) to prevent the target from escaping.
To facilitate the understanding, the center and size of the multi-quadrotor system are defined as the evaluation functions.
Meanwhile,
![](cpb_29_5_050701/cpb_29_5_050701_ieqn34.gif)
![](cpb_29_5_050701/cpb_29_5_050701_ieqn35.gif)
Considering the existence of the displacement vector, the general objective of hunting problem can be expressed as
The masters have already reserved vector constraints that contain formation and position information, while the displacement vectors of the slaves are difficultly obtained. Thus the objective can be rewritten with bearing information as
![](cpb_29_5_050701/cpb_29_5_050701_ieqn36.gif)
![](cpb_29_5_050701/cpb_29_5_050701_ieqn37.gif)
Note that if the slaves could keep the bearing structure, the displacement constraints will be satisfied sequentially. In general, the maximum cruising flight speed of commercial quadrotor Vmax ≈ 20 m/s and the effective distance of protocols l = 200 m. Hence, one can consider bearing congruency under the condition of ω*(t) ∈ [−0.1,0.1] (rad/s) in this paper.[24]
In this section, to solve the hunting problem, distributed protocols for masters and slaves are designed respectively. First, we define the state for masters and slaves.
Then the controllers for masters and slaves are defined, respectively, as
![](cpb_29_5_050701/cpb_29_5_050701_ieqn38.gif)
The derivative of error is
Therefore, the tracking error of velocity in the second subsystem can be defined as
The derivative of z2 can be expressed as
The Lyapunov function of the system can be chosen as
![](cpb_29_5_050701/cpb_29_5_050701_ieqn41.gif)
Using upl and the adaptive law
![](cpb_29_5_050701/cpb_29_5_050701_ieqn43.gif)
![](cpb_29_5_050701/cpb_29_5_050701_ieqn44.gif)
According to LaSalle’s invariance principle, except for the system equilibrium point, V > 0,
Considering the bearing constraints of the slaves, we can define the bearing tracking error
When z3 → 0,
The derivative of z3 is
![](cpb_29_5_050701/cpb_29_5_050701_ieqn49.gif)
![](cpb_29_5_050701/cpb_29_5_050701_ieqn50.gif)
![](cpb_29_5_050701/cpb_29_5_050701_ieqn51.gif)
![](cpb_29_5_050701/cpb_29_5_050701_ieqn52.gif)
![](cpb_29_5_050701/cpb_29_5_050701_ieqn53.gif)
![](cpb_29_5_050701/cpb_29_5_050701_ieqn54.gif)
From the mathematical model of quadrotor systems in Eq. (
Assume that [Ux,Uy,Uz]T = U1Re3 is another form of virtual control variables, then
Convert them to
Then the actual control input can be expressed as
In this section, an example is given to verify the effectiveness of the hunting protocol in the paper. MATLAB-R2017a is used to build the model and run the simulation. The initial parameters of all the quadrotors are shown in Table
![]() | Table 1. Simulation parameters. . |
According to the topology in Fig.
Using the definition of projection operator in Eq. (
Then, the hunting trajectory and its performance are shown as follows.
To reflect the characteristics of formation clearly, we capture three fragments in Fig.
![]() | Fig. 6. The trajectory (full lines) and fragments (imaginary lines) of multi-quadrotor system encircling a moving target (red full line). |
We have studied the bearing hunting protocols in a multi-quadrotor system. First, we have modeled the machine vision for obtaining the bearing state among the quadrotors and target. In the process of controller design, we have achieved the collision avoidance of the quadrotors system through hybrid displacement control. Considering the limits of the transmission capability of the data link, we have adopted the hierarchical network based on bearing constraints to reduce the information needed in formation maintenance. We have given some simulations to illustrate the effectiveness of the proposed protocol.
One area for future research is taking event-triggered and sampling communication into account in the hunting process.
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