Abstract
This paper is concerned with the formation problem of multiple quadrotors, and an event-based control strategy is proposed. The communication topology and relative positions of formation are first considered, and then the model of multiple quadrotors system is developed on the special orthogonal group SO(3). By designing the trigger function, certain events are generated for each quadrotor. Then, the formation controller is driven to update its parameters according to the events. The attitude controller on SO(3) is designed for tracking of the command and stabilization. By the proposed method continuous communication is not required between quadrotors, and it is proved that the quadrotors could achieve the desired formation. Simulation illustrates that the proposed event-based formation control method is effective.
1. Introduction
A team of multiple flight vehicles could have more flexibility, reliability, and efficiency over single vehicle, especially in difficult tasks like search, mapping, and attack missions. Therefore, formation control of multiple vehicles has attracted considerable attention for study [1–3]. As a typical unmanned aerial vehicle (UAV), a quadrotor has simple mechanical and flexible maneuverability. Therefore, quadrotor is usually used as the platform for formation control [4], and various types of implementation and applications such as surveillance [5], fault diagnosis [6], and fault-tolerant control [7] have also been investigated.
As a representative approach, formation control problem in several works [8–10] is studied through kinematics. That is, the quadrotor is considered as a point, and its model is double integrators with position and velocity. The quadrotors formation is studied in [9] through the second-order multiagent system. Similar model is adopted in [10], and the consensus protocol is used for formation. However, the control system of the quadrotor is nonlinear. Its thrust and attitude are coupled. Thus, the dynamics of each quadrotor cannot be easily neglected. Formation control of multiple quadrotors with attitude dynamics can be found in [11–13]. The consensus protocol is used for position control in [11], and it is then transformed as the attitude commands in the form of Euler angles. In [12], the thrust and attitude controllers were studied through dynamic surface control method, with both leader-follower and leaderless cases. The potential functions were proposed for the quadrotors formation in [13]. It also uses repulsive functions to guarantee obstacle avoidance.
Nonetheless, in the referred works [11–13] the dynamics of quadrotor is described with pitch/yaw/roll angles, and then the attitude commands and controllers are developed separately for each angle. It should be noticed that when the pitch angle is 90 degrees, the attitude system described by Euler angles would become singular [14]. This would cause the failure of the controller, so the methods using pitch/yaw/roll angles may not be applicable for large maneuver of quadrotors. To solve this problem, control methods on special orthogonal group for single quadrotor were proposed in [15]. In this singularity could be avoided by proper design, and the attitude can be modeled and controlled coherently. For single quadrotor, a globally defined model on is introduced in [15, 16], and based on it the geometric tracking controller is provided. In [17] the proportional-derivative control on was presented with disturbance compensation. The attitude dynamics was regarded as second-order systems on in [18], and the coordinated attitude control of rigid bodies is studied with undirected communication topology.
It should be also noticed that continuous information exchange between quadrotors is required in the above methods, and the controller is formulated by the real-time states of its neighbors. In practical situations, the communications between quadrotors are difficult to maintain continuous. Event-based control strategy provides a solution that the update of the controller is event-triggered rather than driven by the pass of time [19, 20]. In [21, 22] event-triggered consensus was studied for the multiagent system with single-integrator dynamics. The event-based control for general linear multiagent was discussed in [23]. It provides two sufficient conditions for consensus, with or without continuous communication, and significantly decreases updates of the controller. The event was defined to be triggered at discrete sampling time instants in [24], but a clock for synchronization is necessary in each agent.
Thus, it is suitable to introduce the event-based method to the formation control of quadrotors. Though there are established works on the event-based cooperative control strategy, these methods mainly focus the geometric motion of general agents. It is noted that the event-based formation control with dynamics of quadrotors on has not been fully studied. Extending basic multiagent systems and single quadrotor to the formation control of multiple quadrotors is not a trivial task. Due to the different structure of attitude dynamics on , the event-based design and formation stability analysis would become more complex.
Motivated by these ideas, this paper focuses on the multiple quadrotors system, and the dynamics of each quadrotor on is considered. The event-based control strategy in multiagent system is introduced for quadrotors formation. Based on local information, a triggering function with time-varying threshold is proposed to generate the events. Since the states of other quadrotors cannot be obtained during nontriggering time, this paper employs the states at last triggering time instants to build dynamical models and construct the event-based formation controller. To avoid singularity, the attitude controller is formulated with the desired commands on . Then, the stability of the formation and dynamics of quadrotors are analyzed together as an entire closed-loop system. In contrast to existing methods that depend on continuous communications, the proposed event-based method only requires information exchanges when certain events are triggered. It reduces the communication loads and would be more advantageous for application. A simulation with five quadrotors formations is given to illustrate the effectiveness of the proposed method.
2. Preliminaries
2.1. Modeling of the Quadrotors Formation on SO(3)
The model of a quadrotor is illustrated in Figure 1. The coordinates of the inertial frame are denoted by , and the directions of the three axes of the body frame are represented by , , and .
The attitude describes the orientation from the body frame to inertial frame, and this rotation in three dimensions can be expressed by a rotation matrix . Consider the special orthogonal group:
Each attitude can be mapped to an element in , and this mapping is bijective. Therefore, in this paper we consider to describe the attitude of the quadrotor. For a vector , define the hat map as
Its inverse operation is called the vee map, and it is defined as
Let , , and . According to the definition of , the directions of can be described by ; that is, , . Then, the thrust in inertial frame is . Based on these discussions, we then present the model of the th quadrotor:where and are the position and velocity, respectively, is the mass, is the gravity acceleration, is the total thrust, is the inertia matrix, is the angular velocity, and is the control moment, .
Note that the total thrust and moment satisfywhere is the distance between the center of mass and center of rotor on the plane. is a positive constant and represents the moment produced by the th rotor. It is learned that the matrix in (5) is invertible when . Note that by (5) each can be calculated for given and . Therefore, the total thrust and moment are considered as the control inputs in this paper.
2.2. Communication Topology and Formation of Multiple Quadrotors
The communication topology indicates the information exchange between quadrotors, and it is described by graph theory. In this paper the information link is considered to be directed. Let be a directed graph with nodes; each node represents one quadrotor. is the vertex set and is the edge set. A directed edge from to is denoted by the pair , and it means that the th quadrotor can receive messages from the th quadrotor. In this case the th quadrotor is said to be the neighbor of th quadrotor. The neighbor set is denoted by , where . The adjacency matrix of is a matrix, whose th entry is greater than zero if is an edge of and zero if it is not. The Laplacian is defined as follows:
A directed tree is a directed graph, where every node has exactly one parent except for the root, and the root is the only node with no parent but has a direct path to every other node. A spanning tree of a digraph is a directed tree formed by graph edges that connects all the nodes of the graph.
In the three-dimensional space, the formation of multiple quadrotors can be described by the relative positions between each other. As shown in Figure 2, let be the center of the formation; is the vector that denotes the distance from the th quadrotor to . Therefore, the relative position of these two vehicles is . By selecting certain , it would formulate the desired formation.
3. Event-Based Formation Control Design
In this section, the event-based control strategy for quadrotors formation is presented. Each quadrotor communicates with its neighbors only when it satisfies its own trigger conditions. In nontrigger time instants, the quadrotors need no communication, and it maintains to be functional with its own controllers. Moreover, the trigger conditions of each quadrotor are allowed to be different in the proposed method, and thus the multiple quadrotors system is triggered asynchronously.
3.1. Design of the Event-Based Formation Control
Define , and then the dynamics of position in (4) can be rewritten in a state-space form as follows:where
The event-based formation control for the thrust is designed aswhere “·” is the inner product, is the control gain, is the kth triggering time instant of the th vehicle, and is the state of jth vehicle in the th controller.
The proposed controller uses both the states and its neighbors . Note that the th quadrotor can only use the discrete value . At nontrigger times (), the real-time values of its neighbors are unavailable and the th controller works independently. Thus, we consider constructing virtual models and to replace and .
To analyze the formation control system (7) with controller (9), denote
Note that . Then, we have
Thus, system (7) can be expressed as
Let , , , , and . And denote the errors as and . Then, we haveThus, let , , , and , where is -dimensional vectors, . Then, (15) can be rewritten aswhere
Next, we are to discuss the selection of the control gain . If the graph contains a spanning tree, then , , where are the eigenvalues of . Then, let , and consider the following algebraic Riccati equation:where is a given matrix and is the solution of the equation. Choose , . Then, from (18) it can be verified that is Hurwitz. Further, it is known that is also Hurwitz. Denoting the largest eigenvalue of as , it is learned that .
Based on the previous analysis, consider the bounds of (16). Then, we have
Note that, for the Kronecker product of two matrices and , the norm of it satisfies . Thus,where , , and .
Following a similar method, we have where , , and .
Combining (20) and (21), (19) can be estimated to be
3.2. Design of the Trigger Function
The trigger function is used to generate the events. When the states in the th quadrotor satisfy some conditions at , an event is triggered and the controllers are updated. Thus, the trigger function of th quadrotor is designed aswhere , is a constant and , is the maximum number of the neighbors among quadrotors, , , , , and , , , , and are given constants.
Therefore, the trigger time instants are defined as follows:when , the th quadrotor updates its controller. From (23) it can be learned that after the update the trigger function became less than zero again. The trigger function keeps being negative between two successive trigger times.
By models (4) and (22), it is known that if the attitude subsystem is stable, then . Thus, the trajectory/formation control of the quadrotor depends on the stability of its attitude. Therefore, before conducting stability analysis of the event-based control method, in the next section we would first discuss the attitude subsystem of the quadrotor.
Remark 1. in the trigger function is designed as a time-varying threshold. It could regulate the frequency of triggering the event. In our method is set to be in the form of . Thus, the threshold is relatively large to prevent excessive event-triggering when the system starts; and it became smaller for better accuracy as the system converges.
4. Attitude Control Design on SO(3)
The thrust with both its magnitude and direction determines the flight trajectory. Note that the magnitude of the thrust is given by , and the direction is described by . Therefore, the thrust in the inertial frame can be expressed as . In order to achieve the desired formation, the attitude needs to be adjusted to provide the correct direction. In this section the attitude commands are first discussed, and then the attitude controllers on are proposed to track the commands and stabilize the attitude subsystem.
4.1. Design of the Attitude Command
The attitude command of th vehicle is denoted by , where . Thus, based on the thrust command in (9), we would design as follows:where is defined in (11). It should be noticed that , which means that , , and are orthogonal to each other. Once two of the column vectors in are chosen, the third vector is then fixed. After designing the third body axis , there is one remaining degree of freedom to adjust the attitude. Thus, we have , .
The attitude error between and on is then defined as and the tracking error of angular velocity is
Next, we will discuss the dynamics of and . By noticing and , the derivative of is calculated as follows:
By (26) it can be calculated that , where and .
4.2. Attitude Controller on SO(3) with Stability Analysis
The attitude controller is designed as
We present the following theorem to reveal the stability of the attitude dynamics.
Theorem 1. The attitude of the quadrotor is described in (4), and the command is given in (25). Then, for positive constant scalars and , the tracking error of the quadrotor is stable with controller (29).
Proof. Consider the Lyapunov candidate aswhere , , and .
Note that , , and . Denote and we then haveThus, it is verified that is positive definite. Calculate its derivative and we haveTherefore, for proper value that satisfies the constraint in (30), it is proved that by the attitude could track the desired command and the attitude system is stable.
Remark 2. Roughly speaking, the control system of quadrotor consists of outer loop and inner loop. The outer loop refers to the trajectory/formation control, and states of neighbor quadrotors are needed to achieve formation. The inner loop of the quadrotor handles its own attitude dynamics, and it can work without other quadrotors.
5. Stability Analysis of the Event-Based Formation Control System
After designing the thrust controller (9), the trigger function (23), the attitude command , and controller (29), in this section we discuss the stability of the closed-loop system. The following theorem is presented.
Theorem 2. Consider the quadrotor system described by (4), and the formation is given by . If the communication topology contains a spanning tree, then by the event-based controller (9) and attitude controller (29) on SO(3), the quadrotors will converge to the desired formation; that is, , where .
Proof. Invoking the trigger function (23), we have Then, we are to prove that Denote the Dini derivative of as , and it can be verified thatFor , note that there exist and so that . Then, (35) can be rewritten as where . Then, (34) can be verified by solving inequality (36). Then, from (33) and (34) we haveNote that ; the sum term in (37) is bounded asand it implies thatNext, let , where , , and . Invoking Cauchy- Schwarz inequality, we have The sum from to of (37) can be expressed asBy combining (38)–(40), rewrite (41) asThen, we obtain the fact thatChoose a Lyapunov function as and calculate its derivative. From (22), it is obtained thatBy substituting (43) into (44), is rewritten asTo further estimate the upper bound of , we would begin with . From the definition of , it is learned thatIt is noted by (12) that , and we have Theorem 1 implies that the attitude subsystem is stable, . Denote , and thus we have .
Following a similar approach in (15), (16), and (22), denote and , where is defined in (11). Then, considering (46) and (47), is estimated asBy combining (45) with (48), can be written asTherefore, from (49) we know that is stable in the sense of ultimately uniformly bounded (UUB) and . Then, . This proves that the quadrotors could achieve the desired formation.
Remark 3. It is noted that, at nontrigger times , the th quadrotor could only use , namely, the value of its neighbors at the last trigger time . A typical approach in previous works [23] is to use the zero-order-hold model of . Thus, the controller is formulated with the static value of during . Instead, the proposed event-based controller (9) generates the states based on to use it dynamically.
6. Simulation
To verify the effectiveness of the proposed method, in this section simulation is conducted with five quadrotors. The parameters of the quadrotors are as follows: = 3 kg, kg·m2, . The initial conditions are given in Table 1.
The topology between quadrotors is shown in Figure 3, and the weights of the edges in the graph are , , , , , and .
The desired formation is described by , , , , , and , as shown in Figure 4.
Choose . By solving the Riccati equation (18), the control gain in (9) is obtained as follows:
The parameters of attitude controller (29) are selected as and . Then, the simulation is conducted from 0 s to 20 s. Simulation results of trajectories of the quadrotors are shown in Figures 5–7. The trajectories in the - plane and - plane are given in Figures 6 and 7, respectively. Symbol “Δ” describes the initial position of each quadrotor. The figures illustrate that the five quadrotors achieve the desired formation stably.
The thrusts of quadrotors are illustrated in Figure 8. As a characteristic of the event-based control method, the vertices of the curves indicate the triggering time instants. Note that each quadrotor only communicates with its neighbors at triggering time instants, and then the controller forms its value according to the updated states. This shows that the updates of the thrust controller are driven by the events, not by the pass of time. From Figure 8 it is also learned that the thrust converges to a constant value for equilibrium.
Note that is used to model the attitude during the formation control design and simulation. To better present the results, is transformed as attitude angles as shown in Figure 9. From (25) it is known that as the thrusts converge to a constant, the attitude commands of the quadrotors would also be the same. It is found that the pitch, yaw, and roll angles of quadrotors reach consensus and the attitudes are stable with the proposed controllers on .
(a) Pitch angles
(b) Yaw angles
(c) Roll angles
The design of in trigger function (23) affects the number of triggered events and control accuracy. To further discuss this, define as the errors between current state and desired formation:
Then, different parameters of the trigger function are chosen for comparative simulations. Under the same initial conditions, the convergence time and number of triggering when reaches 0.01 are recorded. The results are given in Table 2. The time responses of are given in Figure 10. It can be seen that, for relatively small , the quadrotors would converge to the desired formation quicker with more triggering times, and vice versa. Thus, the parameters of in trigger function (23) can be designed to regulate the frequency of triggering and control accuracy.
To illustrate the efficiency of the event-based control method, two time-driven methods [9, 11] are conducted for comparison. A continuous-time controller is used in [9], and in [11] the controller is developed on the Euler angle model. The results of are also added in Figure 10. Roughly speaking, the methods with continuous communications would converge quicker than the event-driven method. In [9] reaches 0.01 at 18.45 s. By the finite time controller in [11], converges fast at the beginning, but the convergence time is 21.22 s. The comparative results demonstrate that the proposed method is effective, and the formation can be achieved with proper speed of convergence and triggering times.
7. Conclusion
This paper proposes an event-based formation control method for multiple quadrotors. With communication topology considered, the formation is described by relative positions between quadrotors, and the control model is established on . Then, the triggering functions are designed to generate events, and by these events the thrust controller updates its states. The attitude command and controller are developed on to avoid singularity, and the closed-loop system stability analysis is presented. In simulation the frequency of triggering events and control accuracy is discussed with different parameters in triggering functions, and this shows that, by the proposed method, the multiple quadrotors could achieve the given formation with stable attitudes. The event-based formation control method does not require continuous communication, and it would be more applicable for quadrotors formation.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work is supported by the National Natural Science Foundation of China (no. 61374012).