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.
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 [
As a representative approach, formation control problem in several works [
Nonetheless, in the referred works [
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 [
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
Motivated by these ideas, this paper focuses on the multiple quadrotors system, and the dynamics of each quadrotor on
The model of a quadrotor is illustrated in Figure
The model of the quadrotor.
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
Each attitude can be mapped to an element
Its inverse operation is called the vee map, and it is defined as
Let
Note that the total thrust and moment satisfy
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
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
The relative positions between quadrotors.
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.
Define
The event-based formation control for the thrust
The proposed controller uses both the states
To analyze the formation control system (
Note that
Thus, system (
Let
Next, we are to discuss the selection of the control gain
Based on the previous analysis, consider the bounds of (
Note that, for the Kronecker product of two matrices
Following a similar method, we have
Combining (
The trigger function is used to generate the events. When the states in the
Therefore, the trigger time instants are defined as follows:
By models (
The thrust with both its magnitude and direction determines the flight trajectory. Note that the magnitude of the thrust is given by
The attitude command of
The attitude error between
Next, we will discuss the dynamics of
By (
The attitude controller is designed as
We present the following theorem to reveal the stability of the attitude dynamics.
The attitude of the quadrotor is described in (
Consider the Lyapunov candidate as
Note that
Thus, it is verified that
Therefore, for proper value
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.
After designing the thrust controller (
Consider the quadrotor system described by (
Invoking the trigger function (
Then, we are to prove that
Denote the Dini derivative of
For
Next, let
The sum from
By combining (
Then, we obtain the fact that
Choose a Lyapunov function as
By substituting (
To further estimate the upper bound of
It is noted by (
Theorem
Following a similar approach in (
By combining (
Therefore, from (
It is noted that, at nontrigger times
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:
Initial conditions of the quadrotors.
Quadrotor | Position/m | Velocity/(m/s) | Pitch | Yaw | Roll |
---|---|---|---|---|---|
Angle/deg | Angle/deg | Angle/deg | |||
1 | | | 5 | 30 | 1 |
2 | | | 1.5 | 10 | 2 |
3 | | | 0 | −10 | 4 |
4 | | | 0.5 | −8 | 1 |
5 | | | −1 | 0 | 2 |
The topology between quadrotors is shown in Figure
Topology of the quadrotors.
The desired formation is described by
The desired formation of quadrotors.
Choose
The parameters of attitude controller (
Trajectories of the quadrotors.
Trajectories from top view.
Trajectories from side view.
The thrusts of quadrotors are illustrated in Figure
The thrusts of the quadrotors.
Note that
The attitudes of the quadrotors.
Pitch angles
Yaw angles
Roll angles
The design of
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
Comparative results with different parameters.
Case | Parameters | Numbers of triggering in Quadrotors 1–5 | Convergence time |
---|---|---|---|
1 | | (63, 64, 63, 62, 63) | 19.12 s |
2 | | (54, 54, 54, 55, 54) | 19.24 s |
3 | | (104, 105, 104, 103, 104) | 18.70 s |
4 | | (47, 48, 47, 46, 47) | 19.29 s |
5 | | (74, 75, 74, 72, 74) | 19.09 s |
Comparisons of
To illustrate the efficiency of the event-based control method, two time-driven methods [
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
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This work is supported by the National Natural Science Foundation of China (no. 61374012).