In this article, an attitude tracking controller is designed for a quadrotor unmanned aerial vehicle (UAV) subject to lumped disturbances. Firstly, the attitude dynamical model of the quadrotor under external disturbances is established. Subsequently, an improved sliding mode control (SMC) strategy is designed based on the linear extended state observer (LESO). In this control scheme, the SMC will guarantee the sliding surface is finite time reachable and the LESO will estimate and compensate for the lumped disturbances. Then, the robustness and asymptotic stability of the proposed controller are proved by the stability analyses. Finally, three numerical simulation cases and comparative flight experiments validate the effectiveness of the developed controller.
In recent years, researchers and engineers have worked intensively to develop practical aerial robots capable of performing missions with minimum or no human intervention. Among the aerial robots, it is postulated that a quadrotor will be applied in the future extensively for environmental monitoring, agricultural services, urban modeling, mapping and photographing, and wild fore surveillance, to name several applications [
Generally, the flight controller design of the quadrotor can be handled using the following two approaches: termed linear control approach and nonlinear control approach. In the former method, several common control strategies such as linear quadratic regulator [
On the other hand, the nonlinear control approach becomes an alternative way to design the flight controller independent of model information. For instance, Honglei et al. [
Antidisturbance and robustness issues can be critical for the attitude control of quadrotors. As a robust control tool for disturbance rejection, SMC is insensitive to the model parameters and has a fast convergence speed [
The SMC controller based on DOB has provided a novel approach to overcome the model mismatches and external disturbances, rather than the use of high control gains or extensive iterative calculations [
Considering the superiority of LESO, it may be a feasible way to integrate the LESO with SMC to design a novel robust controller. Moreover, there is no generalized method to design SMC based on LESO (SMC-LESO) for attitude tracking control of a quadrotor having both parametric uncertainties and external disturbance. The main contribution of this article is to use the LESO to improve the performance of the classical SMC in terms of disturbance rejection. Specially, the LESO can effectively reduce the chattering effect in SMC through estimating the lumped disturbances. Moreover, the artificial bee colony algorithm has been used for parameter tuning for the proposed controller, which can realize the optimal control. Practically, the proposed controller is investigated through a series of simulation cases and flight experiments.
The outline of this article is given as follows. The attitude dynamical modeling of a quadrotor is described in Section
The quadrotor aircraft considered in this article is displayed in Figure
The configuration of the quadrotor.
Denoting the thrust
The physical parameters of our quadrotor are listed in Table
Physical parameters of the quadrotor.
Parameter | Explanation | Value |
---|---|---|
Distance between propeller and the mass of vehicle | 0.2 m | |
Mass of vehicle | 1.923 kg | |
Gravitational acceleration constant | 9.8 m/s2 | |
Moment of inertia around | 0.094 kg∙m3 | |
Moment of inertia around | 0.094 kg∙m3 | |
Moment of inertia around | 0.086 kg∙m3 | |
Thrust coefficient | ||
Moment coefficient |
Measurement of principal moments of inertia by using a bifilar pendulum.
The goal in this article is to design a robust control law for system (
Assume measurements of all state variables are available, and they meet the conditions that
From equation (
Take roll channel as an example; the uncertain second-order system under the standard consideration is usually an integral chain system, described by [
For the sake of simplicity, system (
The lumped disturbances
System (
In practice, the energy of the lumped disturbances is finite, and its rate of variation is bounded. Hence, assume that the lumped disturbances
In LESO, the lumped disturbance of the system (
The matrix pair
The LESO of the system (
The error of estimation in equation (
Subtracting (
The error of estimation in equation (
The LESO is bounded input and bounded output (BIBO), and the characteristic values of
Assuming the derivative of lumped disturbance
For any
In that case,
Using the following
For any
The solution of error in equation (
With Assumption
Assuming the first derivative of lumped disturbance satisfies
SMC is deemed as one of the nonlinear control techniques with the ability to process the coupling disturbances and strong nonlinearities [
A sliding mode surface for the system (
The first derivative of equation (
Substituting system (
To ease chattering and reconcile the need for fast convergence, we design a reaching law as
The SMC-LESO control law is designed as follows, which can ensure the sliding surface is finite time reachable:
Consider the Lyapunov function
This implies that the system state will converge to the desired equilibrium area asymptotically when
The block diagram of the designed control scheme has been shown in Figure
Block diagram of the designed control scheme.
In this section, some comparative cases are considered to demonstrate the main ideas of the article. In Case
To show the efficiency of the proposed control strategy, a comparison between the SMC-LESO and SMC-NESO with the PD-LESO is presented. The control laws of the latter controllers are described in Appendix
In this simulation, the referenced attitude commands are selected by
Observation results of the lumped disturbances.
Figures
Consider the input command of the system (
The goal of this simulation is to compare the ability of disturbance rejection between the NESO and LESO in the presence of unstable disturbances. In system (
Simulation results of Case
The absolute values of mean error.
The evolution curves of the three controllers.
Result based on SMC-NESO.
Result based on SMC-LESO.
Simulation results when
Attitude tracking curves
Disturbances estimation by two observers
Simulation results when
Attitude tracking curves
Disturbances estimation by two observers
For the purpose of validating the proposed control strategy SMC-LESO, some experiments are implemented on a quadrotor testbed as displayed in Figure
Take the roll angle as an example, the quadrotor follows a step reference when the wind disturbances produced by fan are adopted to demonstrate the advantages of the SMC-LESO strategy. Similar to Case
To rigorously test our controller, attitude tracking experiment with unexpected wind gusts is carried out. The experiment is divided into three stages: (1) when
The structure of the quadrotor testbed.
Results from roll angle tracking control when wind gusts exist.
RMS, AME, and STD of steady state error for roll angle tracking.
Controller | AME | STD | RMS |
---|---|---|---|
SMC-LESO | 0.62 | 0.38 | 0.33 |
SMC-NESO | 0.89 | 0.64 | 0.71 |
LADRC | 0.70 | 0.33 | 0.49 |
Snapshots of attitude tracking.
Attitude tracking in the experiment.
Roll response
Pitch response
Yaw response
Control torques in the experiment.
Roll controller response
Pitch controller response
Yaw controller response
This article presented an improved sliding mode control based on linear extended state observer for the attitude control of a quadrotor. A LESO is designed to estimate the parametric uncertainties, external disturbance, and complex nonlinear dynamics, while a linear SMC control law is developed to compensate for the lumped disturbance. Results have shown that the SMC-LESO is asymptotically stable, and reachability of the system stability is guaranteed. Three simulation cases show the superiority of the proposed approach over the SMC-NESO and LADRC on control performance, particularly regarding the tracking accuracy, converge rate and robustness. Moreover, real-time experiments on the X450 quadrotor testbed verifies the effectiveness of the proposed algorithm. A future work will be focused on the expansion of the controller so that not only attitude but also for other elements of translational motion, namely, position and velocity, are collectively controlled by this controller. Furthermore, other control strategies will be researched for trajectory tracking control of the quadrotor, such as finite time control and self-adaptive control.
The NESO is designed as
where
The control law of SMC-NESO is designed as
For LADRC, the design of LESO is similar to that in system (
Then, system (
Similarly, the
Flow chart of ABC in parameter tuning (explain of the variables can be found in Ref. [
The results of parameter tuning for three control strategies are listed in Tables
Optimal parameters of SMC-LESO.
Parameters | Value | Parameters | Value |
---|---|---|---|
497.8 | 16.8 | ||
358.2 | 18.2 | ||
500 | 10.6 | ||
13.4 | 9.5 | ||
12.2 | / | / |
Optimal parameters of SMC-NESO.
Parameters | Value | Parameters | Value |
---|---|---|---|
29.8 | 9.6 | ||
1000 | 26.8 | ||
1000 | 726.5 | ||
0.1 | 36.5 | ||
999.3 | 11.9 | ||
799.8 | 14.5 | ||
0.1 | 0.36 | ||
847.0 | 0.002 | ||
1000 | 0.63 | ||
1.1 | 0.004 | ||
0.002 | / | / |
Optimal parameters of LADRC.
Parameters | Value | Parameters | Value |
---|---|---|---|
500.0 | 14.3 | ||
486.7 | 14.0 | ||
405.1 | 11.4 |
The parameters of the aerial robot we used haven been shown in the paper. The data of simulation and experiments can be provided if necessary.
The authors declare that they have no conflicts of interest.
This work is supported by the Anhui Key Laboratory of Mine Intelligent Equipment and Technology, Anhui University of Science and Technology (Grant No. 201902007); the National Natural Science Foundation of China (Grant No. 52005231); the Foundation Research Project of Jiangsu Province (the Natural Science Fund No. BK20170315); and Changzhou Sci&Tech Program of China (Grant No. CJ20179017).