A reliable nonlinear dynamic model of the quadrotor is presented. The nonlinear dynamic model includes actuator dynamic and aerodynamic effect. Since the rotors run near a constant hovering speed, the dynamic model is simplified at hovering operating point. Based on the simplified nonlinear dynamic model, the PID controllers with feedback linearization and feedforward control are proposed using the backstepping method. These controllers are used to control both the attitude and position of the quadrotor. A fully custom quadrotor is developed to verify the correctness of the dynamic model and control algorithms. The attitude of the quadrotor is measured by inertia measurement unit (IMU). The position of the quadrotor in a GPS-denied environment, especially indoor environment, is estimated from the downward camera and ultrasonic sensor measurements. The validity and effectiveness of the proposed dynamic model and control algorithms are demonstrated by experimental results. It is shown that the vehicle achieves robust vision-based hovering and moving target tracking control.
As an emerging platform for unmanned aerial vehicle (UAV) research, the quadrotor has recently gained most attention from the community. With some specific capabilities, such as vertical take-off and landing (VTOL), hovering, fly alone or in team, autonomously fly, it has been envisaged for a wide of applications including military reconnaissance, search and rescue, meteorological survey, environmental monitoring, and wireless mobile senor networks [
Although the quadrotor has a series of advantages, it is an absolutely unstable and underactuated dynamic system with sophisticated nonlinearity and strong coupling. Moreover, it is easily affected by near-surface airstream. Because of these difficulties, the intensive study on dynamical modeling, analysis, and advanced control of quadrotor needs to be done to improve the flight quality. In particular, the actuator dynamic and aerodynamic effects must be investigated to establish a reliable dynamic model of quadrotor. Control method dealing with the nonlinearity and coupling property of quadrotor has to be proposed for precise flight control. Bouabdallah and Siegwart mentioned the importance of actuator dynamic and analyzed forces and moments caused by aerodynamic effects. But they simplified the dynamic model and omitted those effects [
The main contributions of this paper are the following. First, a reliable nonlinear dynamic model is presented based on the analysis of actuator dynamic, aerodynamic effect, and rigid body dynamic. The gyroscope effect of the rotors is considered by dividing the quadrotor into body part and rotor part. It makes the dynamic model more reliable to take actuator dynamic and aerodynamic effect into account. Second, the PID controllers with feedback linearization and feedforward control are proposed to control both the attitude and position of the quadrotor. The dynamic model is explicitly expressed as a cascade system of three subsystems to be suitable for the backstepping method. The control algorithms are realized on a fully custom quadrotor and vision-based autonomous indoor moving target tracking flight is achieved.
This paper is structured as follows. In Section
Most researchers used to regard the whole quadrotor as a rigid model [
The coordinate system is defined in Figure
Body-fixed frame and Earth-fixed frame.
Actuator dynamic describes the relationship between rotor speed and actuator voltage. The latter is our real control input. Basically, the actuator response speed is most interested by designers. Based on Kirchhoff laws and the law of rotation, the simplified actuator dynamic model is [
Actuator delay is curial especially when the attitude control loop runs at a low frequency.
Aerodynamic effect dramatically increases with the variation from equilibrium state. Some literatures show excellent performance on a test bench [
Motor 1 is the front of the quadrotor; speed-up motor 3 and speed-down motor 1 will result in forward flight.
During forward flight, the advancing blade has a higher velocity relative to the free stream and the retreating blade sees a lower effective airspeed. This brings imbalance of lift and results in the propeller plane deflecting from position 1 to position 2. The deflection angle
The deflection of the propeller plane causes an extra moment on the
Air friction is relative to the velocity of quadrotor and can be expressed as
Theoretical and experimental results demonstrate that aerodynamic effect is not trivial even with moderate speed and will be crucially important in aerobatic flight.
We separate the quadrotor into two portions, the body part and the rotor part. The body part includes the frame structure and equipments. The rotor part includes motors and propellers. Manifestly, the relative position between the rotor part and the body part varies as the rotor spins. Hence we cannot assume the whole quadrotor as a rigid body. The mathematical model is based on following assumptions. The body part and rotor part are rigid, respectively. The quadrotor is symmetric. Thrust and drag are proportional to the square of propeller’s speed. Actuator dynamic is identical. The center of gravity (CoG) coincides with the body fixed coordinates origin.
Apply Newton-Euler equation to body part:
Here we employ the Euler angle representation of orientation. We consequently rotate about
Let
According to the third assumption, rotor thrust
Utilizing the analysis of aerodynamic effect, the total force and moment on the body part are
Let
Hence
First, we define a general PID controller with feedforward control
Then we apply feedback linearization to design the attitude and position controllers. Finally, we describe the backstepping control scheme.
The outputs of the position controllers are
Substituting (
This guarantees asymptotic stability and has robustness to some uncertainties. Solving (
The outputs of the position controllers are the inputs of the attitude controllers. The outputs of the attitude controllers are
The outputs of attitude controllers are the inputs of the actuator controllers. The outputs of the actuator controllers are
Backstepping control scheme.
The nonlinear dynamic model and control algorithms are verified on a fully custom quadrotor. First, the design and manufacture of our fully custom quadrotor is briefly described. Then, results of different experiments are discussed consequently.
We developed a fully custom quadrotor using the optimal design algorithm, shown in Figure
Off-board process of the image.
A fully custom quadrotor and autonomous moving target tracking.
The parameters of the rigid body dynamic are calculated during the design process. The parameters of the aerodynamic effect are estimated by empirical model and data fitting. The actuator dynamic is obtained by system identification. The parameters used in the verification experiments are listed in Table
Parameters used in the verification experiments.
Parameter | Description | Value | Units |
---|---|---|---|
|
Gravity |
|
|
|
Mass |
|
|
|
Distance between CoG and motor |
|
m |
|
Distance between CoG and propeller plane |
|
m |
|
Roll inertia |
|
kg·m2 |
|
Pitch inertia |
|
kg·m2 |
|
Yaw inertia |
|
kg·m2 |
|
Rotor inertia |
|
kg·m2 |
|
Thrust coefficient |
|
N·s2 |
|
Drag coefficient |
|
N·s2 |
|
Motor time constant |
|
s |
|
Velocity to angle constant |
|
rad·s/m |
|
Angle to moment constant |
|
N·m/rad |
|
Active area |
|
|
|
Air density |
|
|
|
Damping coefficient |
|
The hardware in loop (HIL) simulation is executed before flight experiment. Manually change the pitch angle, roll angle, and yaw angle of the quadrotor; the speed command of each motor changes, respectively. Manually change the altitude of the quadrotor; the throttle command for four motors changes. Manually change the position of the quadrotor; the attitude commands change. The simulation results verify the correctness of the dynamic model and control algorithms. The HIL simulation is shown in Figure
The HIL simulation: manually change the quadrotor, the 3D model changes, and the curves show the outputs of the controllers.
Attitude control, altitude control, hovering control, and moving target tracking control experiments are consequently performed. As shown in Figure
Attitude control results.
Altitude control results.
Hovering control results.
Moving target tracking control results.
We aim at precise modeling, analysis, and control of a sophisticated nonlinear system. This paper presented the newest research on quadrotor of our project. First, we analyzed the actuator dynamic and aerodynamic effect of the quadrotor. Then, we established a reliable nonlinear dynamic model of the quadrotor. As the backstepping control algorithm is well fit for the cascaded structured systems such as the quadrotor, we designed a series of PID controllers with feedforward control and feedback linearization using the backstepping method. Real experiments were executed and the effectiveness of the proposed dynamic model and control method is demonstrated by the experimental result. The future works include two directions. Firstly, the quaternion representation of orientation needs to be employed since the Euler angle representation is subject to problematic singularities. The global stable controllers are expected to be proposed based on the quaternion representation. Secondly, more efforts need to be done to promote the moving target tracking system more like a heterogeneous multiagent system. Problems within heterogeneous multiagent system are expected to be the next technical breakthrough.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work is partially funded by the Fundamental Research Funds for the Central Universities (no. HEUCF021318), the Natural Science Foundation of Heilongjiang Province (no. A201312), the Harbin Science and Technology Innovation Talent Youth Fund (no. RC2013QN001007), the National High Technology Research and Development Program of China (no. 2013AA122904), and the National Natural Science Fund (no. 11372080).