In this paper we present a system for automatic control of a quadrocopter based on the adaptive control system. The task is to ensure the motion of the quadrocopter along the given route and to control the stabilization of the quadrocopter in the air in a horizontal or in a given angular position by sending control signals to the engines. The nonlinear model of a quadrocopter is expressed in the form of a linear non-stationary system.
uncertainty and external disturbances. In addition, adaptive controllers are designed for a low-power non-linear quadrotor system to eliminate the tracking error, despite disturbances.
In spite of the fact that the mentioned controllers have the reliable tracking the trajectories, they are just suitable for implementation on unmounted stationary the control units. On the other hand, they need fast and heavy computing devices because of their complex and time-consuming control laws. In addition, they are limited by long-range applications, when the delay of communication with the stationary control unit disrupts the real-time operation of the quadrotor. The conventional proportional integral-derivative (PID controller) is the most preferred control method for on-board implementation of the control unit in autonomous systems. However, it is very di cult to adjust the PID gain to a number of operating points or for di erent continuous trajectories. In addition, PID regulators are not very successful for controlling non-linear systems with reduced voltage. Therefore, several methods are used to develop a PID controller with improved tracking characteristics, which may be suitable for on-board implementation. Fuzzy logic is a powerful tool that is often used to obtain excellent results compared to a conventional PID controller.
Consequently, there is a need for a mathematical model which could describe the control of a quadcopter. The di culty is that a quadrocopter has 6 degrees of freedom, while we can control only 4 parameters: the angular velocities of the engines.
The next important task is to build a stabilizing algorithm. Controlled by four spaced apart engines, a quadrocopter is an unstable dynamic system which, due to the nonlinearity of the mathematical model, must be stabilized by the complex control algorithms. 2
Many modeling approaches have been presented [Erg07],[Bou04] and various control methods have been proposed [Alt02], [Alt03]. First of all, several backstepping controllers have been developed. Madani studied a full-state backstepping technique based on Lyapunov stability [Mad06], [MBe06]. E. Altug presented backstepping control using single and double cameras as visual feedback [Alt02], [Alt03]. Other backstepping control methods were used by Castillo. He used this controller with a saturation function and it performed well under perturbation [Cas06]. Also, Metni used backstepping technique in order to obtain adaptive nonlinear tracking law for quadrotors system [Met07].
Feedback linearization controller was implemented by Altug [Alt02]. A PD controller was designed to control y and and feedback linearization controller was implemented to control x and z. A. Benalleuge presented feedback linearization high-order sliding mode observer for a quadrotor. The algorithm had shown robustness for wind disturbances and noise [ Ben06].
The method of quaternions for position stabilization was presented in [ Tay06]. With compensation of the Coriolis and gyroscopic torques, the controller guaranteed exponential stability while a classical PD controller without compensation of the Coriolis and gyroscopic torques could guarantee only asymptotic stability. A sliding mode disturbance observer was shown [ Bes07] and designed as the robust controller for quadrocopters. This controller showed the robustness for external disturbances, model uncertainties, and engine's errors. The robust adaptive fuzzy control was applied in [ Coz06]. This controller showed a good performance against sinusoidal wind disturbances. Mokhtari presented in [Mok04] robust dynamical feedback controller of Euler angles which used the estimation wind parameters. 3 3.1
0 cos # sin #
1 0 sin # A cos # 0 cos sin 0 sin cos 0 cos cos cos cos
cos ' sin + sin # sin # sin # sin cos ' sin cos cos cos sin + cos # cos + sin # cos #
sin sin # cos cos # cos 1 A The transition from in Earth- xed body to the velocities in Body xed system is de ned by equations:
Transformation angles velocities for the transition from one frame system to an-other one will be de ned in the following way:
sin
sin # cos
cos # cos
1
A
(3)
(4)
(5)
(6)
(7)
After a performance of cross product [Che03], the notation of Newton's second law will be obtained:
Neglecting all forces beside thrust force of propeller [Coo97] T and force of gravity [Lel11], the equation (
Transition to Body xed frame with the help of transfer matrix D, we obtain equation:
equation (13) is rewritten as: i - the velocity of each engine (i = 1; 2; 3; 4). In this case, the
q! g sin ru + g sin pv + g cos ' cos sin ' 1 : m
T = b( 12 22 23 24) u_ = rv q!
g sin v = p! ru + g sin sin ' !_ = qr pv + g cos ' cos ( 12 + 22 + 23 +
24) b m 3.3
M = H_ M = H_ + ! H 0 Iy 0
3.4
The dynamic of the engine [Hof07]: Kirchho 's equation and the second law of Newton represent the equations of quadrocopter's engines.
b { coe cient of viscous friction,
Adaptive control is a method used for automatic control of moving in the real time. It uses the online estimating of external parameters and automatic control. The system of equations describes the kind of control presented: Here Am known matrix, x(t) condition vector of system, b and c known constants, Kx { vector of unknowns. Let x_ (t) = Amx(t) + b(u(t) + KxT x(t));
x(0) = x0 y(t) = CT x(t) Unorm(t) =
kxT x(t) + kgr(t) kg =
1 cT A1mb
x_t(t) = Amx(t) + bkgr(t); xm(0) = x0 ym(t) = CT x(t) e(t) = xm(t) x K_x = x(t)e~T (t)pb; Kx(0) = Kx0 ATmP + P Am = Q
The system of equations (24) describes the work of engines. In order to obtain the transfer-function [Kiv06], the transformation of Laplace was implemented in the system (24):
S(JsS + b)C(s) = KtI(s) (Ls + R)I(s) = u(s) KeSC(s) W (s) =
C_ (s) u(s) =
kt (JsS + b)(Ls + R) + kekt (32) (33)
Here C_ (s)-the velocity of the motor shaft; u(s) - voltage input. The result of modeling is represented on the graphics: 3, 4, 5.
The adaptive control as a control system has been considered. The evaluation has been used in the design of the controller. Comparative results of simulation tests have been carried out in MatLab / Simulink. The
Modeling and PD control of a Quadrotor VTOL vehicle, Intelligent Vehicles results of simulations demonstrate that the quadrocopter is successfully stabilized, keeping the desired position and altitude using the proposed controller, thereby showing the e ectiveness of the proposed system under conditions of uncertainty. [Bou04] S. Bouabdallah, A. Noth, R. Siegwart PID vs LQ Control Techniques Applied to an Indoor Micro
Quadrotor, Intelligent Robots, and Systems, 2004. [Cas06] P. Castillo, P. Albertos, P. Garcia, and R. Lozano \Simple real-time attitude stabilization of a quadrotor aircraft with bounded signals," Proc. of the 45th IEEE Conference on Decision and Control, 2006. Pp. 1533-1538. [Met07] N. Metni and T. Hamel \Visual tracking control of aerial robotic systems with adaptive depth estimation," International Journal of Control, Automation, and Systems, 2007. Vol. 5, no. 1. Pp. 51-60. [ Ben06] A. Benallegue, A. Mokhtari, and L. Fridman \Feedback linearization and high order sliding mode observer for a quadrotor UAV," Proc. of the International Workshop on Variable Structure Systems, 2006. Pp. 365-372. [ Tay06] A. Tayebi and S. McGilvray \Attitude stabilization of a VTOL quadrotor aircraft," IEEE Trans. On
Control Systems Technology, 2006. Vol. 14, no. 3. Pp. 562-571.