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AVIATION AND ROCKET-SPACE TECHNOLOGY
CONTROLLER DESIGN BACKSTEPPING SLIDING FOR QUADROTOR TRAJECTORY
Do Tien Luong
Master,
Department of of Aviation Equipment, Air Force Officer's College, Vietnam, Khanh Hoa E-mail: dotienluong24@gmail. com
ДИЗАЙН КОНТРОЛЛЕРА БЭКСТЕППИНГ - СКОЛЬЖЕНИЕ ДЛЯ ТРАЕКТОРИИ КВАДРОТОРА
До Тиен Лыонг
магистрант, кафедра авиационного оборудования, Колледж офицеров ВВС, Вьетнам, г. фКхань хоа,
ABSTRACT
Backstepping control is also known as Backstepping Control (BSC). Backstepping control method may not be effective when encountering disturbances or when the parameters of the object are uncertain, and the problem transition process can make the control process slower, especially in applications that require fast response. To overcome this problem, Sliding Mode Control (SMC) is applied to automatically compensate all the dynamic model errors and the effects of disturbances. This controller has been proven to work stably and has good quality by Lyapunov theory.
In this paper, we present the mathematical model of the unmanned aerial vehicle and build a control algorithm based on the combination of Backstepping and sliding methods. The simulation results are used on Matlab/Simulink software.
АННОТАЦИЯ
Метод Backstepping может оказаться неэффективным при возникновении помех или когда параметры объекта неопределенны, а процесс перехода к проблеме может замедлить процесс управления, особенно в приложениях, требующих быстрого реагирования. Чтобы преодолеть эту проблему, применяется управление скользящим режимом (SMC) для автоматической компенсации всех ошибок динамической модели и эффектов возмущений. По теории Ляпунова доказано, что этот контроллер работает стабильно и имеет хорошее качество.
В данной работе представлена математическая модель беспилотного летательного аппарата и построен алгоритм управления на основе комбинации методов Backstepping и Slide. Результаты моделирования использованы в программном обеспечении Matlab/Simulink.
Keywords: Backstepping control; Slide mode control; UAV; BSC-SMC; Matlab/Simulink; Quadrotor.
Ключевые слова: Backstepping control; скользящее управление; БПЛА; BSC-SMC; Matlab/Simulink; Quadrotor.
Introduction
Quadrotor Model
Quadrotor is a model of drone with 4 engines attached with propellers arranged evenly at 4 corners as shown in Figure 1[5, 6, 8].
Библиографическое описание: Do T.L. CONTROLLER DESIGN BACKSTEPPING SLIDING FOR QUADROTOR TRAJECTORY // Universum: технические науки : электрон. научн. журн. 2024. 12(129). URL:
https://7universum.com/ru/tech/archive/item/19009
Figure 1. Structure of the quadrotor
Flying up, down, right, left or increasing or decreasing the flight speed depends on changing the speed or rotation direction of the 4 engines mentioned above. In which x, y, z are the coordinates of the quadrotor's center of gravity compared to the original coordinates. Roll, Pitch and Yaw are Euler rotation angles representing the direction of the object, denoted by O , 0 and y respectively.
Kinematic model of quadrotor To describe the dynamic model of the drone, we use the Newton-Euler equation and the convention that the linked coordinate system Bxyz is the coordinate system with the origin located at the center of gravity of the quadrotor and the earth inertial coordinate system Ixyz has the origin located at a point on the earth [7, 9, 10] .
We have the following system of equations describing the quadrotor's dynamics:
Ö = x¥0Jyy J::-i-
J...
' = 'i'/i>. J-': Jxx + Ö.Q + — .U, ; T = 6.(9. J J J
yy yy yy
x = (cos<S>.sinß.cosx¥ + sin<S>.sinx¥Y—-; v = (cos<S>.sinß.sinx¥ - sin<S>.cosx¥Y—-; z = -g +cos<S>.cosß.—.U1
m ' m m
Where m is the weight of the quadrotor, g is the gravitational acceleration, Jr is the internal torque of the rotor, d is the distance from the motor to the center of the quadrotor.
Materials and Methodology
Backstepping-sliding controller design The main content of synthesizing the backstepping-sliding controller is to give the control law for each control channel, with the parameters in the quadrotor's dynamic model being clear. The following article will present the steps to synthesize the backstepping controller according to the tilt angle channel, synthesizing the controller for the remaining channels x, y, z and pitch angle 0, direction angle ¥ is done similarly [1-3]. We have the kinematic equation of the channel with inclined angle O as follows:
. .J -J j . d
Step 1. Design the backstepping controller Let A, =0 and X, = 6 therefore X, = 6 and X^ = O. From this we can deduce the model according to the channel O as:
A1 -A,
A'2 = T.0.
• 7 ■>< .Oil - '' .1.
J.... •/.. •/.. "
(2)
desired Qd tilt angle signal be O , Xx = Q .
• Let Z i be the error between the desired tilt angle value and the actual tilt angle, we have Zx = Q - X.
Therefore Z, = 6rf -X, = 6rf -X2.
Choose a Lyapunov control function V of the form
V (Z ) = f Z2.
Conditions for signal Xl ^ Ф^ when:
= Z, .Z, = zr(èrf —X2) < -C;.Zr:(VZ, * 0) (3)
With C i being a positive constant. Considered X2 as a virtual control signal, the virtual control law ensuring Lyapunov stability criterion is deduced as:
Хъ,=ф„+Сгх, w ith С 1 > 0
(4)
• Let Z 2 be the deviation between the desired value and the actual value X 2 . We have Z2 = X2 - X2d .
. . J -J j . d ■
InferZ2 =X2 -X2d = Ч*.в. »j ""--^.en + — .U2 -C..Z, -Ф„
Substitute X2 into (3) to deduce:
Vl(Zl) = ZlXOJ-X2) = Zj[Ф, -(Z2 + X2d)] = Zj[Ф„ -(Z2 + Cj Zj + Ф,)] ^Vl{Zl)=-Zl.Z2-Cl.Zl
Choose the Lyapunov control function V 2 of the
form V2 (Z, Z2 ) = V +j Z:
Conditions for signal X2 ^ X2d when:
V2=V, + Z2.Z2 < 0; (VZj, Z2 ф 0)
(5)
.. J -J J ■ dU
From the Lyapunov stability criterion ensuring the system is globally asymptotically stable, we have the following condition:
(6)
. . J -J J . du
-C, .Z,2 - Z, Z, + Z„ ,Q¥.e.-z-^ -.en +
J J J
-с, л-ф „)<-Cj.Zj2-c2.z22
From there we have the control law U 2 has the form:
J- .
- d
J
Jr
J..
1П - C; .Zj - Cl ,Z2 - C2 ,Z2 + Zj )
(7)
Step 2. Design the slide controller To design the sliding controller, we still use the above system of equations describing the dynamics of the quadrotor . Find the control law, set the error of the oscillation component as follows [4]:
Z = ®d - X
Select slide surface: S0 = X2 -Xlrf -CXZX Consider the following Lyapunov function: J 2
V =—S2 . Taking the derivative of both sides of the above
Lyapunov function, we get: V = S0.S0
Under the condition V <0 that the system is stable,
we have: F = S0.S0 < 0
We perform the selection of sliding surface control law as follows:
So =-k1.sign(s0)- q-
To synthesize the backstepping-sliding control law, we apply sliding control to the result of synthesizing the backstepping control law. We have:
V = - Zj2 1 2 1
Zj = X id — X1
V2 = 1 Z2 + 2 *Ф
V2 = Zj Z1 + 5ф ¿Ф
Зф = Z2 = X2 — Xid — Cj Zj
We can calculate the slip surface as follows:
V2 = Zi Zi + .ф Уф
=-к1.э1§п(Эф ) - qj ^ф уф = X 2 - Xid - CZi = T в.
Jyy - J,, J
---^ .9Q + — .U. - X id - C (X id - X i)
J„ J„ J„
To synthesize a stable control signal in sliding
mode, it is necessary to ensure the necessary sliding condition s.s < 0. Solving system (8) gives U 2 as follows:
U =—(-K.sign()-q -Tв.Jyy J" + J.вQ+Xiä+C(X^irf-Xi))
d
Combining (1) and (9) we get the system of equations representing the back propagation control model and the control signal of the backstepping-sliding angle channel controller as follows:
J„ J„
(9)
X1 -X2
X2 =g + cos<S>.cosd.—.(-Ul)
J
U2 =-f (-ki .sign(s0 ) - qi .ф - T в d
Jyy - J,, J.
JJ
+-Z- .вп+x irf+с (x id - x i))
Proceeding similarly, we get the backstepping-sliding control law for the remaining channels as follows:
J..
U3 (-k2.sign(se) - q2 îj-Тф .l^ü -Лл. фо+Xsd + C3(XM - X3))
d Jyy Jyy
J„ - J„ J,
J J - J . .
U4 = (-k3 .sign(sT, ) - q3 -^ + X 5 d + C5 (X 5 d - X 5 ))
d J
Ux = — (-k4 .sign(sx ) - q4 sx + X ld + C7 (X ld - X 7 ))
'Uл
Uv = — (-k5 .sign(Sy ) - q5 .y + Xм + C9 (X9d - X9 ))
U,
U, =
i = -—- (-k6.sign(sz) - q6 s + g+Xiirf+Cii(Xiirf- X ii)) cos ф.cose
The BSC-SMC control diagram for quadrotor is shown in figure 2.
Figure 2. BSC-SMC controller diagram for quadrotor Results and discussion
The parameters are chosen as follows: m = 0.6kg , beta0 = 189.63 , beta1 = 6.0612 , beta2 = 0.0122 , d = 25cm , b = 280.19, g = 9.81m / s2,
Ix = 3.8278e~3Nm/rad/s2; Iy = 3.8288e-3Nm/ rad /s2; L = 7.6555e-3Nm/rad/s2 ; J = 2.8385e-5Nm/rad/s2 ;
To increase the visualization and clearly see the quality of the controller, we will survey two cases of quadrotor when not affected by noise and when there is noise.
First, conduct a survey of the Quadrotor's motion trajectory with the PID controller. The simulation results of the Quadrotor's motion trajectory and error are shown in Figure 3-5.
1
\ - V \ * Trajectory T.irliour interference
/ _ ^T rn jec to it iviclj interference
L / \
\ \ / / \ \
s
10 5 0 -5 -10
10 20 30
time(s)
a) Quadrotor's trajectory along the Ox axis
40
b) Quadrotor's trajectory error along the Ox axis Figure 3. Quadrotor's motion trajectory and error along the Ox axis
50
1 □ 5 D -5 -10
y (m) i i
✓ \ Trajectory without interference — ^—Trajectory with interference -
/ f \
\
\ \ / f \ > y
\
ta 20 30
time(s)
a) Quadrotor's trajectory along the Oy axis
40
50
20 30
time(s)
b) Quadrotor's trajectory error along the Oy axis Figure 4. Quadrotor's motion trajectory and error along the Oy axis
15 10 5 0 -5 -10
Z (ill) i 1 ■
f \ Trajectory without interference
— " raj ectorv with int erference
—^ /— \ \ \
\ •y \
\ *
10 20 30
time(s)
a) Quadrotor's trajectory along the Oz axis
40
50
8 4 2 0 -2 -4
(m) )
Eerence Mice
E rror Y.irhour in re i rror vritk üiterfer
/ \ / *
/ N 1 V v. _
r t A / V Sn /
f N > V. s
10
40
20 30
time(s)
b) Quadrotor's trajectory error along Oz axis Figure 5. Quadrotor's motion trajectory and error along the Oz axis
From the simulation results, we can see that when there is no interference, the PID controller basically operates as required, the Quadrotor follows the given trajectory stably with the x, y, z axes taking about 3 — 5s to reset to the desired set value with an error of about 0.1 m. However, when there is interference, we can see that the PID controller cannot meet the requirements of quadrotor control, for the movement trajectory along
the Ox, Oy axes, it is basically not much affected (Figures 3, 4), however, along the Oz axis, there is instability, the Quadrotor does not follow the original trajectory, the error is large (Figure 5).
BSC-SMC controller The simulation results of the Quadrotor's motion trajectory and error are shown in Figure 6-8.
a) Quadrotor's trajectory along the Ox axis
0.02 0.01 0
-0.01 -0,02
-
Error without interference
— h Error with interference
10
20 30
time(s)
40
50
b) Quadrotor's trajectory error along the Ox axis Figure 6. Quadrotor's motion trajectory and error along the Ox axis
V (ftl) . ,__ T r aj e ct o it with out int e rffe r en c e -
- , -✓ \
s ( \ V — Trajectory with interference
f \
\ / s X \
S_____7
10
5 o -5 -10
10 20 30
time(s)
a) Quadrotor's trajectory along the Oy axis
40
50
20 30
bime(s)
b) Quadrotor's trajectory error along the Oy axis Figure 7. Quadrotor's motion trajectory and error along the Oy axis
10
5 0 -5 -10
0.01
-0.01
-0.02
-0.03
z (m> Trijectun witliout interference _ _Tnjecton tritli interference
/ X \
/ t \ v
/ \ *
» \ \ / / \ \
/ ..V
10 20 30
time(s)
a) Quadrotor's trajectory along the Oz axis
40
10
20
30
40
50
111
tm ■■ Error with interference
50
time(s)
b) Quadrotor's trajectory error along Oz axis Figure 8. Quadrotor's motion trajectory and error along the Oz axis
Results show that when there is no interference or when affected by interference, the BSC-SMC controller provides stable quadrotor control. The quadrotor follows the initial trajectory at all three axes Ox, Oy, Oz with small trajectory error.
Thus, using BSC-SMC controller in case of disturbance gives better results than PID controller. Quadrotor always follows the given trajectory at all positions in space.
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