Advances in Mechanical Engineering 2021, Vol. 13(3) 1–11
Ó The Author(s) 2021
DOI: 10.1177/16878140211002723 journals.sagepub.com/home/ade
A hybrid robust adaptive control for a
quadrotor UAV via mass observer and
robust controller
Laihong Zhou , Shunjian Xu, Hong Jin and Huihua Jian
Abstract
The flight stability and safety of the quadrotor unmanned aerial vehicle (UAV) with variable mass are the key problems that limit its application. In order to improve the stability and steady-state control precision of the quadrotor system against slow-varying mass and external disturbance, a new robust adaptive flight control algorithm is developed and ana-lyzed in detail in this paper. Firstly, a mass observer based on adaptive control theory is designed to estimate the real-time mass and correct the mass parameter of the UAV. Then, a hyperbolic tangent function and a proportional integral (PI) controller is added into the attitude controller to eliminate the effect of the external disturbances. Finally, a hybrid robust adaptive controller (HRAC) developed with backstepping control method is used here for the trajectory tracking of quadrotor. The boundedness of the nonlinear system is verified by Lyapunov stability theory and uniformly ultimately bounded theorem. The trajectory tracking simulation experiments are presented in MATLAB/SIMULINK simulation environment. According to the simulation results, the real-time mass of quadrotor can be estimated by HRAC satisfacto-rily under the condition of external disturbances, while the estimate error of mass is only 6.4% of its own. In addition, HRAC can provide a higher trajectory tracking accuracy compared with robust optimal backstepping control (ROBC) and robust generalized dynamic inversion (RGDI). The results suggest a promising route based on the mass observer and hybrid robust controller toward slow-varying mass and the external disturbance as effective robustness control strategy for quadrotor UAV.
Keywords
Quadrotor UAV, mass observer, hyperbolic tangent function, adaptive control, robust control, trajectory tracking
Date received: 16 January 2021; accepted: 17 February 2021
Handling Editor: James Baldwin
Introduction
As a new kind of unmanned aerial vehicle (UAV), quadrotor has been developed as a promising contender in small UAV research field for its compact size, light weight and flexible operation, compared with helicopter and fixed-wing airplane.1–3 Furthermore, quadrotor has been widely concerned and used in military surveil-lance, border interdiction, rescue missions, agriculture services, photography, etc., due to its functions of per-forming vertical taking-off and landing, low-altitude hovering and low-speed cruising.4–9Though quadrotor has many advantages compared with other vehicles, the
application of quadrotor has been restricted because the quadrotor system is underactuated (six control out-puts and only four control inout-puts), high nonlinear, strongly coupled and has a time-varying nature.
School of Mechanical and Electrical Engineering, Xinyu University, Xinyu, Jiangxi, China
Corresponding author:
Shunjian Xu, School of Mechanical and Electrical Engineering, Xinyu University, Xinyu, No. 2666 Yangguang Road, Gaoxin District, Xinyu, Jiangxi Province, 338004, P. R. China.
Email: [email protected]
Creative Commons CC BY: This article is distributed under the terms of the Creative Commons Attribution 4.0 License (https://creativecommons.org/licenses/by/4.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages
Therefore, the improvement of the flight control system appears to be quite important and challenging.10–12
Recently, many different control methods like linear control, nonlinear control, and intelligent control are used for the control system of quadrotor UAV. For example, a linear matrix inequality (LMI) based robust quadrotor control algorithm by Ryan and Kim13; a fuzzy backstepping algorithm has been used to study the problem of trajectory tracking according to Yacef et al.,14; Nikolakopoulos and Alexis’s15studied an atti-tude control through switching networked; a distur-bance observer based on sliding mode control is proposed by Besnard et al.16 to eliminate the effect of the external disturbances; Shirzadeh et al.17has studied a tracking problem of moving targets by neural net-work. The flight control system of quadrotor has been improved greatly with the efforts of scientists in the past few years. However, most of these researchers con-centrate mainly on the control technology of quadrotor with constant mass instead of taking the mass change into account.
When the quadrotor is used to carry out some spe-cial tasks such as pesticide spraying, pollination, and firefighting, its mass could change significantly during the release of the carried substances with a difference of more than two third of its initial mass. At the same time, quadrotor can be affected by the external distur-bances caused by air stream as well. Due to these fac-tors, a huge control error usually occurs during the quadrotor flight, with the side-effect of some undesir-able consequences. Therefore, it is very important to design a high performance control algorithm for the quadrotor UAV system with slow-varying mass to ensure the flight safety and flight quality of the UAV. One focus of this study is to supplement the previous studies, where the mass of the quadrotor is assumed to be constant despite the mass change. While the effects of the mass varying on quadrotor has been overly ignored. The mass changes are viewed as disturbances or model errors such as an active model control scheme proposed in literature,18in which the model error was used to describe the load variation. This scheme is only applicable to mass mutation systems. However, for the slow-varying mass system, the control performance is not ideal. The steady state error is large, because there is no real-time estimation of the slow variable quality parameters of the UAV. Real-time estimation of qual-ity parameters is the key difficulty to improve the con-trol performance of slow-variable quality UAV concon-trol system at present.
Therefore, in this paper, in order to solve the prob-lems caused by the slow-varying mass and the external disturbance, a new hybrid robust adaptive flight con-trol algorithm is designed and developed for the quad-rotor UAV. The design process of the control method is as follows: Firstly, a mass observer based on adaptive
control theory was designed for the quadrotor UAV with slow-varying mass to estimate the real-time mass and correct the mass parameter of the quadrotor. Secondly, a hyperbolic tangent function of the second kind of error and a PI controller was added into atti-tude controller to eliminate the effect of the external disturbances. At last, a hybrid robust adaptive control-ler (HRAC) was developed with backstepping control method and the boundedness of the nonlinear system was verified by Lyapunov stability theory and uni-formly ultimately bounded theorem. Simulation results show that HRAC can provide a higher trajectory track-ing accuracy compared with robust optimal backstep-ping control (ROBC)19and robust generalized dynamic inversion (RGDI).20 Because the external disturbance can be eliminated effectively and the real-time mass of the quadrotor can be estimated satisfactorily using HRAC with an estimate error of only 6.4% of its own mass. In this paper, the estimation accuracy of the quality parameters using the quality observer is obvi-ously improved in the proposed control algorithm. In addition, the anti-interference ability of the system is also improved by using the hybrid robust controller. Then, the steady-state control accuracy of the system is improved. Therefore, according to the results of this study, HRAC is proved to be a promising technology to be applied effectively in controlling quadrotor UAV with slow-varying mass under the external disturbance.
This paper is organized as follows: a detailed dynamics model of quadrotor is presented in Section 2. Design process of classical backstepping controller is described in Section 3. The hybrid robust adaptive con-troller for the quadrotor UAV are proposed in Section 4. The trajectory tracking simulation results are pre-sented in Section 5. Conclusions are given in Section 6.
System modeling of quadrotor UAV
As shown in Figure 1, an earth fixed frame E (xe, ye,
ze) and a body fixed frame B (xb, yb, zb) are used to
study the system motion of quadrotor. j =½x, y, zT and h =½f, u, cT are the vectors of linear displacement and angular displacement in earth fixed frame E, where the roll angle (f), the pitch angle (u) and the yaw angle (c) are the three Euler angles around x-axis, y-axis, and z-axis, respectively. The linear velocity and the angular velocity of the airframe are denoted as V =½u, v, wT and X =½p, q, rT in the body fixed frame B. The rela-tion of velocities in the earth fixed frame E and the body fixed frame B can be given as
_j = RV _ h= N X
ð1Þ where, R and N are the translation matrices which can be written as
R= CcCu CcSuSf ScCf CcSuCf+ ScSf ScCu ScSuSf+ CcCf ScSuCf CcSf Su CuSf CuCf 0 @ 1 A ð2Þ N= 1 SfTu CfTu 0 Cf Sf 0 Sf=Cu Cf=Cu 0 @ 1 A ð3Þ
sin(), cos(), and tan() are denoted as S(), C(), and T(),
respectively.
The three Euler angles (roll, pitch, and yaw) of quad-rotor produced by varying the four quad-rotor speeds reason-ably are shown in Figure 1. Then, the corresponding movements of quadrotor arose in the direction of x, y, and z, and the flight movements of quadrotor are shown in Figure 2.
Considering the properties like the structure of the quadrotor which has a very strong rigidity and symme-try, or its usual operation at low speed and limited atti-tude angles, some reasonable assumptions listed as
follows are made first in order to simplify the modeling as well as to accommodate the controller design21,22:
Assumption 1: The quadrotor system has a rigid body and strictly symmetric structure.
Assumption 2: The quadrotor’s center of mass coin-cides exactly with the body fixed frame origin. Assumption 3: The blade flapping for the propellers can be neglected.
Assumption 4: The three Euler angles are bounded as p 2\f\ p 2, p 2\u\ p 2, andp\c\p. Using the Newton–Euler approach, the dynamic equations of quadrotor can be written as below:
m _j = Ff+ Fd+ Fg
I _X + X 3 I X = Mf Md
ð4Þ where Ff, Fd, and Fg are the thrust force, the aerody-namic drag force and the gravitational force respec-tively; X 3 I X , Mf, and Mdare the gyroscopic torque, rotor torque, and aerodynamic torque, respectively.
Substituting equations (1)–(3) into (4), and consider-ing assumptions 1–4, the final dynamic model of the quadrotor system can be formulated as
€x €y €z € f € u € c 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 = CfCcSu+ SfSc 1 mU1 CfSuSc CcSf 1 mU1 CuCf 1 mU1 _u _c Iy Iz Ix _ f _c Iz Ix Iy _u _f Ix Iy Iz 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 + kdx m _x kdy m _y kdz m_z l IxU2 l IyU3 1 IzU4 2 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 5 + 0 0 g kdmx Ix _ fJr Ix _uvr kdmy Iy _u + Jr Iy _ fvr kdmz Iz _c 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 ð5Þ
where Jr is the rotor inertia and the control inputs are given as follows:
Figure 1. Configuration frame scheme of quadrotor UAV.
U1= b(v21+ v22+ v23+ v24) U2= b(v24 v22) U3= b(v23 v21) U4= d(v21+ v23 v22 v24) 8 > > < > > : ð6Þ
vi, i = 1, 2, 3, 4 is the speed of the rotor i and vris vr= v2+ v4 v1 v3 ð7Þ The control input vector is given as
U= u½ x uy U1 U2 U3 U4T ð8Þ where uxand uyare the two virtual control inputs which
are defined as
ux= CfCcSu+ SfSc uy= CfSuSc CcSf
ð9Þ
Design process of classical backstepping
controller
The aerodynamic drag force Fd and the aerodynamic torque Mdare ignored in this section because of the low speed of quadrotor, while the external disturbance is taken into account. Then the nonlinear dynamic equa-tion of quadrotor system is represented as19
€ X= f (X) + g Xð ÞU + fd ð10Þ where f (X) = 0 0 g a1_x3_x5+ a2vr_x3 a3_x1_x5+ a4vr_x1 a5_x1_x3 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 , g Xð Þ = g xð Þ1 0 0 0 0 0 0 g xð Þ2 0 0 0 0 0 0 g xð Þ3 0 0 0 0 0 0 g xð Þ4 0 0 0 0 0 0 g xð Þ5 0 0 0 0 0 0 g xð Þ6 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 = 1 mU1 0 0 0 0 0 0 1 mU1 0 0 0 0 0 0 1 mCfCu 0 0 0 0 0 0 b1 0 0 0 0 0 0 b2 0 0 0 0 0 0 b3 2 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 7 5 ð11Þ
the abbreviations are given as a1= Iy Iz Ix a2=Jr=Ix a3= Iðz IxÞ Iy a4= Jr Iy a5= Ix Iy Iz 8 > > > > < > > > > : , b1= l=Ix b2= l Iy b3= 1=Iz 8 < : ð12Þ
Remark 1.The control input U1 and the parameters m, b1, b2, and b3 are nonzero, and Cf.0, Cu.0 according to Assumption 4, the diagonal elements of g Xð Þ are nonzero.
The state vector can be expressed as X= x½ 1 x2 x3 x4 x5 x6T
= x½ y z f u cT ð13Þ
and the external disturbance can be expressed as fd= f½ d 1 fd 2 fd 3 fd 4 fd 5 fd 6T ð14Þ The desired reference trajectory is defined as Xd= x½ 1d x2d x3d x4d x5d x6dT. Based on Lyapumov theory, the design process of classical back-stepping control (CBC) is given as follows:
Step 1. The tracking error is introduced as
E1= Xd X = ½ e1 e3 e5 e7 e9 e11T ð15Þ then the derivative of E1can be obtained as
_
E1= _Xd _X ð16Þ Consider the Lyapunov candidate V1as
V1= 1 2E
T
1E1 ð17Þ
where the V1is positive definite and the time derivative
is
_
V1= ET1E_1= ET1 X_d _X
ð18Þ For the purpose of stabilizing E1, a stabilizing
func-tion is designed as
a1= _Xd+ K1E1 ð19Þ _
X is substituted for a1, then the equation (18) can be rewritten as
_
V1ðE1Þ = ET1 X_d _Xd K1E1
= K1ET1E1ł 0 ð20Þ where K1= k½ 1 k3 k5 k7 k9 k11 is the parame-terized vector, and each parameter is a positive constant.
Step 2. The second tracking error is introduced as
E2= _X a1= _X _Xd K1E1 =½ e2 e4 e6 e8 e10 e12T
ð21Þ the derivative of E2can be obtained as
_
E2= €X _a1= f Xð Þ + g(X)U €Xd K1E_1+ fd ð22Þ The second Lyapunov candidate V2is considered as
V2= 1 2 E T 1E1+ ET2E2 ð23Þ the derivative of V2with respect to time is
_ V2ðE1, E2Þ = E1TE_1+ ET2E_2 = ET1X_d _X+ ET2X€ _a1 = ET 1ðE2 K1E1Þ + ET2 f Xð Þ + g(X)U €Xd K1E_1+ fd = K1ET1E1 ET1E2+ ET2 f Xð Þ + g(X)U €Xd K1E_1+ fd ð24Þ Step 3. In order to stabilizingE2, the control law of control inputs U is given as
U= g Xð Þ1 E1 f Xð Þ + €Xd+ K1E_1 fd K2E2
ð25Þ where K2= k½ 2 k4 k6 k8 k10 k12 is the parame-terized vector, and each parameter is a positive constant.
Substituting (25) into (24), the derivative of V2can
be rewritten as _
V2= K1ET1E1 K2ET2E2ł 0 ð26Þ namely _V2is negative definite. Thus, the nonlinear sys-tem (10) is asymptotically stabilized using the control law (25).
Hybrid robust adaptive controller for the
quadrotor UAV
Design of mass observer and position controller
For the quadrotor system with slow-varying mass, a mass observer based on the adaptive control theory is designed to estimate the real-time mass of the quadro-tor UAV. Because the system mass is only related to the control inputs ux, uy, and U1 from equation (25),
the mass observer is used for position controller of the control system only as shown in Figure 3. The design process of mass observer is given as follows:
Step 1. The system massm of ux, uy, and U1in
equa-tion (25) is replaced by mass estimator ^m, then
ux= ^ m U1 e1+ €x1d+ k1_e1 fd1 k2e2 ð Þ uy= ^ m U1 e3+ €x2d+ k3_e3 fd2 k4e4 ð Þ U1= ^ m CfCu e5+ g + €x3d+ k5_e5 fd3 k6e6 ð Þ 8 > > > > > > > < > > > > > > > : ð27Þ
and the nonlinear control functions g(x1), g(x2), and
g(x3) can further be rewritten as
^ g Xð 1Þ = ^ g xð Þ1 0 0 0 0 0 0 ^g xð Þ2 0 0 0 0 0 0 ^g xð Þ3 0 0 0 0 B @ 1 C A = 1 ^ mU1 0 0 0 0 0 0 m1^U1 0 0 0 0 0 0 1 ^ mCfCu 0 0 0 0 B @ 1 C A ð28Þ
where X1= xð 1 x2 x3Þ. Similar to g Xð Þ, the ^g xð Þ,1 ^
g xð Þ, and ^2 g xð Þ are nonzero as shown in Remark 1.3
Step 2. The observation error of the system mass is defined as
md= m ^m ð29Þ
the derivative of m with respect to time is approximate to zero, due to the slow change of the mass. Thus, the derivative of mdwith respect to time is
_
md= _m _^m = _^m ð30Þ Step 3. The Lyapunov candidate V3is considered as
V3= 1 2 X6 i = 1 e2i + m 2 d 2mkm ð31Þ the derivative of V3can be obtained as
_ V3= X6 i = 1 ei_ei+ mdm_d mkm = e2ð€x1 €x1d k1_e1 e1Þ + e4ð€x2 €x2d k3_e3 e3Þ + e6ð€x3 €x3d k5_e5 e5Þ X i = 1, 3, 5 kie2i =md m½e2ðk2e2 e1 k1_e1 €x1dÞ + e4ðk4e4 e3 k3_e3 €x2dÞ + e6ðk6e6 e5 k5_e5 €x3d gÞ _^mkm X 6 i = 1 kie2i ð32Þ
Step 4. In order to make the observation error con-vergence, the derivative of ^m with respect to time is designed as
_^
m = kmem ð33Þ
where km.0 is a mass adaptive parameter and em= e2ðk2e2 e1 k1_e1 €x1dÞ
+ e4ðk4e4 e3 k3_e3 €x2dÞ + e6ðk6e6 e5 k5_e5 €x3d gÞ
ð34Þ
Substituting (33) into (32), we have _
V3= X6 i = 1
kie2ił 0 ð35Þ
namely _V3 is negative semi-definite. Therefore, accord-ing to Lyapunov stability theorem, the mass observa-tion error md1 is asymptotically stabilized using the adaptive law (33).
The main external disturbance comes from the tor-ques around x, y, and z-axis, because the disturbing forces on the three axes are small in practical applica-tion of quadrotor. Thus, the disturbing forces fd1, fd2,
and fd3 can be ignored when the mass variation of
quadrotor is large. Therefore, the position controller, namely control law of control inputs ux, uy, and U1can
be rewritten as follows: ux= ^g xð Þ1 1ðe1 f xð Þ + €x1 1d+ k1_e1 k2e2Þ uy= ^g xð Þ2 1ðe3 f xð Þ + €x2 2d+ k3_e3 k4e4Þ U1= ^g xð Þ3 1ðe5 f xð Þ + €x3 3d+ k5_e5 k6e6Þ 8 < : ð36Þ
Substituting (36) into (24), the _V2ðei, ei + 1Þ, i = 1, 3, 5 can be obtained as follows:
_ V2ðe1, e2Þ = k1e21 k2e22ł 0 _ V2ðe3, e4Þ = k3e23 k4e24ł 0 _ V2ðe5, e6Þ = k5e25 k6e26ł 0 8 < : ð37Þ
Design of attitude controller
Due to the existence of the unpredictable external dis-turbance, a hybrid robust adaptive controller which is comprised of a PI controller and the hyperbolic tangent function of the second kind of error is added into atti-tude control law. The improved control law can be rep-resented as follows: U2= g xð Þ4 1 e7 f xð Þ + €x4 4d+ k7_e7 k8e8+ l1e7+ l2p1+ l7q1 ð Þ U3= g xð Þ5 1ðe9 f xð Þ + €x5 5d+ k9_e9 k10e10+ l3e9+ l4p2+ l8q2Þ U4= g xð Þ6 1ðe11 f xð Þ + €x6 6d+ k11_e11 k12e12+ l5e11+ l6p3+ l9q3Þ 8 < : ð38Þ
where l1, l3, and l5are the parameters of proportional, l2, l4, and l6are the parameters of integral, l7, l8, and l9 are the parameters of the hyperbolic tangent func-tion. The integral of ejðj = 7, 9, 11Þ with respect to time piði = 1;3Þ and the hyperbolic tangent function of ejðj = 8, 10, 12Þ qiði = 1;3Þ can be given as pi= Ðt 0ejð Þdtt ði = 1;3, j = 7, 9, 11Þ qi= ee2 ee2 ee2+ ee2 ði = 1;3, j = 8, 10, 12Þ 8 < : ð39Þ
HRAC proposed in this study is composed of a posi-tion controller (36) and an attitude controller (38). The position controller is designed by adding the mass observer (33) on the basis of the backstepping control law (25). The attitude controller is designed by adding PI controller and the hyperbolic tangent function on the basis of the backstepping control law (25).
Theorem 1: Considering Assumptions 1–4, if the sys-tem error is controlled with position controller (36) and attitude controller (38), the solutions to the nonlinear system of quadrotor (equation (10)) will be uniformly ultimately bounded.
Proof.For position controller: _V2 is negative semi-definite according to the equation (37). Thus, the posi-tion control system will be asymptotically stable according to Lyapunov theorem.
For attitude controller: Taking U2 as an example.
The Lyapunov candidate V2is
V2ðe7, e8Þ = 1 2 e 2 7+ e 2 8 ð40Þ the derivative of V2ðe7, e8Þ with respect to time is
_
V eð 7, e8Þ = k7e27
+ e8ðe7+ f (x4) + g xð ÞU4 2+ d4 €x4d k7_e7Þ
ð41Þ Substituting U2of (38) into (41), the _V eð 7, e8Þ can be rewritten as
_
V eð 7, e8Þ = k7e27 k8e28+ e8ðl1e7+ l2p1+ l7q1+ d4Þ ð42Þ Let A = l2p1+ l7q1+ d4, A is bounded because d4, l7q1, and l2p1 after limiting the amplitude are bounded. Equation (42) can be obtained as
_
V eð 7, e8Þ = k7e27 k8e28+ l1e7e8+ e8A ð43Þ the inequalities below are considered here,
l1e7e8ł l21 2e 2 7+ 1 2e 2 8 ð44Þ e8A ł 1 2ge 2 8+ g 2A 2 ð45Þ
where g is constants, and (43) can be represented as _ V ł 2 k7 l21 2 e27 2 2 k8 1 2 1 2g e28 2 + g 2A 2 ð46Þ Let c = min 2 k7 l21 2 , 2 k8 1 2 1 2g , d =g 2A 2, then _ V ł ce 2 7+ e28 2 + d = cV + d ð47Þ which implies V ł V tð Þ 0 d c ec ttð 0Þ+d c ð48Þ
According to the uniformly ultimately bounded the-orem of the nonvanishing perturbation,23errors e7and
e8are uniformly ultimately bounded. In the same way,
errors e9, e10, e11, and e12 are all uniformly ultimately
bounded with attitude controller (38). Since Xd and _Xd are bounded, the solutions to the nonlinear system of quadrotor (equation (10)) will be uniformly ultimately bounded.24 The quadrotor control scheme is shown in Figure 3.
The overall control performance of HRAC control method is much higher than that of CBC, ROBC, RGDI, and other control methods. This is mainly attributed to the following two aspects: On the one hand, the mass observation results was added in the position controller in HRAC, which can reduce the influence of mass parameter changes on position con-trol. On the other hand, in the HRAC control method, a hybrid robust controller was introduced into the atti-tude controller, which can reduce the influence of exter-nal disturbance on attitude control.
Simulation results
In order to validate the new controller HRAC, the simulation experiments of trajectory tracking by s-function of MATLAB/SIMULINK are implemented using RGDI, ROBC, and HRAC in this section. The physical and control parameters of the quadrotor UAV system are listed in Tables 1 and 2, respectively.
The desired inclined circular trajectory is generated using the following command:
xd= 2 sin 0:5t + 4ð Þ yd= 2 cos 0:5t + 4ð Þ zd= 2 cos 0:5t + 4ð Þ cd= 0 8 > > < > > : ð49Þ
Table 1. Parameters of the quadrotor UAV. Symbol Quantity Value m Mass of the body 1.555 kg l Length of the arm 0.225 m g Gravitational acceleration 9.81 m/s2 Ix Moment of inertia around x-axis 4.6 3 1023kgm2 Iy Moment of inertia around y-axis 4.6 3 1023kgm2 Iz Moment of inertia around z-axis 6.8 3 1023kgm2 b Thrust coefficient 2.8266 3 1025Ns2 d Drag coefficient 3.017 3 1027Ns2 Jr Rotor inertia coefficient 2.7335 3 10
the initial value of positions and Euler angles are cho-sen as x1(0) = 4, x2(0) = 6, x3(0) = 6, x4(0) = 0, x5(0) = 0, x6(0) = 0. The simulation is conducted based on four-order Runge-Kutta method with the sampling time fixed on Dt = 0:01s, and the simulation time is given as t = 20s. The value of variable mass is Dm = 0:02675kg=s, and the external disturbances are given as fx= fy= fz= 0:2N , Mx= My= Mz= 0:15Nm. Considering the practical application, the control per-formance requirements are as follows: the steady-state error of x, y, and z three-axis trajectory tracking is ł0.3 m, and the steady-state error of yaw angle is ł 1 rad. In the simulation process, the expected trajectory of UAV tracking were recorded using RGDI, ROBC, and HRAC control algorithms, respectively. The steady-state tracking accuracy of various control meth-ods is investigated under the influence of quality para-meter change and external disturbance.
Simulation results of trajectory tracking are shown in Figures 4 to 6, the tracking errors and the results of mass estimation are shown in Figures 7 and 8
respectively, Figures 9 to 11 show the control inputs with RGDI, ROBC, and HRAC.
As shown in Figures 4 to 6, the trajectory using HRAC is much closer to the desired trajectory than those using other two control methods, which indicates that HRAC can eliminate the effects from the slow-varying mass and the external disturbance more effec-tively than RGDI and ROBC. This can also been con-firmed by the studies on the tracking errors of trajectory as shown in Figure 7. The tracking errors of trajectory are very large on the z-axis using RGDI and ROBC, which demonstrates that the tracking accuracy on the z-axis can be affected by the varying mass of quadrotor seriously. In contrast, the tracking error on the z-axis using HRAC has been reduced by 57.5% and 71.4% than those using RGDI and ROBC, which indicates that the HRAC method has an excellent robustness to the mass varying. The steady-state track-ing errors of x, y, and z axes ustrack-ing HRAC are 0.12, Table 2. Parameters of the proposed controller.
Symbol Quantity Value k1~6 Control gains in the
position controller
diag(5, 4, 5, 4, 6.5, 6.5) k7~12 Control gains in the
attitude controller
diag(10, 8, 10, 8, 12, 12) l1~6 Control gains in the PI
controller
diag(10, 15, 10, 15, 10, 15) l7~9 Control gains in the
hyperbolic tangent function diag(2, 2, 2) km Mass adaptive parameter 105
Figure 4. Inclined circular trajectory tracking response using RGDI.
Figure 5. Inclined circular trajectory tracking response using ROBC.
Figure 6. Inclined circular trajectory tracking response using HRAC.
0.07, and 0.17 m, respectively and the steady-state error of yaw angle is 0.09 rad, which meets the requirements of control performance indicators.
In the process of simulation experiment, the real-time quality parameters of various control algorithms are also tested. From Figure 8, the real-time mass of the quadrotor that could not be estimated is always 1.555 kg using the RGDI and ROBC. Although there are some vibrations caused by external disturbance, the estimated value of mass using HRAC approaching the real mass incrementally over time with the average esti-mate error in 20 s being 6.4% of quadrotor’s own mass. As shown in Figures 9 to 11, each of the control inputs for the three controllers shows some vibrations at the beginning, and asymptotically gets stable after a second. Certainly there’s an argument to be made that the control inputs of the three controllers are
undersaturated and feasible because of the maximum value of the controller input U1 in the simulation experiments is no more than 20 N (The theoretical value of maximum thrust of the quadrotor is 40 N.).
Conclusions
In this paper, a mass observer is designed based on adaptive control theory, and a new controller HRAC is developed by combining the backstepping control Figure 7. Trajectory tracking errors with RGDI, ROBC, and
HRAC.
Figure 8. The real-time mass of quadrotor system.
Figure 9. Control inputs of the trajectory tracking with RGDI.
Figure 10. Control inputs of the trajectory tracking with ROBC.
method, hyperbolic tangent function and PI control with the designed mass observer. The performance of HRAC controller is achieved by controlling the flight of quadrotor UAV with mass-varying under external disturbance effectively. The boundedness of the non-linear system is verified by Lyapunov stability theory and uniformly ultimately bounded theorem. Simulation experiment results indicate that HRAC can accomplish the trajectory tracking of quadrotor accurately and be used as an effective flight controller in the actual quad-rotor UAV controlling.
Future recommendation
In the future work, the mass observer should be further optimized to improve the accuracy of mass estimation. At the same time, the estimation performance of the system for the abrupt change of quality parameters would be investigated comprehensively.
Declaration of conflicting interests
The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
Funding
The author(s) disclosed receipt of the following financial sup-port for the research, authorship, and/or publication of this article: This research was supported by the Science & Technology Project of Education Department of Jiangxi Provincei, China (GJJ191047; KJLD13100; GJJ191041; GJJ191052; GJJ191060).
ORCID iD
Laihong Zhou https://orcid.org/0000-0002-3434-2848
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