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Hindawi Journal of Robotics Volume 2021, Article ID 5216472, 12 pages https://doi.org/10.1155/2021/5216472 Research Article Disturbance Rejection Trajectory Tracking Control for an Unmanned Quadrotor Based on Hybrid Controllers 1 2 Xiaoming Ji and Zihui Cai Department of Electrical Engineering, Jiangsu College of Safety Technology, Xuzhou 221011, Jiangsu, China Beijing Satellite Manufacturing Co., Ltd, Beijing 100000, China Correspondence should be addressed to Xiaoming Ji; jxm27@163.com Received 9 June 2021; Revised 19 June 2021; Accepted 1 July 2021; Published 10 July 2021 Academic Editor: L. Fortuna Copyright © 2021 Xiaoming Ji and Zihui Cai. +is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. +e purpose of this article is to explore a dual-loop problem regarding the trajectory tracking control of a quadrotor unmanned aerial vehicle, applying a linear active disturbance rejection and conditional integrator sliding mode controller, namely, LARC- CISMC. +e quadrotor system model is derived through Newton–Euler method and consists of two subsystems. +e hybrid controller for position and attitude loops is constructed. An evaluation of the proposed controller is presented in comparison with the linear active disturbance rejection controller. Simulation comparisons and experimental tests illustrate that the proposed controller has a satisﬁed robustness and accuracy under lumped disturbances. in controller design. Moreover, the existing uncertain 1. Introduction conditions (external disturbances such as wind gusts), Recently, with the development of automation and un- modelling errors (internal uncertainty), and structural manned aerial vehicle, the researchers have focused on parameter variation make the design of the ﬂight control ﬂight controller design to enhance the performance of the architecture particularly challenging. Consequently, it is aircraft in operating autonomous ﬂight missions. +ey imperative to develop robust, eﬃcient, and high-per- have been applied in both civilian and military applica- formance control approaches capable of addressing all of tions such as aerial photography, target location, disaster the issues above and improving control performance and management, and agricultural care [1–3]. Quadrotors are stability. special kinds of UAVs that are capable of vertical take-oﬀs Recently, numerous achievements have been reported and landings, hovering at a low altitude, and ﬂying in all about the control methods for the quadrotors. In the closed- directions. In particular, the high levels of automation, loop control system of a microquadrotor, a linear quadratic controller is used to realize the goal of trajectory tracking ﬂexibility, and complexity of intelligent aerial equipment present new challenges to controller design. Signiﬁcantly, control [5]. Ma et al. proposed a model predictive control the control, operation, and interaction of the quadrotors method to solve the trajectory tracking problem under are quite challenging due to their inherently unstable, external disturbances [6]. A linear quadratic Gaussian with highly nonlinear, and underactuated dynamics [4]. +e integral action is applied to stabilize a quadrotor’s attitude external disturbances in the harsh environment are in- with good results in the hover phase [7]. Ansari and Bajodah evitable to impact the stability of the quadrotors, proposed a two-loop robust generalized dynamical inversion which leads to the diﬃcult controller design. Meanwhile, controller for quadrotor attitude and position control, which the controller is sensitive to the coupled dynamics of provides a robust trajectory tracking strategy subjected to the quadrotors. Besides the modelled characteristics, model uncertainties and parametric variations [8]. Despite unmodeled characteristics are another important consideration the fact that the aforementioned control strategies oﬀer a 2 Journal of Robotics (iii) Good robustness is provided against various lum- good balance among control performance, operational costs, and computational complexity, they tend to be restricted by ped disturbances, including modelling errors, ex- ternal disturbances, and noise measurements. the predictive mathematical models’ accuracy. Furthermore, the controllers are also vulnerable to lumped disturbances, An overview of this article can be found as follows. +e where the performance of these strategies will be signiﬁ- quadrotor system model is presented in Section 2. In Sec- cantly reduced under the uncertain conditions. tions 3 and 4, a hybrid controller is developed, as well as Researchers have also focused attention on model-free stability analysis. In Section 5, the results of simulations and control strategies that need no information of the model. For experiments are presented. +e main conclusions are pre- example, Tian et al. [9] proposed a multivariable super- sented in Section 6. twisting sliding mode control (SMC) method for ﬁnite-time attitude control of a quadrotor, and numerical simulation and experimental veriﬁcation illustrate the eﬃciency of the 2. System Description proposed controller. A high-performance trajectory tracking A quadrotor aircraft is driven by the propellers mounted at controller using the backstepping technique is developed for the ends of an X-shaped frame as shown in Figure 1. Each the quadrotor using a disturbance observer [10]. While propeller oﬀers thrust F (j � 1 ∼ 4) and moment relying only on a nominal model and its limits, the dis- j M (j � 1 ∼ 4). +e motion of the aircraft is described with turbance observer can estimate disturbances. Najm and j two coordinate frames, i.e., the earth-ﬁxed coordinate frame Ibraheem [11] used linear active disturbance rejection O X Y Z ({E}) and body-ﬁxed coordinate frame control (LADRC) method to stabilize the altitude and at- E E E E O X Y Z ({B}). +e symbols ϕ, θ, and ψ indicate the Euler titude of a quadrotor. LADRC has a linear extended dis- B B B B angles (roll, pitch, and yaw), respectively. ω (j � 1 ∼ 4) de- turbance observer which can reject the lumped disturbances. notes the propeller speed. However, there are two control loops (attitude loop and Based on the mechanical structure and driven mecha- position loop) in the quadrotor system. A single controller nism, the control signals of the quadrotor are calculated as may not be suitable for the cascade structure. Sometimes, the follows: outputs produced from the position loop may cause un- desirable transients, bringing unexpected damage to the τ � F + F + F + F , ⎧ ⎪ 1 1 2 3 4 system’s components. +erefore, approaches that consider a √� √� √� √� combination of multiple control strategies are favoured by 2 2 2 2 scholars. τ � F − F − F + F L, 2 1 2 3 4 ⎪ 2 2 2 2 Ding et al. [12] combined LADRC and integral back- stepping control to realize trajectory tracking of a multi- (1) √� √� √� √� copter. In the above control system, the LADRC has more ⎪ 2 2 2 2 advantages in rejecting the inner-loop disturbances, and the τ � F + F − F − F L, ⎪ 3 1 2 3 4 2 2 2 2 integral backstepping control can eliminate the static errors in the position loop well. Similarly, Mohd Basri et al. [13] proposed a hybrid controller combining the backstepping τ � M + M − M − M , 4 2 4 1 3 technique and adaptive fuzzy method to realize the complex where L is the distance between a propeller and the mass of trajectory tracking of a quadrotor, which eﬀectively sup- the aircraft and τ � [τ , τ , τ , τ ] is the control signal. 1 2 3 4 presses the time-varying perturbations. To stabilize the dual- +e rigid body dynamical model of the quadrotor is loop state variables of a quadrotor, a novel controller derived by applying Newton–Euler’s equations [15]: combining robust generalized dynamic inversion and adaptive nonsingular terminal sliding mode is proposed ⎧ ⎨ mp € � R τ Z − mgZ + F , 1 B I D (2) [14]. Motivated by the above controllers, we will develop a T T Iq € � τ τ τ − q _ Iq _ , hybrid dual-loop controller for a quadrotor. +e main goal 2 3 4 in this article is to demonstrate and show an improvement in where m is the mass of the aircraft, g is the gravitational the tracking performance of a quadrotor control by a hybrid acceleration, q � [ϕ, θ, ψ] is the Euler angle vector, p � approach. +e main contributions of this article are listed as [x, y, z] is the position vector, I � diag(I , I , I ) is the x y z follows: rotational inertia, and Z and Z are the unit vector in the B I (i) +e principle and composition of the hybrid con- body-ﬁxed coordinate frame and earth-ﬁxed coordinate troller are studied. More specially, a LADRC is frame, respectively. +e matrix R maps from earth-ﬁxed introduced to stabilize the position loop, and a coordinates to body-ﬁxed coordinates, which is governed by CISMC is used to stabilize the attitude loop. yaw-pitch-yaw Euler angles: (ii) +e proposed controller is investigated in terms of cθcψ sϕsθcψ − cϕsψ cϕsθcψ + sϕsψ simulation and real-world application in the context ⎢ ⎥ ⎡ ⎢ ⎤ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ B ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ R � ⎢ cθsψ sϕsθsψ + cϕcψ cϕsθsψ − sϕcψ ⎥, (3) ⎢ ⎥ of quadrotor trajectory tracking. To the best of the ⎢ ⎥ E ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ authors’ knowledge, no reports on the LADRC- −sθ sϕcθ cϕcθ CISMC trajectory tracking control technique for quadrotors are available until now. where sα and cα are the simpliﬁed forms of cos α and sin α. Journal of Robotics 3 F F k k k k 4 1 t t t t ⎡ ⎢ ⎤ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 2 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ω τ ⎢ ⎥ ⎢ √� √� √� √� ⎥ ⎢ ⎥ 1 1 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎡ ⎢ ⎤ ⎥ ⎢ ⎥⎢ ⎥ ⎡ ⎢ ⎤ ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ 2 2 2 2 ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ Z ⎢ ⎥ ⎢ k L − k L − k L k L ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ B ⎢ ⎥ ⎢ t t t t ⎥⎢ 2 ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ M ω ω M ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎥ ⎢ ⎥⎢ ⎥ 4 4 1 1 ⎢ τ ⎢ 2 2 2 2 ⎥⎢ ω ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ 2 ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ 2 ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ � ⎢ ⎥⎢ ⎥, ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎥⎢ ⎢ ⎥ ⎢ √� √� √� √� ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎢ 2 ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ τ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ω ⎥ ⎢ 3 ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ 2 2 2 2 ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ 3 ⎥ ψ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎣ ⎦ ⎢ k L k L − k L − k L ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎢ t t t t ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 2 2 2 2 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 2 ⎢ ⎥ τ ⎥ ⎢ ⎥ ⎢ ⎥ 4 ⎢ ω ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 4 ⎣ ⎦ F B {B} −k k −k k m m m m (5) B Y where k B and k are termed as thrust coeﬃcient and moment t m M ω 2 3 3 mg coeﬃcient, respectively, and L is the distance between the center of the propeller hub and the quadrotor mass counter. +e thrusts and moments applied to the quadrotor can be calculated by the propeller speed as follows: {E} 2 2 ⎧ ⎪ F � ρ k AR ω , ⎨ j t a j E (6) 3 2 M � ρ k AR ω , j a m j E Y Figure 1: Free body diagram of quadrotor. where ρ denotes the density of air, A represents the disk area of the propeller, and R is the radius of the propeller. +e dynamic model (4) can be simpliﬁed when the Furthermore, model (2) can be described into transla- quadrotor ﬂies in hover or in low-speed cruise [15]. At this tional and rotational submodels that denote its linear and case, sin θ ≈ θ, sin ϕ ≈ ϕ, cos θ ≈ 1, and cos ϕ ≈ 1. Mean- angular positions, respectively: while, due to the fact that the small rotary inertia and τ (cψsθcϕ + sψsϕ) symmetrical structure of the quadrotor result in a small and ⎧ ⎪ 1 x € � , ⎪ T insigniﬁcant term q _ Iq _ [16], hence, model (4) can be reduced m − k x _|x _|/m into ⎪ τ (sψsθcϕ − cψsϕ) ⎪ mx € � τ (θ cos ψ + ϕ sin ψ), y € � , ⎪ ⎧ ⎪ 1 ⎪ ⎪ _ _ ⎪ m − k y|y|/m ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ my € � τ (θ sin ψ − ϕ cos ψ), ⎪ ⎪ ⎪ ⎪ τ cϕcθ ⎪ € z � , ⎪ ⎪ ⎪ ⎪ ⎪ m − g − k z _|z _|/m ⎪ z ⎪ ⎪ ⎪ m€ z � τ − mg, ⎪ ⎪ ⎨ ⎪ (4) ⎪ ⎪ ⎪ _ ⎨ τ + θψI − I 2 y z τ ⎪ 2 ⎪ ϕ � , (7) ϕ � , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ _ ⎪ 3 τ + ϕψ I − I € ⎪ ⎪ 3 z x θ � , ⎪ € ⎪ θ � , ⎪ ⎪ I ⎪ y ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ τ ⎪ 4 ⎪ € _ ⎩ ψ � . τ + ϕθI − I 4 x y ⎩ ψ � , +e control structure of the quadrotor is illustrated in where k , k , and k are the air drag coeﬃcient. +e Figure 2. +e system is composed of two loops, a position x y z mathematical relation between the control inputs and an- loop and an attitude loop. Furthermore, the closed-loop gular velocities is given by system has four channels: altitude channel, position channel, 4 Journal of Robotics Altitude z channel ϕ 2 y τ r r Attitude 3 Referenced Position + channel trajectories channel Quadrotor Attitude control loop Position control loop ψ Yaw channel Figure 2: Control structure of the quadrotor. attitude channel, and yaw channel. It is evident that the position x _ � x , s s ⎧ ⎪ 1 2 channel and the attitude channel are interdependent; namely, x _ � x + b u, s s 0 2 3 the outputs of the position channel are the reference of the (10) attitude channel. +e controller design in our work requires no x _ � h, ⎪ s ⎪ 3 information of the dynamical model and relies only on input y � x , s s and output data. Besides, each state variable in (7) is related to the control input as a second-order derivative. _ with x � f added as an augmented state, h � f . s L L Transform model (9) into state-space form as 3. Position Loop Control Strategy x _ � A x + B u + E h , s s s s s s s (11) +e purpose of the position control loop is to stabilize the y � C x , s s d states x, y, z and ensure their tracking errors approach zeros. However, there are lumped disturbances in the position loop where which can reduce the control performance. LADRC in- 0 1 0 corporates lumped disturbances into system states, removes ⎡ ⎢ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ disturbances by a linear extended state observer (LESO), and ⎢ ⎥ ⎢ ⎥ A � ⎢ 0 0 1 ⎥, ⎢ ⎥ s ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ uses feedback control law so that no detailed and accurate 0 0 0 mathematical modelling is required. Consider a second- order system is in its general form as [17] ⎡ ⎢ ⎤ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ x € � f x , x _ , w , t + b(t)u, ⎢ ⎥ ⎢ ⎥ B � ⎢ b ⎥, s s s s s ⎢ ⎥ s ⎢ 0 ⎥ ⎢ ⎥ ⎣ ⎦ (8) (12) y � x , s s where u and y are system input and system output, re- C � 1 0 0 , s spectively, f are the lumped disturbances, x are state s s variables, and b(t) are the control terms. ⎢ ⎥ ⎡ ⎢ ⎤ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥. E � ⎢ 0 ⎥ ⎢ ⎥ s ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ Remark 1 (see [18]). +e lumped disturbances f are an- other type of the term f (x , x _ , w , t) that includes para- s s s s metric variations, environmental disturbances, and complex unmodeled features [18]. Remark 2. Compared with (7), system (9) represents a general class of systems since it is not limited to the integral- chained form. Assumption 1. Suppose that f is bounded and diﬀeren- +en, we design the following disturbance observer, tiable. As a result, one gets ‖f ‖<∞, ‖f ‖<∞, and their L L which is called LESO [19], to estimate the state that contains bounds satisﬁes sup ‖f ‖ � f and sup ‖f‖ � h. L b the lumped disturbances: t>0 t>0 Rewrite (8) as z � A z + B u + L y − y , s s s s s s s s (13) x � f + b(t)u, y � C z , s s s (9) y � x . s s where z � [z , z , z ] denotes the estimation of the s 1 2 3 state [x , x , f ]. +e observer gain L can be calculated A diﬀerential equation can be derived from the above s1 s2 L s equation by rewriting it as follows: as Journal of Robotics 5 l α ω λ � ω , 1 1 o ⎢ ⎥ ⎢ ⎥ ⎡ ⎢ ⎤ ⎥ ⎡ ⎢ ⎤ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ (22) ⎢ ⎥ ⎢ 2 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ L � ⎢ l ⎥ � ⎢ α ω ⎥, (14) ⎢ ⎥ ⎢ ⎥ λ � 2ω . s ⎢ 2 ⎥ ⎢ 2 o ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ 2 c l α ω 3 3 o Next, the stability of LADRC will be proved. +e tracking errors of the position loop are deﬁned as where ω > 0 denotes the observer bandwidth. α , α , and α are o 1 2 3 follows: selected to ensure the eigenvalues of (A − L C ) are in the left s s s complex plane. +e characteristic equation can be given as e � x − z . (23) s s 3 2 3 (15) λ(s) � s + α s + α s + α � s + ω . 1 2 3 o +e error matrix is obtained by subtracting (10) from (9): From equation (15), the coeﬃcients are calculated as e_ � A e + E h , (24) s s s α � 3, where α � 3, (16) −l 1 0 ⎡ ⎢ ⎤ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ α � 1. ⎢ ⎥ ⎢ ⎥ 3 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ A � A − L C � ⎢ −l 0 1 ⎥. (25) ⎢ ⎥ ⎢ ⎥ e s s s 2 ⎥ ⎣ ⎦ −l 0 0 Remark 3. All the eigenvalues λ(s) of A are in the left half- Obviously, the LESO is a bounded-input bounded- plane, and the LESO is a bounded-input bounded-output output (BIBO) stable system since all characteristic poly- (BIBO) system. It is necessary to select an appropriate nomial roots are located in the left half-plane. observer gain L to ensure A is Hurwitz [20]. s s Theorem 1. 1e LADRC design process from (8)–(22) leads Remark 4. Suppose the derivative of f in (10) meets the to a BIBO stable closed-loop system since the control law and condition lim ‖f ‖ � 0; the estimation error in (13) will LESO are stable. t⟶∞ approach zero asymptotically. Recalling (11)∼(14), one has +e disturbance observer tracks the system states with z − r 1 e appropriate observer bandwidth and yields z (t) ⟶ y(t), ⎡ ⎢ ⎤ ⎥ ⎢ ⎥ ⎢ ⎥ _ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ z (t) ⟶ y (t), and z (t) ⟶ f . ⎢ ⎥ 1 ⎢ ⎥ 2 3 L ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ u � −λ −λ −1 ⎢ z ⎥ ⎢ ⎥ ⎢ ⎥ According to the feedback linearization method, we can 1 2 ⎢ 2 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 0 ⎢ ⎥ ⎣ ⎦ oﬀset the total disturbance by simply deﬁning the controller as (26) u − z 0 3 u � , (17) ⎡ ⎢ ⎤ ⎥ ⎢ ⎥ b ⎢ ⎥ ⎢ ⎥ 0 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 1 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ � Qz − ⎢ 0 ⎥ � Qz − G, ⎢ ⎥ s ⎢ ⎥ s ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ where we need to determine the error feedback u and by b ⎢ ⎥ ⎢ ⎥ 0 ⎢ ⎥ 0 ⎣ ⎦ substituting (17) into system (10), one has y € � f − z + u , (18) L 3 0 where Q � (1/b ) −λ −λ −1 . 0 1 2 Rewriting the closed-loop system, one gets where r is the reference. +e estimation error can be ig- nored if the disturbance observer is approximately treated as x _ � A x + B u + E h � A x + B Qz − B G + E h , s s s s s s s s s s s s s s an ideal observer. +en, the relationship between y and u _ z � A z + B u + L y − y s s s s s s s s becomes a simple linear double-integrator: � L C x + A − L C + B Qz − B G. s s s s s s s s s y € ≈ u . (19) (27) Here, the lumped disturbances are estimated and eliminated. And the control law u is +e above system can be written into a state-space formula as follows: u � λ r − z − λ z . (20) 0 1 e 1 2 2 x _ A B Q −B E x G s s s s s s s A second-order plant also has the following transfer � + . z _ L C A − L C + B Q z −B 0 h function: s s s s s s s s s s (28) G � . (21) 2 2 +e controller is BIBO stable if the eigenvalues of (28) s + λ s + λ 2 1 are in the left half-plane. It is obvious that h is bounded Here, the gains can be calculated as [21] since f is diﬀerentiable. L 6 Journal of Robotics According to the design concept of the LADRC, there are x LADRC four independent controllers in the position loop, which is _ depicted in Figure 3. LADRC 4. Attitude Loop Controller Designs y LADRC In the attitude loop, the control goal is to stabilize the states ϕ, θ, ψ and ensure their tracking errors approach zeros. +e CISMC derived from SMC is robust, accurate, and easier to LADRC implement when compared to other controllers [22]. +e ψ integral term in CISMC can increase the accuracy of static Figure 3: Control structure of the position loop using LADRC. error and hold the transient response. Consider a second- order dual-integral system: € where Δ � −k σ + k μ tanh(s/μ) + k e_ + f + bu − x € and x � F + bu, a a 0 0 1 ar (29) x € denotes the referenced signals. ar y � x , a a Consider a Lyapunov function: where x denotes the system variable, u is the control input, (35) V � s , y is the system output, b is the control gain, and F is the a a system function. and its ﬁrst derivative is Design sliding mode surface as V � ss. _ (36) s � k σ + k e + e, _ (30) 0 1 Substituting (34) into the above formula, one has where e and e are the tracking error and its derivative, k and k are positive numbers, and σ denotes the output of the s s V � sΔ − k tanh ≤ |s||Δ| − ks tanh . (37) “conditional integrator” which is described as follows: μ μ σ � −k σ + μ tanh . (31) 0 Suppose the error and its derivative are bounded and satisfy that |s|≤ c (c> μ); then (37) can be rewritten as where tanh(·) is the hyperbolic tangent function which can V≤ |s||Δ| − ks tanh ≤ (|Δ| − k)|s|. (38) degree the chattering well [23] and μ is the boundary and results from the continuous approximation of the hyperbolic tangent function. +e CISMC by means of the conditional Select an appropriate control gain k to satisfy the integrator ensures an improvement in transient response of condition ISMC and recovers the SMC performance without chattering. k≥ ρ + max(|Δ|), (39) Since |s(t)|≥ μ, the integral loop (31) will be converted into an exponential stability system with external inputs ±μ. where ρ is a positive number. In this article, we select ρ � 1.5. Since |s(t)|< μ, the integral loop (31) will be converted into Combining (38) and (39), we can obtain that V≤ 0. an error function: +is implied that the tracking error of the system will σ _ � k e + e. _ (32) asymptotically decrease by using the proposed control law CISMC. A conditional integral sliding mode controller can be introduced to provide robust regulation: Remark 5 (see [24]). In equation (39), the smaller boundary Δ can be obtained by selecting the larger parameters k. +e v � −k tanh , (33) larger parameters k are chosen, the smaller boundary Δ will be reached [24]. where k> 0 represents a control gain which is selected With the design concept of CISMC, the attitude con- suﬃciently large to suppress uncanceled terms in s_ and μ is trollers can be obtained. +e tracking errors of the attitude selected suﬃciently small to recover the performance of ideal loop are expressed as SMC without an integrator. +e stability analysis of the conditional integral sliding e ϕ − ϕ 1 d ⎡ ⎢ ⎤ ⎥ ⎢ ⎥ ⎢ ⎥ ⎡ ⎢ ⎤ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ mode controller will be proved as follows. ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ θ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ θ − θ ⎥ ⎢ ⎥ � ⎢ ⎥. (40) ⎢ ⎥ ⎢ ⎥ ⎢ e ⎥ ⎢ d ⎥ ⎢ 1 ⎥ ⎣ ⎢ ⎥ ⎣ ⎦ Substituting (29), (31), and (32) into the derivative of the ψ − ψ sliding mode surface, one gets e d +e conditional integrator sliding mode surfaces de- s � Δ − k tanh , (34) duced by the tracking errors are given using (30): Journal of Robotics 7 ϕ ϕ ϕ ϕ s k σ + k e + e 2 ϕ 0 ϕ 1 1 1 ⎡ ⎢ ⎤ ⎥ CISMC ⎢ ⎥ _ ⎢ ⎥ ⎢ ⎥ ⎡ ⎢ ⎤ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ + ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ϕ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ θ θ θ θ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ θ ⎢ s ⎥ � ⎢ ⎥, (41) ⎢ ⎥ ⎢ ⎥ r ⎢ θ ⎥ ⎢ k σ + k e + e_ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ θ ⎥ τ ⎢ ⎥ ⎢ 0 1 1 1 ⎥ ⎢ ⎥ ⎢ ⎥ 3 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ CISMC ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎢ ⎥ _ ⎢ ⎥ ⎣ ⎦ ψ ψ ψ ψ ψ k σ + k e + e_ 0 ψ 1 1 1 Figure 4: Control structure of the attitude loop using CISMC. ϕ ϕ ⎢ ⎥ ⎡ ⎢ −k σ + μ tanh ⎤ ⎥ ⎢ ⎥ ⎢ 0 ϕ ϕ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ μ ⎢ ⎥ ⎢ ⎥ ⎢ ϕ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 2 2 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ are m � 1.515 kg, I � I � 0.0211 kgm , I � 0.0366 kgm , ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ x y z ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ σ _ ⎢ ⎥ ⎢ ⎥ − 5 − 7 ϕ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ s ⎥ L � 0.63m, k � 1.681 × 10 , k � 2.783 × 10 , ⎢ ⎥ ⎢ ⎥ ⎢ ϕ ⎥ t m ⎡ ⎤ θ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 3 2 ⎢ ⎥ ⎢ −k σ + μ tanh ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ _ ⎥ ⎢ 0 θ θ ⎥ ⎢ σ ⎥ � ⎢ ⎥. (42) ρ � 1.29 kg/m , A � 0.17 m , and R � 0.23 m. It should be ⎢ ⎥ ⎢ ⎥ ⎢ θ ⎥ ⎢ ⎥ a ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎢ μ ⎥ ⎢ ⎥ ⎢ ϕ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ noted that the structural parameters can be identiﬁed by ⎢ ⎥ ⎢ ⎥ _ ⎢ ⎥ σ ⎢ ⎥ ⎢ ⎥ ψ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ direct and indirect measurement approaches [25, 26], and ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ s ⎥ ⎢ ⎥ ⎢ ψ ⎥ ⎢ ψ ⎥ these parameters are derived from the actual quadrotor. ⎣ ⎦ −k σ + μ tanh 0 ψ ψ μ +e purpose of the simulation is to ﬁnd suitable pa- rameters for the controller using adaptive fruit ﬂy opti- +e ﬁrst-order derivative of the sliding mode surfaces mization algorithm (AFOA) [27]. +e references using can be calculated as step signals are set as [x, y, z] � [10 m, 10 m, 10 m] and ∘ ∘ ∘ ϕ ϕ ϕ ϕ [ϕ, θ, ψ] � [60 , −60 , 30 ]. During parameter tuning, the s _ k σ _ + k e _ + € e 0 ϕ 1 1 1 ⎢ ⎥ ⎡ ⎢ ⎤ ⎥ attitude loop is tuned ﬁrst because the position loop ⎢ ⎥ ⎢ ⎥ ⎡ ⎢ ⎤ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ θ θ θ θ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ s _ ⎥ ⎢ ⎥ ⎢ ⎥ � ⎢ ⎥. (43) cannot work independently. +e tuned controller pa- ⎢ ⎥ ⎢ _ _ € ⎥ ⎢ θ ⎥ ⎢ k σ + k e + e ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0 θ 1 1 1 ⎥ ⎣ ⎦ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ rameters of the attitude loop are applied for the position ψ ψ ψ ψ ψ k σ _ + k e_ + € e 0 1 1 1 loop. In addition, an improved error function [28] is introduced as an evaluation index to evaluate the opti- +e control laws of the attitude loop can be obtained by mized result: synthesizing (7), (42), and (43): ∞ ∞ s F(t) � α 2 t|e(t)|dt + α (u(t)) dt + α t + α o, ϕ ϕ ϕ ϕ 1 2 3 s 4 ⎣ ⎦ ⎡€ ⎤ ⎡ ⎢ I ϕ + k σ − k μ tanh − k e_ + v ⎤ ⎥ 0 0 ⎢ ⎥ ⎢ x r 0 ϕ 0 ϕ 1 1 ϕ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ μ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ϕ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ (45) ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ U ⎢ ⎥ ⎢ ⎥ 2 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ where e(t) denotes the tracking error of the system state, t ⎢ ⎥ ⎢ ⎥ ⎡ ⎢ ⎤ ⎥ ⎢ ⎥ s ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ s ⎥ ⎢ ⎢ 2 ⎥ ⎢ ⎥ ⎢ ϕ ⎥ ⎢ ⎥ ⎥ ⎢ θ θ θ θ ⎥ ⎢ ⎢ ⎥ ⎢ ⎥ ⎢ ⎣€ ⎦ ⎥ ⎢ ⎥ ⎢ ⎡ ⎤ ⎥ ⎢ ⎥ ⎢ ⎥ denotes the settling time, o denotes the overshoot, and ⎢ ⎥ ⎢ I θ + k σ − k μ tanh − k e_ + v ⎥ ⎥ ⎢ ⎥ ⎢ U ⎥ ⎢ y r 0 θ 0 θ 1 1 θ ⎥ s ⎢ ⎢ ⎥ � ⎢ ⎥, ⎥ ⎢ ⎥ ⎢ 3 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ μ ⎥ ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ ⎢ ϕ ⎥ ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ ⎥ α , α , α , and α are the weight factors. ⎣ ⎦ ⎢ ⎥ ⎢ ⎥ 1 2 3 4 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ Lumped disturbances simulated by random signals are ⎢ ⎥ U ⎢ ⎥ ⎢ ⎥ 4 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ added to the output ports of the position loop. +e simulation ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎢ ⎥ s ⎥ ⎢ 2 ⎥ ⎢ ψ ⎥ ⎢ ψ ψ ψ ψ ⎥ ⎣ ⎦ time of the parameters tuning in attitude loop and position ⎣ ⎦ ⎡ ⎤ € _ I ψ + k σ − k μ tanh − k e + v z r 0 ψ 0 ψ 1 1 ψ loop is 5 s, and sampling time is 1 ms. +e simulation results are summarized in Figures 5–8. We observe that the AFOA can help LADRC-CISMC to ﬁnd suitable control parameters. −k tanh ⎡ ⎢ ⎤ ⎥ ⎢ ϕ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ More speciﬁcally, Figure 5 shows the system’s attitude re- ⎢ μ ⎥ ⎢ ⎥ ⎢ ϕ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ sponse, while the static tracking errors approach zero. It in- ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ v ⎢ ⎥ ⎢ ⎥ ϕ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ dicates the LADRC-CISMC presents a good tracking ⎢ ⎥ ⎢ ⎥ ⎡ ⎢ ⎤ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ s ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ϕ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ performance during the transient response. Figure 7 shows the ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ −k tanh ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ v ⎥ ⎢ θ ⎥ ⎢ ⎥ � ⎢ ⎥. ⎢ ⎥ ⎢ ⎥ ⎢ θ ⎥ ⎢ ⎥ ⎢ ⎢ μ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ϕ ⎥ ⎢ ⎥ ⎢ ⎥ system’s position response, wherein x, y, z track the reference ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎥ ⎢ ⎥ ⎦ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ well despite existing lumped disturbances. +e results dem- ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ v ⎢ ⎥ ⎢ ⎥ ψ ⎢ ⎢ ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ onstrate that the LADRC-CISMC is capable of superior per- ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ s ⎥ ⎢ ⎥ ⎢ ψ ⎥ ⎣ ⎥ formance under lumped disturbances. Figures 6 and 8 show −k tanh the iteration curves of AFOA. As can be observed, the IITAE index has the ability to perform in-depth data mining, whereas (44) the AFOA is capable of helping LADRC-CISMC to obtain +e control structure of the attitude loop using CISMC is desired tracking performance. Furthermore, the simulation depicted in Figure 4. provides a set of probable control parameters that can be manually adjusted during the experimental testing phase. +e optimized control parameters of the proposed 5. Simulation and Experimental Results controller by AFOA are listed in Table 1. 5.1. Simulation Results. In this section, a simulator devel- oped with the MATLAB/SIMULINK environment has been used to test the proposed hybrid controller for a 5.2. Experimental Results. In this section, an experimental quadrotor system. +e main parameters of the quadrotor ﬂight test has been conducted to validate the performance of 8 Journal of Robotics 80 200 –20 –40 –60 –80 0 12345 Time (s) 010 20 30 40 50 ϕ ϕ Iteration Figure 8: Iteration curve of the LADRC of the position loop. Figure 5: Step response of the attitude loop. Table 1: Tuned control parameters of the LADRC-CISMC. Parameters Value Parameters Value ω 58.06 ω 677.72 o−x c−x ω 41.02 ω 520.10 o−y c−y ω 16.45 ω 553.41 o−z c−z ϕ ϕ k 1.47 k 4.69 0 1 k 64 μ 0.93 ϕ ϕ θ θ k 1.25 k 2.78 0 1 k 77 μ 0.82 θ θ ψ ψ k 1.77 k 7.19 0 1 k 53 μ 0.41 ψ ψ our proposed controller. As depicted in Figure 9, the ex- perimental platform includes an X450 quadrotor, a remote controller, a ground operational station, a global posi- 02 10 0 30 40 50 tioning system (GPS), and a ﬂight controller called Pix- Iteration hawk. +e quadrotor is equipped with a ﬂight controller that includes an embedded processor, sensors and pro- Figure 6: Iteration curve of the LADRC of the attitude loop. pulsion components, and a wireless data link connected to a device capable of transmitting data wirelessly. Ground station programs process ﬂight data for information ex- traction and retrace data regarding the quadrotor’s status. +e quadrotor’s actual position is collected by GPS. Meanwhile, the quadrotor adopted HOBBYWING eagle 20A electronic speed controllers (ESC), SUNNYSKY A2212-980 kV brushless motors, 2200 mAh lithium battery, and 9450 carbon ﬁbre propellers. +e experimental data are collected at a frequency of 50 Hz, which is transformed into the ground operational station. +e proposed hybrid controller has been embedded into the Pixhawk hardware from the simulation platform by a plug-in named PX4 autopilots support [29]. We use a Lissajous curve in the form of [sin(0.5t), sin(0.25t), 9.5]m [30] as the reference tracking 0 1 25 34 trajectory. +e wind gusts are measured by an AR816 digital Time (s) anemometer, and the maximum gust measured in the ex- x x periments is 3.5 m/s. Meanwhile, the performance of the proposed LADRC-CISMC compared to the PID is dis- cussed. +e optimized control parameters of the PID are Figure 7: Position response under step input. listed in Table 2. Figures 10–16 depict the results of 20 Attitude (degrees) Position (m) IITAE IITAE Y (m) Journal of Robotics 9 Remote Ground control station Flight tests Commands Flight data Sensor Flight ESC data control processing algorithms Control instruction receiving and processing Motor Flight controller Figure 9: Frame of the experimental platform. Table 2: Tuned control parameters of the LADRC. seconds of experimental data collection (LADRC-CISMC is called controller 1; PID is called controller 2). It is shown in Parameters Value Parameters Value Figure 10 that the quadrotor can follow a predetermined ω 500.00 ω 54.55 o−x c−x trajectory. However, the position controlled by PID ﬂuc- ω 499.10 ω 50.70 o−y c−y tuates obviously compared to the hybrid controller in the ω 386.45 ω 23.23 o−z c−z presence of the lumped disturbances. It indicates that the ω 222.78 ω 91.67 o−ϕ c−ϕ PID control method has no disturbance rejection ability. ω 310.02 ω 95.00 o−θ c−θ Due to the delay in GPS signals, the actual position is slightly ω 457.24 ω 78.11 o−ψ c−ψ diﬀerent from the predetermined trajectory. Nevertheless, our controller performs better than PID. Figures 11–16 show the behavior of the quadrotor’s pose, and it is evident a reduction in position and Euler angles amplitude in our 9.8 proposed controller compared to PID. 9.6 Furthermore, two evaluation functions of maximum error 9.4 e and mean square error e [31, 32] are proposed to max mse 9.2 evaluate the performance of the LADRC-CISMC (controller 1) and the LADRC (controller 2), which are described as follows: 0.5 0.5 ⎧ ⎪ e � maxl , ⎪ max i –0.5 -0.5 –1 –1 �������� � Reference (46) ⎪ Controller 1 ⎪ 1 e � l , Controller 2 ⎪ mse i i�1 Figure 10: 3D tracking trajectory of the quadrotor. where l is the gap between the referenced and experi- mental trajectories at i sampling time and N is the 16.81% less than that of controller 2, respectively. It sampling length. +e calculated results are listed in indicates that the proposed controller has a higher Table 3. +e maximum error value of controller 1 in the tracking precision than the LADRC. position loop is 67.68%, 87.56%, and 22.17% less than that It can be concluded from the above comparison ex- of controller 2, respectively. +e mean square error of perimental results that the LADRC-CISMC can achieve controller 1 in the position loop is 55.67%, 87.56%, and greater performance than PID in trajectory tracking. X (m) Z (m) 10 Journal of Robotics 1 4 0.5 –2 –0.5 –4 –1 –6 05 10 15 20 0 5 10 15 20 Time (s) Time (s) Reference Controller 1 Controller 1 Controller 2 Controller 2 Figure 14: θ attitude quadrotor response. Figure 11: x position quadrotor response. 1 4 0.5 –1 –0.5 –2 –3 –1 –4 01 5 0 15 20 01 5 10 5 20 Time (s) Time (s) Reference Controller 1 Controller 1 Controller 2 Controller 2 Figure 15: ϕ attitude quadrotor response. Figure 12: y position quadrotor response. 0.5 9.8 9.6 9.4 –0.5 9.2 0 5 10 15 20 –1 05 10 15 20 Time (s) Time (s) Reference Controller 1 Controller 1 Controller 2 Controller 2 Figure 13: z position quadrotor response. Figure 16: ψ attitude quadrotor response. Z (m) Y (m) X (m) ψ (degrees) ϕ (degrees) θ (degrees) Journal of Robotics 11 Table 3: Comparison of tracking errors of the two controllers. u : Error feedback Controller t : Settling time State Index o : Overshoot Controller 1 Controller 2 α , α , α , α : Weight factors. 1 2 3 4 e 0.1734 0.2562 max e 0.0736 0.1322 mse e 0.1970 0.2250 Data Availability max e 0.1105 0.1262 mse +e data used to support the ﬁndings of this study are e 0.0217 0.0979 max available from the corresponding author upon request. e 0.0079 0.0470 mse Conflicts of Interest Moreover, the proposed hybrid controller exhibits a higher ability to reject lumped disturbances. +e authors declare that they have no conﬂicts of interest. 6. 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Journal of Robotics – Hindawi Publishing Corporation
Published: Jul 10, 2021
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