Disturbance observer-based backstepping control of tail-sitter UAVs

actuators Article Disturbance Observer-Based Backstepping Control of Tail-Sitter UAVs Nihal Dalwadi 1,† , Dipankar Deb 1,*,† , Mangal Kothari 2,† and Stepan Ozana 3,†   Citation: Dalwadi, N.; Deb, D.; Kothari, M.; Ozana, S. Disturbance Observer-Based Backstepping Control of Tail-Sitter UAVs. Actuators 2021,10, 119. https://doi.org/ 10.3390/act10060119 Academic Editor: William MacKunis, Muhammad Rehan Received: 7 April 2021 Accepted: 29 May 2021 Published: 3 June 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 1Department of Electrical Engineering, Institute of Infrastructure Technology Research and Management (IITRAM), Ahmedabad 380026, India; [email protected] 2Department of Aerospace Engineering, Indian Institute of Technology, Kanpur 380026, India; [email protected] 3Department of Cybernetics and Biomedical Engineering, Faculty of Electrical Engineering and Computer Science, VSB-Technical University of Ostrava, 17. listopadu 2172/15, 708 00 Ostrava-Poruba, Czech Republic; [email protected] *Correspondence: dipankar[email protected] † These authors contributed equally to this work. Abstract: The application scope of unmanned aerial vehicles (UAVs) is increasing along with commensurate advancements in performance. The hybrid quadrotor vertical takeoff and landing (VTOL) UAV has the benefits of both rotary-wing aircraft and fixed-wing aircraft. However, the vehicle requires a robust controller for takeoff, landing, transition, and hovering modes because the aerodynamic parameters differ in those modes. We consider a nonlinear observer-based backstepping controller in the control design and provide stability analysis for handling parameter variations and external disturbances. We carry out simulations in MATLAB Simulink which show that the nonlinear observer contributes more to robustness and overall closed-loop stability, considering external disturbances in takeoff, hovering and landing phases. The backstepping controller is capable of decent trajectory-tracking during the transition from hovering to level flight and vice versa with nominal altitude drop. Keywords: quadrotor tail-sitter UAV; nonlinear observer; backstepping control; trajectory-tracking 1. Introduction Different variants of unmanned aerial vehicles (UAVs) have received attention in recent times due to potentially diverse types of applications, including surveillance, exploration, and transportation, to name a few. Hybrid vertical takeoff and landing (VTOL) air vehicles with qualities of both rotary-wing and fixed-wing aircraft can hover like rotarywing aircraft or fly with high speed like fixed-wing aircraft. Consequently, hybrid VTOL UAVs can achieve a few missions that are regularly unthinkable for either fixed-wing or rotary-wing elevated robots alone [ 1 ]. There are different types of VTOL aircraft such as tail-rotor, tail-sitter, tilt-wing, and extra-propulsion [ 2 ]. The tail-sitter is the simplest one because it does not require supplementary actuators to perform the VTOL maneuver. Many researchers have explored small-sized tail-sitters. For instance, Bapst et al. [3] proposed a twin-rotor tail-sitter VTOL aircraft containing a flying wing with two rotors and elevons and a single controller for all flight modes and validated their work through outdoor experiments. Forshaw et al. [ 4 ] presented a concept of twin helicopter rotor tail-sitters. In [ 5 ], the researchers proposed a full-altitude controller for hovering, transition, and level flight. Oosedo et al. [ 6 ] designed a quadrotor tail-sitter UAV with high accuracy in altitude control in both hovering and level-flight modes. Later, Oosedo et al. have provided strategies for optimal transition from hovering to level flight [ 7 ] through normal transition, minimizing the transition time, and minimizing the transition time with constant altitude. Wang et al. [8] designed and implemented a low-cost quadrotor tail-sitter UAV with half Actuators 2021,10, 119. https://doi.org/10.3390/act10060119 https://www.mdpi.com/journal/actuators Actuators 2021,10, 119 2 of 24 the power consumed in a typical quadrotor, as per flight test results with all VTOL maneuvers. In [ 9 ], a VertiKUL quadrotor tail-sitter UAV with no controlling surface, but operating in all three modes, was designed. Figure 1illustrates a quadrotor tail-sitter UAV with four tilted rotors to provide the lift force in vertical flight mode and the thrust during level flight. A hybrid quadrotor tail-sitter VTOL UAV can switch between hovering mode to level-flight mode and vice versa by rotating the aircraft’s pitch angle about almost 90◦as shown in Figure 2. Figure 1. Quadrotor tail-sitter UAV. Mathe et al. [ 10 ] listed some generic low-cost platforms and application fields of air vehicles using vision and control methods while emphasizing the sensor suites used for railway inspection. Trotta et al. [ 11 ] proposed network architecture and supportive optimization frameworks allowing UAVs to execute city-scale video monitoring of points of interest validated via imitation of a city environment with live traffic updates from a real bus transportation network using a UAV scheduler and Mixed Integer Linear Programming (MILP) techniques. Otto et al. [ 12 ] provided a literature review of optimization methods to civil applications of UAVs by describing drone applications and outline features applicable to operations planning, and providing insights into emerging modeling methodologies. Other researchers demonstrated (i) the level of throughput provided to a set of areas and (ii) the amount of energy exchanged with the grid by the ground sites for UAV-aided cellular networks [ 13 ]. The J-MATE model designed for optimal energy and throughput through revenue and cost components for large-problem instances shows out-performance of earlier methods. Several control strategies are present in the literature for trajectory-tracking. In [ 14 ], model prediction-based cascaded control is presented for trajectory-tracking of a VTOL tail-sitter UAV in the hovering mode, with simulation conducted in a HIL (hardware in the loop) environment. Lyu et al. [ 15 ] presented a hierarchical control method to achieve autonomous flight with vertical takeoff, hovering, transition, level flight, and landing of a quadrotor tail-sitter UAV. Flight tests with manual and fully autonomous flight modes show a minimal altitude drop between different flight modes. Li et al. [ 16 ] presented a robust nonlinear controller for flight mode transition between hovering to level-flight mode where tail-sitter aircraft model with uncertainties including nonlinear terms, external disturbances, and parametric uncertainties. Actuators 2021,10, 119 3 of 24 Figure 2. Takeoff and landing of tail-sitter quadrotor UAV. Zhou et al. [ 17 ] proposed novel trajectory planning algorithms for a UAV under the constraints of system positioning accuracy while correcting the error during the flight process of a UAV. For the shortest path under the multiple constraints and minimum errors, a genetic algorithm (GA) helps to validate the results experimentally. Dynamic modeling, control law design, and hardware implementation are provided after deriving the dynamic model using the Newtonian method [ 18 ]. The control is designed for both modes— hovering and level-flight mode—to control the vehicle. The approach is implemented on a low-cost DSP-based Embedded Flight Control System (EFCS) for autonomous flight. Zhou et al. [ 19 ] presented a combined control framework for a quadrotor tail-sitter UAV that deals with hovering and level-flight modes and allows continuous transition between these modes as per the directed velocity. The controller is also used to study the UAV’s equilibrium state, mainly during a wind gust. Swarnkar et al. [ 20 ] presented the development of a 6-DOF flight dynamics model, with a comprehensive description of wing aerodynamics, prop wash modeling, and flight dynamics. Quaternions represent the aircraft’s attitude to avoid singularity related to Euler angles, and a nonlinear controller uses a dynamic inversion method for the whole flight regime. Lyapunov-based control provides [ 21 ] trajectory-tracking for fixed-wing MAV. Simulation done in MAV3Dsim validates the efficacy of the control law. Brezoescu et al. [ 22 ] applied an adaptive backstepping scheme on fixed-wing UAV in the existence of unknown crosswind, and adaptive laws are proposed for disturbance estimation and validated through simulation results. Hajiloo et al. [ 23 ] presented nonlinear dynamics of single rotor spherical UAV and backstepping controller design based on it that works well for trajectory-tracking. Espinoza et al. [ 24 ] designed a controller based on backstepping and sliding modes implemented on fixedwing UAV and studied which controller performance is more suitable for UAV. Sartori et al. designed a backstepping controller for fixed-wing UAVs on micro-controller and experimental data logged, endorsing the applicability controller [ 25 ]. Lungu et al. [ 26 ] presented a backstepping and dynamic inverse-based automatic landing system for fixed-wing UAV with wind gusts and atmospheric disturbances. Rubi et al. review the relevant path following algorithms for quadrotors [ 27 ]. The simulation results with two control-oriented algorithms (Feedback Linearization and Backstepping) and two geometric algorithms (Nonlinear Guidance law (NLGL) and CarrotChasing) help to solve the path following problem. The backstepping method achieved the best performance in terms of path distance and yaw error and the best behavior out of the path and at high velocities. Lyu et al. [ 28 ] presented a control method with disturbance observer (DOB) to improve the hovering accuracy in crosswind flow. A nonlinear flight control method is designed for a fixed-wing UAV with an extended state observer (ESO) [ 29 ]. A multiple observer-based anti-disturbance control scheme uses disturbance observer-based (DO) and extended state observer (ESO)-based controller. Actuators 2021,10, 119 4 of 24 Experiments carried out for the payload oscillation disturbance and hybrid disturbances, robustness, and effectiveness are compared with the PID control method [30]. In this paper, we address two major issues for tail-sitter quadrotor UAVs: (i) Trajectorytracking (ii) Compensate the effect of external disturbance on a tail-sitter UAV. For these, we present • a robust controller for tail-sitter UAVs development using the backstepping technique; • a nonlinear disturbance observer for both periodic and wind-gust-type disturbances. A combination of nonlinear observer and backstopping control law ensures robustness for all three modes—Quadrotor mode, Transition mode, and level-flight mode—and ensures a robust approach for the whole flight envelope. Lyapunov stability analysis provides overall closed-loop stability and robustness. The controller’s performance is demonstrated through trajectory-tracking simulations with applied disturbance in quadrotor, takeoff, and landing phases. The rest of this paper is organized as follows: Problem formulation and quadrotor tail-sitter dynamics are presented in Section 2. In Section 3, we design a nonlinear observer to estimate the external disturbances. Next, a nonlinear observer-based backstepping controller is designed for hovering mode, takeoff, landing phase, and level-flight mode in Section 4. Simulation results are presented to demonstrate the efficacy of controller in Section 5and concluding remarks are presented in Section 6. 2. Problem Formulation Next, we describe tail-sitter UAV dynamics and control objectives. Figure 3illustrates a quadrotor tail-sitter UAV with four tilted rotors to provide the lift force in vertical flight mode and the thrust during level flight. For control, the desired trajectory command is given to the quadrotor tail-sitter manually or by an upper-level motion planner. The tail-sitter UAV produces no significant lift and drag in a takeoff phase and landing phase. Therefore, we assume that the tail-sitter UAV acts as a quadrotor. The four input signals [U1U2U3U4] control the vehicle’s motion as done in the quadcopter. The tail-sitter UAV can switch between hovering mode to level-flight mode and vice versa by rotating the aircraft’s pitch angle about almost 90 ◦ degree. In transition phase, control over x and y positions are disabled, and the objective is to maintain altitude and orientation. After completing transition, the vehicle enters the level-flight mode, and acts as a fixed-wing UAV. Please note that the role of yaw and roll is reversed in quadcopter and fixed-wing modes, respectively. Figure 3. Controller Block Diagram. Actuators 2021,10, 119 5 of 24 A mathematical model of the quadrotor tail-sitter UAV is developed using the Newtonian or Lagrangian approach. As per [ 31 ], for u , v and w as the X , Y and Z directional body-axis velocities, and p,qand ras the angular velocities, the flight dynamics are ˙ u=rv −qw +gcos θcos ψ−Lcos α+Dsin α m(1) ˙ v=pw −ru +gcos θsin ψ(2) ˙ w=qu −pv +gsin θ−Lsin α+Dsin α m− 4 ∑ i=1 Ti m(3) ˙ p=qr(Iyy −Izz) Ixx −Irr Ixx 4 ∑ i=1 (−1)iΩi+l(T4−T2) Ixx (4) ˙ q=pr(Izz −Ixx) Iyy +Irq Iyy 4 ∑ i=1 (−1)iΩi+l(T1−T3) Iyy (5) ˙ r=pq(Ixx −Iyy) Izz +1 Izz 4 ∑ i=1 (−1)iQi, (6) where Ixx , Iyy and Izz are the fuselage moment of inertia around each axes, Ir is the propeller gyroscopic effect, m is the fuselage mass, g is the gravitational acceleration, l is the distance from the motor to the center of gravity, φ is the roll angle, θ is the pitch angle, and ψ is the yaw angle, Ωiis the propeller revolution speed of i-th rotor, such that V=qV2 x+V2 y+V2 z,α=arctanVz Vx,T=CTρΩ2d4 L=1 2ρV2SCl(α),D=1 2ρV2SCd(α),Q=CQρΩ2d5, where S is the wing area, V is the velocity, ρ is the air density, d is rotor diameter, c is the drag coefficient, k is a constant, α is an Angle of Attack (AoA), L is the lift force, and D is the drag force. T and Q is trust and torque produced by propellers, d is the diameter of the propeller. Clis a lift coefficient, Cdis a drag coefficient, CTis the thrust coefficient and CQ is the torque coefficient. For conventional tail-sitter quadrotor CT , CQ , ρ , d are constants, trust T and Q are proposal to propeller revolution speed so input of [U1U2U3U4] can be expressed as U1=k(Ω2 1+Ω2 2+Ω2 3+Ω2 4) U2=kl(Ω2 4−Ω2 2) U3=kl(Ω2 1−Ω2 3) U4=c(−Ω2 1+Ω2 2−Ω2 3+Ω2 4) Ω= (Ω2+Ω4−Ω1−Ω3), where Ω is representing the overall residual propeller angular speed, k is thrust constant, c is torque constant and lis length of arm. Although there are no significant aerodynamic forces and moments created during the vertical takeoff phase, hovering mode, and landing phase, the total forces and moments are only due to thrust. Therefore, the quadrotor tail-sitter UAV is considered a quadrotor for analysis purposes. The dynamical equations for quadrotor mode in the hybrid frame as given in [32] are Actuators 2021,10, 119 6 of 24 ¨ φ=Iyy −Izz Ixx ˙ θ˙ ψ−Ir Ixx ˙ ψΩ+U2 Ixx +dφ(7) ¨ θ=Izz −Ixx Iyy ˙ φ˙ ψ+Ir Iyy ˙ θΩ+U3 Iyy +dθ(8) ¨ ψ=Ixx −Iyy Iyy ˙ φ˙ θ+U4 Izz +dψ(9) ¨ x= (cos φsin θcos ψ+sin φsin ψ)U1 m+dx(10) ¨ y= (cos φsin θsin ψ−sin φcos ψ)U1 m+dy(11) ¨ z=−g+cos φcos θU1 m+dz. (12) State vector can be defined for position (10)–(12) and attitude subsystem (7)–(9) as, X= [P˙ P O ˙ O]T∈ R12 , where P= [x y z]T , O= [φ θ ψ]T , and dp=dxdydz , do=dφdθdψare the external disturbances. The control objectives of this study are • Design a nonlinear disturbance observer for model uncertainty as well as wind gust (external) disturbances while in takeoff phase, hovering mode and landing phase. • To design control laws using backstepping technique for quadrotor tail-sitter UAVs to track the given trajectory. 3. Nonlinear Observer Design The external disturbance, such as wind field, has a significant effect on tail-sitter quadrotor UAVs’ stability, particularly in the takeoff and landing phase. In this section, we develop a nonlinear observer to estimate the uncertainties using an observer proposed in [ 33 , 34 ]. The following assumption is assumed for the disturbances d used during backstepping controller design and stability analysis. Assumption 1. The disturbance and derivative of disturbance are bounded: || ˙ dp(t)|| ≤ Dp,|| ˙ do(t)|| ≤ Dot>0, where Dpand Doare positive constants. Similarly, a nonlinear disturbance observer proposed by Yang et al. [ 35 ] and Viswanath et al. [36], can be implemented for both the position and attitude subsystems: ˙ np=−Lpnp−LpLp˙ P+G+1 mUp,ˆ dp=np+Lp˙ P, (13) ˙ no=−Lono−LoLo˙ O+Φ(O,˙ O)−Uo,ˆ do=no+Lo˙ O, (14) where Up=R(O)E3U1 , Uo=Ψ(O)[U2U3U4]T ( Ui , i= 1, . . . , 4) are shown in Figure 3, and ˆ dj , j=p , o is the disturbance estimation, nj is the observer state vector, ζj=LjI3×3 , ζj> 0 are the tunable gain matrices, G= [ 0 0 −g]T , m = mass, R(O) = rotation matrix and E3 is unit vector basis associated with the earth fixed frame (I) . Ψ(O)=[IEM(O)]−1 where EM(O)= Euler matrix. Lemma 1 ([ 37 ]) . For a smooth system ˙ x=f(x) , x∈Rn , with f( 0 ) = 0and a Lyapunov candidate function V(V( 0 ) = 0 ) , let x( 0 )∈C⊂Rn . Along any trajectory x:R+→Rn , starting in C, the following differential disparity is satisfied with β>0 d dt{V(x(t))}<−αV(x(t)) + β,∀t≥0with x(0)∈C, (15) Actuators 2021,10, 119 7 of 24 where αas a tunable positive parameter. Proposition 1 ([ 37 ]) . Under Assumption 1, for an adequately large T∗ there exist appropriate observer gains Lj> 0, j=p , o ,for prescribed asymptotic estimation of observers (13) and (14) for every e>0there exist L∗ jfor all Lj≥L∗ j, the observer errors satisfy ||edj(t)||2≤e,∀t≥T∗,j=p,o. (16) Proof. We rewrite (7)–(12) as ¨ P=G+Up m+dp,¨ O=Φ(O,˙ O) + Uo. (17) By differentiating (13) and using (17), we obtain ˙ ˆ dp=˙ np+Lp¨ P=−Lpnp−LpLp˙ P+G+Up m+LpG+Up m+dp =−Lpnp+Lp˙ P+Lpdp=−Lpedp. (18) Similarly, we can show that ˙ ˆ do=−Loedo, (19) for error terms edj=ˆ dj−dj , j=p , o , and using (18) and (19), the error derivatives are expressed as ˙ edj=−˙ dj−Ljedj. (20) For a positive definite function defined in terms of error term edj, given by V1j=eT djedj, (21) and using Assumption 1, (20) and the inequality − 2 eT dj ˙ dj≤ ||edj||2+|| ˙ dj||2 , the time derivative of V1jcan be expressed as ˙ V1j=2eT dj˙ edj=−2eT djLjedj−2eT dj ˙ dj≤ −2eT djLjedj+||edj||2+|| ˙ dj||2 ≤(−2Lj+1)V1j+D2 j. (22) It turns out that the inequality (22) takes the form of (15) with α=− 2 Lj+ 1 and β=D2 j , i.e., for j=p , o such types of α , β exist. A lower bound on the observer gains Lj indicated by L∗ j , i.e., L∗ j≤Lj ensures that (− 2 Lj+ 1 )≤(− 2 L∗ j+ 1 ) , and so ˙ V1j(t)≤ (−2L∗ j+1)V1j(t) + D2 j. Therefore, to ensure V1j(t) = ||edj||2≤e , ∀t≥T∗ ; j=p , o , we can choose L∗ j such that (−2L∗ j+1)e+D2 j=0, that is L∗ j=1 2 D2 j e+1!. (23) Actuators 2021,10, 119 8 of 24 4. Backstepping Control Design This section develops a robust controller using a backstepping technique [ 38 – 40 ] for the takeoff phase, hovering, transition, level-flight modes, and landing phase. The quadrotor tail-sitter UAV is an underactuated system used in most vehicles that need to control altitude and position using only four inputs [7]. 4.1. Quadrotor Mode We formulate nonlinear observer-based control law for the quadrotor mode using the backstepping method. Using Equations (10)–(12) let us consider position subsystem as ˙ PQ1=PQ2,˙ PQ2=−g+1 mUp+dp. (24) For position tracking, the error is defined as e1=PQ1d−PQ1 . The time derivative is given as ˙ e1=˙ PQ1d−˙ PQ1=˙ PQ1d−PQ2. (25) Lyapunov function candidate for position subsystem is chosen as VQP1=1 2eT 1e1. (26) Now, with velocity tracking error defined as e2=PQ2d−PQ2, and using (25), we get ˙ e1=˙ PQ1d−PQ2d+e2, (27) where PQ2dis virtual input designed to stabilized ˙ e1, such that PQ2d=˙ PQ1d+c1e1,˙ PQ2d=¨ PQ1d+c1˙ e1, (28) where c1is positive definite matrix, and by substituting (28) into (27), we obtain ˙ e1=e2−c1e1. (29) Lyapunov function candidate is chosen as VQP2=1 2eT 1e1+1 2eT 2e2. (30) Taking time derivative of (30) and using (28), we obtain ˙ VQP2=eT 1˙ e1+eT 2˙ e2=eT 1(−c1e1+e2) + eT 2(¨ PQ1d+c1˙ e1−˙ PQ2). (31) Substituting (24) into (31), we obtain ˙ VQP2=−eT 1c1e1+eT 1e2+eT 2¨ PQ1d+c1˙ e1−−g+1 mUp+dp =−eT 1c1e1+eT 2e1+¨ PQ1d+c1˙ e1+g−1 mUp−dp. (32) Now, the Control law for the position sub system can be defined as Up=me1+c1˙ e1+g+¨ PQ1d+c2e2−ˆ dp, (33) Actuators 2021,10, 119 9 of 24 where c2 is a positive definite matrix, Using (12), three components of UP : [U1 , Ux , Uy] are given as U1=Up cos φcos θ, (34) Ux=cos φsin θcos ψ+sin φsin ψ cos φcos θU1, (35) Uy=cos φsin θsin ψ−sin φcos ψ cos φcos θU1. (36) To compensate for the disturbance dp to achieve improved robustness, we employ the nonlinear disturbance observer (13). Theorem 1 ([ 37 ]) . For the error subsystem (25) with the disturbance observers (13) and (14), and the control signals (33) and (34), there exist positive definite matrices c1,c2and Lp, resulting in 0<||e1||2+||e2||2≤e,∀t≥T∗(37) for tracking errors e1,e2, and a chosen adequately large e. Proof. Lyapunov Function candidate can be defined as, VQP =VQP1+VQP2. (38) Considering (31), (33), from (38), we obtain ˙ VQP =˙ VQP1+˙ VQP2=−eT 1c1e1+eT 1e2+eT 2(¨ PQ1d+c1˙ e1−g+1 mUp+dp) −eT dp(Lp−1 2I3×3)edp +1 2D2 p =−eT 1c1e1−eT 2c2e2+eT 2(ˆ dp−dp)−eT dp(Lp−1 2I3×3)edp +1 2D2 p ≤ −eT 1c1e1−eT 2c2e2−1 2eT 2e2−eT dpLpedp +1 2D2 p <δ1VQP +D2 p, (39) where δ1=min{2λmin(c1), 2(λmin(c2)−1 2), 2(λmin(Lp−1))}. The above gains are chosen to deliver any scale of the tunable λ> 0 so that (37) follows from Lemma 1. Yaw angle can be directly measured by the sensor and desired roll angle ( φd ) and pitch angle ( θd ) can be calculated using position and attitude subsystem. Reference trajectory for the attitude subsystem can be defined as Od= [φd , θd , ψd]T . The desired angles φdand θdcan be obtained using (35) and (36) such that UxCφdCθdCψd= (CφdSθdC2 ψd+SφdSψdCψd)U1, (40) UyCφdCθdSψd= (CφdSθdS2 ψd−SφdSψdCψd)U1. (41) Adding Equations (40) and (41) and dividing by Cφdand Cθd, we obtain UxCψd+UySψd= (tan θd)U1, (42) φdand θdobtained from Equations (40)–(42): Actuators 2021,10, 119 16 of 24 0 50 100 150 200 250 300 350 400 -2 0 2 Roll (rad) Desired Roll angle Actual Roll angle 0 50 100 150 200 250 300 350 400 -2 0 2 Pitch (rad) Desired Pitch angle Actual Pitch angle 0 50 100 150 200 250 300 350 400 Time (seconds) -2 0 2 Yaw (rad) Desired Yaw angle Actual Yaw angle Figure 8. Attitude tracking by backstepping controller. Figure 9shows the three-dimensional trajectory-tracking, and Figure 10 shows the velocity profile during the same. It is observed that the proposed controller facilitates effective tracking of the desired trajectory. Figure 9. Trajectory-tracking. 0 50 100 150 200 250 300 350 400 -5 0 5 X Axis (m/s) 0 50 100 150 200 250 300 350 400 -10 0 10 20 Y Axis (m/s) 0 50 100 150 200 250 300 350 400 Time (seconds) -4 -2 0 2 4 Z Axis (m/s) Figure 10. Velocity profile. 5.2. Quadrotor Mode with External Disturbance This section simulates the takeoff phase, hovering mode, and the landing phase with external disturbance. As shown in Figure 4, for this simulation only quadrotor mode, 0 <t< 20 (takeoff phase) and 401 <t< 441 (hovering and landing phase) considered. For robustness against disturbance, we design a nonlinear disturbance observer. We evaluate the disturbance observer’s performance and the disturbance impact on the position and attitude subsystem of UAV. Simulation is performed for 60 s, in which t=0 to t = 20 s UAV commanded to takeoff with 2 m/s constant velocity, t = 20 to t=40s , tail-sitter is in hovering mode and t = 40 to t = 60 s it commanded to landing. During this Actuators 2021,10, 119 17 of 24 period, two types of external disturbance are applied to it. First, periodic disturbances [dxdydz]=[ 1 +sin 2 t 1 +sin 2 t 1 +sin 2 t] and [dφdθdψ]=[sin 2 tsin 2 tsin 2 t] are applied. Figure 11 shows the position tracking when a periodic disturbance is applied during flight. There is no significant change in position. Figure 12 shows the attitude tracking, there is no substantial change in the attitude subsystem of the UAV. These results validate the proposed backstepping controller performance. 0 10 20 30 40 50 60 0 0.5 1 1.5 X Axis (m) Desired X position Actual X position 0 10 20 30 40 50 60 0 0.5 1 1.5 Y Axis (m) Desired Y position Actual Y position 0 10 20 30 40 50 60 Time (seconds) 0 50 Z Axis (m) Desired Z position Actual Z position Figure 11. Position tracking when disturbance [1 +sin(2t)1+sin(2t)1+sin(2t)] applied. 0 10 20 30 40 50 60 -1 0 1 2 Roll (rad) Desired Roll angle Actual Roll angle 0 10 20 30 40 50 60 -1 0 1 2 Pitch (rad) Desired Pitch angle Actual Pitch angle 0 10 20 30 40 50 60 Time (seconds) 0 0.1 0.2 Yaw (rad) Desired Yaw angle Actual Yaw angle Figure 12. Attitude tracking when disturbance [sin(2t)sin(2t)sin(2t)] applied. Figures 13 and 14 show the disturbance observer performance for the position and attitude subsystems, and the error in estimating these disturbances are shown in Figure 15 and Figure 16 respectively. We observe an error in millimeters in Figure 15. 0 10 20 30 40 50 60 0 2 4 X Axis (m) 0 10 20 30 40 50 60 0 2 4 Y Axis (m) 0 10 20 30 40 50 60 Time (seconds) 0 2 4 Z Axis (m) Figure 13. Disturbance observer outcome for position subsystem with periodic disturbance. Actuators 2021,10, 119 18 of 24 0 10 20 30 40 50 60 0 2 4 Roll (rad) 0 10 20 30 40 50 60 0 2 4 Pitch (rad) 0 10 20 30 40 50 60 Time (seconds) 0 2 4 Yaw (rad) Figure 14. Disturbance observer outcome for attitude subsystem with periodic disturbance. 0 10 20 30 40 50 60 -1 0 1 ex (m) 0 10 20 30 40 50 60 -1 0 1 ey (m) 0 10 20 30 40 50 60 Time (seconds) -1 0 1 ez (m) Figure 15. Estimation error of position subsystem when [1 +sin( 2 t) 1 +sin( 2 t) 1 +sin( 2 t) ] applied to position subsystem. In Figure 16, there is a small error in the attitude subsystem. Figure 17 shows the trajectory-tracking error with observer and without observer. it can be seen that there are small (0.05 m) steady-state errors in the z axis during takeoff and landing and during hovering mode there is no error. Tracking error with observer in x-y axis is much better then without observer, which validates the designed nonlinear observer’s performance with periodic disturbance and from Figure 18 it can be seen that there is a very small tracking error of attitude subsystem when periodic disturbance is applied. 0 10 20 30 40 50 60 -1 0 1 e (rad) 0 10 20 30 40 50 60 -1 0 1 e (rad) 0 10 20 30 40 50 60 Time (seconds) -1 0 1 e (rad) Figure 16. Estimation error of attitude subsystem when [ sin( 2 t)sin( 2 t)sin( 2 t) ] is applied to the attitude subsystem. Actuators 2021,10, 119 19 of 24 0 10 20 30 40 50 60 -0.5 0 0.5 1 1.5 X Axis (m) Tracking error with observer Tracking error without observer 0 10 20 30 40 50 60 -1 0 1 Y Axis (m) Tracking error with observer Tracking error without observer 0 10 20 30 40 50 60 Time (seconds) -0.1 0 0.1 Z Axis (m) Tracking error with observer Tracking error without observer Figure 17. Position tracking Error when [1 +sin( 2 t) 1 +sin( 2 t) 1 +sin( 2 t) ] is applied to the attitude subsystem. 0 10 20 30 40 50 60 -2 0 2 Roll (rad) Tracking error with observer Tracking error without observer 0 10 20 30 40 50 60 -2 0 2 Pitch (rad) Tracking error with observer Tracking error without observer 0 10 20 30 40 50 60 Time (seconds) -0.1 0 0.1 0.2 Yaw (rad) Tracking error with observer Tracking error without observer Figure 18. Attitude tracking error when [ sin( 2 t)sin( 2 t)sin( 2 t) ] is applied to the attitude subsystem. Next, we apply the Von Karman wind gust model (turbulence model) as an external disturbance. External disturbances such as wind gusts have more impact in the takeoff, landing hovering mode in actual scenarios. We present simulation work for the takeoff, landing and hovering phase with a Von Karman turbulence model that evaluates the proposed nonlinear controller’s performance. Figure 19 shows the position tracking with wind gust, providing small changes in the positioning subsystem during takeoff, landing and hovering. 0 10 20 30 40 50 60 0 0.5 1 X Axis (m) Desired X position Actual X position 0 10 20 30 40 50 60 0 0.5 1 Y Axis (m) Desired Y position Actual Y position 0 10 20 30 40 50 60 Time (seconds) 0 20 40 Z Axis (m) Desired Z position Actual Z position Figure 19. Position tracking with Von Karman wind turbulence model. Similarly, Figure 20 shows small fluctuations in the attitude subsystem. Actuators 2021,10, 119 20 of 24 0 10 20 30 40 50 60 -2 0 2 4 Roll (rad) Desired Roll angle Actual Roll angle 0 10 20 30 40 50 60 -2 0 2 4 Pitch (rad) Desired Pitch angle Actual Pitch angle 0 10 20 30 40 50 60 Time (seconds) 0 0.05 0.1 Yaw (rad) Desired Yaw angle Actual Yaw angle Figure 20. Attitude tracking with Von Karman wind turbulence model. Figures 21 and 22 show the observer’s performance during estimation of disturbance. These results validate the designed nonlinear observer’s performance. 0 10 20 30 40 50 60 -4 -2 0 2 X Axis (m) 0 10 20 30 40 50 60 -2 0 2 4 6 Y Axis (m) 0 10 20 30 40 50 60 Time (seconds) -4 -2 0 2 Z Axis (m) Figure 21. Disturbance observer outcome for position subsystem with Von Karman wind turbulence model. 0 10 20 30 40 50 60 -0.5 0 0.5 Yaw (rad) 0 10 20 30 40 50 60 -0.5 0 0.5 Pitch (rad) 0 10 20 30 40 50 60 Time (seconds) -0.5 0 0.5 Yaw (rad) Figure 22. Disturbance observer outcome for attitude subsystem with Von Karman wind turbulence model. Figures 23 and 24 show the performance of disturbance observer for position and attitude subsystem with Von Karman wind turbulence model. It can be seen that the error is small. Actuators 2021,10, 119 21 of 24 0 10 20 30 40 50 60 -0.5 0 0.5 ex (m) 0 10 20 30 40 50 60 -0.5 0 0.5 ey (m) 0 10 20 30 40 50 60 Time (seconds) -0.5 0 0.5 ez (m) Figure 23. Estimation error of position subsystem when Von Karman wind turbulence model applied to position subsystem. 0 10 20 30 40 50 60 -0.05 0 0.05 e 0 10 20 30 40 50 60 -0.05 0 0.05 e 0 10 20 30 40 50 60 Time (seconds) -0.05 0 0.05 e Figure 24. Estimation error of attitude subsystem when Von Karman wind turbulence model is applied to the attitude subsystem. Figures 25 and 26 show the small tracking errors in position and attitude subsystems with and without observer where there is a small (0.05 m) steady-state error in the z -axis during takeoff and landing, with wind gust applied. 0 10 20 30 40 50 60 -1 0 1 X Axis (m) Tracking error with observer Tracking error without observer 0 10 20 30 40 50 60 -1 0 1 Y Axis (m) Tracking error with observer Tracking error without observer 0 10 20 30 40 50 60 Time (seconds) -0.1 0 0.1 Z Axis (m) Tracking error with observer Tracking error without observer Figure 25. Position tracking error when Von Karman wind turbulence model is applied to the attitude subsystem. Actuators 2021,10, 119 22 of 24 0 10 20 30 40 50 60 -2 0 2 Roll (rad) Tracking error without observer Tracking error with observer 0 10 20 30 40 50 60 -2 0 2 Pitch (rad) Tracking error without observer Tracking error with observer 0 10 20 30 40 50 60 Time (seconds) -0.05 0 0.05 0.1 Yaw (rad) Tracking error without observer Tracking error with observer Figure 26. Attitude tracking error when Von Karman wind turbulence model is applied to the attitude subsystem. Figure 27 shows the comparison between nonlinear observer-based backstepping controller (NDO-BC) and nominal backstepping Controller (BC) when periodic disturbance and wind gust acting on tail-sitter while tracking trajectory [ 1 +sin( 0.5 t) 1 +cos( 0.5 t) 2 t] in quadrotor mode. It can be seen that nonlinear observer-based backstepping controller performance is far better then the nominal backstepping controller. 0 1 100 2 Z Axis (m) (a) Y Axis (m) 200 1 0 X Axis (m) 0 -1 -1 Desired path BC NDO-BC 0 100 12 Z Axis (m) (b) Y Axis (m) 1 200 X Axis (m) 00 -1 -1 Desired path BC NDO-BC Figure 27. Comparison of nonlinear disturbance observer-based backstepping controller and nominal backstepping controller when (a) periodic disturbance (b) wind gust applied in Quadrotor Mode. 6. Conclusions In this paper, a backstepping controller is designed for quadrotor tail-sitter UAVs for an autonomous flight with takeoff, hovering mode, level-flight mode, and landing phase. The nonlinear disturbance is designed to estimate external disturbance for hovering mode. Three different types of external disturbance are applied. The Von Karman wind gust model is applied as an external disturbance during the landing and takeoff phase. We formulated an appropriate dynamical model considering wind gust disturbances in MATLAB/Simulink and carried out simulation work. Simulation results show that the presented backstepping controller effectively controls all phases of autonomous flight. During transition maneuver, there is negligible altitude drop and successfully tracking of the commanded trajectory. When different external disturbances are applied to the UAV in landing, takeoff phase, and hovering mode, it maintains its position, which demonstrates the proposed nonlinear-based backstepping controller effectiveness. Author Contributions: Conceptualization, N.D. and D.D.; methodology, D.D. and N.D.; software, N.D.; validation, N.D.; formal analysis, D.D.; writing—original draft preparation, N.D. and M.K.; writing—review and editing, D.D., M.K. and S.O.; supervision, D.D.; funding acquisition, S.O. All authors have read and agreed to the published version of the manuscript. Actuators 2021,10, 119 23 of 24 Funding: This research was funded by the European Regional Development Fund in the Research Centre of Advanced Mechatronic Systems project, within the Operational Programme Research, Development and Education, grant number: CZ.02.1.01/0.0/0.0/16_019/0000867. Institutional Review Board Statement: Not Applicable. Informed Consent Statement: Not Applicable. Conflicts of Interest: The authors declare no conflict of interest. References 1. Kita, K.; Konno, A.; Uchiyama, M. Transition between Level Flight and Hovering of a Tail-Sitter Vertical Takeoff and Landing Aerial Robot. Adv. Robot. 2010,24, 763–781. [CrossRef] 2. Zhang, F.; Lyu, X.; Wang, Y.; Gu, H.; Li, Z. Modeling and Flight Control Simulation of a Quadrotor Tailsitter VTOL UAV. In Proceedings of the AIAA Modeling and Simulation Technologies Conference, Grapevine, TX, USA, 9–13 January 2017; American Institute of Aeronautics and Astronautics: Reston, VA, USA, 2017. [CrossRef] 3. 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