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Abstract

Mini aerial vehicles have gained a lot of significance in the past five decades. The future defense strategies mainly rely upon mini aerial vehicles. In maneuvering a mini aerial vehicle, control of parameters like pitch angle, roll angle, side slip angle, etc., are very crucial and important. Due to geometry and weight constraints, controlling a mini aerial vehicle is very difficult. Mini aerial vehicles also gets easily disturbed from its path because of slight disturbance of wind. As weight constraint is imposed on mini aerial vehicle, low cost sensors are adopted. Low cost sensors gives noisy feedback, which still makes a challenging task to design a controller to track pitch angle for a mini aerial vehicle. The present work attempts to develop a flight simulator for Black Kite mini aerial vehicle and to design a proportional-integral controller to track pitch angle by taking feedback from an observer, which takes consideration of noisy sensor inputs.

Keywords

Mini aerial vehicle pitch angle flight simulator observer and proportional-integral controller

Introduction

Aerodynamics is the branch of fluid dynamics, which particularly deals with the movement of fluids over solid objects. The word aerodynamics had achieved a great significance after the success of Wright brother’s first flight in 1903. Aeronautical engineers use the fundamental principles of aerodynamics to design, build, and operate the aircrafts. Aerodynamics also plays an important role in other numerous applications. It plays a vital role in vehicle design by improving the overall efficiency of the vehicle. It is used to predict forces and moments acting in sailing. It is also used to calculate the wind loads acting in the design of buildings and bridges.

Aerodynamic forces acting on an aircraft or MAV helps us to understand the dynamics of it. When an aircraft or MAV moves relative to the air, aerodynamic force is generated on the rearward of the wing due to the aerofoil shape of a wing. The force generated acts at an angle with the direction of relative motion of air. When this force is resolved into components, it gives arise to lift and drag. Lift is the force component perpendicular to relative motion of air and helps the aircraft to glide over air, whereas drag is the force component parallel to the relative motion of air and offers resistance to the relative motion of aircraft. Apart from lift and drag, thrust and weight are the two other forces acting on an aircraft. Thrust is an aerodynamic force, which is used to generate forward movement of an aircraft, whereas weight is a body force and it arises because of gravity.

The journey of Unmanned Aerial Vehicle (UAV) began in the 19th century and the development of radio controlled UAVs started in the 20th century.¹ In the past eight decades since their introduction in 1922, the unmanned flight systems application has grown rapidly. The first combat use of UAVs by the US Air Force was carried out in the year 1944 against Japan.² Today UAV has grown as a frontier research and development area. They have a variety of applications, which include Intelligence, Surveillance, Target Acquisition and Reconnaissance (ISTAR), Combat, Vertical Take-Off and Landing (VTOL), Radar and Communication Relay and Aerial Delivery and Re-supply. UAVs are generally classified based on their performance specifications and their mission aspects.

In the present thesis, the mini aerial vehicle (MAV), which is considered for this problem weighs up to 1 kg. The flying platform considered in this thesis has a mass of 0.3 kg. These systems are classified according to many features such as weight, endurance, and range, maximum altitude, wing loading, engine type, and power/thrust loading, etc.³ These systems pose a variety of challenging research potential including autonomous take-off and landing. Beard et al. report about autonomous technologies of a small UAV with respect to the development of hardware and software architectures.⁴ Parra et al. bring out the development of a MAV through wind tunnel tests. They established a nonlinear model and a control law for attitude control using dynamic inversion technique.⁵ Similarly, numerous works are reported with respect to the advancement of small UAV using dynamic inversion and also neural networks.^6,7 Researchers report numerous works related to the state estimation concentrated more towards the attitude estimation.

A notable work from William E. Green and Paul Y. Oh presents about the autonomous hovering of a fixed wing micro air vehicle.⁸ They have developed a Proportional-Derivative (PD) controller in case of autonomous hovering of a fixed-wing micro aerial vehicle. Another attractive research capability of UAVs is the cooperative missions. Cummings and Mitchell bring out the operational scheduling strategies for multiple UAVs.⁹ These types of mission aspects demand good controller, which indeed require accurate feedback quantities. Buschmann et al. discuss about the application of MAV for meteorological purposes. They have implemented a feedback control structure with autopilot in the Carolo aircraft.¹⁰ The present article aims at designing a proportional-Integral (PI) controller to track pitch angle in Black Kite MAV by taking feedback from extended Kalman filter (EKF) observer. EKF observer is used to predict pitch angle from noisy sensor inputs.

Flight simulation

Simulation

Simulation is defined as the process of imitation of real world system in a virtual world over time. In order to simulate a particular system, first a mathematical model which represents the system function and behavior should be developed. The model represents the system, but the simulation represents the operation of the system. Simulations are used where the real system can’t be engaged or it is dangerous to engage a real system. Simulations find a lot of application in engineering domain like safety engineering, testing, cost effective analysis, and inventory management. In present situation, flight simulation of Black Kite MAV is developed in Simulink environment. The mathematical model of MAV includes kinetics and dynamics of system in terms of the mathematical equations. These equations are incorporated into Simulink environment with the help of in-built mathematical functional blocks and solved to obtain the solution for steady state flight and then later implemented for flight simulation with disturbance.

Aerodynamic forces and moments

In order to address the problem of design of a PI controller to track pitch angle in MAV, it is essential to understand the dynamics and kinetics of a MAV. Aerodynamic forces and moments play a crucial role in navigating a MAV. The aerodynamic forces and moments used in our simulation are listed as follows, which involves lift, drag, and moments with coefficients C_L, C_D, C_Y and C_l, C_m, and C_n respectively.

Lift L = q_{dyn} {SC}_{L}

(1)

Drag D = q_{dyn} {SC}_{D}

(2)

Sideforce Y = q_{dyn} {SC}_{Y}

(3)

Rollingmoment \bar{L} = q_{dyn} {SbC}_{l}

(4)

Pitchingmoment M = q_{dyn} {\bar{S}}_{c} {bC}_{m}

(5)

Yawingmoment N = q_{dyn} {SbC}_{n}

(6)

The longitudinal data of MAV used in our simulation is obtained from National Aerospace Laboratories (NAL), Bangalore. The data were obtained in the form of plots through actual wind tunnel test on Black Kite MAV.¹¹ It was converted into equations and used in our simulation. The different equations used in simulations are listed as follows:

C_{L} = 0.1748 + 2.453 α - 1.691 α^{2} + 29.986 α^{3} - 49.245 α^{4} - 0.3638 δ_{e}^{2} + 0.7405 δ_{e}

(7)

C_{D} = 0.08712 - 0.05593 α + 3.482 α^{2} + 0.2258 δ_{e}^{2} + 0.1471 δ_{e}

(8)

C_{Y} = - 0.7678 β + (0.15 / V) * (- 0.124 p + 0.366 r) - 0.02956 δ_{a} + 0.1158 δ_{r}

(9)

C_{l} = - 0.0618 β + (0.15 / V) * (- 0.5045 p + 0.1695 r) - 0.09917 δ_{a} + 0.006934 δ_{r}

(10)

C_{m} = 0.0385 - 0.59977 α - 1.27402 α^{2} - 0.1587 δ_{e}^{2} + 1.5894 δ_{e}

(11)

C_{n} = 0.006719 β + (0.15 / V) * (- 0.1585 p - 0.1112 r) - 0.003872 δ_{a} - 0.0826 δ_{r}

(12)

The coefficient of lift, drag, and side forces have been specified with respect to wind axis. These are converted into body axis C_X, C_Y, and C_Z through following transformations:

[C_{X} C_{Y} v] = [- \cos α 0 \sin α 010 \sin α 0 \cos α] [C_{D} C_{Y} C_{L}]

(13)

The net aerodynamic forces acting along x-, y-, and z-axis on MAV are listed as follows, respectively:

F_{ax} = q_{dyn} {SC}_{X}

(14)

F_{ay} = q_{dyn} {SC}_{Y}

(15)

F_{az} = q_{dyn} {SC}_{Z}

(16)

The current work has been carried out using the Black Kite MAV developed by National Aerospace Laboratories (NAL), Bangalore. The aircraft has an overall length of 0.3 m and an overall height of 0.155 m. The weight of the aircraft is 2.943 N. The aspect ratio of wing is 1.45.

Thrust/propulsion force

Thrust is defined as the net force required to generate forward movement of a MAV. In our problem, a single motor is used to generate thrust. As only single motor is placed at the center of flying platform to generate thrust, there will be no moments resulting from the thrust and also thrust acts along x-axis and there won’t be any thrust component along y- and z-axis, respectively.

Thrust = \frac{4}{π^{2}} ρ n^{2} D^{4} [- 0.0948 J_{ad}^{2} + 0.058 J_{ad} + 0.0761]

(17)

Gravitational force

Gravitational force is the net force acting on the MAV due to gravitational pull. As gravitational force acting on MAV is inclined at an angle, it is resolved into components along x-, y-, and z-axis, respectively. The gravitational forces acting on a MAV along x-, y-, and z-axis are listed as follows:

F_{gx} = - W \sin θ

(18)

F_{gy} = W \cos θ \sin φ

(19)

F_{gz} = W \cos θ \cos φ

(20)

The net forces acting on a MAV along each axis is the vector sum of the aerodynamic, gravitational, and thrust force acting in respective direction and they are listed as follows:

F_{X} = F_{ax} + F_{px} + F_{gx}

(21)

F_{Y} = F_{ay} + F_{py} + F_{gy}

(22)

F_{Z} = F_{az} + F_{pz} + F_{gz}

(23)

State dynamics vector

State is generally defined as the parameter, which we are interested in simulation. In present flight simulation, the states which we are interested are velocity, angle of attack, side slip angle, pitch, roll, yaw, roll angle, pitch angle, yaw angle, latitude, longitude, altitude, velocity along body x-, y-, and z-axis, respectively. The derivative of these states is known as state dynamic vector and these are derived from Newton’s laws of motion. The aerodynamic forces of the aircraft mainly depend upon the true airspeed V, angle of attack, and side slip angle. So it is more convenient to use these quantities instead of the linear velocity components u, v, and w along the body axis. Since V, α, andβ can be expressed in terms of u, v, w, and vice versa, both sets of variables can be applied for solving the equations of motion. The true airspeed is also expressed in the following form:

V = \sqrt{u^{2} + v^{2} + w^{2}}

(24)

u = V \cos α \cos β

(25)

v = V \sin β

(26)

w = V \cos β \sin α

(27)

u^{•} = rv - qw + \frac{F_{X}}{m}

(28)

v^{•} = pw - ru + \frac{F_{Y}}{m}

(29)

w^{•} = qu - pv + \frac{F_{Z}}{m}

(30)

By differentiating equation (24) and substituting equations (25), (26), and (27), we will get state dynamic velocity vector, $V^{•}$

V^{•} = u^{•} \cos α \cos β + v^{•} sin β + w^{•} \sin α \cos β

(31)

Now substituting equations (28), (29), and (30) in equation (31), the terms involving the vehicle rotational rates p, q, and r turn out to be identically zero and the resulting equation becomes,

V^{•} = \frac{1}{m} (F_{X} \cos α \cos β + F_{Y} sin β + F_{Z} \sin α \cos β)

(32)

Similarly, $α^{•}$ and $β^{•}$ are also defined as follows:

α^{•} = \frac{1}{Vm \cos β} ((- F_{X} \sin α \cos β + F_{Z} \cos α)) + q - (p \cos α + r \sin α) \tan β

(33)

β^{•} = \frac{1}{Vm} (- F_{X} \cos α \sin β + F_{Y} \cos β - F_{Z} \sin α \sin β) + p \sin α - r \cos α

(34)

So far, we have derived differential equations for the true airspeed, angle of attack, side slip angle, and the rotational velocity components. However, to solve the equations of motion, it is also necessary to know the attitude of the aircraft relatively to the Earth. The attitude of the aircraft with respect to the Earth is defined by the Euler angles. The kinematic relations which determine the time-derivatives of these Euler angles are given by the following equations:

φ^{•} = p + (q \sin φ \tan θ) + (r \cos φ \tan θ)

(35)

θ^{•} = q \cos φ - r \sin φ

(36)

ψ^{•} = (q \sin φ \sec θ) + (r \cos φ \sec θ)

(37)

p^{•} = (9.91 e - 3) pq + 0.94 qr + 666.58 l + 106.6 n

(38)

q^{•} = - 0.022 p^{2} - 0.78 pr + 0.022 r^{2} + 609.75 m

(39)

r^{•} = - 0.588 pq - (9.91 e - 3) qr + 106.59 l + 4422.33 n

(40)

The position of the aircraft with respect to the Earth-fixed reference frame is given by the coordinates x-, y-, and z, which are defined by the following derivative equations:

x^{•} = u \cos θ \cos ψ + v [(\sin φ \sin θ \cos ψ) - (\cos φ \sin ψ)] + w (\cos φ \sin θ \cos ψ + \sin φ \sin ψ)

(41)

y^{•} = u \cos θ \sin ψ + v [(\sin φ \sin θ \sin ψ) + (\cos φ \cos ψ)] + w (\cos φ \sin θ \sin ψ - \sin φ \cos ψ)

(42)

z^{•} = - u \sin θ + v \sin φ \cos θ + w \cos φ \cos θ

(43)

Often the altitude of the aircraft is used instead of the coordinate z. The relationship between the time derivatives of H and z is simple and is given as,

H^{•} = - z^{•}

(44)

The above derived differential equations namely, three force equations, three moment equations, and six kinematic relations can be combined in a nonlinear vector equation and written as follows:

X^{•} = f {X, F (t), M (t)}

(45)

Here “X” is defined as the state vector and is written as follows:

X = [V α β pqr φ θ ψ xyz uvw] T

(46)

The above equation represents the dynamic model of the rigid body. These equations plays an important role to understand the dynamic behavior of a MAV and also build the flight simulation. Using this dynamic model, a flight simulation has been done in Simulink to generate the true values of the states.

TRIM conditions—Steady state flight

Trim refers to achieving a steady state flight by balancing forces and moments acting on it. In a simple sense, it can be achieved by making lift equal to drag, thrust equal to weight and provided that there is no pitching moment.

The flight simulation has been carried out about a steady wing level trim condition. A code has been written in Simulink for trimming the aircraft in longitudinal direction and supplies these initial conditions to the flight simulation when the simulation begins at time t = 0 for first iteration. For later part of simulation, the values at time, t − 1 are used in simulation at time, t. The trim algorithm solves the aircraft equations of motion in order to balance the forces and moments. This is achieved by equating $V^{•}$ , $α^{•}$ and $β^{•}$ equations to zero and simplifying them with conditions θ = α, p = 0, q = 0, r = 0, and β = 0.

V^{•} = \frac{1}{m} {F_{X} \cos α \cos β + F_{Y} \sin β + F_{Z} \sin α \cos β)

(47)

α^{•} = {\frac{1}{Vm \cos β}} ((- F_{X} \sin α + F_{Z} \cos α)) + q - (p \cos α + r \sin α) \tan β

(48)

q^{•} = - 0.022 p^{2} - 0.78 pr + 0.022 r^{2} + 609.75 m

(49)

Since β = 0 and θ = α, the equation (2.47) may be written as,

\frac{1}{m} (F_{X} \cos α + F_{Z} \sin α) = 0

(50)

The forces in the above equation are same as the forces in equations (21), (22), and (23). Note that “Fpz” is zero here as there is no contribution from the thrust towards the force along body z-axis. Using the above equations and simplifying equation (50) yields

Thrust * \cos α = C * q_{dyn} * S

(51)

Similarly, the equations (48) and (49) are equated to zero and simplified. This yields the following relations:

- C * q_{dyn} * S - Thrust * \sin α + W = 0

(52)

C_{m} = 0

(53)

It may be noted that C_L, C_D, and C_m are functions of α and $δ_{e}$ . Thrust is a function of V and n. Now by solving equations (51), (52), and (53) by considering them as simultaneous algebraic equations in α, n, and $δ_{e}$ for a given V, we obtain the trim values of α, n, and $δ_{e}$ . Various control surface inputs ( $δ_{e}, δ_{a}, δ_{r}, δ_{T}$ ) are given about the trim condition and the true states of the aircraft are generated using the complete six DOF equations for estimation. It is noteworthy to recall that the description of aerodynamic coefficients such as C_L, C_D, C_m, and thrust are given above.

State integration

The state dynamic vector presented earlier in this chapter has been used in simulating the true states of the MAV. State dynamic vector consists of 15 first-order partial differential equations. Each equation represents a state with a function of state dynamic vector. Different methods are available for integrating the state dynamic vector. The different types of solver available in Simulink are fixed step solver and Variable step solver.

In case of fixed step solver, the step size remains constant throughout the simulation, whereas in the case of variable step solver, the step size varies throughout the simulation depending on the dynamics of the model. In particular, a variable-step solver increases or reduces the step size to meet the error tolerances. The various types of methods available in explicit fixed step solver are Euler’s method, Heun’s method, Bogacki-shampine, fourth-order Runge-Kutta method, etc., similarly, the various types of methods available in variable step solver are Dormand Prince, stiff/NDF, ADAMS, stiff/mod. Rosenbrock, Bogacki-Shampine, mod. Stiff/Trapezoidal, etc. In our present scenario, we used fourth-order Runge-Kutta method to solve state dynamic vector and to generate true states of MAV.

Flight simulation—Results

A flight simulation has been built in Simulink and Simulated with the initial conditions obtained from trim algorithm in order to achieve a steady state flight and the results have been verified from the data obtained from NAL, Bangalore. The present simulation is performed at V = 10 m/s and at a height, H = 150 m. Upon simulation, different true states of MAV like velocity, angle of attack, side slip angle, pitch, roll, yaw, roll angle, pitch angle, yaw angle, latitude, longitude, altitude, velocity along body x-, y-, and z-axis are generated in the form of plots. In case of steady state flight, all the states except velocity, angle of attack, pitch angle, latitude, altitude, velocity along body x- and z-axis are equal to zero and the pitch angle is equal to angle of attack. Later, the variation in states is also observed by disturbing the elevator through an impulse input of magnitude 0.03 radian from 5 to 6 s.

EKF observer

A major problem in the control system is internal states of many systems cannot be directly observed and therefore state feedback is not possible, which in turn makes highly impossible to design a controller for a particular state parameter. An observer is an algorithm that reconstructs the system states from measurements and the mathematical model of the dynamic system. Observers use the available measurements to estimate the states of a dynamic system, which is essential for feedback control purposes when all the states are not measurable. The main assumption behind this concept is that the model of the dynamic system and inputs to it are known.

The Kalman filter provides an efficient computational procedure to estimate the states of a linear system. It has advantages in the form of minimum variance estimator and also operates recursively. But for a system based on a nonlinear model, there is no unique solution to nonlinear filtering. The EKF is one of the variant of Kalman filter based on linearization of the nonlinear model about the most recent state estimate and covariance. Linearization about the most recent estimate helps to minimize the errors. Kalman filter has numerous application in current technology. A common application of Kalman filter is found in navigation, guidance, and control of vehicles. Furthermore, Kalman filter is widely used in time series analysis, which includes signal processing and econometrics. Kalman filter algorithm works in a two-step process. It is also known as prediction-correction algorithm. In prediction step, Kalman filter provides the estimate of current state variables with other uncertainties. When the outcome of the next measurements are observed, these estimates are updated using a weighted average. The more weight being given to estimates gives state with higher certainty. The outputs of the accelerometer along the body axes at the centre of gravity, usually called as the specific forces are considered in the method of estimation of the attitude angles. The accelerometer output equations are as given in equation (60). These specific forces are taken from the flight simulation and a zero mean white noise is added. An EKF estimation algorithm has been implemented. The state vector and measurement vector are written as follows:

Statevector x = [θ φ]^{T}

(54)

Controlvector u = [pq r] T

(55)

Measurementvectory = [A_{x} A_{y} A_{z}]^{T}

(56)

In the attitude estimation stage, the pitch and roll angles are estimated. The kinematic models for these two states are as quoted in the equations (35) and (36), which are rewritten in vector form as follows:

Now we know that $α = θ$ and β = 0 for the steady wing level flight, which reduces the equations

Also, it has been assumed that the rate of change of component velocities are small, which brings out,

(u^{•} v^{•} w^{•}) \approx 0

(59)

Based on the above equations (58) and (59), we write the measurement model for this stage as follows:

(A_{x} A_{y} A_{z}) = (\frac{Vq \sin θ}{g} + \sin θ \frac{V (r \cos θ - p \sin θ)}{g} - \cos θ \sin φ \frac{- Vq \cos φ}{g} - \cos θ \cos φ)

(60)

Using above state vector and measurement vector model, observer is designed through Kalman filter approach. A zero mean white Gaussian noise with specific standard deviation is added to accelerometer and gyroscope sensors along all directions and pitch angle is predicted from these noisy sensor input and the predicted pitch angle is used to design controller to track pitch angle in a MAV.

Controller design

Controllers plays a vital role in lot of industrial applications. It can be used to regulate speed, temperature, pressure, and other process variables. A controller continuously calculates the error value defined as the difference between measured process variable and a desired set point. It attempts to minimize the error over time by adjusting process variables. In today’s industrial application various controllers like proportional (P), PI, and proportional-integral-derivative (PID) are widely used. The combination of three terms in PID are sometime interpreted as proportional controller is the accumulation of present errors, integral controller is the accumulation of past errors, and derivative controller predicts the future errors. In our present problem, we have developed a PI controller to track pitch angle in a MAV. The error value calculated through comparator is given to PI controller, which in turn controls the elevator to achieve the desired pitch angle.

u (t) = K_{p} e (t) + K_{i} \int_{0}^{t} e (t) d t

(61)

The output value produced by proportional term is proportional to the current error and the response can be adjusted by multiplying error value with proportional gain constant, Kp. The output value produced by integral term is proportional to both magnitude and duration of the error. The integral term is the sum of the instantaneous error over time and gives the accumulated errors that have been corrected previously. The accumulated error is then multiplied by the integral gain constant, Ki. The desired target of the process can be achieved by tuning the gain constants of both integral and proportional terms. The well trade-off between both terms considering its advantages and disadvantages gives a well-tuned fine controller. In our problem, we have adopted Ziegler–Nichols Closed-Loop tuning method, which is one of the oldest and the finest methods to tune PID controller and its principle can be applied to PI controller also.

Ziegler–Nichols closed-loop tuning

Ziegler–Nichols method was developed by John G. Ziegler and Nathaniel B. Nichols and it is one of the heuristic method of tuning of P/PI/PID controllers. In this method, the integral gain constant and derivative gain constant are adjusted to zero and the proportional integral constant is tuned until it achieves a maximum value, which is also known as ultimate gain, K_u at which output of the control loop is stable and the oscillations are consistent. The time required to complete one full oscillation at steady state is recorded as oscillation period, T_u. Using these two parameters, ultimate gain and oscillation period, the other gain constants can be defined as follows (Table 1).

Table 1.

Gain constant parameters by Ziegler–Nichols method.

Controller type	K_p	K_i	K_d
Proportional (P)	0.50K_u	–	–
Proportional-integral (PI)	0.45K_u	$1.2 \frac{K_{p}}{T_{u}}$	–
Proportional-integral- derivative (PID)	0.60K_u	$2.0 \frac{K_{p}}{T_{u}}$	$\frac{K_{p} T_{u}}{8}$

Results

The major focus of the problem is to design a controller to track pitch angle. In order to achieve this, a steady state flight simulation based on mathematical mode of Black Kite MAV is built in Simulink and noisy sensor outputs are taken into consideration to predict pitch angle. The predicted pitch angle is taken as negative feedback and a PI controller is designed. Figures 3 and 4 show the pitch rate and pitch angle in steady state flight, whereas Figures 5 and 6 show the variation of pitch rate and pitch angle with disturbance in elevator. Figure 7 shows the comparison of predicted pitch angle and pitch angle obtained from flight simulation. The designed controller is tested with various inputs like ramp, sine, and step input. The controller works very well when the individual input like sine, ramp, or step given but it exhibits minor noise in case of combined input of all these. The following figures illustrates the results when the controller is tested with different inputs. It gave us confidence to estimate and design controller to other attitude angles and also to extend this approach for designing other type of controllers.

Figure 1.

Black Kite MAV.

Figure 2.

Flight simulation of Black Kite mini aerial vehicle in Simulink.

Figure 3.

Pitch rate in steady state flight.

Figure 4.

Pitch angle in steady state flight.

Figure 5.

Pitch rate with disturbance in elevator.

Figure 6.

Pitch angle with disturbance in elevator.

Figure 7.

Predicted pitch angle vs flight simulation pitch angle.

Figure 8.

Pitch angle tracking for mixed input.

Figure 9.

Pitch angle tracking for pure sine wave.

Footnotes

Acknowledgment

We greatly acknowledge management and faculty of SASTRA University for providing this opportunity.

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) received no financial support for the research, authorship, and/or publication of this article.

Appendix 1

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Design of a proportional-integral controller to track pitch angle in a mini aerial vehicle