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Abstract

In the present study, a desired reference trajectory was autonomously tracked by means of a quadrotor unmanned aerial vehicle with a self-tuning fuzzy proportional integral derivative controller. A proportional integral derivative controller and a fuzzy system tuning gains from proportional integral derivative controller are applied to stabilize the quadrotor, to control the attitude and to track the trajectory. Inputs of fuzzy logical controller consist of the speed required for the distance between the current position of unmanned aerial vehicle and the defined reference point and differences between orientation angles and variance in differences. Outputs of fuzzy logical controller consist of the proportional integral derivative coefficients which produce pitch, roll, yaw and height values. The fuzzy proportional integral derivative control algorithm is real-time applied to the quadrotor in MATLAB/Simulink environment. Based on data from experimental studies, although both classical proportional integral derivative controller and self-tuning fuzzy proportional integral derivative controller have accomplished to track a defined trajectory with the aircraft, the self-tuning fuzzy proportional integral derivative controller has been able to control with less errors than the classical proportional integral derivative controller.

Keywords

Quadrotor proportional integral derivative fuzzy self-tuning trajectory tracking

Introduction

Unmanned aerial vehicles are very complicated high-tech systems that require the cooperation of aeronautical, electronic, mechanical, electro-mechanical and software disciplines.¹ These vehicles have become primary and critical research areas and been widely used for military and civil purposes such as aerial surveillance, data collection, remote sensing, reconnaissance, search and rescue, fire detection, damage assessment, communication transfer, mine dragging, detection of hazardous substances and logistics. Other advantages with using unmanned aerial vehicle include lower loss costs, being superior over a manned aircraft in terms of loss of life and absence of limitations such as human-related fatigue or operating hours as with a manned vehicle.² As a complement of systems, unmanned aerial vehicle’s basic components include an aircraft, ground control station and data terminal. Although currently a large part of applications using such tools can be manually conducted by a user at the ground control station, the number of systems that can autonomously control the flight is growing increasingly.³

One of the most preferred unmanned aerial vehicles is a four-rotor helicopter, also known as quadrotor. This aircraft has superior features such as ability to take-off and land vertically, hover in a stable state in the air and perform quick maneuvers. However, it has a number of limitations such as having a complex and difficult control and limited capacity of energy and load.⁴

The synchronized rotational speed for four motors that are mounted per corner equidistant from the center is the key to control the quadrotor.⁵ There are many controller designs and applications in the literature to control unmanned aerial vehicles, e.g., proportional integral derivative (PID)/PD controller, fuzzy controller, sliding mode controller, neuro-fuzzy controller and image-based controller. To control quadrotor, an intelligent system has been developed by Santos et al.⁶ based on fuzzy logic. They intended to re-stabilize roll, pitch and yaw angles by applying the height and certain values of axis angles, as initial conditions, to the simulation model they created using parameters of the quadrotor they designed. The results of simulation demonstrated good performance of the intelligent fuzzy controller they developed using an input structure similar to inputs of PID controller. Dydek et al.⁷ developed a direct and indirect model reference adaptive control system for a low-cost and light-weight quadrotor unmanned aerial vehicle platform. They used Draganflyer V Ti, four rotor helicopter, and its parameters for their study. The validity of approach to adaptive controlled trajectory tracking was tested by simulations and indoor flight tests. The adaptive controller is observed to increase the strength of quadrotor against parametric uncertainties. Al-Younes and Jarrah⁸ compared the performance of backstepping fuzzy logic controller with the performance of backstepping least mean square controller in terms of attitude stabilization control of a quadrotor. Parameters of backstepping controller were obtained using fuzzy logic in the study, and the backstepping fuzzy logic controller delivered an outstanding performance. Amoozgar et al.⁹ used an adaptive PID controller for fault-tolerant control of quadrotor in the presence of actuator faults. They used Quanser Qball-X4 unmanned aerial vehicle as the platform of quadrotor and a fuzzy inference scheme for real-time tuning of PID controller gains. The method they proposed appears to be efficient according to the results. Altug et al.¹⁰ presented a control method for an autonomous quadrotor using visual feedback as the primary sensor. HMX-4 quadrotor is used in their study. They used a ground camera in their system to calculate the position and the motion. Tests were performed by two different control methods. The first method they used was a series of mode-based feedback linearizing controllers, and the other method was using backstepping control law. These control methods were tested by simulations in MATLAB/Simulink environment, and backstepping controller delivered more successful results. Bouabdallah and Siegwart¹¹ studied backstepping and sliding-mode control methods in quadrotor control. They used OS4 quadrotor to test control methods, simulations and actual system. They found that sliding-mode controller caused drift in sensor due to its switching nature. On the other hand, the backstepping controller was capable of controlling orientation angles in the presence of high disturbance effects. Erginer and Altuğ¹² compared the use of hybrid fuzzy logic PD controller for controlling quadrotor with classical PD controller. They used Draganflyer III quadrotor to perform tests on the real system. When the results of simulation and application were compared, the fuzzy PD controller which is a more suitable choice for non-linear systems delivered a better performance against disturbing effects than that of classical PD controller. Sharma and Barve¹³ compared the performance of classical PID controller with the performance of fuzzy logic controller in terms of ensuring flight stabilization of quadrotor. The tests performed in simulation environment indicated that fuzzy logic controller stabilized the aircraft faster than PID controller did. Mahony et al.¹⁴ presented modeling, estimation and control of quadrotor. After modeling the aircraft in MATLAB/Robotic toolbox, the position was estimated using data obtained from inertial measurement unit and the aircraft was controlled using image-based control technique. Gomez-Balderas et al.¹⁵ described a quadrotor tracking a ground moving target using switching control method. The architecture of the quadrotor includes an embedded camera. Images from this camera were processed in real time by an on-board computer to estimate the position based on image.

In this study, a flight control system is designed which is required for a quadrotor to fly autonomously and to direct maneuvers of the aircraft. Autonomous flight tasks include ability to take-off and land vertically, hover in a certain position in the air and follow a certain trajectory. Development of control algorithms is the main step of the study. Therefore, use of a quadrotor with predefined parameters for the existing hardware allows quick design of control algorithms. In the study, MATLAB/Simulink environment was applied in order to control quadrotor tracking a defined reference trajectory in the space coordinate system by means of self-tuning fuzzy PID controller. The PID gains scheduling-based control algorithm is used for attitude stabilization and position control of quadrotor. The tuning of PID gains is performed by a fuzzy logic controller. The experimental study section provides the comparison of results from tracking a trajectory with classical PID controller and with self-tuning fuzzy PID controller.

Mathematical model of quadrotor

Development of small-sized light-weight electronic components that increase the speed and reliability in communication technology and improve command control algorithms and data transmission systems has made quadrotor unmanned aerial vehicles popular among researchers, academicians and military users. This type of aircraft has been used by researcher in universities for the purposes of testing and evaluation in many areas such as flight control theory, navigation, real-time systems and robotics. It is also used by security forces for military purposes including tracking, reconnaissance and search and rescue in cities.

A quadrotor is a helicopter with four rotors mounted symmetrically. Thus, the quadrotor movements are similar to a conventional helicopter. The difference is that movement is achieved by varying each of four motor speeds to obtain a desired effect that causes movement. This aircraft is able to take-off and land vertically and hover in a certain position in the air as well as flying at low speeds. Other advantages of this aircraft include high maneuverability, possibility to design the unmanned version in smaller size, reduced mechanical complexity due to absence of moving parts except propellers, easy validation of platform in a laboratory and low cost.

Vertical motion of quadrotor is a result of simultaneously increasing or decreasing the rotational speeds of all rotors. The motion along any direction on the lateral axis is obtained by decreasing the rotational speeds of rotors along the desired direction of motion and increasing the rotational speeds of rotors opposite to the desired direction of motion. The longitudinal motion and yaw motion are achieved by moment of rotors that rotate at different speeds in the same manner as lateral motion.¹⁶ Figure 1 gives the motions of the quadrotor schematically.

Figure 1.

Basic motion of a quadrotor. CCW: counterclockwise.

Axis of the quadrotor motion with six degrees of freedom (6-DOF) is shown in Figure 2. The reference system and abstract control inputs (f_i) are associated with the model of aircraft. Inertial and body reference axes are represented with x, y, z and i, j, k, respectively.

Figure 2.

6-DOF model of a quadrotor with the reference system.

The pitch motion (θ) is produced in the direction of x axis as a result of rotational motion around the y axis. The rolling motion (φ) is produced in the direction of y axis as a result of rotational motion around the x axis. The yaw motion (ψ) is the rotational motion around the z axis. Thus, the quadrotor rotates in the direction of vertical axis. The quadrotor ascends or descends on the z axis with throttle. Thus, the quadrotor is enabled to fly to any given point in space either by a sequence of different motions or by combining several motions together.

The association between the axis of body reference and the inertial reference axis of aircraft is established through position vectors x, y, z and Euler angles θ, φ, ψ. The aircraft is assumed to be rigid and symmetrical. However, the center of mass, center of gravity and origin point of body axis are assumed to be coincident. In these conditions, equations (1) to (3) provide mathematical expression of $T z, y, x$ matrix to describe transformation from body axis to inertial axis. Herein and in subsequent uses, the mathematical notations c and s represent cosine function and sine function, respectively.

T z, y, x = T z . T y . T x

(1)

T z, y, x = [c ψ - s ψ 0 s ψ c ψ 0001] . [c θ 0 s θ 010 - s θ 0 c θ] . [1000 c φ s φ 0 s φ c φ]

(2)

T z, y, x = [c θ c ψ c ψ s θ s φ - s ψ c φ c ψ s θ c φ + s ψ s φ c θ s ψ s ψ s θ s φ + c ψ c φ s ψ s θ c φ - c ψ s φ - s θ c θ s φ c θ c φ]

(3)

The thrust and moments in opposite direction of rotation are produced in the vertical axis of aircraft as a result of rotational motion of each rotor of quadrotor. Rotors are reciprocally matched to balance the total moment. Rotors 1 and 3 produce a counterclockwise moment, and rotors 2 and 4 produce a clockwise moment. It is experimentally observed that these moments are subject to forces produced linearly at low speeds. There are four input forces and six outputs (x, y, z, θ, φ, ψ). Equation (4) provides mathematical expression of input forces of quadrotor. They are also called “Control Inputs.”

[U 1 U 2 U 3 U 4] = [b b b b 0 - b 0 b - b 0 b 0 - d d - d d] . [Ω_{1}^{2} Ω_{2}^{2} Ω_{3}^{2} Ω_{4}^{2}]

(4)

In equation (4), U₁ represents the thrust force acting on quadrotor body, U₂ represents the force causing roll motion, U₃ represents the force causing pitch torque and U₄ represents the force causing yaw torque. $Ω i$ indicates angular velocity of rotors (i = 1, 2, 3, 4), b indicates the thrust factor and d indicates the drift factor. The complete Euler–Lagrange mathematical model of the system with assumptions that the structure of quadrotor is fully rigid and symmetrical, and that the center of mass and center of body are on the same axis is given in equation (5).

{\overset{··}{x} = (c ψ s θ s φ + s ψ c φ) . \frac{U 1}{m} \overset{··}{y} = (s ψ s θ c φ + s ψ c φ) . \frac{U 1}{m} \overset{··}{z} = - g + (c θ c φ) . \frac{U 1}{m} \overset{··}{φ} = \overset{\cdot}{ψ} \overset{\cdot}{θ} (\frac{I y - I z}{I x}) - \frac{J}{I x} \overset{\cdot}{θ} (- Ω 1 + Ω 2 - Ω 3 + Ω 4) + \frac{l}{I x} U 2 \overset{··}{θ} = \overset{\cdot}{ψ} \overset{\cdot}{φ} (\frac{I x - I z}{I y}) + \frac{J}{I y} \overset{\cdot}{φ} (- Ω 1 + Ω 2 - Ω 3 + Ω 4) + \frac{l}{I x} U 3 \overset{··}{ψ} = \overset{\cdot}{θ} \overset{\cdot}{φ} (\frac{I x - I y}{I z}) + \frac{1}{I y} U 4

(5)

Equation (5) calculates acceleration produced on x, y, z axes and angular accelerations. Here, J represents the inertia of one rotor; $I x, y, z$ are the inertia of body of quadrotor on x, y, z axes, respectively; l is the distance of rotor to the center of mass; g is the gravitational acceleration and m is the mass of aircraft.^17,18

Control methods for quadrotor

The aircraft is controlled by various controlling structures which are included to the developed mathematical model of the quadrotor. To design such controllers, U₁, U₂, U₃ and U₄ are used which are the inputs of the system moving the aircraft. These inputs control the vertical motion (h) on the z axis, the motion on the y axis, (roll angle—φ), the motion on the x axis, (pitch angle—θ) and the yaw angle (ψ) around the z axis, respectively. The numerical values to assign to these inputs are calculated by algorithms of the controller that is designed by the same inputs. As provided in equations, U_i inputs are obtained from angular velocity of propellers. This angular velocity is achieved by rotation of motor. Due to model-based system design, φ roll angle, θ pitch angle, $\overset{\cdot}{ψ}$ yaw rate and $\overset{\cdot}{h}$ vertical velocity, which represent U_i inputs, are used to control the aircraft instead of rotational angle of motors while creating algorithms of controller.

PID control

A PID controller is a feedback controller that is widely used in different types of systems, robotics and automation industry throughout the world due to its simple and robust structure and performance that often gives satisfactory results. The output equation of PID controller which is a type of linear controller is provided in equation (6).¹⁹

U PID = k p e (t) + k i \int e (t) dt + k d \frac{de (t)}{dt}

(6) where k_p, k_i and k_d are proportional, integral and derivative gains, respectively, e(t) is the error caused by the difference between the reference and response of the system. The proportional gain is used to control the rise time of system response. The integral gain is used to eliminate steady-state error. The derivative gain allows reducing the amount of overflow and developing a transient response. Table 1 shows the effect of such increased parameters on a controlled system.²⁰

Table 1.

Effects of independent P, I and D tuning.

Closed-loop response	Rise time	Overshoot	Settling time	Steady-state error	Stability
Increasing k_p	Decrease	Increase	Small increase	Decrease	Degrade
Increasing k_i	Small decrease	Increase	Increase	Large decrease	Degrade
Increasing k_d	Small decrease	Decrease	Decrease	Minor change	Improve

The success of PID controller depends on proper selection of gain parameters. The PID controller in the literature can be divided mainly into two categories. In the first category, the controller’s parameters are unchanged during control after they have been tuned or chosen in certain optimal ways. At that point, the most well-known method is tuning PID gains with Ziegler–Nichols formula. The classical PID controllers in this category have a simple structure but cannot always efficiently control systems according to changing conditions and may need frequent on-line retuning. In addition, since classical PID controller has a fixed gain, its performance is limited within a range wider than operating points. Controllers in the second category have the structure of a classical PID controller, but their parameters can be adapted on-line. Certain knowledge like the structure of plant model is required here. Controllers in this category can be defined as dynamic or adaptive.²¹

The literature includes studies in which PID controllers are applied in quadrotor and an autonomous flight is performed. OS4 quadrotor aircraft was hovered in air efficiently by PID controller.^22,23 However, test results suggested that the stability of quadrotor was not fully obtained and a slight steady-state error appeared. Particularly for unexperienced beginners, a controller has been designed in order to make flight control easy for a quadrotor and to ensure a stationary hover in air at a fixed height from the ground.²⁴

Self-tuning fuzzy PID control

As a PID controller is based on the linear model, non-linearity in system brings uncertainty and degraded performance. This eliminates learning ability and adaptability necessary to overcome non-linearities and uncertainties. Thus, conversion of proportional, integral and derivative gains of PID controllers into a self-tuning structure, in order to adapt on-line, increases the success rate in control of non-linear platforms.^25,26 This is the point where we can include fuzzy logic, a knowledge-based control method, in control algorithm in order to determine PID parameters according to changing conditions.^27,28

The input signal goes through fuzzification, inference and defuzzification stages to find the fuzzy control output corresponding to an input in the fuzzy logic controller. The fuzzification, inference and defuzzification stages are used in order for the crisp matching between the rule base and knowledge base. The fuzzy values may range from 0 to 1. This is a significant advantage for fuzzy logic. Furthermore, the fuzzy logic does not require the mathematical model of the system to which it will be applied. It should be enough to have only expert knowledge for designing the fuzzy controller. Therefore, the fuzzy logic is a control method that is easier to understand compared to conventional control methods.²⁹

Using the universe of fuzzy reasoning and discourse to regulate PID gains based on the classical PID controller enable the inclusion of robustness and adaptability features of fuzzy system into the created control algorithm. Figure 3 shows block diagram of self-tuning fuzzy PID control algorithm used in the study.

Figure 3.

Basic structure of self-tuning fuzzy PID control algorithm. PID: proportional integral derivative.

The term self-tuning indicates the controller characteristics of tuning its control parameters on-line automatically so as to have the most suitable values of those gains obtained, which results in optimization of the process output. The design of controlling rules for a self-tuning fuzzy PID controller is based on theoretical and experimental analysis. The gains k_p, k_i and k_d can be tuned by adjusting other controlling parameters and coefficients on-line, which increases the precision of control resulting in better performance than the classical PID controller. The main advantages of this controller are to avoid overflow and reduce settling time.

The self-tuning fuzzy PID controller describes how to find the fuzzy relationship between the gains k_p, k_i and k_d and the e(t) and de(t), and to modify these three gains to meet different requirements according to principles of fuzzy control.³⁰ The input of self-tuning fuzzy PID controller is error e(t), change of error de(t) and the output is the control signal transmitted to the system. PID gains are tuned by the fuzzy logic controller where such parameters are assumed to be variable in the range of [ $k p \min$ , $k p \max$ ], [ $k i \min$ , $k i \max$ ] and [ $k d \min$ , $k d \max$ ], respectively, during adjustment of gains k_p, k_i and k_d. Each parameter range is determined by tests performed on the PID controller in order to achieve applicable rule bases with high inference efficiency. The parameters k_p, k_i and k_d are normalized to between zero and one for convenience through formulas provided in equation (7).

{k p' = (k p - k p \min) / (k p \max - k p \min) k i' = (k i - k i \min) / (k i \max - k i \min) k d' = (k d - k d \min) / (k d \max - k d \min)

(7)

In order to adjust gains, the fuzzy rules are created as follows: (if) error e(t) is A_i and change of error de(t) is B_i, (then) $k p'$ gain C_i, $k i'$ gain D_i and $k d'$ gain are E_i (i = 1, 2, … m, m is total number of rules). A_i, B_i, C_i, D_i and E_i represent the fuzzy sets. These fuzzy sets are characterized by linguistic terms through membership functions created. After fuzzification using membership functions and rule bases, defuzzification is carried out according to the selected method. Finally, obtained parameters $k p'$ , $k i'$ and $k d'$ are inserted into equation (8) where appropriate to find gains of the controller.³¹

{k p = (k p \max - k p \min) k p' + k p \min k i = (k i \max - k i \min) k i' + k i \min k d = (k d \max - k d \min) k d' + k d \min

(8)

The literature includes studies performed by a simulator, which has delivered successful results utilizing the self-tuning fuzzy PID controller to achieve attitude stabilization and motion control of a quadrotor.^32,33 The contribution of our study to the literature is that the trajectory is autonomously tracked in real-time by a quadrotor using a self-tuning fuzzy PID controller.

Autonomous trajectory tracking of quadrotor

The AR.Drone 2.0 which is manufactured by Parrot is used in this study. The AR.Drone is a low-cost quadrotor. Open application programming interface and Wi-Fi control features make the aircraft used in the study very suitable for allowing quick development of algorithms. A MATLAB/Simulink interface is developed to transmit data to the AR.Drone and to read data from the AR.Drone. The developed Wi-Fi interface allows quick performance of system identification, control and guidance tests. Figure 4 shows a block representation of the developed system.

Figure 4.

Block representation of the developed system. UDP: User Datagram Protocol.

The control of quadrotor to autonomously reach reference waypoints is achieved by means of controllers provided in the block representation. In the first stage of the study, a classical PID control algorithm and in the second stage, a self-tuning fuzzy PID control algorithm is employed in these blocks in order to track trajectory. The accuracy of control and guidance algorithms designed with a dynamic model of the AR.Drone was tested in real-time on the testbed given in Figure 5. The experimental study section details the application data obtained.

Figure 5.

Testbed for real-time applications.

Platform of the AR.Drone quadrotor

The main reasons for using the Parrot AR.Drone 2.0 in this study include outstanding maneuverability, an environment to develop open source software, low-cost and availability of support for spare parts. The AR.Drone 2.0 has a cross type quadrotor design. Total flying weight is 380 g with indoor hull (Figure 6(a)) and 420 g with outdoor hull (Figure 6(b)).

Figure 6.

The AR.Drone 2.0 with outdoor hull and indoor hull: (a) outdoor hull and (b) indoor hull.

The AR.Drone 2.0 is powered by an 11.1 -V, 1500-mAh Li-Po battery with discharge rate of 10 C. The charging status is continuously monitored by the embedded system. The aircraft is activated by four three-phase brushless DC motors with 14.5 W and 28,500 r/min. The motor drivers incorporate sensors to have status information on stopping and turning of motor. An 8-MIPS AVR processor is available for each driver. This aircraft is suitable for academic studies due to its electronic and mechanical features.

The mainboard of AR.Drone 2.0 contains 1 GHz 32 bit ARM Cortex A8 processor and TMS320DMC64X digital signal processor of 800 MHz with unshared video support. The system also has a 200-MHz, 1-GB DDR2 RAM memory. With such features, it is able to run a 32-bit operating system. The mainboard contains modules to communicate with the outer world. These modules particularly include IEEE 802.11 wireless communication protocol allowing TCP/IP communication. This protocol uses the same 2.4 GHz frequency as Wireless Local Area Network that is widely used today. The communication between the aircraft and the controller (tele-operator or the master computer) is achieved by employing three different User Datagram Protocol (UDP) communication channels through Wi-Fi connection in a coverage of 100 m. Initial commands transmitted to the aircraft via command channel are land, take-off, adjust limits, calibrate sensors, change cameras, pitch, roll, yaw and adjust vertical velocity. Status data of the aircraft are transmitted by NavData channel to receive feedback. This data include the current state of the aircraft (flying, landing, taking-off and calibration status of the aircraft) and sensor data (current pitch angle, roll angle, yaw rate, height, battery level and velocity on all axes). Finally, visual data from cameras are transmitted through stream channel. The communication distance can be easily increased by router and expander relays. Another communication module is the USB 2.0 port included on the aircraft. If required, any USB devices, e.g. GPS and flash memory, can be inserted to the aircraft.

The location of front camera in the front of AR.Drone 2.0 directly affects the location of aircraft rotor. The image of 1280 × 720 pixel resolution captured by the front camera is transmitted at 30 FPS via wireless communication protocol using H.264 video coding. Another camera (vertical camera QVGA) is located under the aircraft and transmits an image of 320 × 240 pixel resolution at 60 FPS via IEEE 802.11 wireless communication protocol. The vertical camera is used, in the navigation algorithms, to measure the horizontal (longitudinal and lateral) velocity. Two different algorithms are used to estimate the horizontal velocity. One tracks local interest points over different frames and calculates the velocity from the displacement of these points. It provides a more accurate estimate of the velocity and is used when the aircraft’s speed is low and there is enough texture in the picture. The second algorithm estimates the horizontal speed by computing the optical flow on pyramidal images. It is the default algorithm during flight. It is less precise but more robust since it does not rely on highly textured or high-contrast scenes.

The mainboard, serving the key role for flying, receives status info on the aircraft from navigation board, which contains an inertial measurement system with 6-DOF. This inertial measurement system consists three-axis gyroscope of 2000°/s precision, three-axis accelerometer with ± 50 mg precision and three-axis magnetometer with 6° precision. In order to support this system, an ultrasonic sensor together with a pressure sensor provides altitude measurements for automatic altitude stabilization and assisted vertical speed control. The navigation board sensors give feedback to the AR.Drone 2.0. Euler angles of the AR.Drone are calculated using these data which is used to achieve attitude stabilization.³⁴

Data communication

The AR.Drone quadrotor can be controlled by a computer with IEEE 802.11 wireless connection. AT commands are transmitted through the port No.5556 in UDP packets. This port is listened to by the embedded system of AR.Drone. The embedded system processes the transmitted commands every 30 ms to allow a softer motion of the aircraft. The system assumes that the wireless connection is disconnected when there is a gap greater than 2 s between two sequential command lines, and then switches to emergency landing mode. The quadrotor provides two data flows to the client it is connected. One of them is cruise data (navdata) and the other is video broadcast. The cruise data include angle values from inertial measurement system, pressure and altitude values from ultrasonic sensor, camera images and speed data. Cruise data are transmitted by the aircraft to the client in UDP packets through port No.5554 in accordance with predefined stream sequence. Unlike other communications, video broadcast is transmitted through port No.5555 via TCP. The embedded system starts video broadcast once the client is connected.^35,36 Internal Wi-Fi subsystem in the aircraft is used to measure signals for identification. This process is shown in Figure 7. In this figure, the server is represented by the AR.Drone and the client is represented by the computer.

Figure 7.

Communication stream.

In the present study, MATLAB/Simulink is used to communicate with the aircraft. First, the internet connection of the aircraft is chosen from the list of available wireless connections. The communication is established through UDP protocol using UDP block in Simulink. Once the communication is established, control signals are transmitted to the aircraft. At the same time, the aircraft starts transmitting data to the computer. This data exchange between the aircraft and the MATLAB/Simulink model in the computer is executed at 16 Hz frequency. The data received by MATLAB are read and recorded. Thus, the aircraft is directly controlled wirelessly via Simulink using Real-Time Windows Target for real-time experiments. Wireless connectivity which is established in this way enables rapid guidance and control.

Dynamics and self-tuning fuzzy PID control of the AR.Drone quadrotor

A dynamic model of the Parrot AR.Drone 2.0 can be derived using experimental input–output data, structure of equations for aircraft motions and system identification techniques.^37,38 The dynamics of AR.Drone are described by transfer functions provided in equation (9). Transfer functions G_φ, G_θ, G_ψ and G_h represent linear dynamics derived using experimental input and output data from aircraft motions.

{G φ = \frac{φ}{φ r} G θ = \frac{θ}{θ r} G ψ = \frac{ψ}{ψ r} Gh = \frac{h}{h r}

(9)

In equation (9), φ, θ and ψ represent roll, pitch and yaw motions of the aircraft, respectively, where h represents the height of the aircraft. The reference commands φ_r, θ_r, $\overset{\cdot}{ψ}$ _r and $\overset{\cdot}{h}$ _r are roll angle, pitch angle, yaw rate and vertical velocity, respectively. The desired motion is achieved by entering reference values into the internal controller of the vehicle as input, which are floating-point values scaled in the range [−1, 1]. The output values φ_m, θ_m, ψ_m, h_m, u_m and v_m measured by inertial sensors and vision algorithms of the aircraft indicate roll, pitch and yaw angles in radian, the height in meters and velocities in m/s on the longitudinal and lateral axes, respectively. The AR.Drone uses inertial information from its IMU for estimating the state of the aircraft. It fuses the IMU data with information from the vision algorithms and an aerodynamics model to estimate the velocity of the aircraft. Figure 8 shows reference input data and measured output data for the quadrotor.

Figure 8.

Inputs and outputs of the AR.Drone.

A guidance module is inserted into the internal controller for the aircraft to follow the waypoints. A mathematical model of aircraft dynamics is used to design internal controllers C_φ, C_θ, C_ψ and C_h with guidance module. The controllers C_u, C_v, C_ψ and C_h shown in Figure 9(a) and (b) are used to track the longitudinal velocity u, lateral velocity v, yaw angle ψ and height h, respectively. The guidance module of the aircraft shown in Figure 9(c) directs the aircraft from current inertial position [X, Y] ^T to the desired position [X_ref, Y_ref] ^T with reference speed commands [u_ref, v_ref] ^T .

Figure 9.

Components of the AR.Drone control and guidance: (a) yaw and height control, (b) velocity control and (c) waypoint control.

Figure 10 shows schematic of positions, orientations and velocity vectors for the aircraft within x–y planes of the local coordinate system.

Figure 10.

Coordinate frames and angles.

The distance P_τ between the aircraft and the target is defined by equation (10) using body axis i, j, and k.

P τ = [e i e j] = T . [X ref - X Y ref - Y]

(10) where T is coordinate transformation that brings a vector from the inertial frame to the moving body frame, and its formula is provided in equation (11).³⁹

T = [cos (ψ) sin (ψ) - sin (ψ) cos (ψ)]

(11)

The results of P_τ expression are multiplied by PID coefficients k, determined by either the classical PID controller or the fuzzy logic controller, to obtain longitudinal and lateral velocities provided in equation (12).

[u ref, v ref]^{T} = k . P τ

(12)

In order to ensure the AR.Drone quadrotor follows the desired trajectory and reaches the reference waypoints with minimum oscillation and error, self-tuning fuzzy PID control algorithm is proposed in the study, which is the content of controllers represented by C_φ, C_θ, C_ψ and C_h blocks. The basic structure of fuzzy logic controller used for tuning PID gains is shown in Figure 11.⁴⁰

Figure 11.

Fuzzy logic controller.

Inputs of fuzzy logic controller shown in Figure 11 are the error e(t) and change of error de(t), and outputs are gains k_p, k_i and k_d. Seven fuzzy values are selected for linguistic expression of e(t) and de(t) inputs of the controller: NB, NM, NS, Z, PS, PM and PB. Negative big is represented as NB, negative medium as NM, negative small as NS, zero as Z, positive small as PS, positive medium as PM and positive big as PB. For linguistic expression of outputs k_p, k_i and k_d of the controller, seven fuzzy values are selected: VVS, VS, S, M, B, VB and VVB. Very very small is represented as VVS, very small as VS, small as S, medium as M, big as B, very big as VB and very very big as VVB. Input and output membership functions used for adjusting gains of PID controllers have an identical form, whereas value ranges of fuzzy sets are different. These ranges are determined using trial-and-error approach. Figure 12 shows defined membership functions.

Figure 12.

Membership functions for all inputs and outputs: (a) e(t) and de(t) inputs and (b) k_p, k_i and k_d outputs.

The membership functions provided in Figures 12(a) and (b) are composed of triangle-shaped memberships. The width of fuzzy sets is selected from inputs to be [−1, 1] for e(t) error and to be [−10, 10] for change of error de(t). In case the inputs value falls outside of these value ranges, a saturation point is used to keep values in this range. The value range for outputs is selected as [0.2, 0.7] for proportional gain k_p, as [0.001, 0.01] for integral gain k_i, and as [0.1, 0.15] for derivative gain k_d.

The set of linguistic rules is the basic part of the fuzzy controller. In most cases, it is easy to convert expert opinions into these rules, and any number of rules can be created to define motions of the controller. In the study, outputs k_p, k_i and k_d have one rule base each. Using the facility to set rules provide by the fuzzy inference system editor in the MATLAB fuzzy logic toolbox, a total of 49 rules are created for three fuzzy logic controllers of three rule bases. These rules are based on expert opinions and experience. Rules provided in Tables 2 to 4 can be interpreted as follows: for example, according to rule tables if the error e(t) is NL and the change of error de(t) is PS, the gain k_p will be VL, the gain k_i will be VL and the gain k_d will be VS. The output of fuzzy logic system is fuzzy and cannot be directly applied to the dynamic system. A defuzzification process is needed to convert these fuzzy outputs into numbers that represent the same fuzzy outputs. The control signal should be continuous, and any change at input should not result in any significant changes at output. The defuzzification algorithm should be clear and the process to determine the output signal should be defined clearly. Therefore, the center of gravity defuzzification method is selected in this study.

Table 2.

Fuzzy rules for k_p.

Change of error (de)	Error (e)
Change of error (de)	NB	NM	NS	Z	PS	PM	PB
NB	M	S	VS	VVS	VS	S	M
NM	B	M	S	VS	S	M	B
NS	VB	B	M	S	M	B	VB
Z	VVB	VB	B	M	B	VB	VVB
PS	VB	B	M	S	M	B	VB
PM	B	M	S	VS	S	M	B
PB	M	S	VS	VVS	VS	S	M

NB: negative big; NM: negative medium; NS: negative small; Z: zero; PS: positive small; PM: positive medium; PB: positive big; VVS: very very small; VS: very small; S: small; M: medium; B: big; VB: very big; VVB: very very big.

Table 3.

Fuzzy rules for k_i.

Change of error (de)	Error (e)
Change of error (de)	NB	NM	NS	Z	PS	PM	PB
NB	M	S	VS	VVS	VS	S	M
NM	B	M	S	VS	S	M	B
NS	VB	B	M	S	M	B	VB
Z	VVB	VB	B	M	B	VB	VVB
PS	VB	B	M	S	M	B	VB
PM	B	M	S	VS	S	M	B
PB	M	S	VS	VVS	VS	S	M

Table 4.

Fuzzy rules for k_d.

Change of error (de)	Error (e)
Change of error (de)	NB	NM	NS	Z	PS	PM	PB
NB	M	B	VB	VVB	VB	B	M
NM	S	M	B	VB	B	M	S
NS	VS	S	M	B	M	S	VS
Z	VVS	VS	S	M	S	VS	VVS
PS	VS	S	M	B	M	S	VS
PM	S	M	B	VB	B	M	S
PB	M	B	VB	VVB	VB	B	M

Experimental results

Real-time control applications are performed on the AR.Drone 2.0 quadrotor with the dynamic model and controllers created in MATLAB/Simulink environment. The main objective of this study is to enable the aircraft to follow a defined trajectory with algorithm of an improved controller. When the aircraft takes off, the initial position and orientation of the aircraft is set to zero. Then the aircraft is informed about the reference waypoints. The waypoints information which are necessary for trajectory tracking includes five data. These are X_ref, Y_ref, h_ref, yaw angle and waiting time at that point. The relative position between vehicle and target is calculated in MATLAB/Simulink. And the controllers provide to track desired parameters. The aircraft’s position and velocity is measured by its inertial sensors and vision system. The performance of classical PID controller to follow a defined trajectory is compared with the performance of self-tuning fuzzy PID controller. In addition to these two control algorithms, fuzzy logic controller is also tested alone for controlling the aircraft to follow the defined trajectory. However, the performance results are less successful than the first two control methods.

The first stage of experiments involved tests on following the trajectory of the aircraft controlled by classical PID controller. The gains of classical PID controller are determined using Ziegler–Nichols method according to analyses for system response performed on the aircraft model in MATLAB/Simulink environment. Hence, PID gains are selected as k_p = 0.25, k_i = 0.014 and k_d = 0.12. In the next stage, the self-tuning fuzzy PID controller is used for the aircraft to follow the defined trajectory where fuzzy logic system is used to tune PID gains.

Figures 13 to 15 show graphical representation of data obtained during real-time trajectory tracking by classical PID control and self-tuning fuzzy PID control methods. These trajectories are formed by sequential points having a certain distance between each other and created different patterns. Tables 5 to 7 show errors produced during trajectory-tracking using the above methods. For both of the control algorithms, experiments are performed indoor at a constant height of 1 m and in absence of wind effect.

Figure 13.

Equilateral trajectory tracking with PID control and self-tuning fuzzy PID control. PID: proportional integral derivative.

Table 5.

Performance comparison table: equilateral trajectory.

Algorithm	Maximum error (m)	Total error (m)
PID	0.32	8.02
Fuzzy-PID	0.15	4.35

PID: proportional integral derivative.

For the equilateral trajectory in Figure 13, the performance of classical PID and the performance of self-tuning fuzzy PID controllers are compared in case of following reference points at equal distance to each other by performing cross motions. The aircraft that is held a total of 24 s at defined reference points completed the flight track in 40 s with classical PID controller and in 39 s with self-tuning fuzzy PID controller. Table 5 shows comparison between control methods, according to the amount of error for following the equilateral trajectory. The aircraft with PID controller completed four reference points of the trajectory with maximum and total errors of 0.32 and 8.02 m, respectively. The same trajectory is followed by self-tuning fuzzy PID controller and maximum and total errors were determined as 0.15 and 4.35 m, respectively.

For the square trajectory in Figure 14, the performance of classical PID and the performance of self-tuning fuzzy PID controller are compared in case of following defined reference points by performing longitudinal and lateral motions. The aircraft that was held for a total of 48 s at defined reference points completed the flight track in 68 s with classical PID controller and in 66 s with self-tuning fuzzy PID controller. Table 6 shows comparison between control methods, according to the amount of error for following the square trajectory. The aircraft completed eight reference points of the trajectory with PID controller and maximum and total errors are determined as 0.39 and 12.17 m, respectively. The same trajectory is followed by self-tuning fuzzy PID controller, and maximum and total errors are determined as 0.18 and 6.94 m, respectively.

Figure 14.

Square trajectory tracking with PID control and self-tuning fuzzy PID control. PID: proportional integral derivative.

Table 6.

Performance comparison table: square trajectory.

Algorithm	Maximum error (m)	Total error (m)
PID	0.39	12.17
Fuzzy PID	0.18	6.94

PID: proportional integral derivative.

Table 7.

Performance comparison table: circular trajectory.

Algorithm	Maximum error (m)	Total error (m)
PID	0.45	13.24
Fuzzy-PID	0.16	5.64

PID: proportional integral derivative.

Figure 15.

Circular trajectory tracking with PID control and self-tuning fuzzy PID control. PID: proportional integral derivative.

For the circular trajectory in Figure 15, the performance of classical PID and the performance of self-tuning fuzzy PID controller are compared in terms of following defined reference points by performing short-distance motion. The aircraft that is held for a total of 64 s at defined reference points completed the flight track in 102 s with classical PID controller and in 95 s with self-tuning fuzzy PID controller. Table 7 shows comparison between control methods, according to the amount of error for following the circular trajectory. The aircraft completed 16 reference points of the trajectory with PID controller, and maximum and total errors are determined as 0.45 and 13.24 m, respectively. The same trajectory is followed by self-tuning fuzzy PID controller, and maximum and total errors are determined as 0.16 and 5.64 m, respectively.

Figures and error tables clearly show that the flight performance is improved and successful outcomes are achieved when fuzzy system is used to adjust PID gains. The aircraft is expected to follow the trajectory as close as possible to the linear route that represents the shortest path between the reference points. However, inertial effects resulting from the structure of aircraft and sensor drifts cause a number of deviations in the real-time experiment. In addition, the battery levels are measured in the end of the experiments and it is observed that the flights are completed using approximately 25% of energy for classical PID control algorithm and 20% of energy for self-tuning fuzzy PID control algorithm. The video for significant parts of experimental study can be found at https://youtu.be/aL1FMFS7RnY.

Conclusions and evaluation

MATLAB/Simulink-based dynamic model of quadrotor and real-time trajectory tracking control applications were successfully performed using classical PID controller and self-tuning fuzzy PID controller and are presented in this paper. Although the literature includes many studies on self-tuning fuzzy PID control of dynamic systems, this method has only recently been used to control quadrotor. We can classify that the studies used this control method as performed in simulation environments and real-time applications. In the presented study, the novelty is using this method for the real-time trajectory tracking.

First, a classical PID controller is used for autonomous following of the trajectory by this aircraft. Initially, gain values of PID controller are determined by Ziegler–Nichols method. In the next stage, the fuzzy controller is used to adjust gains of PID controller. With these controllers, the performances of the aircraft to track a defined trajectory with minimum oscillation are observed in real-time and compared.

The dependency of fuzzy controller on the quadrotor aircraft model to the aircraft is much less compared to conventional controllers. The fuzzy controller is not linear. Therefore, it is more suitable for non-linear systems such as a quadrotor aircraft. Experimental results indicate that the self-tuning fuzzy PID controller has an outstanding performance compared to the conventional controllers in terms of the amount of error.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study was supported by Karabük University within the scope of Scientific Research Projects with KBÜ-BAP-13/2-DR-008 code.

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Real-time trajectory tracking of an unmanned aerial vehicle using a self-tuning fuzzy proportional integral derivative controller

Abstract

Keywords

Introduction

Mathematical model of quadrotor

Control methods for quadrotor

PID control

Self-tuning fuzzy PID control

Autonomous trajectory tracking of quadrotor

Platform of the AR.Drone quadrotor

Data communication

Dynamics and self-tuning fuzzy PID control of the AR.Drone quadrotor

Experimental results

Conclusions and evaluation

Footnotes

Declaration of conflicting interests

Funding

References