ATTITUDE AND ALTITUDE CONTROL OF TWO WHEEL TRIROTOR HYBRID ROBOT

Transcription

1 ATTTUDE AND ALTTUDE CONTROL OF TWO WHEEL TRROTOR HYBRD ROBOT A MASTER S THESS in Mechatronics Engineering Atilim University by HUSEN ALWAF JAN 213

2 ATTTUDE AND ALTTUDE CONTROL OF TWO WHEEL TRROTOR HYBRD ROBOT A THESS SUBMTTED TO THE GRADUATE SCHOOL OF NATURAL AND APPLED SCENCES OF ATLM UNVERSTY BY HUSEN ALWAF N PARTAL FULFLLMENT OF THE REQUREMENTS FOR THE DEGREE OF MASTER OF SCENCE N MECHATRONCS ENGNEERNG JANUARY 213 ii

3 Approval of the Graduate School of Natural and Applied Sciences, Atılım University. Prof. Dr. İbrahim Akman Director certify that this thesis satisfies all the requirements as a thesis for the degree of Master of Science. Prof. Dr. Abdulkadir Erden Head of Department This is to certify that we have read the thesis Attitude and Altitude Control of Two Wheel Trirotor Hybrid Robot submitted by HUSEN ALWAF and that in our opinion it is fully adequate, in scope and quality, as a thesis for the degree of Master of Science. Asst. Prof. Dr. Kutluk Bilge Arıkan Co-Supervisor Asst. Prof. Dr. Bülent İrfanoğlu Supervisor Examining Committee Members Asst.Prof.Dr. Fuad ALEW (AT.U. MECE. Dept.) Asst.Prof.Dr. HAKAN TORA (AT.U. E.E. Dept.) Asst.Prof.Dr. K. ATLGAN TOKER (AT.U. M.E. Dept.) Asst. Prof. Dr. Bülent İrfanoğlu Asst. Prof. Dr. Kutluk Bilge Arıkan (AT.U. MECE. Dept.) (AT.U. MECE. Dept.) Date: / /213

4 declare and guarantee that all data, knowledge and information in this document has been obtained, processed and presented in accordance with academic rules and ethical conduct. Based on these rules and conduct, have fully cited and referenced all material and results that are not original to this work. Name, Last name: Husein ALWAF Signature: iv

5 ABSTRACT ATTTUDE AND ALTTUDE CONTROL OF TWO WHEELED TRROTOR HYBRD ROBOT Alwafi, Husein M.S., Mechatronics Engineering Department Supervisor: Asst.Prof.Dr. Bülent İrfanoğlu Co-Supervisor: Asst.Prof.Dr. Kutluk Bilge Arıkan January 213, 59 pages Two Wheel Tri-Rotor (2W3R) hybrid robot that can move on ground, hover and navigate in air, is a novel system studied in this thesis. Physical structure of the system had been built in Flying Robotics Laboratory by undergraduate students as a course project in Mechatronics Department at Atılım University. Core of this thesis is to design controllers to stabilize and control the system on its hovering conditions. Stabilization and control of roll, yaw, pitch, and the altitude dynamics using the propulsion units are studied. Nonlinear equations of motion of the physical system are first derived, and then state feedback linearization technique and Linear Quadratic Regulator (LQR) are used, and control systems are developed in Matlab Simulink. Keywords: Hybrid robot, linear quadratic regulator, feedback linearization, attitude, altitude. v

6 ÖZ İKİ TEKERLEKLİ ÜÇ ROTORLU MELEZ ROBOTUN YÖNELİM VE İRTİFA DENETİMİ Alwafi, Husein Yüksek Lisans, Mekatronik Mühendisliği Bölümü Tez Yöneticisi: Yrd. Doç. Dr. Bülent İrfanoğlu Ortak Tez Yöneticisi: Yrd. Doç. Dr. Kutluk Bilge Arıkan Ocak 213, 59 sayfa İki tekerlekli üç döner kanatlı (2W3R) hibrid robot bu tezde incelenen kara ve havada gezinebilen, yeni bir sistemdir. Sistemin fiziksel yapısı Uçan Robotlar Laboratuvarında bir ders projesi olarak lisans öğrencileri tarafından Atılım Üniversitesi Mekatronik Mühendisliği Bölümünde yapılmıştır. Bu tez çalışmasının temel konusu sistemin havada stabil olarak gezinebilmesi için durum geri beslemeli denetimci tasarımı ve uygulanmasıdır. Yalpa (roll), yunuslama (pitch) ve sapma (yaw) ve irtifa dinamiklerinin eyleyiciler kullanılarak stabilizasyonu ve denetimi çalışılmıştır. İlk olarak doğrusal olmayan hareket denklemleri yazılmış, daha sonra durum geribesleme ile doğrusallaştırma ve doğrusal kuadratik regülatör kullanılmış ve Matlab Simulink içinde kontrolcüler geliştirilmiştir.. Anahtar Kelimeler: Melez robot, doğrusal kuadratik regülatör, durum geri beslemeli doğrusallaştırma, yönelim, irtifa. vi

7 To My Family vii

8 ACKNOWLEDGEMENTS Firstly, would like express my gratitude to the members of Mechatronic Department, for their support. Successful completion of this project would not have been possible without the guidance by Dr. Kutluk Bilge ARKAN and Dr. Bülent İRFANOĞLU. Thanks to my parents, for their unconditional love and support, would not have made it this far without their sacrifices. Not forgetting my true friends, who have always been supportive of my work and lending a hand at times of needs? Special thanks to my wife for her unlimited support. Finally, a word of appreciation goes out to all my friends and colleagues at the University for showing interest in my work. viii

9 TABLE OF CONTENTS ABSTRACT.. Öz.... DEDCATON. ACKNOWLEDGMENTS. TABLE OF CONTENTS... LST OF FGURES... LST OF ABBREVATONS NOMENCLATURE. v vi vii viii ix xi xiii xv CHAPTER 1 NTRODUCTON Aim and Scope of Thesis Outline of Thesis CHAPTER 2 LTERATURE SURVEY Unmanned Aerial Vehicle (UAV) Two-wheel Robots Hybrid Robot. 12 CHAPTER 3 PHYSCAL SYSTEM.. 15 ix

10 3.1 The Bifiliar Pendulum nertial Measurement Unit Motors Controller CHAPTER 4 MATHEMATCAL MODELLNG Mathematical Model of Propulsion Unit.. 24 CHAPTER 5 CONTROLLER DESGN AND SMULATONS LQR Feedback Linearization Control with PD. 33 CHAPTER 6 DSCUSSON AND CONCLUSON.. 39 REFERENCES.. 41 x

11 LST OF FGURES Figure 1- Two wheel balancing system.. 2 Figure 2- Remote control of quadrotor... 5 Figure 3- Different structures and designs 6 Figure 4- ALV quadrotor 8 Figure 5- Commercially available quadrotor.. 8 Figure 6- Delta (trirotor) UAV 9 Figure 7-2TEA vehicles.. 1 Figure 8- JOE vehicles 1 Figure 9- Balancing with fuzzy PD control. 11 Figure 1- Segway HT vehicle Figure 11- The ibot in action, ibot climbing. 12 Figure 12- ATHLETE and Walking Quadrotor 13 Figure 13- Twin Rotor Two Wheeled Hybrid System 13 Figure 14- nclinometer, accelerometer and MU.. 14 Figure 15-2W3R hybrid systems Figure 16- Systems overview. 16 Figure 17- Oscillation of the system 17 Figure 18-3DM-GX2 MU 18 Figure 19- BLDCM, BDCM and ESC 19 xi

12 Figure 2- MF624 DAQ Card Figure 21- Reaction Torque and tilting... 2 Figure 22- proposed tri-rotor hybrid robot. 21 Figure 23- Propulsion unit thrust measurement setup.. 24 Figure 24- Hand Type Tachometer. 24 Figure 25- LQR based control 27 Figure 26- LQR simulation 29 Figure 27- Variation of phi angle ( ) with time. 31 Figure 28- Variation of pitch angle ( ) with time 31 Figure 29- Variation of yaw angle ( ) with time 31 Figure 3- Variation of Altitude angle (z) with time.. 32 Figure 31- Variation of control inputs with time.. 32 Figure 32- Sample trajectory control 33 Figure 33- Feedback linearization and PD. 35 Figure 34- Feedback linearization and PD Simulink 36 Figure 35- Detailed Feedback linearization and PD 36 Figure 36- Variation of roll angle ( ) with time. 36 Figure 37- Variation of pitch angle ( ) with time 37 Figure 38- Variation of yaw angle ( ) with time.. 37 Figure 39- Variation of Altitude. 37 xii

13 LST OF ABBREVATONS A/D D/A DAQ DC DSP ESC FBD /O EEE MU LQR MEMS PC PC PD PT PWM RTWT 2W2R UAV UGV Analog/Digital Digital/Analog Data Acquisition Direct Current Digital Signal Processing Electronic Speed Controller Free Body Diagram nput / Output The nstitute of Electrical and Electronics Engineers nertial Measurement Unit Linear Quadratic Regulator Micro Electro Mechanical Systems Personal Computer Programmable nterface Controller Proportional ntegral Derivative Personal Transporter Pulse Width Modulation Real Time Windows Target Two Wheeled Twin Rotor Hybrid Robotic Platform Unmanned Aerial/Air Vehicle Unmanned Ground Vehicle xiii

14 USB VTOL - - Universal Serial Bus Vertical Take-off and Landing xiv

15 NOMENCLATURE The following are the list of variables used in this thesis: - Angular acceleration components expressed in body reference frame (rad/ - Acceleration (m / - Angular acceleration (radian/ p, q, r - Angular velocity components expressed in body reference frame (rad/s) d - Drag factor,, - Euler anglers (rad) - Friction coefficient - Force in x direction (N) - Force in y direction (N) - Force in z direction (N) - Gravitational acceleration (m / - Gear Ratio - nertia Matrix - nertia Matrix Elements (kg ) - Mass of body (kg) -Torque produced by DC Motor (Nm) - Terminal Voltage (Volt) - Linear displacement of body in x direction (m) - Linear displacement of body in y direction (m) - Linear displacement of body in Z direction (m) - Linear acceleration of body in x direction (m / - Linear velocity of body in x direction (m/s) xv

16 - Linear acceleration of body in y direction (m / - Linear acceleration of body in Z direction (m) - Linear velocity of body in y direction (m/s) - Mass (kg) - Moment (Nm) - Normal Force (N) R - Rotation Matrix -Terminal resistance of DC motor (Ohm) -Torque Constant b - Thrust factor -Voltage applied (Volt) xvi

17 CHAPTER 1 NTRODUCTON Autonomous mobile robotics has been one of the active research areas in robotics studies all around the world. Utilization of autonomous robots which can move both on ground and in air offer navigational advantages when compared to those robots that can move only either on ground or in air. Generally, wheeled or legged locomotion are the two typical choices for ground motion, on the other hand, there is no choice other than flying if navigation in air required. Therefore, researches on Unmanned Aerial Vehicles (UAV) will be one of the dominant factors in shaping the future of hybrid robots that can navigate both on ground and in air. These hybrid robots may be used in many applications for various needs. The UAVs stability problems and their design as agile vehicles make these systems interesting in academic studies. Today, Quadrotor UAVs can be defined as standard platforms. Besides quadrotors, new designs and systems are appearing as research platforms. As robots are required to operate in more challenging environments the idea of hybrid robot which combines more than one type of locomotion to be able to operate in different environments arises. The idea of combining the motion capabilities of two-wheeled balance system and triple rotors flying platform is the starting point of this study. Two-wheel three-rotor (2W3R) platform is one of the hybrid robotic systems in the Flying Robotics and Robotic Vehicles Research Laboratory of Atılım University 1. Hybrid robots in our laboratory are the novel systems which can navigate on ground and hover and navigate in air whenever necessary. 2W3R is the second version of hybrid robots of concern. n this version wheels are not actuated by motors. Therefore they don t contribute to control actions neither in air nor on ground. n the first version which given in Figure 1, namely Two-Wheeled

18 Twin-Rotor (2W2R) platform [1] wheels have dc motors which generate traction force on ground and also produce moment. Ground motion and attitude stabilization of the hybrid robot are studied. For ground motion, LQR and Error Space Approach controllers are designed and implemented. The control system is developed in Matlab/Simulink and real time implementation is achieved by using Simulink Real Time Windows Target. n the second version of the hybrid platform two of the three rotors can be tilted by servo actuators to maintain yaw stability and maneuverability. Tilting rotors belong to the propulsion units with dc brushed motors. The third one is actuated by a brushless dc motor. Brushed ones have comparatively less thrust potential than the brushless one. Center of gravity is placed close to the brushless unit to compensate for the difference in thrust capacities. Figure 1-2W2R Hybrid robotic platform [1] 1.1 Aim and Scope of Thesis The main aim of this study is to construct mathematical model and design control systems for the air mode of the 2W3R (two wheeled three rotor) hybrid robotic system. Hybrid robots are the systems which can navigate on ground and hover and navigate in air whenever necessary. Two of the rotors have tilting ability to maintain yaw control. 2

19 Details of the scope are listed below: 1- Deriving the mathematical model. Flying mode of the hybrid system is studied in this thesis. t is assumed that the platform stands stable on the ground as an initial condition. Altitude and attitude dynamics (roll, pitch and the yaw) of the platform are modeled. Calculations of the physical parameters are done previously. 2- Design of controller systems. Altitude and attitude dynamics of the platform are controlled. Control systems are designed using linear quadratic regulators and feedback linearization techniques. Physical implementation of the designed controllers is not included in this thesis. Control systems do not consider the transition from ground mode to air mode. The system is assumed as standing on the ground stably. 1.2 Thesis Outline The thesis is organized as follows: Chapter 2 presents the literature survey and previous researches and work which are related to our system. Chapter 3 deals with the physical system of the system and describes the used components in details. Chapter 4 explains how to derive the mathematical model and describe the movements of the system. Chapter 5 deals with the definition of the control algorithms and simulations. Chapter 6 presents the conclusions that have been taken regarding the results and what can be done in the future. 3

20 CHAPTER 2 LTERATURE SURVEY Prior to undertaking research projects it is critical to conduct literature review in order to provide the researcher with much needed information on the technology available and methodologies used by other research counterparts around the world on the topic. This chapter provides a condensed summary of literature reviews on key topics related to hybrid flying robot. n this study, we consider the attitude stabilization problem of the hybrid robot which can move on ground by two wheels and hover and maneuver in air by propellers. Therefore, the main topic of the study is related to the two wheel balancing systems, flying aircraft vehicles and hybrid robots. 2.1 Unmanned Aerial Vehicle (UAV) Rotorcrafts have witnessed an incredible evolution in the last years. University students and researchers continuously work to introduce more robust controllers and modeling techniques, so that they can provide detailed and accurate representations of real-life flying robots. This section introduces some of these works presented in recent years. The ability of design a vertical takeoff and landing (VTOL)-UAV, which is highly maneuverable and extremely stable, is an important contribution to the field of aerial robotics since potential applications are tremendous (high buildings and monuments investigation, rescue missions, film making, etc.). n practical applications, the position in space of the UAV is generally controlled by an operator through a remote control system using a visual feedback from an onboard camera, while the attitude is automatically stabilized via an onboard controller. 4

21 Figure 2- Remote control of quadrotor [2] The attitude controller is an important feature since it allows the vehicle to maintain a desired orientation and, hence, prevents the vehicle from flipping over and crashing when the pilot performs the desired maneuvers. The attitude control problem of a rigid body has been investigated by several researchers and a wide class of controllers has been proposed. This is a particularly interesting problem in dynamics since the angular velocity of the body cannot be integrated to obtain the attitude of the body. Different structures found in literature, starting with single rotor like the classical helicopter or the tilting rotor aircraft. Four-rotor rotorcrafts are also very popular [3]. Other multi-rotorcrafts have even a large number or rotors as the one in [4] which has eight rotors. A quadrotor is one of the famous UAV s. The basic motions of a rotary aircraft are generated by varying the rotor speeds of all rotors, thereby changing the lift forces. Several control algorithms guaranteeing asymptotic stability and, under certain initial conditions, local exponential stability is found. Aiming at more efficient configurations in terms of size, flight range, autonomy and payload capacity, various types of rotorcrafts are designed. Figure 3 shows some different structures and designs of rotary aircrafts with different sizes. 5

22 Figure 3- Different structures and designs of rotary aircrafts [2][4] A lot of different methods have already been studied to achieve autonomous flights. LQR, PD, H and others are used in researches in order to develop robust and agile flying robots. The following are some of the researches and thesis related to these controllers. n [3] design of a small unmanned indoors quadrotor aircraft is studied at low translational speeds around the hovering condition, where the aerodynamic forces on the airframe are disregarded. For state estimation and noise filtering a Kalman filter is implemented. Linear control techniques such as PD, LQ are employed and compared with each other in terms of flight trajectory reference tracking and parametric and model uncertainty. On the OS4 project [5], the researcher compared classical PD and PD controllers for orientation stabilization with modern LQ adaptive optimal control, even when he realized that the latter one yielded only average results, due to modeling imperfections 6

23 The author in [6], has presented a system model with DC motors by using the Lyapunov Function's non-linear control technique for stabilizing the aircraft's orientation (Euler angles), he compared the real system's behavior with a respective simulation. To design this controller, the model has been linearized around the hover situation. Hence, the gyroscopic effects haven t been taken into consideration in the controller design. The closed loop model has been simulated on Simulink with the full non linear model. The controller parameters have been adjusted with this more complete model. The simulation has led to satisfactory results. The quadrotor attitude stabilizes itself after 3 seconds. n [7] LQR in the attitude loop is used. At low thrust levels, the control was satisfactory but at higher thrust levels, performance was degraded due to vibrations. Apply lower costs on attitude deviations by varying the matrix Q but this degrades tracking performance is a solution to this problem. A good compromise has to be found. An interesting performance comparison among four control techniques: LQR, LQR with gain scheduling, feedback linearization and sliding mode control are made in [8]. Experimentally, they verified that LQR with gain scheduling presented more robustness in light of modeling uncertainties whereas for an accurately modeled system a better performance was achieved with the sliding mode approach. Recently, Autonomous Locomotion ndividual Vehicle (ALV) quadrotor UAV shown in Figure 4 is simulated in [9] has recently proposed a model to simulate the By using a model built in Matlab and Simulink he was able to test the capabilities of the ALV, where he designed a Kalman filter to estimate the quadrotor state by modeling its real sensors. After that, he designed and implemented LQR controller to stabilize the rotorcraft. Positioning control was aided either through a joystick or a simulated on-board camera. A Simulink real-time simulation of the system was successfully integrated with the FLGHTGEAR flight simulator, by using the joystick a user can pilot the quadrotor to a certain target, and, after locking on the target, the rotorcraft performs autonomous flight through established waypoints. 7

24 Figure 4- ALV quadrotor [9] n [1], a vehicle control system for a quadrotor Micro-UAV is presented based on a combined control strategy including feedback linearization shown in Figure 5. Both an inner-loop attitude controller and an outer-loop velocity controller have been developed during the proposed work in order to cope with the nonlinear dynamic behavior of the vehicle. The dynamic model of the quadrotor is derived and implemented in a Matlab/Simulink simulation model. With the help of that simulation, the nonlinear vehicle control system is tested and its efficiency demonstrated. Figure 5- Commercially available quadrotor [1] The reduction in the number of rotors from four to three allows to obtain more compact vehicles (backpack able UAVs) for rapid deployment, as well as longer flight autonomy. A novel configuration for a three-rotor mini Unmanned Aerial Vehicle (UAV) is proposed in [11]. Firstly, the Mathematical model of the vehicle s attitude is obtained 8

25 through the Newton-Euler formulation. And then a control law which is robust with respect to dynamical couplings and adverse torques is proposed. Figure 6 shows the structure of the vehicle. The vehicle tilts simultaneously the three rotors to stabilize the yaw dynamics. Figure 6- Delta (trirotor) UAV [11] 2.2 Two Wheeled Robots n recent years, two wheel self-balancing vehicles took place in literature. The basic idea for a two-wheeled dynamically balancing robot is pretty simple; drive the wheels in the direction that the upper part of the robot is falling. f the wheels can be driven in such a way as to stay under the robot's center of gravity, the robot remains balanced. n practice this requires two feedback sensors a tilt or angle sensor to measure the tilt of the robot with respect to gravity, and wheel encoders to measure the position of the base of the robot. Two wheeled balancing robot need a good controller to maintain itself in upright position without the needs from external forces. Nowadays, various controllers are implemented on two wheeled balancing robot, one of them a Linear Quadratic Regulator such as the research which is done at mechatronics Laboratory at Atilim University, [12], namely 2TEA, which is shown in Figure 7. 9

26 Figure 7-2TEA vehicles [12] Pole-Placement Controller which is achieved in [13] in his thesis, the response of two types of controller (LQR and Pole placement) on the two wheel balancing system are compared. And Proportional ntegrated Derivative Controller such as the researchers at the ndustrial Electronics Laboratory at the Swiss Federal nstitute of Technology who built a scaled down prototype of a Digital Signal Processor controlled twowheeled vehicle based on the inverted pendulum with weights attached to the system to simulate a human driver. Figure 8- JOE vehicles [13] A linear state space controller utilizing sensory information from a gyroscope and motor encoders is used to stabilize this system. 1

27 Figure 9 shows two-wheeled self-balancing robot with a fuzzy PD control method is described and analyzed in [14], the control scheme consists of a fuzzy logic-based pre-compensator followed by fuzzy PD control. Figure 9- Balancing with fuzzy PD control [14] Different structure of two wheel balancing system has been designed and implemented depending on the application and the desired task. One commercially available two wheeled robot is called SEGWAY HT (Dean Kamen, 21) shown in Figure 1, t is invented by Dean Kamen. An extra feature this robot has is that it is able to balance while a user is standing on top of it and navigate the terrain with it. However, this uses five gyroscopes and a few other tilt sensors to keep it balanced. Figure 1- Segway HT vehicle [12] 11

28 The ibot, in Figure 11, is an inverted pendulum vehicle developed by Dean Kamen and other engineers in 199 s as a medical device similar to a wheelchair. The ibot provides several unique advantages over a traditional electric wheelchair, which are aimed to allow a more normal and independent day-to-day lifestyle for the handicapped. Figure 11-The ibot in action, ibot climbing [15] stairs 2.3 Hybrid Robots n literature most of the hybrid robots govern legged and wheeled locomotion. Combine wheeled and a legged motion on a hybrid design is one of the researches that took place in literature survey. California nstitute of technology in [16] proposed All-Terrain Hex-Limbed Extra- Terrestrial Explorer (ATHLETE) the vehicle concept is based on six Degrees-of- Freedom (DOF) limbs, each with a 1 DOF wheel attached. ATHLETE in Figure.12 uses its wheels for efficient driving over stable, gently rolling terrain, but each limb can also be used as a general purpose leg. n the latter case, wheels can be locked and used as feet to walk out of excessively soft, obstacle laden, steep, or otherwise extreme terrain. ATHLETE is envisioned as a heavy-lift utility vehicle to support human exploration of the lunar surface, useful for unloading bulky cargo from stationary Landers and transporting it long distances. The walking quadrotor aerial vehicle is shown in Figure 12. This type of hybrid robot is capable of both aerial and terrestrial locomotion. A unique compliant mechanism 12

29 Figure 12- ATHLETE and Walking Quadrotor [16] design makes it possible to use a single actuator set for both walking and flying. This is advantageous because it reduces both the total weight of the system and the control system complexity. Recently, in [17] the llinois nstitute of Technology (T) developed the HyTAQ (Hybrid Terrestrial and Aerial Quadrotor), where it has a cylindrical outer protective cage, which is attached to the quadrotor using a shaft connected by a couple of rotating joints, which allows the HyTAQ to fly or roll over the ground, as and when the need arises. The quadrotor will deliver the thrust required for aerial and terrestrial locomotion. When the HyTAQ is on the ground, the rolling cage would mean the robot needs to overcome just rolling resistance to move forward, and it will jump over an obstacle that it finds difficult to roll over. n 21, Two-Wheeled Twin Rotors Hybrid Robotic Platform [1] shown in Figure 13 is designed. After deriving the nonlinear equations of motion of the physical system, they are linearized in state space form, for ground motion, LQR and Error Space Approach controllers are designed and implemented. Figure 13- Two Wheeled Twin Rotor Hybrid System [1] 13

30 n order to implement the control systems, two ways are followed in the literature. mplementing designed controllers using embedded controller systems is one of these ways. The second way is implementing control systems with using PC and data acquisition, nearly all of the Literature used Matlab Simulink software. As a sensing system, the accelerometer, inclinometer, gyroscope, encoder and nertial measurement units MU as shown in Figure 14, which is a combining of some accelerometers and gyroscopes are used in literature to find out the states of the systems. \ Figure 14- From the left, inclinometer, accelerometer and MU There is a lot of control techniques used in literature; the control design of the nonlinear systems such as our systems can be subdivided into techniques which attempt to treat the system as a linear system in a limited range of operation and use (well-known) linear design techniques for each region. Most of the researches use LQ or LQR for the stabilization of the two wheeled balance robot system and UAV. As the optimal controller, instead of LQR type of controllers, pole placement technique is another popular control system for the stabilization of the systems. Traditional PD control technique is designed and implemented in some researches also. Feedback linearization is another technique used in some papers. After the stabilization of the system, the trajectory control is applied with a PD controller commonly but some other control systems are designed and implemented for the trajectory control. As robust controller, sliding mode based techniques are used to design the robust controllers on two wheeled balancing systems [18]. n some systems Fuzzy Logic Controllers is used, such as in [14]. 14

31 CHAPTER 3 PHYSCAL SYSTEM The physical system of 2W3R includes structure, motors, motor controllers, batteries, sensors, controller hardware and software. The final design of the system is shown in Figure 15. Figure 15-2W3R hybrid systems The systems overview is demonstrated in Figure 16. The sensors detect the system information and send it to the computer via acquisition card where the real time control algorithm runs. After data processing, the motor controller is activated to generate signal which actuates the motors. Actuated motors rotate the wheels and the propellers to make the system moves. Mechanical structure of the systems was constructed as a term project by undergraduate students. As shown in Figure 15, chassis is made of aluminum in order to have the lightest design. 15

32 Figure 16-Systems overview The 2W3R has three propellers one of them (the front one) is fixed propeller connected to brushless DC motor and rotates in opposite direction to the other two propellers which connected to the brushed DC motors, the latest propellers are tilted by servomotors with an angle ( ) to maintain yaw control and to get flexible maneuverability. 3.1 The Bifiliar pendulum The moment of inertia of the body about each axis of rotation should be calculated. Bifiliar pendulum technique is applied. The inertia matrix in general is described in equation (3.1). However, a diagonal inertia tensor is assumed in the thesis. xx yx zx yy xy zy zz xz yz (3-1) First, the system is hung to make free oscillations around the rotation axis. Two ropes with same length (L) were used to hang the system, and they were tied from the points which were approximately same distance (R) to rotation axis. Second, the system was rotated by a certain small angle, and then released free to oscillate about 16

33 the axis. Third, the response of the system was observed by MU. The oscillation period (T n ) was measured. To determine the mass moment of inertia (J) about rotation axis, the formula in (3-2) is used. J Tn mgr 2 L 2 (3-2) Where: m g= Gravitational acceleration ( 2 ) s m=mass of the system (Kg) Figure 17 shows the response of the system the axes Figure 17- Oscillation of the system The inertia values about the rotation axes are determined as following: xx =.75 kgm 2, yy =.8 kgm 2 and zz =.1425 kgm 2 17

34 The detailed information about the elements of the physical system is described in the following. 3.2 nertial Measurement Unit (MU) An nertial Measurement Unit is used in self-balancing robots and flying aircrafts, to measure orientation, angular velocity and linear accelerations. n our systems MU is used to measure Euler angles and angular rate. Micro Strain 3DM-GX2 shown in Figure 18 is employed as MU. t combines three angular rate gyros with three orthogonal DC accelerometers, three orthogonal magnetometers and multiplexer. MU connects to computer via RS-232. Sampling rate of sensor is 1 Hz in this system. The output of the MU is a packet, explained in the technical data sheet. Figure 18-3DM-GX2 MU 3.3 Motors n our systems we use brushless DC motors (BLDCM) and brushed DC motors (BDCM) motors. The BLDCM differs from the conventional Brushed DC Motor (BDCM) in their concept essentially in that the commutation of the input voltage applied to the armature's circuit is done electronically, whereas in the later, by a mechanical commutator (brush). Figure 19 shows the used brushed and brushless motors and their Electronic Speed Controller (ESC) which is used to control the brushless electrical motors. 18

35 3.4 The controller Figure 19- BLDCM, BDCM and ESC The major aim is to design and develop a prototype system that is presented as a hybrid robot. Therefore, the physical implementation should be easy. The preferred controller software is Matlab Simulink. The real-time controller software is generated in Matlab Simulink by using Real-time Windows Target Block set. The collected data from encoder and MU sensor are processed and the control input is calculated and applied to the motor driver units as PWM signals. To get inputs from system and give outputs to the system the Humusoft MF624 DAQ Card which shown in Figure 2 is used. Figure 2- MF624 DAQ Card 19

36 CHAPTER 4 MATHEMATCAL MODELLNG Mathematical modeling is a vital step in development and control of a dynamic system. n fact, the model allows the engineer to analyze the system. The dynamic models are widely used in control design. This is especially important for aerial robots where the risk of damage is very high as a fall from a few meters can seriously damage the platform. Thus, the possibility to simulate and tune a controller before implementing it on the real machine is highly appreciable. n this thesis the attitude and altitude dynamics are modeled. The structure of 2W3R (two wheel trirotors) hybrid robot basically comprises of three rotors attached at the ends of virtual a triangle shape. Where the main rotor is fixed brushless DC motor rotates clockwise direction and the other two motors get tilted with certain angle by servomotors connected on their motor axes as shown in Figure 21, the tilted motors are DC brushed motors rotate counter clockwise. Reactive torque Generated torque Generated force Figure 21- Reaction Torque and tilting angle t s possible to control the pitch, roll and yaw attitude of the vehicle. Then, its displacement is produced by the total thrust of the three rotors whose direction varies 2

37 according to the attitude of the system. Positive directions of translation and rotational motions of the platform and wheels are shown in Figure 22. Following assumptions are made in deriving the mathematical model of the system: 1- The structure supposed to be rigid. 2- nertia tensor supposed to be diagonal. 3- COG and body fixed frame origin are assumed to be coinciding. 4- Thrust is proportional to the square of propeller s speed 5- The propellers are supposed to be rigid. Figure 22- Proposed tri-rotor hybrid robot n our thesis attitude and altitude dynamics of 2W3R is modeled using Lagrange equations. The equations of motion are developed in terms of the rotational parameters of 6-DOF using generalized co-ordinates in a vector q as presented below: q X Y Z T (4-1) A Lagrangian is obtained by modeling the energy of the system, where the difference between kinetic and potential energy is taken. The kinetic energy of the system is 21

38 modeled for both translational and rotational motions. The potential energy of the system is related only to the altitude of the system. The Lagrangian (L) can be expressed as: Where: L: Lagrangian K trans K rot L K K P (4-2) trans : Kinetics Energy due to translational motion : Kinetics Energy due to rotational motion P: Potential Energy rot By assuming that the system has mass m and diagonal inertia tensor with the elements,,, we can obtain the kinetic energy and the potential energy expression as follows. K 1 2 x y ( sin ) z 1 2 K trans ( 2 x y rot x y z ) z ( cos sin cos ) ) 1 2 ( sin cos cos P mgz (4-3) Where: m is the robot total mass g is gravitational acceleration The following Lagrange equation with an external generalized force is used: d dt L L q q (4-4) Where: : The external forces and moment vector acting on the body The generalized forces acting on the system are forces M x M y M T z. F x F y F z T and moments By using Euler- Lagrange equation to find the Lagrangian we can find the equations of rotation motion of the system: 22

39 23 x x x z y M ) ( y y y x z M ) ( (4-5) z z z y x M ) ( m )F cos (cos g Z z z y x M and M M, are the resultant moments of the thrusts about body axes, b b b z y x,,, and F z is the resultant force in z b axis, as shown on Figure 22. They are defined as follows. ) cos ( )sin ( )sin ( ) cos ( )sin ( ) cos ( M M M F F L F F h F F L F L M M F F L M M M z y x (4-6) n above equation L 1, L 2, L 3, and h are the corresponding distances that are given in Figure 22. F 1,F 2 and F 3 are the thrust forces produced by the propulsion units. M 1, M 2 and M 3 are drag moments due to the rotation of the propellers. They are assumed to be proportional to the thrust forces generated by the propellers as follows. M i =k i F i, i=1, 2, 3 (4-7) k i is the force to drag moment conversion constant and depends on the propeller characteristics. β is the tilting angle of rotors 1 and 2. Rotors are tilted by using fast response servo dc motors. Servo motors have internal closed-loop control architecture. t is modeled by a first order differential equation given as below. V 4 is the input to the closed-loop servo motor. a β and b β are the relevant model parameters of the motor. V 4 b a (4-8)

40 4.1. Mathematical Model for propulsion unit. An algebraic model that relates the propeller thrust to motor voltage has been obtained experimentally by using a test setup in Figure 23. The test setup is employed in the Flying Robotics Laboratory of the Mechatronics Engineering Department at Atılım University. This setup has a bar, which is mounted to the setup with bearings. When the motor rotates, a force generated on the scale device surface is measured. Brushless motors in the system are driven by Electronic Speed Controllers (ESC) that makes the brushless motor behave like a continuous servo motor. Figure 23- Propulsion unit thrust measurement setup [19] n this setup, there is a motor and a propeller in one side of the scale and a payload is assembled to the other side of the scale. Under indoor conditions, voltage is applied to the propulsion unit is increased from to 1 V. With.5 V increments, then the thrust of the propeller is measured by the force sensor. n addition the angular velocity of the propeller is measured by a hand tachometer which is shown in Figure 24. Figure 24- Hand Type Tachometer 24

41 Linear models that approximate the generated thrust as a function of voltage input to the dc motor are listed below. Note that the third equation belongs to the propulsion unit with brushless dc motor. F k 1 v1v1 F k 2 v1v 2 F k V k 3 v3 3 v4 (4-9) By experimental means we get k v1.2, k v and k v

42 CHAPTER 5 CONTROLLER DESGN AND SMULATON This thesis presents the design of optimal Linear Quadratic Regulator (LQR) and feedback linearization with a linear PD controller. Controllers are designed and optimized on mathematical model and simulations. 5.1 Linear Quadratic Regulator (LQR) Design Nonlinear state equations are linearized around the hovering condition. Linear timeinvariant state equations are formed as below. A and B matrices are found by using the Taylor series expansion of the nonlinear state equations. x Ax Bu (5-1) The LQR based control architecture is given in Figure 25. An outer integral loop is designed to track the reference inputs for Euler angles, (,,) and altitude, Z, System output, y, is given as follows. y Du Z T Cx (5-2) The state vector is formed as below: x Z Z (5-3) T nput vector consists of the voltage inputs to three propulsion units and input, V 4, to servo motors. They provide angle that is applied in opposite directions with the same amplitude to both of the tilting rotors. nput vector, u, is given as below. 26

43 u V V V V4 (5-4) The controllability of the physical system is examined by checking the controllability matrix Co as: Co B BA BA 2... BA 8 (5-5) And it is found that Controllability matrix is full rank, i.e., 2W3R is fully controllable for all states. Figure 25- LQR based control architecture Both of the inner and outer loops are constructed by designing an LQR for the augmented system which is formed by using the following augmented state vector. x A x x, x x 1 x 11 x 12 x T 13 (5-6) Following error states are regulated to make the platform track the reference attitude and altitude. t x1 R dt t 27 x11 R dt

44 x t R 12 dt x t R Z 13 z dt (5-7) R j represents the relevant reference input for the corresponding output. Augmented state equations are given as follows. x A A C 9x4 B 9x4 xa u 4x4 4x4 4 R (5-8) mxnand j are the zero matrix of size mxn and identity matrix of size jxj respectively. R is the reference input vector given below. R R R R T R z (5-9) Control input to the augmented system is applied in the following form. K K A K i u K A x A (5-1) (5-11) Controller gain matrices K and Ki are optimized by minimizing the following quadratic cost function below. J x T A Qx A u T R udt u (5-12) Q and R are the weighting matrices that are tuned by simulations. Optimal gain matrices are calculated using the solution, P, of the following algebraic Riccati equation. Where: A T P PA T PBR 1 u B T P Q (5-13) 28

45 A A C 9x4 4x4 B B 4x4 K A R 1 u B T P (5-14) Designed controller is applied in a Simulink model, Figure 26. Simulations are performed on the model assigning different initial conditions and various forms of reference inputs. Clock t To Workspace6 states 3*pi/18 R_phi rad rad deg deg dy1 dy2 u control rad deg rad deg rad deg rad deg rad deg rad deg rad deg beta and alt states angular velocities Euler angles rad deg y1 *pi/18 R_theta 3*pi/18 rad deg dy3 1 s -K2* u x' = Ax+Bu y = Cx+Du Linear 2W3R C* u rad deg rad deg y2 y3 R_psi dz -K1* u y6 5 R_z Figure 26- LQR simulation The following linear state space system is obtained when we consider the hovering as operating point: 29

46 A (5-15).74 B (5-16) By choosing an appropriate Q and R matrices (Q=7*eye (8) and R=.1*eye (4)) matrices, the gain matrices K and K i are obtained as below: K (5-17) Ki (5-18) Following figures present the system output due to nonzero initial conditions and nonzero altitude and yaw angle references ( altitude =5m; and 3 ). 3

47 Figure 27- Variation of roll angle ( ) with time Figure 28- Variation of pitch angle ( ) with time Figure 29-Variation of yaw angle ( ) with time 31

48 Figure 3-variation of Altitude with time Figure 31-variation of control inputs with time Settling times and rise times of the responses are adjusted according to the previous flight experience in Flying Robotics Laboratory by tuning the Q and R u matrices. Control signals are within the limits of the actuator signals. This control architecture can be utilized to navigate the 2W3R in air by assigning appropriate references for Euler angles and altitude. Sample simulation is performed to present the trajectory control of the vehicle keeping the altitude at 5 meters, Figure 32, where Z (altitude)=5m. 32

49 system trajectory 6 5 Z position (m) Y position (m) X position(m) 1 Figure 32- Sample trajectory control 5.2. Feedback Linearization Control with Linear PD Action: Feedback linearization is the exact linearization of the nonlinear system by utilizing state or output feedback [2]. The central idea is to algebraically transform nonlinear system dynamics into (fully or partly) linear ones, so that linear control techniques can be applied. Nonlinear state equations and output equations are given as below: x f ( x) g( x) u y h( x) (5-19) t is required to find the relative degree of each output and following state feedback below. v n is the new control input, fb ( x) and ( x) fb are smooth functions defined in a neighborhood of some point n x R and ( x ) fb u ( x) ( x) v i fb fb n (5-2) u i : below is determined by the controller and it is converted to the input vector, u, to the platform. 33

50 u i M x M y M z F z T (5-21) u T 1 u i (5-22) t is assumed that servo motors are faster than the dynamics of the platform and therefore modeled as ideal actuators. This assumption reduced the state vector and modified the input vector which is given below: x Z Z T (5-23) u V 1 V 2 V 3 T (5-24) Relative degree of each output is 2 for 2W3R. v n1, v n2, v n3 and v n4 are components of the new control input vector and output vector of the feedback linearized system is given as below. y y y y v v v v n1 n2 n3 n4 (5-25) n order to simplify the design procedure, we have approximated the M x and M y as: M x F F cos 2 1 L1 (5-26) M y F L F F 1 2 cos L (5-27) Considering the input/output linearization and relative degree for each output term, second derivatives of each output is presented below. 34

51 35 z z y x z y x z y x y x z x z y F M M M m g Z 2 1 cos cos (5-28) Using the Equation (5-28) above, feedback linearizing input to the system is given below. General schematic of the applied control is given in Figure 33. n fb fb i v x x u ) ( ) ( ) ( x fb (5-29) 1 2 fb ) x ( Designed controller is applied on the system model in Simulink, Figure 34. Detailed view of feedback linearization and PD controller blocks are presented in Figure 35. Figure 33- Feedback linearization and PD

52 fb lin+pid for roll dynamics fb lin+pid for pitch dynamics Md disturbance moment Md Mx Mx Md phi1 F1 phi theta ksi -K- -K- -K- phi theta ksi My My F2 -K- Ref Alt. Md Mz Fz Ref Alt.1 fcn Mz F3 Fz beta control allocation u_v NL model of 2W3R second set of states x z dz -K- -K- fb lin+pid for yaw dynamics -K- reference z fb lin+pid for altitude dynamics dphi dtheta dksi z -K- dz Figure 34- Feedback linearization and PD controller phi PD(s) PD Controller vn1 x Mx dtheta Subtract dksi y-z Figure 35- Detailed view of Feedback linearization and PD block Simulink model Simulations are performed on the model assigning different initial conditions and various forms of reference inputs. Following figures show the simulation results. Figure 36- Variation of roll angle ( ) with time 36

53 Figure 37- Variation of pitch angle ( ) with time Figure 38- Variation of yaw angle ( ) with time Figure 39- Variation altitude with time 37

54 n this simulation non-zero initial conditions are given for Euler angles. Roll and pitch angles are regulated at zero degree while the yaw angle reaches the desired nonzero reference heading of 3 degrees. Applied reference altitude is 5 meters. Figures show that outputs track the desired angles successfully. n fact PD control is sufficient for the feedback linearized system. However, in physical implementations disturbances may affect the system dynamics and integral action might be required. 38

55 CHAPTER 6 DSCUSSON AND CONCLUSON A novel platform 2W3R is introduced in this study. t is a hybrid platform which can navigate on ground and in air with two passive wheels and three rotors. Two of the rotors can be tilted via fast response servo motors. Two different control architectures are designed for the air mode considering the attitude and altitude dynamics of 2W3R. The first one is based on linearization of the attitude and altitude dynamics of the system about an operating point, i.e. hovering condition. nner LQR loop stabilizes the system while the outer loop makes the platform track reference attitude and altitude signals. Weighting matrices are tuned to reach the desired transient performance of the closed-loop system. Second control architecture uses feedback linearization principle with outer PD loops for each of the dynamics. As the nonlinear output dynamics is linearized by state feedback, PD design is simple and performed by pole placement technique according to the desired settling time and maximum overshoot criterion. Even PD control seems to be enough for the system during simulations, in real implementations there will be some disturbing effects and modeling uncertainties that will deviate the system performance from the simulation results. To overcome these effects integral action is also utilized, i.e. PD control action is preferred. Both of the control systems give successful results. They can be utilized to guide the platform in 3D space by sustaining reference attitude and altitude inputs. Feedback linearization with a PD control can be discussed as more general control architecture and may perform better in maneuvers with large deviations from hovering condition. Prior to the embedded control implementation on physical system, tethered control is used for tuning the control parameters. Weighting matrices and PD parameters will be tuned for linear and nonlinear architectures respectively. Since onboard battery 39

56 pack is utilized in real system, power consumption for control action is critical. During tuning there will be trade off in transient response performance and amplitude of control inputs. 2W3R is an experimental system for the sake of academic research purposes. These architectures will be implemented on the physical system to perform basic indoor tests. Robust architectures should be implemented for outdoor navigation. Designing robust controllers with disturbance rejection ability is another ongoing study for 2W3R. 4

57 REFERENCES 1. K. Doğanç, 21, Design of a Two Wheeled Twin Rotored Hybrid Robotic Platform, M.Sc. Thesis, Department of Mechatronics Engineering, Atılım University, Ankara, Turkey. 2. M. Oliveira, 211, Modeling, dentification and Control of Quadrotor Aircraft, M.Sc. Thesis, Department of Control, Czech Technical University, Prague, Czech Republic. 3. Tayebi and S. McGilvray, 26, Attitude Stabilization of a VTOL Quadrotor Aircraft, Proceedings of the EEE Transactions on Control Systems, pp , Orillia, Canada. 4. H. Romero, S. Salazar-Cruz, A. Sanchez, R. Lozano, 27, A New UAV Configuration Having Eight Rotors, Proceedings of the 46th EEE Conference on Decision and Control. pp , New Orleans, USA. 5. S. Bouabdallah, A. Nothand and R. Siegwart, 24, PD vs LQ Control Techniques Applied to an ndoor Micro Quadrotor ntelligent Robots and Systems, Proceedings of the 24 EEE/RSJ nternational Conference, Vol.3, pp , Lausanne, Switzerland. 6. S. Bouabdallah, P. Murrieri and R. Siegwart, 24, Design and Control of an ndoor Micro Quadrotor,, Proceedings of the 24 EEE nternational Conference on Robotics and Automation, No. 21, Vol.5, pp , New Orleans, USA. 7. G. Hoffmann., D. Rajnarayan, S. Waslander, D. Dostal, J. and J. Tomlin, 24, The Stanford Test Bed of Autonomous Rotorcraft for Multi Agent Control", Proceedings of the Digital Avionics Systems Conference, Vol.2, Stanford University, CA, USA. 8. K. Stol, and V. Kermen, 27, Control of 3DOF Quadrotor Model, Proceedings of the nternational Conference on Robot motion and control 27, pp.29-38, Auckland, New Zealand. 41