Designing, Modeling and Control of a Tilting Rotor Quadcopter

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2 Designing, Modeling and Control of a Tilting Rotor Quadcopter A Dissertation submitted to te Graduate Scool of te University of Cincinnati in partial fulfillment of te requirements of te degree of Doctor of Pilosopy in te Department of Electrical Engineering and Computing Systems of te College of Engineering and Applied Sciences by Alireza Nemati M.S. Sarood University of Tecnology August 27 Committee Cair: Manis Kumar, P.D. Ali Minai, P.D.

3 An Abstract of Designing, Modeling and Control of a Tilting Rotor Quadcopter by Alireza Nemati Submitted to te Graduate Faculty as partial fulfillment of te requirements for te Doctor of Pilosopy Degree in Electrical Engineering University of Cincinnati Marc 216 Te aim of te present work is to model, design, control, fabricate and experimentally study quadcopter wit tilting propellers. A tilting quadcopter is an aerial veicle wose rotors can tilt along axes perpendicular to teir respective axes of rotation. Te tilting rotor quadcopter provides te added advantage in terms of additional stable configurations, made possible by additional actuated controls, as compared to a traditional quadcopter witout titling rotors. Te tilting rotor quadcopter design is accomplised by using an additional motor for eac rotor tat enables te rotor to rotate along te axis of te quadcopter arm. Conventional quadcopters, due to limitation in mobility, belong to a class of under-actuated robots wic cannot acieve any arbitrary desired state or configuration. For example, te veicle cannot over at a defined point at a tilted angle. It needs to be completely orizontal in order to over. An attempt to acieve any pitc or roll angle would result in forward (pitc) motion or lateral (roll) motion. Tis proposed tilting rotor concept turns te traditional quadcopter into an over-actuated flying veicle allowing us to ave complete control over its position and orientation. In tis work, a dynamic model of te tilting rotor quadcopter veicle is derived for flying and overing modes. Te model includes te relationsip between veicle orientation angles and rotor tilt-angles. Furtermore, linear and nonlinear controllers ave been designed to acieve te overing and navigation capability wile aving any desired pitc and/or roll orientation. In te linear approac, te four independent speeds of te propellers and teir rotations about te axes of quadcopter arms ave been considered as inputs. In order to start tracking a desired trajectory, first, overing from te initial starting point is needed. Ten, te orientation ii

4 of te veicle to te desired pitc or roll angle is obtained. Subsequently, any furter cange in pitc or roll angles, obtained using a linear controller, result in motion of te quadcopter along te desired trajectory. Te dissertation ten presents a nonlinear strategy for controlling te motion of te quadcopter. Te overall control arcitecture is divided into two sub-controllers. Te first controller is based on te feedback linearization control derived from te dynamic model of te tilting quadcopter. Tis controls te pitc, roll, and yaw motions required for movement along an arbitrary trajectory in space. Te second controller is based on two Proportional Derivative (PD) controllers wic are used to control te tilting of te quadcopter independently along te pitc and te yaw directions respectively. Te overall control enables te quadcopter to combine tilting and movement along a desired trajectory simultaneously. Furtermore, te stability and control of tilting-rotor quadcopter is presented upon failure of one propeller during fligt. On failure of one propeller, te quadcopter as a tendency of spinning about te primary axis fixed to te veicle as an outcome of te asymmetry about te yaw axis. Te tilting-rotor configuration is an over-actuated form of a traditional quadcopter and it is capable of andling a propeller failure, tus making it robust in one propeller failure during te fligt. Te dynamics of te veicle once te failure accrued is derived and a controller is designed to acieve overing and navigation capability. Te dynamic model and te controller of te veicle were verified wit te elp of numerical studies for different fligt scenarios as well as failure mode. Subsequently, two different models of te veicle were designed, fabricated and tested. Experimental results ave validated te dynamical modeling and te fligt controllers. iii

5 Copyrigt 216, Alireza Nemati Tis document is copyrigted material. Under copyrigt law, no parts of tis document may be reproduced witout te expressed permission of te autor. iv

6 Acknowledgments First and foremost, I would like to express my sincere gratitude to my advisor Professor Manis Kumar for te continuous support of my P.D. study and related researc, for is patience, motivation, generosity, entusiasm and immense knowledge. I appreciate all is contribution of time, ideas, encouragement and financial support tat e provided. His continuous guidance elped me carry out researc and write tis tesis. During my study, e was not only a great advisor for problems related to my work, but also an exceptional consultant for oter situations I faced. I wis to express my sincere tanks to Professor Ali Minai, my tesis advisor, for is valuable guidance and encouragement extended to me. I am tankful to Dr. Kelly Coen for all is motivations e gave not only to me but also to te oter members of our team, for is guidance and suggestions. Besides my advisors, I would like to tank Professor Raj Batnagar, and Professor Rui Dai, for teir insigtful comments, assistance and encouragement trougout my project. I addition, a tank you to Younes Keradmand, wo was my boss before I started my P.D. for 5 years. I really appreciate all te advise e provided me like a fater and encouragements e gave like a friend during my career. A tank you also to professor Abdulla afje wo as inspired me te professionalism. My appreciation also extends to my fellow lab mates. Balaji R Sarma, Baisravan Homcauduri and Ruoyu Tan wo elped me settle in te lab for first couple of monts. I would like to tank my friends at te Cooperative Distributed Systems Laboratory for te wonderful times, long meetings and exciting brainstorming discussions we ad. Two of my colleagues cannot remain nameless, Moammad Sarim and Moammadreza Radmanes for te stimulating discussions, for te sleepless nigts we were working togeter before competitions and deadlines, for te travels we made for different conferences and for all te fun we ad in te v

7 last couple years. Te final words in acknowledgments are usually reserved to tose dearest to te autor. I do not wis to break tis tradition. Witout te love and support from my family, I would not ave come tis far. vi

8 Table of Contents Abstract iii Acknowledgments v Contents vii List of Tables x List of Figures xi List of Abbreviations xiv 1 Introduction Objectives Approaces Motivation for tis work Contribution Publications Organization of Tesis History of te Quadcopters A Brief History of Quadcopters Te Early History of Quadcopters History of Tilt Rotor Vertical Takeoff Veicles History of Tilt Rotor Quadcopter Current Quadcopters Quadcopters wit Tilting Mecanism vii

9 3 Dynamic Modeling Traditional Quad-rotor Tilting Rotor Quadcopters Control System Linear Controller Design Proportional Derivative Control Nonlinear Control Feedback Linearization PD Based Tilting Angle Controller Stability Analysis Fault Tolerant Fligt Introduction Fault Detection Dynamic Modeling Tilting Rotor Quadcopters wit One Propeller Failure Controller Design Linearization and Stability Analysis Hardware design Description of te Prototype Central body Tilting Mecanism Drive System Motors Batteries Avionics Control Board Communications viii

10 7 Numerical Simulations and Experimental Results Numerical Simulations Simulation set-up Tilt-Rotor Quadcopter Simulation results Feedback Linearization Numerical Simulation Fault Tolerant Numerical Simulation Hover Fligt Tracking a Trajectory Preliminary Experimental Results Hovering on te spot Trajectory Line Box Conclusions and future work Conclusions Future Works References 14 ix

11 List of Tables 6.1 Specifications of te Prototype Veicle s component mass details Power output from ESC, 4S LiPo, 1x47 Propeller x

12 List of Figures 2-1 Gyroplane Te Oemicen De Botezat Convertawings, Model A George Lebergers 193 tilting propeller vertical take-off flying macine Tree-view drawing of te Focke-Acgelis FA-269 convertiplane Te Bell XV-3, during fligt testing XV-15 taking off Te V-22 Osprey, during transition fligt X-19 in overing fligt Bell X-22A Te Quad-TiltRotor concept, University of Patras CAD design of te CQTR, Istanbul Commerce University Te Configuration of te QTR UAV under researc a) Quadcopter mode b)flying wing mode, Beiang University CQTR wit integrated actuators in different fligt configurations Overview of developed CQTR, Nion University CAD model of te quadcopter wit tilting propellers, Max Planck Institute Veicle prototype on te ball joint rig fligt test, Cranfield University Scematic diagram sowing te coordinate systems and forces acting on te quadrotor Coordinate Frames and Free body diagram of Tilting Quadcopter Te block diagram of position and orientation control algoritm xi

13 4-2 Hovering wit tilted arms Free body diagram of tilt-rotor quadcopter upon propeller failure Veicle s Position Veicle s Orientation Te CAD model of te tilting quadcopter HS-587MH HV Digital Micro Servo Te CAD model of components of te tilting mecanism Te real model of te first model of tilting Quadcopter Te CAD model of transparent tilting mecanism Te CAD model of tilting mecanism mounted on te arm wit te servo and te motor Te CAD and te actual model of te prototype kv brusless DC motors Te reference (commanded) pitc and roll angle Te actual trajectory followed by te UAV in 3-dimensions Te actual orientation of te veicle in 3 directions Te angle of eac arm during simulation Te speed of eac rotor Te enlarged view of te figure Quadcopter trajectory in tree-dimensional space Position and orientation of te quadcopter: altitude vs. time (top left), x-position vs. y-position (top rigt), pitc vs. time (bottom left), and yaw vs. time (bottom rigt) Te reference roll (bottom) and actual roll angle (top) during te fligt Inputs generated by te proposed feedback linearization metod Te actual trajectory followed by te UAV in 3- dimensions Veicle s trajectory in X,Y and Z Veicle s trajectory in X,Y and Z xii

14 7-14 Te actual orientation Te actual trajectory followed by te UAV in 3- dimensions Veicle s trajectory in X,Y and Z Veicle s trajectory in X,Y and Z Te actual orientation Angular velocity around Z axis Snapsot of te overing wit te tilted angle Veicle s trajectory in 3-dimension Te actual orientation of te veicle along te tree directions Veicle s trajectory in 3-dimension Te actual orientation Veicle s position versus time grap along te 3 directions Veicle s trajectory in 3-dimension Te actual orientation Veicle s position versus time grap along te 3 directions xiii

15 List of Symbols ACAH ESC FEM FPGA IMU MV PWM RC rpm Rx Tx SP UART GW lipo D QTW QTR UAV VTOL V/STOL Attitude Command Attitude Hold Electronic Speed Controller Finite Element Metod Field Programmable Gate Array Inertial Measurement Unit Measured values Pulse Widt Modulation Remote Control Revolutions per minute Receive Transmitt Setpoints for te controller Universal Asyncronous Receiver/Transmitter Gross Weigt litium polymer Tree dimensional Quad Tilt Wing Quad Tilt Rotor Unmanned Air Veicle Vertical TakeOff and Landing Vertical/Sort TakeOff and Landing xiv

16 Capter 1 Introduction Tis capter discusses te objectives, approaces, applications and contributions of te researc. Te list of publications resulting from te present work are also provided. 1.1 Objectives Te primary objective of tis work is to model, design, control, fabricate and experimentally study quadcopter wit tilting propellers. Te tilting propellers are expected to result in more stability as well as te ability to follow any arbitrary trajectory in a smooter manner as compared to te conventional quadcopters and also add te capability to continue te mission in te case of one propeller failure. In addition to tis advantage, te tilting mecanism turns te conventional quadcopter, wic is an under-actuated veicle, to an over-actuated robot wic allows full control over a wide range of state-space. In order to incorporate tis new mecanism, wic makes te dynamics of te quadcopter igly nonlinear, tis researc focuses on developing novel control mecanisms in order to acieve te desired fligt requirements as well as make te veicle robust to one propeller failure during te fligt. 1.2 Approaces In order to accomplis te objectives set fort for tis study, te new equations of motion for te tilting quadcopter are first derived matematically. Te derived equations is used to design 1

17 a linear and nonlinear controller. Te numerical simulation of te platform is programmed in bot Simulink and MATLAB for solving te igly nonlinear dynamic equations of motion. Te simulation environment is ten used to verify te performance of te developed control mecanisms. Once te simulation results indicate acievement of desired performance after running wit different fligt situations, a 3D model of te new platform is designed using SolidWorks. Te designed 3D model is used to make and 3D print different parts and fabricate te veicle. Several attempts are made to improve te platform design. Eac attempt focused on factors suc as te weigt, te robustness and te functionality of te platform. Several set of real-world fligt tests are carried out to evaluate te capability of te platform in different fligt conditions. Te actual fligt tests ave validated te derived equations for te dynamic model as well as te designed control systems. 1.3 Motivation for tis work Quadcopters are one of te most popular designs for miniature aerial veicles (MAVs) [56]due to teir vertical take-off and landing capability, simplicity of construction, maneuverability, and ability to negotiate tigt spaces making it possible for use in cluttered indoor areas. Due to tese capabilities, quadcopters ave recently been considered for a variety of applications bot in military and civilian domains [21, 49, 18, 12]. In particular, quadcopter MAVs ave been explored for applications suc as surveillance and exploration of disasters [11, 33, 67] (suc as fire, eartquake, and flood), searc and rescue operations [17, 73], monitoring of azmat spills [2], and mobile sensor networks [19, 72]. Blimps [91], fixed-wing planes, single rotor elicopters, bird-like prototypes [28], coaxial dual rotor elicopters [53], quad-rotors [68, 69, 7, 71] and tilting rotor quadcopters [51, 55, 54, 26, 61, 78] are examples of different configurations and propulsion mecanisms tat ave been developed to allow 3D movements in aerial platforms. Eac of tese as advantages and drawbacks. Tis dissertation focuses on quadcopters or quad-rotors wic consist of four rotors in total, wit two pairs of counter-rotating, fixed-pitc blades located at te four corners of te aircraft. Tis kind of design as two main advantages over te comparable vertical takeoff and 2

18 landing (VTOL) Unmanned Aerial Veicles (UAVs) suc as single rotor elicopters. Firstly, quad-rotors do not require complicated mecanical linkage for rotor actuation. Quad-rotors utilize four fixed pitc rotors te variations of wose speeds form te basis of te control. It results in simplified design and maintenance of te quad-rotors. Secondly, te use of four individual rotors results in teir smaller diameters as compared to te similar main rotor of a elicopter. Te smaller te rotors te less is stored kinetic energy associated wit eac rotor. Tis diminises te risk posed by te rotors if it comes in contact wit any external object. Furtermore, by securing te rotors inside a frame, te protection of rotors during collisions is acieved. It allows indoor fligts in obstacle-dense environments wit lower risk of quad-rotor damage, and iger operator and surrounding safety. Tese benefits ave resulted in safe test fligt by inexperienced pilots in indoor environments and recovery time in case of collisions. In particular, vertical, low speed, and stationary fligt are well-known caracteristics of a quad-rotors. Structurally, quad-rotors can be made in a small size, wit a simple mecanics and control. Toug, as a main drawback, te ig energy consumption can be mentioned. However, te trade-off results are very positive. Tis configuration can be attractive in particular for surveillance, for imaging dangerous environments, and for outdoor navigation and mapping. Conventionally, te quad-rotor attitude is controlled by canging te rotational speed of eac motor. Te front rotor and back rotor pair rotates in a clockwise direction, wile te rigt rotor and left rotor pair rotates in a counter-clockwise direction. Tis configuration is devised in order to balance te moment created by eac of te spinning rotor pairs. Tere are basically four maneuvers tat can be accomplised by canging te speeds of te four rotors. By canging te relative speed of te rigt and left rotors, te roll angle of te quad-rotor is controlled. Similarly, te pitc angle is controlled by varying te relative speeds of te front and back rotors, and te yaw angle by varying te speeds of clockwise rotating pair and counterclockwise rotating pair. Increasing or decreasing te speeds of all four rotors simultaneously controls te collective trust generated by te robot. One of te basic limitations of te classical quad-rotor design is tat by aving only 4 independent control inputs, i.e., te 4 propeller spinning velocities, te independent control of te six-dimensional position and orientation of te quad-rotor is not possible. For instance, a 3

19 quad-rotor can over in place only and if only wen being orizontal to te ground plane or it needs to tilt along te desired direction of motion to be able to move. Tilting rotor quadcopter concept as evolved to solve tese basic limitations of a quad-rotor. Tilt-design makes te dynamics of te quadcopter more complex, and introduces additional callenges in te control design. However, tilting rotor quadcopter, designed by using additional four servo motors tat allows te rotors to tilt, is an over-actuated system tat potentially can track an arbitrary trajectory over time. It gives te full controllability over te quad-rotor position and orientation providing possibility of overing in tilted configuration. Anoter application of te tilting platform lies in its ability to recover during a failure situation. In conventional quadcopters, if one of te propellers fails, due to its inerent dependency on te symmetry of te platform, te veicle becomes entirely uncontrollable. However, in te proposed platform, if one of te motors completely fail, by using te tilt mecanism of one of te tree remaining motors, te unbalanced momentum can be compensated. Moreover, providing additional actuation would make te quadcopter more robust to disturbances wic can be rejected more effectively because of te enanced maneuverability of te quadcopter wit tilting design. Tere is a lot of interest recently in developing small aerial veicles tat can carry umans for transportation in an autonomous manner. To ave te quadcopter to be operational for suc purposes, it needs to be scaled up to be able to carry more payloads. Due to safety issues of te uman passenger, it sould be robust to external disturbances and unpredictable situations. To reject external disturbances, agility becomes an important issue. Te problem wit scaled up quadcopter is te eavy weigt. Once te weigt scales up, te inertia will increase and larger moments will be needed to create te same angular acceleration on a ligter veicle. As a consequence, longer propellers would be needed. Terefore, a stronger motor is required to rotate te new propellers wic add more weigt to te veicle. Tis would make te veicle to become more sluggis. Altoug te tilting mecanism can not make a big difference in trottle, it is expected tat te orientation agility can be increased in big size veicles due to te tilting mecanism. 4

20 1.4 Contribution Tis dissertation focuses on designing, fabricating, modeling and controlling a quadcopter wit tilting propellers. Te contribution of te work lies in following: Te matematical representation of te quadcopter dynamics wit tilting rotors as been derived in order to be used in system analysis and control design. Appropriate control tecniques ave been designed for igly nonlinear dynamics of te quadcopter wit tilting propellers. Based on te dynamic equation of motion of tilting rotor quadcopter, te dynamic model of quadcopter wit one motor failure as been derived and te control tecnique as been designed in order not only to maintain te stability of te veicle after te failure, but also to continue fligt mission. Two different platforms ave been fabricated for te quadcopter wic were designed in SolidWorks environments and some parts ave been printed by using 3D printer. Te numerical simulations and experimental results ave validated te matematical representation as well as designed control tecniques. 1.5 Publications Journals Alireza Nemati and Manis Kumar. Control of Microcoaxial Helicopter Based on a Reduced-Order Observer, Journal of Aerospace Engineering, 41574, 215. Moammadreza Radmanes, Manis Kumar, Alireza Nemati and Moammad Sarim. Dynamic Optimal UAV Trajectory Planning in te National Airspace System via Mixed Integer Linear Programming, Proceedings of te Institution of Mecanical Engineers, Part G: Journal of Aerospace Engineering, ,

21 Alireza Nemati and Manis Kumar. Dynamic Modeling and Control of a Quadcopter wit Tilting Rotors, submitted to IEEE Transactions Aerospace and Electronic Systems, 215. Book Capters Manis Kumar, Moammad Sarim and Alireza Nemati. Autonomous Navigation and Target Geolocation in GPS Denied Environment, Multi-Rotor Platform Based UAV Systems. Wiley Publising, Manis Kumar, Alireza Nemati, Anoop Satyan and Kelly Coen. Real-time Video and FLIR Image Processing for Enanced Situational Awareness, Multi-Rotor Platform Based UAV Systems. Wiley Publising, Proceedings Alireza Nemati and Manis Kumar. Modeling and Control of a Single Axis Tilting Quadcopter, American Control Conference (ACC), pp IEEE, 214. Alireza Nemati and Manis Kumar. Non-Linear Control of Tilting Quadcopter Using Feedback Linearization Based Motion Control. Dynamic System and Control Conference (DSCC), pp. V3T48A5-V3T48A5, ASME, 214. Moammadreza Radmanes, Alireza Nemati, Moammad Sarim and Manis Kumar. Fligt Formation of Quad -copters in Presence of Dynamic Obstacles using Mixed Integer Linear Programming, Dynamic Systems and Control Conference. ASME, 215. Alireza Nemati, et al. Autonomous Navigation of UAV troug GPS-Denied Indoor Environment wit Obstacles, American Institute of Aeronautics and Astronautics, AIAA SciTec, DOI: / , 215. Moammad Sarim, Alireza Nemati and Manis Kumar. Autonomous Wall-Following Based Navigation of Unmanned Aerial Veicles in Indoor Environments. American Institute of Aeronautics and Astronautics, AIAA SciTec, DOI: / ,

22 Moammad Sarim, Alireza Nemati, Manis Kumar and Kelly Coen. Extended Kalman Filter based Quadrotor State Estimation based on Asyncronous Multisensor Data, Dynamic Systems and Control Conference. ASME, 215. Moammadreza Radmanes, Manis Kumar, Alireza Nemati and Moammad Sarim. Solution of Traveling Salesman Problem wit Hotel Selection in te Framework of MILP- Tropical Optimization, accepted in American Control Conference (ACC), IEEE, 216. Alireza Nemati, Neal Soni, Moammad Sarim, and Manis Kumar. Design, fabrication and control of a tilt rotor quadcopter. In ASME 216 Dynamic systems and control conference. American Society of Mecanical Engineers, 216. Alireza Nemati, Rumit Kumar, and Manis Kumar. Stabilizing and control of tilting-rotor quadcopter in case of a propeller failure. In ASME 216 Dynamic systems and control conference. American Society of Mecanical Engineers, 216. Intellectual property Alireza Nemati, Medi Hasemi and Manis Kumar, World Frame Based Radio Controller(RC) for Multi-copter UAVs Filed for provisional patent, October 215 Alireza Nemati, Manis Kumar and Rumit Kumar. Fault Tolerance Quadcopter. Provisional Patent Has Been Filed by University of Cincinnati, February Organization of Tesis Tis dissertation consists of eigt capters. Tis includes Introduction as te first capter. Capter 2 is a literature review tat provides a brief istory of te conventional as well as tilting quadcopters. Capter 3 reports on te dynamic model of te tilting quadcopter and considers te nonlinearities tat add to te equation due to additional control inputs. Capter 4 presents a combined linear and nonlinear controller wic is used to control desired orientation during te fligt. Te dynamic model of te veicle and te proposed control tecnique once te failure occurs is presented in capter 5. 7

23 Te ardware design process is described in detail in Capter 6. Results from numerical simulation and experimental studies carried out to verify te modeling and control of tilting rotor quatcopters following a reference trajectory wit simultaneous control of bot pitc and roll angles are discussed in Capter 7. Te conclusions and future works are presented in Capter 8. Tis capter summarizes te dissertation, discusses te contributions and also outlines directions for future works to be pursued. 8

24 Capter 2 History of te Quadcopters 2.1 A Brief History of Quadcopters Te Early History of Quadcopters A Quadcopter or Quadrotor is multi-rotor mecatronic device capable of Vertical Takeoff and Landing (VTOL) tat is lifted or propelled by four independently rotating rotors. Te idea beind Quadcopters was first developed in te early 19s. Tere were very few unique and momentous quadcopter designs developed trougout te 2t century. Te earliest ideas for a quadcopter were designed and test piloted by Louis Breguet, Etienne Oemicen, George DeBotezat, and D.H. Kaplan. Te first successful fligt of a quadcopter aerial veicle was in 197. Tis device, named te Gyroplane (Figure (2-1) 1 ), was built by Breguet broters and consisted of a 55p Renault engine and two forward-tilting 2-blade rotors. It was reported to ave multiple successful fligts during te summer of 198. However it s mobility and range of fligt were very limited. 1 ttps://en.wikipedia.org/wiki/breguet-ricet-gyroplane 9

25 Figure 2-1: Gyroplane. In te 192s, Etienne Oemicen, was able to construct te first stable VTOL quadcopter wic e named Oemicen II (Figure (2-2) 2 ). It consisted of a single 18p Gnome engine powering four rotors, a complex steel-tube framework of cruciform layout, five smaller propellers mounted orizontally to provide lateral stability, and an additional pair of propellers tat were mounted to te nose of te craft for steering. Te last pair of propellers provided forward trust. Tis design made tousands of successful fligts during te mid 192s and even establised a world record of flying one kilometer in seven minutes and forty seconds. Almost all quadcopters in te 192s were unable to sustain a controlled fligt and ad to use te Ground Effect to sustain fligt limiting tese designs to stay low to te ground. Figure 2-2: Te Oemicen2 Around te same time, George DeBotezat designed and built te Flying Octopus (Figure (2-3) 3 for te United States Army Air Corps. Te 1678kg X saped structure supported 2 ttps://en.wikipedia.org/wiki/c389tienne-oemicen 3 ttps://en.wikipedia.org/wiki/pescara-model-3-helicopter 1

26 four 8.1m diameter six-blade rotors; one on eac end of te 9m long arms. At eac end of te lateral arms, two smaller propellers wit variable pitc supplied trust and enabled yaw control. After working on is design for a little over two years, DeBotezat was able to develop a fairly capable quadcopter. Tis design was able to carry a payload of up to 4 people including te pilot. However, te design was considered to be flawed as it was under-powered, unresponsive and very fragile. Te craft was only capable of reacing an altitude of around 5m rater tan te 1m desired by te army. Figure 2-3: De Botezat Early quadcopters typically contained a single engine positioned in te center of te fuselage tat drove te four rotors via belts or safts. Tese belts and safts were eavy and more importantly broke down often. In addition, te four rotors of te quadcopter were ever so sligtly different from one anoter, so te quadcopter was not naturally stable during fligt. Running all rotors at te same speed did not produce a stable fligt and eac rotor ad to be constantly adjusted to sustain a stable fligt. In te early 19s, wit te absence of any digital computers or sensors, flying a quadcopter required a monumental workload for te pilot making te early quadcopters very inefficient and not practical for transportation. Tese early quadcopters designs also included multiple additional rotors located on different locations of te quadcopter for additional stability, making tese designs not true quadcopters. As materials and engineering practices evolved over te century, numerous improvements were made by bot increasing te power of te motors and reducing te overall weigt of te designs. During te early 195s, D.H. Kaplan, worked on and test piloted te Convertawings Model A 11

27 Quadcopter (Figure (2-4) 4 ). Kaplan s design featured four rotors and ad a two motor layout wit te rotors positioned in an H configuration. Kaplan s macine may be considered te first true quadcopter as it was capable of sustaining a controlled fligt witout te use of te ground effect or any additional propellers. Te 2, 2 pounds craft ad a muc simpler design tan previous quadcopters due to te fact tat control was obtained by varying te trust between te individual rotors eliminating te need for complex cyclic-pitc-control systems and additional rotors on te sides of te fuselage. Tis design first flew in Marc of 1956 wit great success. Te design, in particular its control system, was a precursor of a majority of te current vertical takeoff aircraft designs tat incorporate tandem wings. Figure 2-4: Convertawings, Model A History of Tilt Rotor Vertical Takeoff Veicles Tilt Rotors combined te properties of a elicopter wic included Vertical Take Off and Landing (VTOL), overing, and vertical, forward, and lateral fligt, wit te desirable properties of a fixed-wing aircraft including long range fligt, low power consumption and te ability to carry eavier payloads. Te first design tat resembled a modern tilt rotor device was patented in May of 193 by George Leberger (Figure (2-5)) 5. 4 ttps://en.wikipedia.org/wiki/de-botezat-elicopter 5 ttp://istory.nasa.gov/monograp17.pdf 12

28 Figure 2-5: George Lebergers 193 tilting propeller vertical take-off flying macine. Toug tis design never amounted to a prototype, it was te first step in making a functional VTOL capable tiltrotor veicle and inspired te design of te Focke-Acgelis FA-269 (Figure (2-6) 6 ) trail-rotor convertiplane project in Germany during World War II[43]. 6 ttp://istory.nasa.gov/monograp17.pdf 13

29 Figure 2-6: Tree-view drawing of te Focke-Acgelis FA-269 convertiplane A prototype of tis aircraft was built in 1943 and consisted of two puser propellers tat tilted below te two wings for takeoff and landing. However, te project was discontinued after te allies destroyed a full scale mock-up of tis design and muc of te researc during a bombing in A few years later, variants of tis tilt rotor configuration surfaced again in te design studies at Bell and McDonnell Douglas. Te Bell XV-3 (Figure (2-7) 7 ) was a tiltrotor aircraft designed by Bell in te 195s[15]. Its first successful fligt was in August It was te first aircraft to successfully transition between elicopter and fixed wing for normal fligt. Te XV-3 was powered by a single 45p radial engine tat propelled te aircraft at a maximum speed of 296 km/. Te craft ad a maximum altitude of 46 meters. Tis aircraft was a proof of concept and made over 1 successful transitions before it was severely damaged in a wind tunnel accident and te design was scraped. 7 ttp://ntrs.nasa.gov/arcive/nasa/casi.ntrs.nasa.gov/24875.pdf 14

30 Figure 2-7: Te Bell XV-3, during fligt testing. Te data and experience collected during tis trial were key to te development of te Bell- XV15 (Figure (2-8) 8 ) and te V-22 Osprey (Figure (2-9) 9 )[57, 43]. Figure 2-8: XV-15 taking off 8 ttps://en.wikipedia.org/wiki/bell-xv-3 9 ttps://en.wikipedia.org/wiki/bell-boeing-v-22-osprey 15

31 Figure 2-9: Te V-22 Osprey, during transition fligt Bot tese designs followed te same principles as te Bell XV-3 and ad many successful fligts over teir life History of Tilt Rotor Quadcopter In respect to quadcopter tilting macines, tere ave been two early designs tat stand out, te Curtiss X-19 and te and te Bell X-22. Te Curtiss X-19 (Figure (2-1) 1 ) built in 196s was a passenger plane tat consisted of two sets of tin wings eac wit a 3 bladed rotor tat could rotate 9 degrees [3]. Wit its massive 2,2 p engines, it could carry up to 55kg of cargo or 4 passengers along wit te two crew members. Two turbosaft engines were oused in te rear fuselage and powered te four rotors. Te aircrafts first fligt was in 1963 and was capable of flying up to 523 Km and reaced a maximum speed of 65 Km/. Two prototypes of te X-19 were built but te project was canceled after te first prototype crased during its second fligt. 1 ttps://en.wikipedia.org/wiki/curtiss-wrigt-x-19 16

32 Figure 2-1: X-19 in overing fligt Te Bell X-22 was built a couple years after te Curtiss X-19 and is considered to be one of te most versatile and longest lived of te many VTOL aircrafts tat ave been developed[3]. It is similar to te X-19 in tat it as four wings eac wit teir own 3 bladed propeller eac able to rotate 9 degrees but instead of aving 2 motors, te Bell X-22 ad four 125 p motors eac powering teir own rotor. Te design was able to carry up to six passengers and two pilots and reaced a maximum speed of 57 km/our wit a range of up to 716km. Te two prototypes of te Bell X-22 were used for many years by bot NASA and te US Navy for V/STOL and performed very well. One is still on display at te Niagara Falls Aerospace Museum in New York (Figure (2-11) 11 ). More modules of quadcopters are currently being developed for te US Army Corps Including te Bell Boeing Quad Tiltrotor (QTR). It is currently under study and was first designed in Te Bell Boeing Quad Titlrotor is predicted to be able to carry up to 8 passengers wit a cruise speed of 52 km/our. Figure 2-11: Bell X-22A 11 ttps://en.wikipedia.org/wiki/bell-x-22 17

33 2.2 Current Quadcopters Quadcopters are one of te most popular designs for miniature aerial veicles (MAVs) due to teir vertical take-off and landing capability, simplicity of construction, maneuverability, and ability to negotiate tigt spaces making it possible for teir use in cluttered indoor areas. Due to tese capabilities, quadcopters ave recently been considered for a variety of applications bot in military and civilian domains. In particular, quadcopter MAVs ave been explored for applications suc as surveillance and exploration of disasters (suc as fire, eartquake, and flood), searc and rescue operations, monitoring of azmat spills, and mobile sensor networks[37] [64][14]. Blimps, fixed-wing planes, single rotor elicopters, bird-like prototypes, coaxial dual rotor elicopters, quad-rotors, tilting rotor quadcopters are examples of different configurations and propulsion mecanisms tat ave been developed to allow 3D movements in aerial platforms [39] [5] [9] [5]. Eac of tese designs ave advantages as well as drawbacks. Tis work focuses on quadcopters or quad-rotors wic consist of four rotors in total, wit two pairs of counter-rotating, fixed-pitc blades located at te four corners of te aircraft. Tis kind of design as two main advantages over te comparable vertical takeoff and landing (VTOL) Unmanned Aerial Veicles (UAVs) suc as single rotor elicopters. Firstly, quad-rotors do not require complicated mecanical linkage for rotor actuation. Quad-rotors utilize four fixed pitc rotors te variations of wose speeds form te basis of te control. It results in simplified design and maintenance of te quad-rotors. Secondly, te use of four individual rotors results in teir smaller diameters as compared to te similar single rotor of a elicopter. Smaller rotors imply less stored kinetic energy associated wit eac rotor. Tis diminises te risk posed by te rotors if it comes in contact wit any external object. Furtermore, by securing te rotors inside a frame, te protection of rotors during collisions is acieved. Tis configuration allows indoor fligts in obstacle-dense environments wit lower risk of quad-rotor damage, and iger operator and surrounding safety. Tese benefits ave resulted in safe test fligts by inexperienced pilots in indoor environments and lesser recovery time in case of collisions [25]. In particular, stable, vertical, low speed and stationary fligts are well-known caracteristics of a quad-rotor. Structurally, quad-rotors can be designed in a small size, wit simple mecanics and control. Te quadcopters ave been found to be an attractive coice in particular for surveillance, for 18

34 imaging dangerous environments, and for outdoor navigation and mapping [59], [22]. Te major drawback, owever, is ig energy consumption due to te use of four rotors. Conventionally, te quad-rotor attitude is controlled by canging te rotational speed of eac motor. Te front rotor and back rotor pair rotates in a clockwise direction, wile te rigt rotor and left rotor pair rotates in a counter-clockwise direction. Tis configuration is devised in order to balance te moment created by eac of te spinning rotor pairs. Tere are basically four maneuvers tat can be accomplised by canging te speeds of te four rotors. By canging te relative speed of te rigt and left rotors, te roll angle of te quad-rotor is controlled. Similarly, te pitc angle is controlled by varying te relative speeds of te front and back rotors, and te yaw angle by varying te speeds of clockwise rotating pair and counterclockwise rotating pair. Increasing or decreasing te speeds of all four rotors simultaneously controls te collective trust generated by te robot [9]. One of te basic limitations of te classical quad-rotor design is tat by aving only 4 independent control inputs, i.e., te 4 propeller spinning velocities, te independent control of te six-dimensional position and orientation of te quad-rotor is not possible. For instance, a quad-rotor can over in place only and if only wile being orizontal to te ground plane or it needs to tilt along te desired direction of motion to be able to move. Tilting rotor quadcopter design as been developed to solve tese basic limitations of a quad-rotor. In next section, tere are several examples of recent development about tilting concept. Recently, tere as been a renewed interest in quadcopters from obbyists, universities, and corporations across te globe. Te renewed interest is due to te many significant tecnological advances in sensors and micro-controllers over te past decade tat ave allowed tese once large macines to be miniaturized to fit in te palm of a and and be autonomously controlled. A significant number of quadcopters ave been introduced for bot military and civilian use as a result of partnersips between companies and universities tat ave enabled tis quadcopter UAV revolution. Many companies suc as AeroQuad, ArduCopter, DJI, and Parrot AR.Drone ave sparked te interest of obbyist; Coupled wit te DIY (Do It Yourself) and open source movement, Quadcopter UAVs are more popular and progress in te sector is advancing faster tan ever before. 19

35 2.2.1 Quadcopters wit Tilting Mecanism Quadcopters wit tilting propellers are divided into two categories. Quadcopters wic ave te capability of flying bot as a conventional quadcopter as well as a fixed wing aerial veicle by tilting all propellers in te same direction by te same amount of angle and te one wic is capable of making a tilt in any individual propellers independently. In te first type of tilting quadcopter( Convertible Quad Tilt Rotor (CQTR)), all propellers are canging teir orientation simultaneously wit te same amount of angle. However, variety of platform, mecanism and control metods ave been used by different researcers and universities. Since all propellers tilt wit te same angle, te dynamic equation of te aircraft does not cange a lot and te complexity of te equation is not te most critical obstacle tat needs to be tackled for te designers. Te ability of transition between vertical take off and over fligt to orizontal fligt is one te most difficulties te needs to be consider for CQTR. Wit tis ability, aircraft will be able not only to take off and land at any inappropriate area, but also by taking an advantage of aerodynamic sape of te wing, it will be able of flying orizontally in long distances. Te fligt mode tat makes te transition between vertical and orizontal fligt as been gaining remarkable interest among researcers. Numerous innovative CQTR platform ave been studied in very last few years. Papacristos et al [62] from university of Patras, Greece, ave focused on ybrid model predictive fligt mode conversion control of CQTR. Teir aircraft platform (sown in Figure (2-12) 12 ) Figure 2-12: Te Quad-TiltRotor concept, University of Patras. capable of flying bot as a quadcopter as well as fix wing aircraft. Tey ave developed an innovative control sceme based on ybrid systems teory. An approximation of complete 12 ttp:// 2

36 nonlinear dynamics as been derived and used as a model for control during autonomous midfligt conversion. Altoug tey ave not flown te aircraft, but by using simulation studies its been sown tat teir proposed strategy exceeds te functionality of te fligt-modes conversion. Te standard NACA2411 airfoil as been selected for te design. Te wings wit total span of 1 meter are mounted on te tilting mecanism. Te wings are capable of rotating 9 degrees angle. Anoter researc group as focused on design and control of gas-electric ybrid CQTR wit morping wing in order to extend te overing fligt up to 3 ours or up to 1 ours of orizontal fligt [13]. Tey ave minimized te mecanical morping wings and aerodynamic cost for bot ig speed and low speed fligts. A variety of novel features ave been used in teir concept Figure (2-13) 13. Figure 2-13: CAD design of te CQTR, Istanbul Commerce University. A gas engine-battery ybrid propulsion is used due to capability of carrying eavy payload and very long fligt duration, plus a carbon composite wings not only to reduce te weigt but also for andling bot ig speed and low speed fligt. Te V-type structure of te aircraft allows to extension of te rear rotors from te center of te mass wic ensure te full coverage of te wing area by te rotor for preventing stalls as well as minimum required speed for orizontal fligt. For te transition, two servo motors ave been located next to saft of rear and front rotors. Te veicle s take off weigt is 2 kg and te wing-span including te larger winglets is 2.5 meters. Numerical simulation as been used to validate and understand te fligt beavior and performance of te aircraft. By placing four rotors in two axes wit almost te same level, rotor s trust will not cover te most, even if like te previous work, te rotors 13 ttp://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=

37 of one te axis extend from te center. Te researcers in Beiang University, Cina, [65] ave proposed a prototype concept of CQTR tat place te rotors of te front and rear axis of te orizontal mode in two different levels. In teir concept, for cange in configuration, te two front rotors tilt down to -9 degree, wile te two rear rotor tilt up to 9 degree. After te transition is done, all rotors will be facing front wit exactly te same angle but in two different levels. Figure (2-14) 14 sows te configuration of te CQTR in bot quadcopter and flying wing mode. Figure 2-14: Te Configuration of te QTR UAV under researc a) Quadcopter mode b)flying wing mode, Beiang University. Tey ave focused on te trajectory tracking control for overing and acceleration fligt of CQTR by using dynamic inversion. Teir scenario is based on 4 strategies as follows: takeoff and reac a certain altitude, find an optimal transition trajectory to not only minimize te transition time but also not to lose altitude, te next scenario is to keep flying in fixed wing mode and te ability to cange te altitude. During te cruise fligt, te rotor speed and te forces allocated between te rotors and te wings maintain te needed trust to control altitude and attitude. Attitude Command Attitude Hold (ACAH) metod is being used for attitude control strategy. Te last scenario is to lower te speed and keep tilting back te rotors to teir original angle during te takeoff. Altoug tey ave built a prototype version of te veicle wit dimension of 1.8 meter for te wing span and gross weigt of 5.2kg, but numerical simulation is used to evaluate teir proposed control system. Several researces ave introduced many interesting prototype of CQTR, but tere are few 14 ttp://comb.buaa.edu.cn/publications/pateers/214/48.tml 22

38 groups tat validate teir control systems by experimental results. Hancer et al [23] presented a prototype CQTR equipped wit robust position controller to track desired trajectory under aerodynamic and external wind disturbances, as sown in Figure (2-15) Figure 2-15: CQTR wit integrated actuators in different fligt configurations. Dryden model as been used to model wind effects wic is included in dynamic model. Tese disturbances are being estimated by using disturbance observer wic is commonly used in motion control systems. Parametric uncertainties and nonlinear term are also added to external disturbances as a total disturbance. Performance of te overing fligt is verified wit te experimental results. Te transition from overing fligt to orizontal cruise fligt wic is te most critical part of te experiment is not tested in real world, but trajectory tracking performance is confirmed wit numerical simulations. Mikami and Uciyama [47] from Nion University ave validated teir concept by numerical simulation as well as experimental results. Due to strong nonlinearities of dynamical beavior, a linearization metod witout any approximation as been applied to teir control strategy. Tis proposed control strategy is being used in bot translational and rotational controllers. Figure (2-16) 17 sows an overview of developed CQTR 15 ttp://researc.sabanciuniv.edu/15316/1/cdc1.pdf 16 ttp://people.sabanciuniv.edu/munel/publications/journalpapers/mecatronics-212.pdf 17 ttp://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=

39 Figure 2-16: Overview of developed CQTR, Nion University. Overall weigt of te prototype is.48 kg wit te lengt and wing span of.8 and.7 meter, respectively. Altoug, due to weigt limitation and small size of te battery, te endurance of te veicle is not longer tan 15 minutes, but te experimental results verified te validity of fligt control strategy. Te veicle is equipped wit micro-computer, flow sensor, radio module, IMU and ultra-sonic sensor. Te experimental results sow tat te transition from over fligt to orizontal cruise fligt as been accomplised. Even-toug above mentioned air-crafts are using a tilting mecanism for te propellers, but te under actuation problem of te robot still remains, and in addition to tat, tese additional dynamic of convertible quad tilt rotor is not making te big problem for te control systems. Also by canging te angle of all propellers simultaneously, translational motion can not be acieved independently witout tilting te aircraft. If te aircraft s arm ad te capability of independently tilting about teir own axis, te dependency of translation and orientation problem, could be solved and te system would not be called an under actuated robot. Tere ave been some attempts to carryout tis problem by several groups. Ryll et al. [76] from Max Plank Institute, Germany, proposed a tilting quadcopter wit 8 independent control inputs tat allows te aircraft for independent attitude and position control. Tey ave added four additional servo motors to allow te propellers to tilt about teir axes ttps:// BC3BCltoff/358f3f a4cbc4bc2d2c2d3df3fc6be77 24

40 Figure 2-17: CAD model of te quadcopter wit tilting propellers, Max Planck Institute. Te linearized compensation control based on te quadcopter s dynamic as been used to acieve 6 DOF of motion control. Te tracking of an arbitrary trajectory was te main focus of te group. Tey ave simplified te dynamic model in order to ave a suited model for control design. Since 8 independent inputs are available, teir proposed control design is over actuated. As in many output tracking control tecniques, an appropriate way tey solved te problem was to place te output feedback linearization metod. Numerical simulation was te way tey applied te controller presented in [76]. Te tracking performance of te controller was validated and te capability of te proposed metod to avoid te singularities was guaranteed. Anoter paper as been publised from te same autors [77] to validate te proposed strategies. Altoug te prototype wic is called omnicopter as been tested on a testing gimbals, but teir results sow te full controllability over 6 DOF body pose in space. Anoter group [81] as tested teir prototype on te ball joint rig. Ball joint rig is a device tat can be attaced to te underneat of te quadcopter and it can let te quadcopter to orient around and follow te desired orientation witout an actual fligt ttps:// Al-Riani/28bdc9144a153c132b37382be32e94d83b1abfed/pdf 25

41 Figure 2-18: Veicle prototype on te ball joint rig fligt test, Cranfield University. Te main proposal of te prototype was to improve te performance and fault tolerance of quadcopter veicles. Tey ave proposed dual axis tilting propellers wic enables gyroscope torque, differential trusting and trust vectoring. Not only te matematical representation of te model was modeled and verified by te experimental results, but also a control system was developed based on PD controller and validated troug test on a ball joint rig fligt test. Tilt-design makes te dynamics of te quadcopter more complex, and introduces additional callenges in te control design. However, tilting rotor quadcopter, designed by using additional four servo motors tat allow te rotors to tilt, is an over-actuated system tat potentially can track an arbitrary trajectory over time. It gives te full controllability over te quad-rotor position and orientation providing possibility of overing in a tilted configuration. Tis work presents a matematical dynamic modeling of te tilting rotor quadcopter wic provides a description of te dynamical beavior of te quadcopter as a function of te rotational speeds of eac of te four rotors and teir respective tilt-angles. Te developed matematical representation of te tilting rotor quadcopter can be used to obtain te position and orientation of te quad-rotor. Te same model can furter be used to develop a linear and nonlinear control strategy via wic te speeds of te individual motors and te respective tilt-angles can be manipulated to acieve te desired motion and configuration. 26

42 Capter 3 Dynamic Modeling Unlike traditional quad-rotor models, wic ave only four rotatory propellers as te veicle s inputs, in tilting rotor quadcopters, tere are four more servo motors attaced to te eac arm tat adds one degree of freedom to eac of te propellers, resulting in te tilting motion along teir axes. In tis capter, first te dynamic model of a traditional quad-rotor is described, ten, te equations of motion of a tilting rotor quadcopter are presented. 3.1 Traditional Quad-rotor Figure (3-1) scematically sows te coordinate system and forces acting on a traditional quad-rotor. In te 3 dimensional space, te world-frame (E) denotes te fixed reference frame wit respect to wic all motions can be referred to and te body-frame (B) is a frame attaced to te center of mass of te veicle. Te rotation of eac rotor causes an aerodynamic force or trust tat acts perpendicular to te plane of rotation of te rotor. In addition to te forces, eac rotor produces a moment perpendicular to te plane of propeller rotation. Te moment produced by a propeller on te veicle is directed opposite to te direction of rotation of te propeller, and terefore to cancel out rotation along te Z-axis, te moments for rotor 1 and 3 are set in clockwise ( Z B ) direction and for rotor 2 and 4 are set in counter clockwise (Z B ) direction. Based on NASA Standard Airplane [35], Euler angle transformations are defined by ψ, θ and φ wic respectively represents te eading, attitude and bank angles also referred to as yaw, pitc and roll angles. Combined transformation matrix from body coordinate to te 27

43 world coordinate is obtained by tree successive rotations. Te first rotation is about X axis, followed by anoter rotation about Y axis and te last rotation is about Z axis. For eac rotation, transformation matrix can be written as: 1 R x = cosφ sinφ sinφ cosφ cosθ sinθ R y = 1 sinθ cosθ cosψ sinψ R z = sinψ cosψ 1 Resultant transformation matrix will be given by: cψcθ cψsθsφ sψcφ cψsθcφ + sψsφ R EB = sψcθ sψsθsφ + cψcφ sψsθcφ cψsφ sθ cθsφ cθcφ (3.1) were cψ and sψ denote cos(ψ) and sin(ψ) respectively, and similarly for oter angles. By obtaining veicle s vertical forces in te world frame and writing te equations of motion based on te Newton second law along te X, Y and Z axes, we can write: mẍ = mÿ = m z = F i (sψsφ + cψsθcφ) C 1 ẋ F i (sψsθcφ cψsφ) C 2 ẏ (3.2) F i (cθcφ) mg C 3 ż were m is te total mass of quad-rotor, g is te acceleration due to gravity, x, y and z are quadcopter position in world frame coordinate, C 1, C 2 and C 3 are drag coefficients. Note tat 28

44 te drag forces are negligible at te low speed. F i,(i = 1, 2, 3, 4) are forces produced by te four rotors as given by te following equation: F i = K f ω 2 i (3.3) were ω i is te angular velocity of i t rotor and K f is a constant. In addition, Euler equations are written in order to obtain angular accelerations of te veicle given by: I x φ = l(f 3 F 1 C 1 φ) I y θ = l(f 4 F 2 C 2 θ) (3.4) I z ψ = M 1 M 2 + M 3 M 4 C 3 ψ were l is distance of eac rotor from te veicle s center of mass. I x, I y and I z are moment of inertia along x, y and z directions respectively. C 1, C 2 and C 3 are rotational drag coefficients. M i, (i = 1, 2, 3, 4) are rotors moment produced by angular velocity of rotors and given by: M i = K m ω 2 i (3.5) were ω i is te angular velocity of i t rotor and K m is a constant. Figure 3-1: Scematic diagram sowing te coordinate systems and forces acting on te quadrotor During a overing fligt, te quad-rotor not only as zero acceleration and velocity but also 29

45 needs to ave zero pitc and roll angles, i.e. r = r, θ = φ =, ψ = ψ, ṙ =, θ = φ = ψ =. At tis nominal over state, te produced force form eac propellers must satisfy: F i = 1 (mg) (3.6) 4 and ence motor speeds are given by: ω i = ω = mg 4k f (3.7) 3.2 Tilting Rotor Quadcopters For a tilting rotor quadcopter, four oter variables are added representing te angles of te quad-rotor arms. Adjustment of tese angles results into improved veicle maneuverability and capability for overing at a tilted angle. To illustrate te motion of te tilting rotors quadcopter, a scematic diagram sowing te forces/moments acting and coordinate frames used in te modeling is provided in Figure (3-2). As it can be seen from tis figure, te propellers are free to tilt along teir axes. Te planes sown wit dased lines are te original planes of rotation wit zero tilt angles for te respective propellers. Similarly, te planes sown wit te rigid lines are te tilted planes of rotation for te respective propellers. θ i, (i = 1, 2, 3, 4) is te tilted angle of te corresponding propellers. It may be noted tat te forces generated by te propellers are perpendicular to tese respective planes of rotations. 3

46 Figure 3-2: Coordinate Frames and Free body diagram of Tilting Quadcopter Te equation governing te acceleration of te center of mass is: ẍ F 1 sθ 1 F 3 sθ 3 C 1 ẋ m ÿ = R EB F 2 sθ 1 F 4 sθ 3 C 2 ẏ z mg F 1 cθ 1 + F 2 cθ 2 + F 3 cθ 3 + F 4 cθ 4 C 3 ż Using te rotational matrix in (1), equations of motion in world-frame can be rewritten as: mẍ = F 1 sθ 1 cψcθ F 3 sθ 3 cψcθ F 4 sθ 4 cψsθsφ + F 2 sθ 2 cψsθsφ + F 4 sθ 4 sψcφ F 2 sθ 2 sψcφ + F 1 cθ 1 cψsθcφ + F 2 cθ 2 cψsθcφ + F 3 cθ 3 cψsθcφ + F 4 cθ 4 cψsθcφ + F 1 cθ 1 sψsφ + F 2 cθ 2 sψsφ + F 3 cθ 3 sψsφ + F 4 cθ 4 sψsφ C 1 ẋ mÿ = F 1 sθ 1 sψcθ F 3 sθ 3 sψcθ F 4 sθ 4 sψsθsφ + F 2 sθ 2 sψsθsφ F 4 sθ 4 cψcφ + F 2 sθ 2 cψcφ + F 1 cθ 1 sψsθcφ + F 2 cθ 2 sψsθcφ + F 3 cθ 3 sψsθcφ + F 4 cθ 4 sψsθcφ F 1 cθ 1 cψsφ F 2 cθ 2 cψsφ F 3 cθ 3 cψsφ F 4 cθ 4 cψsφ C 2 ẏ 31

47 m z = F 1 sθ 1 sθ + F 3 sθ 3 sθ F 4 sθ 4 cθsφ + F 2 sθ 2 cθsφ + F 1 cθ 1 cθcφ + F 2 cθ 2 cθcφ + F 3 cθ 3 cθcφ + F 4 cθ 4 cθcφ mg C 3 ż (3.8) Similarly, te angular accelerations are determined by Euler equations: I x φ = l(f 3 cθ 3 F 1 cθ 1 C φ) 1 + (M 1 sθ 1 M 3 sθ 3 ) + (M 2 + M 4 ) I y θ = l(f 4 cθ 4 F 2 cθ 2 C θ) 2 + (M 4 sθ 4 M 2 sθ 2 ) + (M 1 + M 3 ) I z ψ = l(f 1 sθ 1 + F 2 sθ 2 + F 3 sθ 3 + F 4 sθ 4 C ψ) 3 + (M 1 cθ 1 M 2 cθ 2 + M 3 cθ 3 M 4 cθ 4 ) (3.9) were M i, (i = 1, 2, 3, 4) are te tilting moments created by te four servo motors attaced to te end of eac arm to enable teir tilting motion. It may be noted tat tese moments are negligible because te moments produced by te servo motors are used to tilt te arms wic are connected to te main body via mecanical bearings. Neglecting bearing friction, te moments transmitted to te main body of te quadrotor are negligible. Based on te dynamic model presented above, we propose te following two Teorems. Note tat witout lost of generality, yaw angle is assumed to be zero in following teorems. Teorem 1: Considering te dynamics of te tilting rotor quadcopter given by Equations (3.8) and (3.9), and assuming te relationsip between te tilting angles of te four rotors θ 1 = θ 3 and θ 2 = θ 4 and all rotors aving equal rotational speeds, te quadcopter, at an equilibrium overing state, acieves a roll angle φ given by φ = θ 1 /2 wen te pitc angle is zero, and a pitc angle θ given by θ = θ 2 /2 wen te roll angle is zero. Proof: In tilt-overing, te arm angles of te first and tird propellers are tilted by θ 1 and θ 3 = θ 1, respectively. Tis produces a roll angle φ of te veicle, and, te equations for linear 32

48 motion of te quadcopter is given by: mẍ mÿ m z F 1 s(θ 1 φ) + F 3 s( θ 3 φ) F 2 sφ F 4 sφ = + F 1 c(θ 1 φ) + F 2 cφ + F 3 c( θ 3 φ) + F 4 cφ mg (3.1) For overing, te accelerations ẍ, ÿ, and z sould all be equal to zero. Using te equation corresponding to te acceleration in X direction, and noting tat F 1 = F 2 = F 3 = F 4 since rotational speeds of all rotors are te same, te angle φ can be obtained as: φ = θ 1 2 (3.11) Similarly to te equation (3.1), if te second and fourt arms are tilted, te equations of motion can be written as: mẍ mÿ m z = F 1 sθ F 3 sθ 3 + F 2 s(θ 2 θ) + F 4 s( θ 4 θ) + F 1 cθ + F 2 c(θ 2 θ) + F 3 cθ + F 4 c( θ 4 θ) mg (3.12) Similar to above, te angle θ resulted from tilting of te second and fort arms, is given by: θ = θ 2 2 (3.13) Teorem 2: Considering te dynamics of te tilting rotor quadcopter given by Equations (3.8) and (3.9), and assuming te relationsip between te tilting angles of te four rotors θ 1 = θ 3 and θ 2 = θ 4, te motor speed needed for veicle for overing wit a tilt angle is given by: ω i = ω = mg 4k f c θ 1 2 wen θ = and ω i = ω = mg 4k f c θ 2 2 wen φ = (3.14) 33

49 Proof: In overing wit roll angle and zero pitc angle, te acceleration along z axis is zero, z =. Terefore, using te tird row of Equation (3.1), we get: cos( θ 1 2 ) F i = mg (3.15) Based on Equation (3.3), and noting tat eac rotor s angular speed is te same (i.e., F 1 = F 2 = F 3 = F 4 ), te angular speed is given by: ω i = ω = mg 4k f c θ 1 2 (3.16) Similarly, considering overing wit pitc angle and zero roll angle, te Equation (3.12) gives: cos( θ 2 2 ) F i = mg (3.17) Now similar to above, te angular speeds of te rotors is given by: ω i = ω = mg 4k f c θ 2 2 (3.18) 34

50 Capter 4 Control System 4.1 Linear Controller Design In tis capter, te control strategy of te tilting rotor quadcopter is presented. Te aim of te control strategy is not only control te position of te veicle to follow an arbitrary trajectory in 3 dimensions, but also to ave control over te orientation of te veicle in overing as well as during trajectory tracking Proportional Derivative Control Te controller inputs are four independent speeds of propellers and teir rotations about te axes of quadcopter arms. Referring to Figure (3-2) and te two Teorems, it is assumed tat θ 1 = θ 3 and θ 2 = θ 4. It may be noted tat tese constraints, in fact, make te over-actuated system into fully actuated system (two inputs to tilt te rotors anoter four inputs for teir rotational speeds make te total number of independent control inputs to be six). For 6 DoF quadcopter, tis results into complete control over its position and orientation. Te dynamic model of te tilting rotor quadcopter, described in (3.8) and (3.9), is used to design te PD controllers for orientation adjustment and trajectory tracking. Figure (4-1) sowes te block diagram of te control algoritm for orientation and position control during te fligt. 35

51 Figure 4-1: Te block diagram of position and orientation control algoritm To start tracking a specific trajectory, first, overing from te initial starting point is necessary. Ten, te orientation of te veicle to a specific pitc or roll angle is obtained. In [46], te relationsip between te rotational speeds of te motors and te deviation of te orientations from nominal vectors for overing and navigation is described in detail for conventional quadcopter. Similar to tat approac for te tilting rotor quadcopter, te rotational speeds are observed as: ω des 1 ω des 2 ω des 3 ω des 4 = ω + ω f ω φ ω θ ω ψ (4.1) were ω des i, (i = 1, 2, 3, 4) are te desired angular velocities of te respective rotors. Te overing speed, ω, is calculated from Teorem II. Te proportional- derivative laws are used to control ω φ, ω θ, ω ψ and ω f wic are deviations tat result into forces/moments causing 36

52 roll, pitc, yaw, and a net force along te z B axis, respectively, wic are calculated as: ω φ = k p, φ(φ des φ) + k d, φ(p des p) ω θ = k p, θ(θ des θ) + k d, θ(q des q) ω ψ = k p, ψ(ψ des ψ) + k d, ψ(t des t) (4.2) were p, q and t are te component of angular velocities of te veicle in te body frame. Te relationsip between tese components and derivatives of te roll, pitc and yaw angles are provided below [34]. φ p θ = T q ψ t 1 tanθ.sinφ tanθ.cosφ T = cosφ sinφ secθ.sinφ secθ.cosφ Te relationsip between te tilt angles of individual rotors, given by θ des i, i = 1, 2..4, and te reference pitc and roll angles is given by : θ des 1 θ des 2 θ des 3 θ des 4 = φ des 2θ des φ 1 1 θ (4.3) were φ des and θ des are reference roll and pitc angles and φ and θ are orientation deviations. Figure (4-2) sows te orientation of te veicle wit respect to te tilted propellers. A proportional-derivative controller is used to control te orientation deviation using te reference 37

53 orientation values as: φ θ = k p, φ (φ des = k p, θ (θ des φ) + k d, φ ( φ des p) θ) + k d, θ ( θ des q) (4.4) Figure 4-2: Hovering wit tilted arms Te matematical model of DC servo motor is obtained by te following first order transfer function tat relates te motor angular velocity (rad/s) to input voltage (V) as: Ω(s) V(s) = K τs + 1 (4.5) were τ represents te time constant of te system, and K represents te steady state gain value. Te angular position of te servo motor can be obtained by integrating te motor angular velocity. Te transfer function relating te angular position (rad) and input voltage (V) can be obtained as: Θ(s) V(s) = K s(τs + 1) = K τs 2 + s (4.6) Te above equation represents a second order transfer function. So, tis system is identical to a second order actuation system. Suc systems exibit a transient response wen tey are subjected to external inputs or environmental disturbances. It sould be noted tat, te transient response caracteristics are one of te most important factors in system design. In general, 38

54 transfer function of a 2nd order system wit input, u(t) and output, y(t) can be expressed as: y(s) u(s) = kω 2 s 2 + 2ζω + ω 2 (4.7) Te TGY-21DMH servo motor used in tilt rotor quadrotor system is similar to an actuator system wit damping ratio (ζ) =.7, and it as an angular speed of 8 rad/s wen operated at 6V and 6.98 rad/s wen operating at 4.8V. Te natural frequency for matematical model is considered to be as 16 rad/s by considering a factor of safety equal to 2, te DC gain as been considered as unity. In order to ave te quad-rotor track a desired trajectory r i,t, te command acceleration, r des i is calculated from proportional-derivative controller based on position error, as [46]: ( r i,t r des i ) + k d,i (ṙ i,t ṙ i ) + k p,i (r i,t r i ) = (4.8) were r i and r i,t (i = 1, 2, 3) are te 3-dimensional position of te quad-rotor and desired trajectory respectively. It may be noted tat ṙ i,t = r i,t = for over. During te fligt of a tilting quadcopter, te orientation of te veicle needs to be set at specific pitc or roll. Tis can be obtained by linearizing te equation of motion tat correspond to te nominal over states. Te nominal over state (φ = φ des = θ 1 /2, θ = θ des = θ 2 /2, ψ = ψ T, θ = ψ = φ = ) corresponds to equilibrium overing configuration wit te reference pitc or roll angles. Te cange of te pitc or roll angles are supposed to be small during fligt. By linearizing Equation (3.8) about tese nominal overing states, desired pitc and roll angles to cause te motion can be derived as given by te following equations : r des 1 = 2g(A θ des + B φ des + C) r des 2 = 2g(D θ des + E φ des + F) r des 2 = 8k f ω mg ω F (4.9) 39

55 were A = s(2φ des + c(φ des B = s(2θ des )c(ψ T)s(θ des ) + s(2θdes)c(ψ T)s(φ des )c(ψ T)cθ des cφdes + c(2θ des )cθdes )c(ψ T)c(θ des )c(φdes )c(ψ T)s(θ des )s(φdes) + s(2θdes)s(ψ T)s(φ des ) ) c(2φ des c(2θ des )c(ψ T)s(θ des )s(φdes ) )c(ψ T)s(θ des )s(2φdes) + c(2φdes)s(ψ T)c(φ des ) + c(2θ des )s(ψ T)c(φ des ) C = s(2φ des )c(ψ T)c(θ des ) + s(2θdes)c(ψ T)s(θ des )s(φdes ) s(2θ des )s(ψ T)c(φ des ) + c(2φdes)c(ψ T)s(θ des )c(φdes ) + c(2θ des )c(ψ T)s(θ des )s(φdes) + c(2φdes)s(ψ T)s(φ des + c(2θ des )s(ψ T)s(φ des ) D = (2φ des + c(2θ des + c(2φ des E = s(2θ des + (2φ des + (2θ des )c(ψ T)s(θ des ) + s(2θdes )s(ψ T) )s(ψ T)c(θ des )s(φdes ) )s(ψ T)c(θ des )s(φdes ) )s(ψ T)s(θ des )c(φdes ) (2φdes )s(ψ T) )s(ψ T)s(θ des )c(φdes ) )s(ψ T)s(θ des )c(φdes) c(2φdes)c(ψ T)c(φ des ) ) c(2θ des )c(ψ T)c(φ des ) F = s(φ des + s(θ des + c(2θ des )c(ψ T)c(θ des ) + s(2θdes)s(ψ T)s(θ des )s(φdes ) )c(ψ T)c(φ des ) + c(2φdes)s(ψ T)s(θ des )s(φdes ) )s(ψ T)s(θ des )s(φdes) c(φdes)c(ψ T)c(φ des ) c(θ des )c(ψ T)c(φ des ) G = c( θ 1 2 )c(θ 2 2 ) were φ des and θ des are respectively te desired deviation in roll and pitc angles from te nominal overing values (φ des and θ des respectively) tat are needed for position control wen 4

56 te orientation is set to be given by te nominal overing values. Equation (5.14) represents a pair of two coupled linear equations wic are to be solved to obtain te φ des and θ des. Te final desired pitc or roll angles are calculated by: φ des = φ des + φ des θ des = θ des + θ des (4.1) Te desired speeds of te individual rotors are calculated by Equation (5.12). Equation (5.12) is obtained after determination of Equations (5.13) to (4.1). 4.2 Nonlinear Control Te four rotational velocities of te rotors are te inputs of te veicle, but in order to simplify te equations of motion wic are described in (3.8) and (3.9), new artificial input variables are defined as te following. It may be noted tat we assume tat te tilting appens only along te roll direction. u 1 = (F 1 + F 2 + F 3 + F 4 )/m u 2 = l(f 3 F 1 )/I x u 3 = l(f 4 F 2 )/I y u 4 = k(f 1 cosθ 1 F 2 cosθ 2 + F 3 cosθ 3 F 4 cosθ 4 )/I z (4.11) were k is force/moment scaling factor. Te equations of motion of te veicle, considering small angle assumption and tilting 41

57 along only roll direction (ence, θ 1 = θ 3 and θ 2 = θ 4 = ), can be obtained as: ẍ = 1 2 sinθ 1cosθ + u 1 cosθ 1 cosφ cosφsinθ ÿ = u 1 cosθ 1 sinφ cosφ u 1 cosθ 1 cosφ sinφ z = mg 1 2 u 1sinθ 1 sinθ + u 1 cosθ 1 cosφ cosφcosθ u 1 cosθ 1 sinφ sinφcosθ φ = (cosθ 1 + ksinθ 1 )u 2 θ = u 3 ψ = ku 4 (4.12) were φ is te te desired roll angle tat te quadcopter is supposed to tilt during te fligt Feedback Linearization A brief review of nonlinear control using feedback linearization metod [86, 94, 8, 79] is presented ere. Among te two fundamental design tecniques for feedback linearization, i.e., Input-Output linearization and Input-State linearization, we utilize te Input-Output linearization tecnique in wic we differentiate te output of te systems as many as times as needed so tat te input of te system appears in te last derivative[32, 1]. Tis tecnique is a systematic way to linearize globally part of, or all, te dynamics of system [29]. Te following paragraps explains ow te new/syntetic input v is cosen in order to yield te following transfer function from te syntetic input to te output y [36]: Y(s) V(s) = 1 s γ were γ, te relative degree, is te last derivative of output so tat te pysical input appears in te equation. If tis order is less tan te system order (n), ten tere will be internal dynamics in te feedback linearized system. In cases wen internal dynamics appears, te stability of tese dynamics sould be also be considered. Here we consider a nonlinear system in te 42

58 following form: ẋ = f (x) + g(x)u y = (x) (4.13) were x( R n ) is te state vector, u( R m ) represents te control inputs, and y( R p ) stands for te outputs, f and g are smoot vector fields, and is a smoot scalar function. A smoot function is defined as an infinitely differentiable function. Te control design process is to find an integer γ and a state feedback control law, u = α(x) + β(x)v (4.14) were α and β are smoot functions. Tis control law exactly linearizes te map between te transformed input v and te output y and yields a linear system (linear from te syntetic input v to te output y). Te above idea can be implemented by successively differentiating te output as: y (γ) = L γ f (x) + L gl γ 1 f (x)u (4.15) were y (γ) represents te γ t derivative of y, L k f (x) is called te Lie derivative of Lk 1 f (x) wit respect to field f. Here, if L g L γ 1 f (x) is bounded away from zero for all x, te control law is given by u = 1 L g L γ 1 f ( Lγ f + v) (4.16) Te functions α(x) and β(x) in (4.14) can be obtained as: α(x) = β(x) = L γ f (x) L g L γ 1 f (x) 1 L g L γ 1 f (x) (4.17) 43

59 In order to facilitate te design block, assuming F(x) = L γ f (x) G(x) 1 = 1 L g L γ 1 f (x) (4.18) Eq. (4.16) can ten be written as: u = G 1 (x)(v F(x)) (4.19) We see tat te above inversion-based control law as te capability to sape te output response by simply designing te new control v to get te closed-loop linear system wic finally yields te desired output. y (γ) = v (4.2) Once linearization as been acieved, any furter control objective suc as pole placement may be easily met using te linear controls teory. For te nonlinear quadcopter system under consideration in tis section, in order to make te system feedback linearizable, one may consider coosing x, y and z as te output variables. It can easily be seen tat u 2 and u 3 in (4.11), wic are te control inputs representing te pitc and roll angular accelerations of te veicle, do not appear in te equation of te outputs. By successively differentiating of equations of motion till te input terms appear, we can generate te new control input of te system. It can be seen te new input terms appear in te fourt derivatives of te outputs as obtained from (4.12): ] x (4) = f (x) x + [g(x) x1 g(x) x2 g(x) x3 u ] y (4) = f (x) y + [g(x) y1 g(x) y2 g(x) y3 u ] z (4) = f (x) z + [g(x) z1 g(x) z2 g(x) z3 u (4.21) 44

60 were f (x) x = u 1 θsinθ 1 sinθ + u 1 θcosθ 1 cosφ cosφcosθ u 1 φcosθ 1 cosφ sinφsinθ u 1 φ θcosθ 1 cosφ sinφcosθ) g(x) x1 = 1 2 sinθ 1cosθ + cosθ 1 cosφ cosφsinθ g(x) x2 = 1 2 u 1sinθ 1 cosθ u 1 cosθ 1 cosφ cosφsinθ g(x) x3 = u 1 cosθ 1 cosφ cosφsinθ f (x) y = 2u 1 φcosθ 1 sinφ sinφ 2u 1 φcosθ 1 cosφ cosφ g(x) y1 = cosθ 1 sinφ cosφ cosθ 1 sinφ sinφ g(x) y2 = g(x) y3 = u 1 cosθ 1 sinφ cosφ + u 1 cosθ 1 sinφ sinφ f (x) z = 2 u 1 θsinθ 1 cosθ 2u 1 θcosθ 1 cosφ cosφsicθ 2u 1 φcosθ 1 cosφ sinφcosθ + 2u 1 θ φcosθ 1 cosφ sinφsinθ + 2u 1 θcosθ 1 sinφ sinφsinθ 2u 1 φcosθ 1 sinφ cosφcosθ + 2u 1 θ φcosθ 1 sinφ cosφsinθ g(x) z1 = 1 2 sinθ 1sinθ + cosθ 1 cosφ cosφcosθ + cosθ 1 sinφ sinφcosθ g(x) z2 = 1 2 u 1sinθ 1 sinθ u 1 cosθ 1 cosφ cosφcosθ + u 1 cosθ 1 sinφ sinφcosθ g(x) z3 = u 1 cosθ 1 cosφ cosφcosθ + u 1 cosθ 1 sinφ sinφcosθ [ ] T u = u 1 u 2 u 3 45

61 were u is te control inputs wic control te x, y and z. We coose u as: 1 g(x) x1 g(x) x2 g(x) x2 u = g(x) y1 g(x) y2 g(x) y3 g(x) z1 g(x) z2 g(x) z3 f (x) x + v 1. f (x) y + v 2 f (x) z + v 3 (4.22) Te output equation is now given by: x (4) v 1 y (4) z (4) = v 2 v 3 (4.23) We set pseudo inputs terms as: v 1 v 2 v 3 = x (4) d k x1 e (3) x k x2 e (2) x k x3 e x k x4 e x y (4) d k y1 e (3) y k y2 e (2) y k y3 e y k y4 e y z (4) d k z1 e (3) z k z2 e (2) z k z3 ė z k z4 e x (4.24) were e x, e y and e z are errors defined as:e x = x x d, e y = y y d and e z = z z d, [k x1,..., k x4 ], [k y1,..., k y4 ] and [k z1,..., k z4 ] are gains, x d, y d and z d are desired outputs. From (4.24), te error dynamics are given by: e (4) x k x1 e (3) x k x2 e (2) x k x3 e x k x4 e x = e (4) y k y1 e (3) y k y2 e (2) y k y3 e y k y4 e y = e (4) z k z1 e (3) z k z2 e (2) z k z3 ė z k z4 e x = (4.25) A PD controller is also design to control te yaw motion, and is given by: u 4 = ψ d = k ψ1 ( ψ d ψ) + k ψ2 (ψ d ψ) (4.26) were k ψ1 and k ψ2 are derivative and proportional gains respectively. 46

62 4.2.2 PD Based Tilting Angle Controller Tilting rotor quadcopter is designed by using additional four servo motors attaced to te end of eac arm tat allow te rotors to tilt. Tis capability turns te veicle into an overactuated system tat potentially can track an arbitrary trajectory over time. During te fligt, as te non-linear control based on te proposed feedback linearization metod provides te amount of pitc and roll required to track an arbitrary trajectory, a PD controller is also designed to allow te veicle to eiter fly or over wit desired orientation or tilting. Te relationsip between te tilt angles of te individual rotors, given by θ des i, i = 1, 2..4, and te reference pitc and roll angles is given by [51] : θ φ θ 2 θ 3 θ 4 = 1 1 2θ 1 1 φ 1 1 θ (4.27) were φ and θ are reference roll and pitc angles and φ and θ are orientation deviations. A proportional-derivative controller is used to control te orientation deviation using te reference orientation values as: φ = k p, φ (φ φ) k d, φ p θ = k p, θ (θ θ) k d, θ q (4.28) were p, q (and t in te Equation below) are te components of angular velocities of te veicle in te body frame. Te relationsip between tese components and derivatives of te roll, pitc and yaw angles are provided in [74]. It may be noted tat tis PD controller is designed to desirably control bot pitc and roll angles. However, in tis section, for design of feedback linearization metod, we assumed te quadcopter to be tilted just in roll direction. Hence, for te simulation studies carried out at capter 7, we set te reference pitc angle to be zero for tis controller. 47

63 4.3 Stability Analysis Te inerently unstable tilting quadcopter dynamics described in (3.8) and (3.9) can be written in state-space form: Ẋ(t) = f (X(t), U(t)) were U(t) and X(t) are input and state vectors. x 1 = x x 2 = ẋ 1 = ẋ x 3 = y x 4 = ẋ 3 = ẏ x 5 = z x 6 = ẋ 5 = ż x 7 = φ x 8 = ẋ 7 = φ x 9 = θ x 1 = ẋ 9 = θ x 11 = ψ x 12 = ẋ 11 = ψ X = [ x ẋ y ẏ z ż φ φ θ θ ψ ψ ] T U = [F 1 F 2 F 3 F 4 θ 1 θ 2 ] T Te state space model Ẋ(t) = f (X(t), U(t)) is not only non-linear but also igly complicated due to tigt coupling between different terms. In order to reduce te number of complicated derivative terms involved in furter dynamics, te small angle assumption as been applied to differentiation described in (3.8) and (3.9). We ave linearzed te system about an operating overing point wile tilting along pitc or roll direction. Te operating overing point X e is acieved wit te input (U e ) suc tat f(x e (t),u e (t))=. Te linearized system is given by: 48

64 Ẋ(t) = AX(t) + BU(t) Te linearization is carried out via calculating te Jacobian matrices as: A = f X and B = f U calculated at operating point: X e,u e. Tis yields: 1 K 1 K 2 1 L 1 1 A = M 1 M , 49

65 B = A 1 A 2 A 3 A 4 A 5 A 6 B 1 B 2 B 3 B 4 B 5 B 6 C 1 C 2 C 3 C 4 C 5 C 6 D 1 D 3 D 5 E 2 E 4 E 6 F 1 F 2 F 3 F 4 F 5 F 6 (4.29) were: K 1 = 1 m ( F 4sinθ 4 sinθcosφ + F 2 sinθ 2 sinθcosφ F 1 cosθ 1 sinθsinφ F 2 cosθ 2 sinθsinφ F 3 cosθ 2 sinθsinφ F 4 cosθ 2 sinθsinφ) K 2 = 1 m ( F 1sinθ 1 sinθ F 3 sinθ 1 sinθ + F 4 sinθ2cosθsinφ + F 2 sinθ 2 cosθsinφ + F 1 cosθ 1 cosθcosφ + F 2 cosθ 2 cosθcosφ + F 3 cosθ 1 cosθcosφ + F 4 cosθ 2 cosθcosφ) L 1 = 1 m ( F 4sinθ 2 sinφ F 2 sinθ 2 sinφ F 1 cosθ 1 cosφ F 2 cosθ 2 cosφ F 3 cosθ 1 cosφ F 4 cosθ 2 cosφ) M 1 = F 4 sinθ 2 cosθcosφ + F 2 sinθ 2 cosθcosφ F 1 cosθ 1 cosθsinφ F 2 cosθ 2 cosθsinφ F 3 cosθ 1 cosθsinφ F 4 cosθ 2 cosθsinφ) M 2 = F 1 sinθ 1 cosθ F 3 sinθ 1 cosθcosφ F 4 sinθ 2 sinθcosφ F 2 sinθ 2 sinθsinφ F 1 cosθ 1 sinθcosφ F 2 cosθ 2 sinθcosφ) F 3 cosθ 1 sinθcosφ F 4 cosθ 2 sinθcosφ 5

66 A 1 = sinθ 1 + cosθ 1 sinθ, A 2 = cosθ 2 sinθ, A 3 = sinθ 1 + cosθ 1 sinθ A 4 = cosθ 2 sinθ, A 5 = F 1 cosθ 1 + F 3 cosθ 1 F 1 sinθ 1 sonθ + F 3 sinθ 1 sinθ A 6 = F 2 sinθ 2 sinθ F 4 sinθ 2 sinθ B 1 = cosθ 1 sinφ, B 2 = sinθ 2 cosθ 2 sinφ, B 3 = cosθ 1 sinφ B 4 = sinθ 2 cosθ 2 sinφ, B 5 = F 1 sinθ 1 sinφ + F 3 sinθ 1 sinφ B 6 = F 4 cosθ 2 + F 2 cosθ 2 + F 2 sinθ 2 sinφ + F 4 sinθ 2 sinφ C 1 = sinθ 1 sinθ + cosθ 1, C 2 = sinθ 2 sinφ + cosθ 2 C 3 = sinθ 1 sinθ + cosθ 1, C 4 = sinθ 2 sinφ + cosθ 2 C 5 = F 1 cosθ 1 sinθ F 3 cosθ 1 sinθ F 1 sinθ 1 F 3 sinθ 1 C 6 = F 4 cosθ 2 sinφ + F 2 cosθ 2 sinφ F 2 sinθ 2 F 4 sinθ 2 D 1 = lcosθ 1 + ksinθ 1, D 3 = lcosθ 1 ksinθ 1 D 5 = lf 3 sinθ 1 + lf 1 sinθ 1 + kf 1 cosθ 1 kf 3 cosθ 1 E 2 = lcosθ 2 ksinθ 2, E 4 = lcosθ 2 + ksinθ 2 E 6 = lf 4 sinθ 2 + lf 2 sinθ 2 + kf 4 cosθ 2 kf 2 cosθ 2 F 1 = lsinθ 1 + kcosθ 1, F 2 = lsinθ 2 kcosθ 2, F 3 = lsinθ 1 + kcosθ 1 F 4 = lsinθ 2 kcosθ 2, F 5 = lf 1 cosθ 1 + lf 3 cosθ 1 kf 1 sinθ 1 kf 3 sinθ 1 F 6 = lf 2 cosθ 2 + lf 4 cosθ 2 kf 2 sinθ 2 kf 4 sinθ 2 Once te system is linearzied, te controllability of te system in te vicinity of equilibrium points can be analyzed using tools of linear system teory. We analyzed te system s contollability for ranges of values on tiltilng angles along pitc or roll directions. As an example, we provide results for one particular overing point wen F 1 = F 2 = F 3 = F 4 = mg, 4 φdes =, and θ des = 2. 51

67 A = , 1 1 C =

68 B = Te matrices A and B were used to determine te controllability of te system and was found to be controllable. Tis means tat a feedback control law U(t) = K f d X(t) can be designed to stabilize and control te system via pole placement metod were K f d is a 6 12 feedback control gain matrix. 53

69 Capter 5 Fault Tolerant Fligt 5.1 Introduction In tis capter, stability and control of tilting-rotor quadcopter is presented upon failure of one propeller during fligt. Te tilting rotor quadcopter provides advantage in terms of additional stable configurations. On failure of one propeller, te quadcopter as a tendency of spinning about te primary axis fixed to te veicle as an outcome of te asymmetry about te yaw axis. Te tilting-rotor configuration is an over-actuated form of a traditional quadcopter and it is capable of andling a propeller failure, tus making it a fault tolerant system. In tis capter, a dynamic model of tilting-rotor quadcopter wit one propeller failure is derived and a controller is designed to acieve overing and navigation capability. Te simulation results of translational and overing motion are presented. Multicopters wit six or more propellers are also popular as te veicle is able to maintain normal fligt if one of te propellers fails [48]. But multicopters are costly as compared to te quadcopters wile applications are te same. VTOL UAVs are finding more and more applications in civilian domain and tis canging scenario demands new rules and regulations in future[63, 89]. System failures are inevitable during fligt of UAVs. Propeller or motor failure is one of te most common failure in case of quadcopters[6]. Currently, te commercial solution available to deal wit propeller failure is emergency paracute wic assists in emergency landing of quadcopters [75]. Te operational scenario of quadcopters requires te design of controllers capable of fault detection, isolation, and diagnosis [3]. Once te failure occurs, te system must be capable of 54

70 maintaining te stability of te system and complete te mission witout muc compromise in system performance. In passive fault tolerant control system (PFTCS) te control algoritm is designed to acieve a given objective in ealty or faulty situation witout canging its control law [93], wereas In active fault tolerant control system (AFTCS), to preserve te ability of system to acieve te objective, te control law is canged according to fault situation [8, 42]. Te fault diagnosis and identification (FDI) block, also termed as diagnosis unit, consists residual generator and residual evaluation sub-units. A residual is generated by comparing te process output and te model output, if te residual differs from zero. Te residual evaluation compares it to a tresold to decide and indicate fault. Based on te diagnosis result te reconfiguration block as to adapt te controller in suc a way tat te new controller is able to cope wit te faulty process. Fault Detection and Isolation (FDI) system for actuator faults for an exacopter veicle as been presented in [2]. A diagnostic Tau observer is applied to te exacopter nonlinear model to generate residual signals. In te fault-free case, residuals are close to zero, wile in case of a faulty actuator, te value of residuals and fault is detected. Furter, Fault isolation is realized by exploiting te matematical model of te exacopter. By quickly detecting te fault, te control law can be modified to satisfy te closed-loop requirements of te system and tus making it an active fault tolerant control. In [48] te control strategy is presented using periodic solutions for a quadcopter experiencing one, two opposite, or tree complete rotor failures. Te strategy employed is to define an axis, fixed wit respect to te veicle body, and ave te veicle rotate freely about tis axis. By tilting tis axis, and varying te total amount of trust produced, te veicles position can be controlled. Emergency landing procedure of quadcopter as been presented in [38, 41, 24] by using PID and Backstepping control approac respectively. Te strategy is to switc off te propeller aligned on te same quadcopter axis of te failed propeller. Tis action converts te quadcopter configuration into a birotor aerial veicle. Te UAV becomes free to spin in yaw axis wile controlling te remaining attitudes of te UAV and ten emergency landing procedure is exercised. 55

71 Te tilting-rotor quadcopter is an over-actuated [51, 52, 82] form of a traditional quadcopter and it is capable of andling a propeller failure, tus making it a fault tolerant system. A robust, fault tolerant control law and redundant mecanical design of te quadcopter can ensure safe andling of te quadcopter even after te propeller failure. In tis capter, te tilt rotor mecanism and PD control of te quadcopter ave been used to stabilize te quadcopter after te propeller failure and tus control all states of te UAV. 5.2 Fault Detection Fault Detection and Isolation (FDI) system for motor failure is a very important aspect of fault tolerant control for quadcopters. Quadcopters belong to te class of very fragile aircraft and if a motor failure occurs, it leads to igly unstable system dynamics. On failure of one propeller, te quadcopter as a tendency of spinning about te primary axis fixed to te veicle as an outcome of te asymmetry about te yaw axis. Te second major asymmetry is created in te roll or pitc plane depending on te corresponding motor failure. If any one of motor 1 or 3 fails, te asymmetry will be in pitc and yaw plane wereas if motor 2 or 4 fails tere would be a roll and yaw asymmetry. A robust Fault Detection and Isolation (FDI) system can minimize te reaction time for control system reconfiguration and improve te efficiency of fault tolerant control significantly. As suc, tis mecanism plays a key role in FTC. FDI can be implemented wit a current sensor tat can be used to monitor te amount of current supplied to eac quadcopter motor. Te signal from tis sensor can be used to take identify te fault and take furter decision for control system reconfiguration. Tese current sensors fall in category of arduino energy monitors and are very easily available. 5.3 Dynamic Modeling Unlike traditional quadcopter models, wic ave only four rotary propellers as te veicle s inputs, in tilting rotor quadcopters, tere are four more servo motors attaced to te eac arm tat adds one degree of freedom to eac of te propellers, resulting in te tilting motions along teir axes. Te equation of motion of a tilting rotor quadcopter as been discussed in 56

72 previous sections. In tis section, te equations of motion of a tilting rotor quadcopter wit one propeller failure is presented Tilting Rotor Quadcopters wit One Propeller Failure Wen all te propellers of te tilt-rotor quadcopter are working ten it yields a stable configuration as a result of symmetry of forces and moments. Assuming tat one propeller/motor fails during overing fligt of quadcopter wic is located in te pitc plane. Ten, te quadcopter would possess tree working propellers and one failed propeller. Once te failure occurs, te UAV will experience asymmetry about te yaw axis because of M 1, M 3, M 4 moments of working propellers wile M 2 =. Anoter asymmetry would occur in pitc plane as F 2 = and F 4 would still ave some magnitude. Te equations of motion can be modified by putting F 2 and M 2 equal to zero. Once again by using rotational matrix in (3.1), equations of motion in world-frame can be written as: mẍ = F 1 sθ 1 cψcθ F 3 sθ 3 cψcθ F 4 sθ 4 cψsθsφ + F 4 sθ 4 sψcφ + F 1 cθ 1 cψsθcφ + F 3 cθ 3 cψsθcφ + F 4 cθ 4 cψsθcφ + F 1 cθ 1 sψsφ + F 3 cθ 3 sψsφ + F 4 cθ 4 sψsφ C 1 ẋ mÿ = F 1 sθ 1 sψcθ F 3 sθ 3 sψcθ F 4 sθ 4 sψsθsφ F 4 sθ 4 cψcφ + F 1 cθ 1 sψsθcφ + F 3 cθ 3 sψsθcφ + F 4 cθ 4 sψsθcφ F 1 cθ 1 cψsφ F 3 cθ 3 cψcφ F 4 cθ 4 cψsφ C 2 ẏ m z = F 1 sθ 1 sθ + F 3 sθ 3 sθ F 4 sθ 4 cθsφ + F 1 cθ 1 cθcφ + F 3 cθ 3 cθcφ + F 4 cθ 4 cθcφ mg C 3 ż (5.1) It sould be noted tat F 2 terms ave vanised from te equations wic will result in asym- 57

73 metry because of one propeller failure. Similarly, te angular accelerations are determined by Euler equations: I x φ = l(f 3 cθ 3 F 1 cθ 1 C 1 φ) + (M 1 sθ 1 M 3 sθ 3 ) + M 4 I y θ = l(f 4 cθ 4 C θ) 2 + M 4 sθ 4 + (M 1 + M 3 ) I z ψ = l(f 1 sθ 1 + F 3 sθ 3 + F 4 sθ 4 C ψ) 3 + (M 1 cθ 1 + M 3 cθ 3 M 4 cθ 4 ) (5.2) were M i, (i = 1, 2, 3, 4) are te same tilting moments wic are created by te four servo motors attaced to te end of eac arm to cause a tilt angle. Te absence of F 2, M 2 sould be noted in pitc and yaw acceleration equations. Te components of rotor moment M 1, M 3, M 4 would not produce a symmetrical outcome wic represent unstable dynamics of quadcopter upon propeller failure. Te available inputs to stabilize and control tis system are angular speed ω 1, ω 3, ω 4 of tree working rotors and tilt angle θ i, (i = 1, 2, 3, 4) of all rotors. Teorem-III: Considering te dynamics of tilt-rotor quadcopter upon propeller failure given by Equations (5.1) and (5.2), te quadcopter can be stabilized in yaw and pitc plane if fourt rotor is tilted by an angle θ 4 suc tat θ 4 = c 1 [ω 2 4 /(ω2 1 cθ 1 + ω 2 3 cθ 3)]. Figure 5-1: Free body diagram of tilt-rotor quadcopter upon propeller failure Proof: Wen propeller failure occurs te dynamics of te quadcopter are igly non-linear. 58

74 Tus, we ignore te drag forces and moments generated because of rotor tilt for simplification. Tis assumption simplifies te angular acceleration equations and te equations for pitc and yaw acceleration are given by: I y θ = lf 4 cθ 4 + M 4 sθ 4 I z ψ = lf 4 sθ 4 + M 1 cθ 1 + M 3 cθ 3 M 4 cθ 4 (5.3) If te quadcopter as to be stabilized in pitc and yaw plane, θ, ψ sould be zero. Tus, equation (5.3) can be equated to zero to solve for θ 4. lf 4 cθ 4 = M 4 sθ 4 lf 4 = M 4 (sθ 4 /cθ 4 ) lf 4 sθ 4 M 4 cθ 4 = M 1 cθ 1 M 3 cθ 3 (5.4) Substituting te value lf 4 in equation (5.4) and rearranging te equation: M 4 (sθ 4 sθ 4 /cθ 4 ) M 4 cθ 4 = M 1 cθ 1 M 3 cθ 3 M 4 (s 2 θ 4 + c 2 θ 4 )/(cθ 4 ) = M 1 cθ 1 M 3 cθ 3 (5.5) Since, s 2 θ 4 + c 2 θ 4 = 1 te above equation reduces to te following form: cθ 4 = M 4 /(M 1 cθ 1 + M 3 cθ 3 ) θ 4 = c 1 [M 4 /(M 1 cθ 1 + M 3 cθ 3 )] (5.6) Te above expression can be re-written in terms of angular velocity by using equation (3.3) : θ 4 = c 1 [ω 2 4 /(ω2 1 cθ 1 + ω 2 3 cθ 3)] (5.7) Tis condition sould old for attaining a stable configuration after one propeller failure in te tilt-rotor quadcopter. Oterwise, te system can not be stabilized or controlled. In fact, once 59

75 te system is stabilized minor deviation in angular speeds of propellers and rotor tilt angle can be utilized to maneuver te quadcopter Teorem-IV: Once propeller failure occurs, te quadcopter can old a certain altitude if te angular speed of remaining propellers is increased by: ω 1 = ω 1 + ω 2 /3 ω 3 = ω 3 + ω 2 /3 ω 4 = ω 4 + ω 2 /3 (5.8) wic means: ω new = ω + ω 2 /3 (5.9) Proof: ω 2 represents te angular speed of te second propeller at te instant of failure, tis will result in te loss of altitude but te angular speed of tree remaining propellers can be increased by a factor of ω 2 /3 in order to compensate for te loss. On te oter and, an extra compensation component ω 4 /cθ 4 sould come in te equation of fourt rotor to overcome te tilt effect of te rotor. Tus, angular speed of fourt rotor will be iger as compared to angular speed of first and tird rotor. ω 1, ω 3, ω 4 are te increased angular speeds of te propellers in order to old te altitude. Future, te new angular speeds must satisfy equation (5.7) in order to yield a stable configuration. We can conclude tis statement in te form of equation (5.8): θ 4 = c 1 [ω 2 4 /(ω 2 1 cθ 1 + ω 2 3 cθ 3)] (5.1) 6

76 5.4 Controller Design In tis section, te control strategy of te tilting rotor quadcopter in a case of motor failure during te fligt is presented. Two PD controllers are used due to compensate te unbalance moments created by an odd number of propellers, and also to stabilize veicle s orientation and make it functional to continue its mission witout cras. Te veicle originally as eigt independent inputs wic includes four speed of propellers and four tilted angle of eac motor about its axis. In te case of motor failure, two inputs are automatically out of equations. To make te veicle compensate te moments of te veicle, not only te speed of te remaining propellers needs to be controlled individually, but also te tilted mecanism needs to be set in a way tat compensates te moment from te breakdown motor. In tis work, its assumed tat motor two is te one tat stopped working during te fligt. It sould be noted tat te measurement sensor needs to report te failure immediately. Referring to te teorems, te tilting angle of motor 1 and motor 3 needs to be set at te same orientation and te tilted angle of te remaining motors can be immediately set according to Teorem III. To start compensating te unbalanced moment situation after te failure, first, getting back to overing is necessary. Ten, te orientation of te veicle to a specific pitc or roll angle is obtained. In [46], te relationsip between te rotational speeds of te motors and te deviation of te orientations from nominal vectors for overing and navigation is described in detail for conventional quadcopter. Based on te new dynamics equation of te veicle wit tree propellers and one tilted servo motor (motors 1 and 3 are assumed to be level), te following equations are obtained: M B x Lk f ω 1 Lk f ω 3 ω 1 M B y = Lk f ω 4 ω 3 M B z 1 ω 4 (5.11) θ 4 were M B x, M B y and M B z are torque components separated in body frame. It needs to be mentioned tat tese equations are obtained from linearization of equation (5.2) around its nominal 61

77 over states wile first and tird servo motors are assumed not to be tilted. Te rotational speed on eac individual motor and te tilted angle of te rear motor are calculated as: ω des 1 ω des 3 ω des 4 θ des 4 = ω new + ω f ω φ ω θ θ 4 + θ 4 (5.12) were ω des i, (i = 1, 3, 4) are te desired angular velocities of te respective rotors. θ 4 is te tilted angle tat needs to be old for motor 4 to attaining a stable configuration and is calculated in Teorem III. Te overing speed, ω new, is calculated from Teorem IV. Te proportionalderivative laws are used to control ω φ, ω θ, θ 4 and ω f wic are deviations tat result into forces/moments causing roll, pitc, yaw, and a net force along te z B axis, respectively, wic are calculated as: ω φ = k p, φ(φ des φ) + k d, φ(p des p) ω θ = k p, θ(θ des θ) + k d, θ(q des q) θ 4 = k p, ψ(ψ des ψ) + k d, ψ(t des t) (5.13) were p, q and t are te component of angular velocities of te veicle in te body frame. During te fligt of a tilting quadcopter, te orientation of te veicle needs to be set level. Tis can be obtained by linearizing te equation of motion tat correspond to te nominal over states. Te nominal over state (φ = θ =, ψ = ψ T, θ = ψ = φ = ) corresponds to equilibrium overing configuration wit te reference pitc or roll angles. Te cange of te pitc or roll angles are supposed to be small during fligt. By linearizing Equation (3.8) about tese nominal overing states, desired pitc and roll angles to cause te motion can be derived as given by te following equations : r des 1 = g(cψθ des + sψφ des ) sψf 4 sθ 4 r des 2 = g(sψθ des cψφ des ) + cψf 4 sθ 4 (5.14) 62

78 were θ des and φ des are te desired pitc and roll to be added to te nominal over states to move te veicle to desired trajectory r i,t, te command acceleration, r des i proportional-derivative controller based on position error, as [46]: is calculated from ( r i,t r des i ) + k d,i (ṙ i,t ṙ i ) + k p,i (r i,t r i ) = (5.15) were r i and r i,t (i = 1, 2, 3) are te 3-dimensional position of te quad-rotor and desired trajectory respectively. It may be noted tat ṙ i,t = r i,t = for over. 5.5 Linearization and Stability Analysis Te dynamic model of te veicle wit one motor failure can be described wit te differential equations (5.1) and (5.2). Tis inerently unstable tilting quadcopter dynamics can be written in state-space form: Ẋ(t) = f (X(t), U(t)) were U(t) and X(t) are input and state vectors [1]. x 1 = x x 2 = ẋ 1 = ẋ x 3 = y x 4 = ẋ 3 = ẏ x 5 = z x 6 = ẋ 5 = ż x 7 = φ x 8 = ẋ 7 = φ x 9 = θ x 1 = ẋ 9 = θ x 11 = ψ x 12 = ẋ 11 = ψ 63

79 X = [ x ẋ y ẏ z ż φ φ θ θ ψ ψ ] T U = [F 1 F 3 F 4 θ 4 ] T Since te quadcopter will not be used for acrobatic maneuvers after te motor failure, te igly nonlinear and also complicated dynamics, can be simplified wit te small angle assumption [87] to cover overing and moving around wit small deviation in orientation. In order to reduce te number of complicated derivative terms involved in furter dynamics, te small angle assumption as been applied to differentiation described in (5.1) and (5.2). We ave linearized te system about an operating overing state (φ = θ =, ψ = ψ T, θ = ψ = φ = ). Tese operating overing point X e is acieved wit te input (U e ) suc tat f (X e (t), U e (t)) = (5.16) Te linearization of te dynamics will result in A and B matrices, Ẋ(t) = AX(t) + BU(t) were: a i, j = f i(x, U ) X j b i, j = f i(x, U ) U j Te linearization is carried out via calculating te Jacobian matrices [4] yields: 64

80 1 A 2,7 A 2,9 1 A 4,7 1 A = A 6,7 A 6, , A 2,1 A 2,2 A 2,3 A 2,4 A 4,1 A 4,2 A 4,3 A 4,4 B = A 6,1 A 6,2 A 6,3 A 6,4 A 8,1 A 8,2 A 1,3 A 1,4 A 12,1 A 12,2 A 12,3 A 12,4 (5.17) 65

81 were: A 2,7 = 1 m ( F 4sinθ 4 sinθcosφ F 1 cosθ 1 sinθsinφ F 3 cosθ 3 sinθsinφ F 4 cosθ 4 sinθsinφ) A 2,9 = 1 m ( F 1sinθ 1 sinθ + F 3 sinθ 3 sinθ F 4 sinθ4cosθsinφ + F 1 cosθ 1 cosθcosφ + F 3 cosθ 1 cosθcosφ F 4 cosθ 4 cosθcosφ) A 4,7 = 1 m (F 4sinθ 4 sinφ F 1 cosθ 1 cosφ + F 3 cosθ 3 cosφ F 4 cosθ 4 cosφ) A 6,7 = F 4 sinθ 4 cosθcosφ F 1 cosθ 1 cosθsinφ F 3 cosθ 3 cosθsinφ F 4 cosθ 4 cosθsinφ) A 6,9 = F 1 sinθ 1 cosθ + F 3 sinθ 3 cosθcosφ + F 4 sinθ 4 sinθcosφ F 1 cosθ 1 sinθcosφ F 3 cosθ 1 sinθcosφ F 4 cosθ 4 sinθcosφ A 2,1 = sinθ 1 + cosθ 1 sinθcosφ, A 2,2 = sinθ 3 + cosθ 3 sinθcosφ A 2,3 = cosθ 4 sinθcosφ, A 2,4 = F 4 sinθ 4 sinθcosφ, A 4,1 = cosθ 1 sinφ, A 4,2 = cosθ 3 cosφ, A 4,3 = sinθ 4 cosφ cosθ 4 sinφ A 4,4 = F 4 cosθ 4 cosφ + F 4 sinθ 4 sinφ A 6,1 = sinθ 1 sinθ + cosθ 1 cosθcosφ, A 6,2 = sinθ 3 sinθ + cosθ 3 cosθcosφ A 6,3 = sinθ 4 cosθsinφ + cosθ 4 cosθcosφ, A 6,4 = F 4 cosθ 4 cosθsinφ F 4 sinθ 4 cosθcosφ A 8,1 = lcosθ 1 + ksinθ 1, A 8,2 = lcosθ 1 ksinθ 1 A 1,3 = lcosθ 4 + ksinθ 4 A 1,4 = lf 4 sinθ 4 + kf 4 cosθ 4 A 12,1 = lsinθ 1 + kcosθ 1 A 12,2 = lsinθ 3 + kcosθ 3 A 12,3 = lsinθ 4 kcosθ 4 A 12,4 = lf 4 cosθ 4 + lf 4 sinθ 4 66

82 Once te system is linearized, te controllability of te system in te vicinity of equilibrium points can be analyzed using tools of linear system teory [92] A = , 1 1 C = 1 67

83 B = A feedback control law [58] U(t) = K f d X(t) can be designed to stabilize and control te system via pole placement metod were K f d is a 4 12 feedback control gain matrix as below: T K f d =

84 Figures (5-2) and (5-3) sow ow position and orientation of quadcopter wit one failed motor remaining stable X Y Z meter time(sec) Figure 5-2: Veicle s Position φ θ ψ.2 rad time(sec) Figure 5-3: Veicle s Orientation 69

85 Capter 6 Hardware design 6.1 Description of te Prototype Te development of a proposed control tecniques for tilting quadcopter requires te development of an adequate platform for te preliminary experiments. Te primary consideration for te prototype design was to make te quadcopter small and ligtweigt so tat it was able to carry an extra component required for te tilting mecanism during te fligts. Te initial configuration and concept of te tilting quadcoter are presented in Figure (6-1). As te fabricating process was a senior design project, te cost of te veicle was kept as low as possible. Te structural design of te tilting quadcopter can be divided into two parts: te central body were all avionics and components are placed, and te tilting mecanism. Figure 6-1: Te CAD model of te tilting quadcopter 7

86 6.1.1 Central body As te most important mecanical components of te tilting mecanism are located at te center area, tis part not only needs to ave enoug space for all components, but also needs to maintain its symmetry requirement. Te central body or core area needs to ave space for essential component suc as: Four Electronic Speed Controls (ESC) to control eac motor Communication Hardware A power distribution board Autopilot Bearings, and Servo motors. Tere are two prototypes wic ave been designed and fabricated. Te core part in te first prototype is made using two seets of flame-retardant Garolite wic is connected by plastic bolts and spacer. Te flame-retardant Garolite offers excellent strengt, low water absorption and good electrical insulating quality in bot umid and dry conditions wit maximum temperature of 265 F Te material in flame-retardant Garolite is fiberglass-clot wit a flame retardant resin. Te ardness also meets Rockwell M11-M115 wic is standard ardness test. Te reason to use two seets is to ave all bearings, servo motors and electronic parts in between. Four additional polycarbonate round tubes are also attaced to te frame in order to make four arms for te quadcopter. Polycarbonate round tube is a cost effective material wit outstanding mecanical, termal, cemical, pysical, and electrical properties. It is ligt in weigt, as excellent impact strengt, eat resistance to 25 F and most importantly it comes in a round sape wic provides te tilting mecanism wit more degree of freedom to rotate Tilting Mecanism Te most important features of te tilting quadcopter is te fact tat eac arm can rotate about its axis independently. In order to ave tis independency, four additional inputs 71

87 are needed to rotate eac arm to ensure te tilting mecanism. Te servos (Figure (6-2)) are mounted on top of te lower seet of te central core. Eac arm is directly linked to te servo motor using gears. Te gear is ooked around te tube in te middle by two bearings. All eigt bearings (two for eac arm) mount between upper and lower seets. Tis unique design allows te arms to rotate separately wit any desired angle witout mecanical constraint. Figure (6-3) sows ow servos, tubes, bearings and seets are connected. Figure 6-2: HS-587MH HV Digital Micro Servo. Figure 6-3: Te CAD model of components of te tilting mecanism. Table (6.1) and (6.2) summarize te specifications and te masses of te veicle s components of te first prototype. 72

88 Table 6.1: Specifications of te Prototype Dimensions 65cm 65cm witout propellers Gross weigt Motor Driver Battery Propeller Autopilot Motor Servo Communications 2.4kg ESC 3Amp Litium-polymer cells 4S APC propeller Pixawk Motor 85Kv AC HS-587MH Digital servo MAVLink, MAVROS and Wifi Figure 6-4: Te real model of te first model of tilting Quadcopter. After ten test fligts, due to weigt of te veicle, te motors were not strong enoug to andle te tests and te experimental data was not acceptable. To reduce te weigt and also make te tilt mecanism more agile, second prototype was designed in Autodesk Inventor and SolidWorks and fabricated by our group. For te second prototype, te 3D robotics framework was used as te body and four separate tilt mecanisms were designed and built by using a 73

89 3D printer. Te printed 3D mecanism was mounted at te very end of eac arm. Te parts were made in a way tat perfectly fit te arms to avoid any slinging. Te electronics, power, propellers, motors and te auto pilot are same as te first prototype. Table 6.2: Veicle s component mass details. Items Weigt (g) Quantity Subtotal (g) Motors Servos Mecanical Gears Prop adapters Plastic Bolts ESC Battery bearings Landing gear Center plates Controller Battery Holder Arms Wirings and Connectors Total 2291 Tis veicle also as eigt control inputs wic are used for rotating te four propellers and tilting mecanism for te arms. It weigts 1kg less tan te first prototype. Te diameter also does not exceed 55cm. To tilt te motors around te arms, a tilting mecanism as been located at te end of eac arms. Eac tilting mecanism consist of tree separate parts: i) te servo motor older ii) te plate older iii) and te motor plate wic is mounted on te base via a tilting mecanism. Tere are some similar mecanisms available in te market, but te main reason tat makes 74

90 our design more reliable, is te way te servo is connected to te tilting part. In available versions of te tilting mecanism, te servo is screwed to te mecanism wit te same screw wic olds te tilting parts togeter. Te problem comes wen te screw is tigtened enoug to attac te servo to te tilting part to avoid any sliding. Tis also puses te two separate mecanisms towards eac oter and makes it arder to tilt. Wile more force is needed to be able to make te pressed part to rotate, an extra force wic comes from te servo motor, makes te cog loose. Te looseness between te servo and te tilting part contributes to a delay wen te command is received te motor starts tilting. Tis delay adds te nonlinear parts to te control system wic is not easy to be taken into te equation. Te currently designed tilting mecanism as an extra screw wic is placed between te tilting mecanism and te servo cog. By aving tis additional screw, te servo cog can be fastened to te tilting mecanism as muc as needed, wile te two tilting mecanisms are attaced to eac oter wit separate screw. Wit tis design, not only tere is no friction in tilting mecanism, te servo would never get loose. Figure (6-5) sows te transparent CAD model of te tilting mecanism. Figure (6-6) sows ow te tilting mecanism is mounted at te end of quadcopter s arm and ow te servo, tilting mecanism and te motor are connected. In Figure (6-7), bot CAD and actual model are presented. Figure 6-5: Te CAD model of transparent tilting mecanism 75

91 Figure 6-6: Te CAD model of tilting mecanism mounted on te arm wit te servo and te motor Figure 6-7: Te CAD and te actual model of te prototype. As te servo is directly connected to te tilting mecanism wit no gears in between. Te smaller and ligter servo wic produces te lower torque compared to te previous design can also be used. Tis configuration not only eliminates external gears wic were used before, but also reduces te weigt of te servo by alf. One of te drawback of tis mecanism as compared to te previous one, is te freedom of tilting around te axis. In previous design tere was not any mecanical limitation for te tilting mecanism. Altoug te previous design could rotate around te arms up to 36 degrees, it never was our concern. However, tis 76

92 tilting mecanisms is limited to 6 degrees wic meets te criteria of te experiments. 6.2 Drive System Motors A multi-rotor flying veicle is more efficient wen it is ligter. One of te important criteria to coose a suitable motor is to use te ligtest possible motor wic can provide at least twice as muc trust to lift te veicle and as te best response for control system in difficult fligt conditions. Brus-less DC motors afford better efficiency and power density compared to brused DC motor [31]. Due to iger power density, controllability, minimum requirements for maintenance, compact size and ligt weigt [6] tey ave become increasingly popular and most commonly used motors in MAV tecnology. As a result of tese advantages, te 85 KV Brus-less DC (BLDC) motor was selected for te tilting quadcopter. It is also used by most of te 3D Robotics multi-rotors veicles. It delivers a very good level of te trust compared to even bigger size of counterpart 85 KV motors. Figure 6-8: 85 kv brusless DC motors Most brus-less DC motors ave tree terminals as sown in Figure (6-8). Wile tese tree terminals are connected to stator, te permanent magnets are placed on te rotor suc tat te poles are facing te stator. An Electronic Speed Control (ESC) is used to send te command to control te rotation of te motor. Te power output from ESC by using four cells LiPo and 1 47 Propeller are sown in Table (6.3). 77

93 Table 6.3: Power output from ESC, 4S LiPo, 1x47 Propeller. 25% 5% 7% 1% Amp (A) Wattage (W) Trust(gr) Batteries Litium Polymer (LiPo) batteries are currently te preferred power sources for most ligtweigt, ig-current, ig-capacity power storage [7]. Tey offer ig energy-storage/weigt ratio and ig discarge rate [88]. Tese batteries use normal litium ion cemistry wit te polymer separators to provide ig discarge rates. It comes wit different cell numbers. Eac cell provide 3.7 Volts wit internal resistance of approximately.3 Om [4]. 6.3 Avionics Control Board A Control Board needs to be selected tat would maintain list of requirements to be able to ave control over position and orientation. As tere is no control board for tilt mecanism in te market, in order to apply te proposed control strategy, it needs not only to be an open source, but also is required to be able to communicate wit an out source computing devices. Furtermore tree axis gyroscope and tree axis accelerometers are also need to be included. Anoter criteria wic was taken into te account to coose te control board was te reasonable price, te size and te weigt. All tese reasons led us to use PixHawk autopilot [44]. PixHawk is armed wit advanced 32 bit ARM processor, micro SD card for logging, Integrated backup systems and 14 PWM servo outputs wic are ideal for our veicle to be connected to additional tilting servo motors [45]. Anoter specification tat made us to use PixHawk was 78

94 te way it could communicate wit Robot Operating System (ROS) wic is te main coding environment to apply te control tecniques [66]. ROS is a set of software frameworks for robot software development. ROS also provides low-level device control and message passing between processes wic allows to sare data across multiple and commonly specialized processes [16] Communications MAVLink (Micro Air Veicle Link) is a very ligtweigt protocol for communicating wit small unmanned veicle. It is designed as a eader-only message marsalling library [27]. It is mostly used for communication between ground station and small unmanned veicles. It can be used to transmit all fligt data including altitude, attitude, GPS position, air speed, battery status, way points etc[84, 85, 83]. Te transmitted data as te packet structure. Te payload from te packets are called as a MAVLink message wic is identifiable by te ID for eac message. Te stream of bytes tat can be encoded by te ground station can be sent via Telemetry or USB serial. MAVROS is also te package in ROS environment tat provides communication driver wit MAVLink communication protocol for various autopilots including PixHawk. Not only all fligt data is available in ROS, but also te new commands wic are calculated troug control systems can be sent back to te autopilot in real time. 79

95 Capter 7 Numerical Simulations and Experimental Results 7.1 Numerical Simulations Simulation set-up To validate te presented dynamic model and te control metod, numerical simulations of te tilting rotor quadcopter were carried out using te MATLAB. Te discretized versions of te dynamic and te controller equations are solved by te Euler metod. Here, we provide te results from one of te simulation scenarios studied. In tis scenario, te veicle s initial position was (.1,.8, ). Te final position was set to (.8,.3, 1.5). In tis study te SI unit system is used. In tis scenario, te desired pitc or roll angles were modified during te fligt so tat bot of tese angles were simultaneously controlled. Figure (7-1) sows te reference pitc and roll angle. It can be seen tat for time t = sec to t = 1 sec, te reference pitc and roll angles are zero. At t = 1 sec, te reference pitc angle increases from and reaces te value of 18 at t = 4 sec and stays wit te same until te end of trajectory. At t = 4 sec, te reference roll angle is also increased from and reaces te value of 12 at t = 7 sec. 8

96 2 Pitc (degree) time(sec) 15 Roll (degree) time(sec) Figure 7-1: Te reference (commanded) pitc and roll angle Tilt-Rotor Quadcopter Simulation results Te procedure to accomplis te fligt simulation is to ave te veicle take-off from an initial point vertically till te desired eigt, and ten steer to te destination point wit te orizontal fligt. During fligt, te orientation of te veicle is supposed to cange according to reference inputs witout losing te eigt. Te quadcopter trajectory in te tree dimensional space from te initial point to te desired destination is sown in Figure (7-2) Z(m) Y(m).5.2 X(m) Figure 7-2: Te actual trajectory followed by te UAV in 3-dimensions 81

97 Figure (7-3) sows te actual cange in te pitc, roll, and yaw angles of te tilting rotor quadcopter during te fligt. It can be seen tat te cange in te yaw angle is close to zero wile te actual pitc and roll angles closely follow te reference values. θ(degree) φ(degree) ψ(degree) time(sec) time)sec) 4 x time(sec) Figure 7-3: Te actual orientation of te veicle in 3 directions 4 3 θ 1 (degree) 2 θ 2 (degree) time(sec) time(sec) 2 2 θ 3 (degree) 2 θ 4 (degree) time(sec) time(sec) Figure 7-4: Te angle of eac arm during simulation Figure (7-4) sows ow four servomotors modulate te angle of eac arm to follow te referenced orientation commands during te fligt. Figure (7-5) sows ow te speed of four motors canges during te fligt to track te trajectory and maintain te eigt of te veicle. Figure (7-6) sows te enlarged view of a portion of te Figure (7-6). It can be seen from Figure 82

98 (7-6) tat te angular velocities of rotors stabilize close to t = 1sec (till wen te reference tilts in pitc and roll are zero). At t = 1sec, te reference pitc angle is commanded to gradually increase to reac a value of 18 at t = 4sec. Corresponding to tis, te individual rotor speeds can be seen to increase from t = 1sec to t = 4sec. At t = 4sec, te reference roll angle is commanded to gradually increase to reac a value of 12 at t = 7sec. Consequently, tere is furter increase in te rotational speeds of te rotors. Tis increase in motor speed can be explained by Teorem 2. Comparing Equation (3.14) (for tilted configuration) to Equation (3.7) (for non-tilted configuration), te teory predicts te need of more rotor speed in tilted configuration so tat te vertical component of force still balances te weigt in te tilted configuration. Te simulation results sown in Figure (7-6) just confirms te teory ω 1 (rad/s) 1 5 ω 2 (rad/s) time(sec) time(sec) ω 3 (rad/s) 1 5 ω 4 (rad/s) time(sec) time(sec) Figure 7-5: Te speed of eac rotor 83

99 ω 1 (rad/s) ω 2 (rad/s) time(sec) time(sec) ω 3 (rad/s) ω 4 (rad/s) time(sec) time(sec) Figure 7-6: Te enlarged view of te figure Feedback Linearization Numerical Simulation In order to verify te performance of te proposed feedback linearization based control metod of te tilting-rotor quadcopter, numerous numerical simulations were carried out in MATLAB/Simulink environment. Here, we present te results from one of te simulations. Te simulation comprises of te UAV taking off from te initial position located at (5.,.,. ) and reacing te desired destination located (4.5, 1., 1.), via passing te waypoint located at (3., 4., 5.). Also, during te fligt, te orientation of te veicle is supposed to cange according to te reference input (in tis case, te desired roll angle) witout deviating from te desired trajectory. Te reference roll angle is set as follows. During te fligt, at t = 5 sec, te reference roll angle is commanded to gradually increase from o to reac a value of 12 o at t = 1 sec. Te quadcopter is ten supposed to move towards te destination wit te commanded roll angle (refer to te bottom plot of Figure (7-9)). Te quadcopter trajectory obtained from te numerical simulation in te tree-dimensional space from te initial point to te waypoint and ten to te final position is sown in Figure (7-7). 84

100 1 8 ( 4.5, 1., 1. ) z ( meter) ( 3., 4., 5. ) 6 4 ( 5.,.,. ) y ( meter) x ( meter) Figure 7-7: Quadcopter trajectory in tree-dimensional space Figure (7-8) sows te position and orientation of te veicle during te fligt. Figure (7-9) sows te comparison between te reference roll and te actual roll angle te te veicle. z (meter) time(sec) 1 y (meter) x (meter).1 θ (degree) time(sec) ψ (degree) time(sec) Figure 7-8: Position and orientation of te quadcopter: altitude vs. time (top left), x-position vs. y-position (top rigt), pitc vs. time (bottom left), and yaw vs. time (bottom rigt).. 85

101 15 φ (degree) time(sec) 15 φ (degree) time(sec) Figure 7-9: Te reference roll (bottom) and actual roll angle (top) during te fligt u 1 (m/sec 2 ) 1 5 u 2 (rad/sec 2 ) 1 5 u 3 (rad/sec 2 ) time(sec) time(sec) u 4 (rad/sec 2 ) time(sec) 15 x time(sec) Figure 7-1: Inputs generated by te proposed feedback linearization metod. As we cose x, y, and z as te outputs of te feedback linearization metod, and u 1, u 2, u 3, and u 4 as te normalized total lift forces and control inputs for roll, pitc, and yaw respectively, Figure (7-1) sows tat te zero dynamics of te controller are stable ( as te relative degree 86

102 is smaller tan te order of te system) Fault Tolerant Numerical Simulation To validate te presented dynamic model in te case of motor failure and te proposed control tecnique, numerical simulations of te tilting rotor quadcopter were carried out using te MATLAB. Te discretized versions of te dynamic and te controller equations are solved by te Euler metod. We ave completed two different scenarios in order to evaluate te veicle s response in te case of motor failure in following a trajectory as well as over fligt. In bot scenarios, te altitude maintained its desired value Hover Fligt Te first simulation is te overing task in one spot wit motor failure after stable over fligt. Tis scenario sows te performance of te controller and igligts te position control wit motor failure. In te first scenario, mission was started by taking off and overing in te fixed altitude and in one spot. Te initial take of point is located at (.2,.,.). Figure (7-11) sows te 3 dimensional pat of te fligt. As it can be seen, te veicle is flying in te neigborood of te desired spot. Te fligt as te error in te range of.4 meters to over around te spot after one of te motors failed. 87

103 15 position Z position Y position X.4.6 Figure 7-11: Te actual trajectory followed by te UAV in 3- dimensions In tis scenario te veicle flew for 2 seconds. Figure (7-12) sows te position of veicle in eac individual axis. At time t = 2 sec, wen motor number 2 stops working, te altitude of te quadcopter drops and after very sort time of adjusting, it maintained its altitude to te end of te fligt. Te veicle also as small amount of movement along X and Y axis. Figure (7-13) sows ow te oter motors increase te rotational speed in order to compensate te failed motor force to maintain te altitude. Altoug te rotational speed of te motors is increased, but te speed limitation of eac motor, will not let te veicle to maintain te exact previous altitude. 88

104 1 X(m) time(sec).5 Y(m) time(sec) 2 Z(m) time(sec) Figure 7-12: Veicle s trajectory in X,Y and Z ω 1 (rad/s) 1 5 ω 2 (rad/s) time(sec) 1 2 time(sec) ω 3 (rad/s) 1 5 ω 4 (rad/s) time(sec) 1 2 time(sec) Figure 7-13: Veicle s trajectory in X,Y and Z 89

105 θ(rad) φ(rad) ψ(rad) time(sec) time(sec) time(sec) Figure 7-14: Te actual orientation Figure (7-14) sows ow orientation of te veicle cange after te motor failure. As it can be seen from te Figure (7-14), te yaw angle canges rigt after te motor failure and remains in a constant angle. It can easily be notified tat te veicle is not spinning around Z axis after te motor failure Tracking a Trajectory Here, we provide te results from te second simulation we studied. In tis scenario, te veicle s initial position was (.,.,.). Te final position was set to (3., 1., 1.). In tis scenario, It can be seen tat for time t = sec to t = 1 sec, te quadcopter flew wit all propellers working and it reaced its designated altitude at 1 meters. at t = 1 sec, one te motors stopped working and te proposed control tecnique were applied immediately. During te simulation, it is assumed tat rigt after te motor failure, te control system can take over and apply te control tecnique for new faulty system. Te quadcopter trajectory in te tree dimensional space from te initial point to te desired destination is sown in Figure (7-15). 9

106 15 1 position Z position Y position X 3 4 Figure 7-15: Te actual trajectory followed by te UAV in 3- dimensions Figure (7-16) sows te position of veicle in eac individual axis. At time t 1 sec, wen motor number 2 stops working, te altitude of te quadcopter drops by 3 meters and after very sort time of adjusting, it maintained its altitude to te end of te fligt. 4 X(m) time(sec) 2 Y(m) time(sec) 2 Z(m) time(sec) Figure 7-16: Veicle s trajectory in X,Y and Z 91

107 Figure (7-17) sows ow te oter motors beaved after motor 2 failure. As it can be seen from te Figure (7-17), to compensate te lost force from motor 2, all working motors speed up. Altoug te rotational speed of te motors is increased, but still te altitude is decreased to 7 meters due to limitation of te amount of rotational speed eac motor can provide. ω 1 (rad/s) 1 5 ω 2 (rad/s) time(sec) time(sec) ω 3 (rad/s) 1 5 ω 4 (rad/s) time(sec) time(sec) Figure 7-17: Veicle s trajectory in X,Y and Z Once te quadcopter started to adjust te stability after te motor failure, te unbalance moments caused most amount of cange in yaw angle. As it can be seen, te yaw angle started to increase rigt after te motor failure. 92

108 φ(rad) θ(rad) ψ(rad) time(sec) time(sec) time(sec) Figure 7-18: Te actual orientation Figure (7-18) sows te orientation of te veicle before and after motor failure. 1.6 θ 4 (rad) time(sec) t(rad/s) time(sec) Figure 7-19: Angular velocity around Z axis. 93

109 7.2 Preliminary Experimental Results Te tilting rotor quadcopter was tested to find out if it was correctly modeled and controlled. Extensive simulations were carried out to verify te validity of te dynamic model and control sceme. Tis section presents te experimental results obtained wit te second prototype. Te first experiment is te overing task in one spot wit tilted orientation. Tis scenario sows te performance of te controller and igligts te position control wit tilted angle. Te second experiment is intended to demonstrate te performance in tracking a simple trajectory wile te orientation keeps te desired value during te fligt. In bot scenarios, te altitude maintained its desired value Hovering on te spot In te first experiment, te scenario was started by taking off and overing in te fixed altitude and in one spot wile te quadcoter tilts by a desired angle. Figure (7-2) sows te snapsot of te overing wit te tilted angle. Figure 7-2: Snapsot of te overing wit te tilted angle Te tilting starts at t = 8s and continues till it reaces te desired angle at t = 12s. Te veicle overs at te same spot wit.4 rad tilt along te pitc direction till t = 17s and starts to reduce te tilt to te orizontal fligt. At t = 25s it starts to tilt in te opposite direction along te pitc wit te same angle and maintains te orientation till t = 33s wen it starts to tilt back to te original attitude again. 94