AM25 UNMANNED AIR VEHICLE FLIGHT CONTROL

Transcription

1 AM25 UNMANNED AIR VEHICLE FLIGHT CONTROL submitted by LOW JUN HORNG Department of Mechanical Engineering In partial fulfillment of the requirements for the Degree of Bachelor of Engineering National University of Singapore Session 2004/2005

2 Abstract ABSTRACT The objective of this project is to integrate a Global Positioning System (GPS) [5] assisted autopilot navigation system onto an existing platform to enable it to fly autonomously from waypoint to waypoint. It is required therefore to find the appropriate Proportional, Integral, and Derivative (PID) gain values to set for the given autopilot system. The original Unmanned Air Vehicle (UAV) platform was given without adequate aerodynamic, propulsion and stability data. Thus studies were made to determine such data by reverse engineering before the values of the gains could be found. The task was challenging as the UAV had an unconventional shape, thus there were no prior literature on how such values could be derived Besides finding the PID values, it was necessary to manufacture a replica of the given UAV. Once again, this task was challenging both due to its highly curved shape, as well as a need to keep the weight of the final replicated UAV low in order for it to be able to takeoff together with the additional weight of the control system which was to be integrated. Together with another student, Mr. Navabalachandran J, a study was done to look into the various materials that could be used, and the various methods that could be used to manufacture the highly curved wing. The replica UAV was then produced. The required aerodynamic data were derived by first doing a Computer Aided Drawing (CAD) of the UAV, followed by doing Computational Fluid Dynamic (CFD) calculations using the CAD model. These data were then used to find to how the given UAV would react in response to certain control input and environmental conditions in the form of the Equations of Motion for the UAV. Once these were National University of Singapore i

3 Abstract found, the transfer functions of the required control loops were derived using Laplace transform of the appropriate Equations of Motion. Using the transfer functions, optimisation was then used to find the most appropriate gains to use for the respective feedback control loops for the flight controls of the UAV (elevator from pitch, aileron from roll). A comparison with various optimisation methods was done and the most appropriate one was chosen. The gains were thus derived using the chosen method. Various tests were done along the way to validate the authenticity of the results derived. Results of the calculations were found to be satisfactory. A paper based on this project [14] was presented at the Republic of Singapore Air Force s (RSAF) Aerospace Technology Seminar on February National University of Singapore ii

4 Acknowledgement ACKNOWLEDGEMENT The author would like to extend sincere gratitude to his project supervisor, Associate Professor Gerard Leng Siew Bing for his guidance, and above all patience in answering all queries pertaining to the project. Also, the authors would like to thank Mr Leong See Kit of Cradence Pte Ltd for allowing us to use his equipment during the course of this project. Thanks also go to Mr Navabalachandran s/o Jayabalan for his assistance during the manufacturing and flight-testing phase of this project. The author will also like to extend his gratitude to the staff of Dynamics Laboratory, for their assistance for the duration of the project, as well as Mr Neo Ken Soon of the Advanced Manufacturing Laboratory for his assistance in the use of the 3D Laser scanner. National University of Singapore iii

5 Table of Contents TABLE OF CONTENTS SUMMARY ACKNOWLEDGEMENT TABLE OF CONTENTS LIST OF FIGURES LIST OF TABLES LIST OF SYMBOLS i iii iv vi viii iv 1. Introduction Problem Definition And Background Organisation of Thesis 3 2. Description of UAV Physical Characteristics Autopilot Control (Micropilot ) 7 3. Testing of GPS 8 4. Building of CAD and Physical Model Building of CAD Model Building of Physical Model Deriving Aerodynamic Coefficients Computational Fluid Dynamics Determining Type of Flow Aerodynamic Coefficient Analysis Semi-Empirical Derivation 20 National University of Singapore iv

6 Table of Contents 6. Derivation And Verification Of Transfer Functions Finding Transfer Functions Response To Controls Verification Of Results Verify Cl And Cd Verify Transfer Function Finding PID Gains Optimisation Methods Optimisation Integration Of Autopilot And Flight Tests Integration Of Equipment Flight Tests Conclusion 38 References 39 Appendix A GPS Test Results A-1 Appendix B CFD Results (Laminar) B-1 Appendix C Roots and Stability Modes C-1 Appendix D Glide Test D-1 Appendix E Complex RF E-1 Appendix F MATLAB Codes F-1 Appendix G Sensitivity Test G-1 Appendix H Motor and Propeller Selection H-1 National University of Singapore v

7 List of Figures LIST OF FIGURES 1.1. Comparison of F-117 and the given UAV D views of the UAV 2.2. The given UAV 2.3. Shape of the UAV (rear view) 2.4. The Micropilot control card 3.1. Setup to test GPS 3.2. Graph of the change in Latitude values and change in Longitude values 4.1. Taking 3D laser scans of one wing 4.2. CAD model of the UAV 4.3. Mesh of the UAV used in FLUENT 4.4. Bottom half of wing made of Glass Reinforced Plastic 4.5. Side profiles of the wing to be cut into ribs 4.6. Ribs mounted on the bottom layer of wing 4.7. The completed model 5.1. Definition of the Stability axes 5.2. Picture of the FLUENT computation display 5.3. Aiflow to see the effects due to velocity 5.4. Aiflow to see the effects due to angle of attack 5.5. Aiflow to see the effects due to control surfaces 5.6. Graph to determine equation of the curvature of the leading edge of wing 6.1. Root locus plots for longitudinal and lateral stability 6.2 UAV in glide test 6.3. Comparison between a sketch of the Dutch roll mode and the glide test National University of Singapore vi

8 List of Figures 6.4. Response of UAV (roll) in response to a unit step input 7.1. Open loop response to step input and Closed loop response to step input 7.2. Closed loop control system of the UAV with PID controller (Longitudinal) 7.3. Open loop response to step input & Closed loop response to step input 7.4. Optimisation methods tree 7.5. Control model set up using SIMULINK 7.6. Response to a step input before and after optimization 7.7. Optimisation using three systems together 8.1. Doing Thrust calculation and Propeller analysis 8.2. GCS control screen 8.3. Doing CG balancing 8.4. GCS control screen and PID gains screen 8.5. Loading commands into the UAV 8.6. UAV in flight 9.1. UAV in flight (front view) A.1 Movement of GPS Receiver B.1 CFD results (Longitudinal) B.2 CFD results (Lateral) C.1. D.1. Typical Root locus plot for aircraft stability Relationship between glide slope and Forces H.1 Thrust calculation and propeller selection setup National University of Singapore vii

9 List of Symbols LIST OF TABLES 4.1. Comparison of Fiber properties (Ref: Hull & Clyne) 4.2. Comparison of Resin properties (Ref: Hull & Clyne) 7.1. Optimised gains for longitudinal and lateral stability 7.2. Optimised gains for longitudinal and lateral stability (3 systems) A.1 Position recorded by GPS and Difference in location distances B.1 Longitudinal Aerodynamic Coefficients of CFD Calculations B.2 Lateral Aerodynamic Coefficients of CFD Calculations G.1 Sensitivity of Longitudinal coefficients G.2 Sensitivity of Lateral coefficients H.1 Comparison of Brushless Motors H.2. Results of Thrust Calculation National University of Singapore viii

10 List of Symbols LIST OF SYMBOLS α β δa δe θ ф ψ ρ μ a a a b C c h lf y h c y C C C C C C D Dt Dy L Lt Ly C ij Angle of Attack Sideslip angle Aileron deflection angle Elevator deflection angle Pitch angle Roll angle Yaw angle Density Viscosity Local Lift curve slope at coordinate h Fin Lift curve slope Local Lift curve slope at spanwise coordinate y Wing Span Mean aerodynamic chord Local Tailplane Chord at coordinate h Function of chord length with respect to Y coordinate Coefficient of Drag Coefficient of Drag at Tailplane Coefficient of Drag at spanwise coordinate y Coefficient of Lift Coefficient of Lift at Tailplane Coefficient of Lift at spanwise coordinate y Aerodynamic coefficient of parameter j with respect to i National University of Singapore ix

11 List of Symbols d Length parameter for Reynold s number, horizontal glide distance g Gravitational acceleration, defined as 9.81m/s 2 h H f I i I ij K K K l l f t d i p L m M N p q r Re S s u V V f t Height Fin span measured perpendicular to the roll axis Moment of Inertia about oi axes Product of Inertia about oi and oj axes Differential Gain value Integral Gain value Proportional Gain value Distance of fin from location of centre of gravity Distance of tail from location of centre of gravity Rolling moment Mass Pitching moment Yawing moment Roll Rate Pitch rate Yaw rate Reynolds Number Wing area Wing semi-span ( b / 2 ), Laplace operator Indicated airspeed Fin volume ratio Tailplane volume ratio National University of Singapore x

12 Chapter 1 Introduction 1. INTRODUCTION This project is an industrial project sponsored by Cradence Services Pte Ltd. This company has developed an Unmanned Aerial Vehicle (UAV) of a rather unconventional shape and structure, resulting in some undesirable flight characteristics. The overall task therefore, to correct its undesirable flight characteristics and thus enable the UAV to achieve stable flight between two given points. This project was done in collaboration with another Final Year student. 1.1 PROBLEM DEFINITION AND BACKGROUND Unmanned Aerial Vehicles (UAVs) are increasingly being used for surveillance purposes, especially in militaries around the world. As they get smaller, weight becomes an important factor and much care is taken to reduce the number of parts of an UAV. Thus justifies the development of a small unconventional flying wing UAV. Research was done to find out how such aircraft can be controlled to produce stable flight. A comparison was made with another unconventional aircraft of a similar structure, the F-117 Stealth Fighter. The F-117 is also essentially a flying wing, and its control surfaces are also in the shape of a V, but found at the back end of its fuselage (the given UAV had control surfaces at the wing tips). Also, the Stealth Fighter was inherently unstable [1], much like the given UAV, and the only way to control it would be to fly by wire, which means its flight is governed by a computerised autopilot control system. National University of Singapore 1

13 Chapter 1 Introduction Tilted Elevons Figure 1.1. Comparison of F-117 and the given UAV The usual method of developing an aircraft, as described in Raymer [2], is to come up with a conceptual design, based on the mission and design requirements, then designing an aerofoil with the suitable aerodynamic properties, followed by doing a sizing and performance optimization, and then integrate it together with the other parts of the aircraft, i.e. controls, propulsion systems, payloads etc. In this project however, a finalized model of the original UAV platform was given without adequate aerodynamic, propulsion and stability data. This disrupts the chain of development as integration of control systems onto the UAV cannot be done without knowledge of such information. The task is made more challenging due to the fact that the given UAV has an unusual shape when compared to conventional aircraft, as estimates for aerodynamic and stability data for conventional aircraft can be derived using established formulae available in flight dynamics textbooks, whereas no literature was found with respect to the given configuration. From S. Kanowitz, et al [4], the estimation of PID gains for their Micro Air Vehicle (MAV) was by doing a piloted flight test to estimate the gains. Even thought the situation was similar, this would not be possible for this project as the given UAV is unstable and difficult to control manually. Another method would have to be devised. The solution therefore, is to use reverse engineering to derive the required values. Many different tests and calculations have to be done in various stages to obtain these National University of Singapore 2

14 Chapter 1 Introduction required data. The stages set out for this project are: i) Testing of GPS, ii) Developing a CAD and physical model of the given UAV, iii) Computational Fluid Dynamic (CFD) and semi-empirical estimation of aerodynamic coefficients, iv) Obtaining an estimated mathematical model (transfer functions) of the UAV, v) Finding the appropriate gains of the required control loops through optimization, vi) Integration of the control system and the model UAV, and vii) actual flight tests. 1.2 ORGANISATION OF THESIS This thesis is composed of 9 chapters. Chapter 1 introduces and defines the objectives of this project. Chapter 2 describes the features of the given UAV. Chapter 3 shows the results of GPS testing of the given autopilot control system. Chapter 4 handles the CAD modeling and manufacturing of a replica of the given UAV. Chapter 5 deals with the Computational Fluid Dynamic (CFD) calculations to derive the aerodynamic coefficients, and thus the Equations of motions of the UAV. Chapter 6 gives and insight into the derivation and verification of the required transfer functions determined from the Equations of motion. Chapter 7 introduces the optimization to estimate the PID gains to be used for the given autopilot control card. Chapter 8 shows the integration of all the equipment into the UAV, and the flight tests that follow. Chapter 9 concludes this thesis. National University of Singapore 3

15 Chapter 2 Description of UAV 2. DESCRIPTION OF UAV The description of the UAV will be done in two parts. The first will focus on the physical features of the UAV, the second will focus on the control system for the UAV. 2.1 PHYSICAL CHARACTERISTICS The given UAV is much smaller than normal aircraft. It measures 0.86m in width by 0.81m in length. It has no landing gears and takes off by means of a hand launch. The original UAV has a weight of less than 1kg, and is supposedly capable of flights up to an altitude of about 500m Figure D views of the UAV National University of Singapore 4

16 Chapter 2 Description of UAV As mentioned in the introduction, the given UAV is rather unconventional. Unlike the normal aircraft configuration (wing, fuselage and tail), the UAV given is almost a flying wing, but unlike a real flying wing, it is equipped with a fuselage to store the electrical systems and the payload. Elevons Figure 2.2. The given UAV Also, instead of having 3 sets of control surfaces on the aircraft (Elevators, ailerons and rudders), there is only one set of control surfaces on the UAV. These are the elevons found at the ends of the wing. These elevons control the pitching and rolling of the aircraft. No rudders have been installed on this UAV. Furthermore, the ends of the wing where the elevons are situated are angled upwards at about 30 degrees to the horizontal to compensate for the lack of the rudder surfaces, acting as a pair of winglets to provide stability to the aircraft. Even then, the lack of real rudder surfaces on the UAV may make the control of the lateral movements of the UAV harder to stabilize as any adverse yaw due to the rolling motion of the UAV might not be cancelled out. National University of Singapore 5

17 Chapter 2 Description of UAV Figure 2.3. Shape of the UAV (rear view) The wing is also highly curved, forming a horizontally extended M shape, looking from the back of the UAV. This makes the mathematical modeling of the UAV more difficult as the shape is rather unconventional. During the manufacturer s test flights, the UAV was found to exhibit some undesirable flight characteristics, probably due to its unconventional shape. It is hoped that these would be corrected by integrating an autopilot control system onto the UAV. National University of Singapore 6

18 Chapter 2 Description of UAV 2.2 AUTOPILOT CONTROL (MICROPILOT ) Figure 2.4. The Micropilot control card To provide autopilot control and stability to the UAV, a Micropilot control card was provided and is required to be integrated into the UAV. It consists of an on-board Global Positioning System (GPS) [5], gyro unit as well as an air data unit. From these units, the UAV flight data i.e. position, turn rate, pitch and roll angles, airspeed, altitude etc are determined and fed back into the corresponding control loops to determine what corrective actions need to be taken to take it to its determined waypoint at a given speed and altitude. It is thus required to find the appropriate Proportional, Integral and Derivative (PID) gains and input them into the program provided with the autopilot card to determine the appropriate amount the control surfaces would correct each situation by to prevent over or under-correction, which may lead to instability in flight. Finding these gains would thus be the main aim of this project. To control the aircraft using the autopilot, it is necessary to know which feedback loops are involved before we are able to find the appropriate gains for each of them. For the Micropilot card, as there is no rudder on this UAV platform, only the feedback loops involving the aileron and elevators are considered. National University of Singapore 7

19 Chapter 3 Testing of GPS 3. TESTING OF GPS It was necessary to test the GPS for its response to a change in location as this is essential in the autopilot control system, as the GPS would provide real time position to the control system, and any delay or Figure 3.1. Setup to test GPS discrepancy in location is detrimental to the control of the UAV. An experimental setup was done on top of block WS2 of the Engineering Faculty in NUS. The GPS was linked up to the laptop and its detected position is continuously recorded into a text file. The GPS was tested by moving it to four different locations, each about five meters apart from each other. At each location, the GPS was kept at that position for a period of about five minutes, then moved to the next location. From the results, it was seen that the GPS responded almost immediately to any change in position, and at a single location, the maximum difference in position reported was less than two meters. Figure 3.2. Graph of the change in Latitude values (left) and change in Longitude values (right) Further information on the values of the differences between the values can be found in Appendix A. National University of Singapore 8

20 Chapter 4 Building of CAD and Physical Model 4. BUILDING OF CAD AND PHYSICAL MODEL To derive the aerodynamic coefficients of the UAV, Computational Fluid Dynamic (CFD) calculations would have to be done. To do so, a CAD model would first have to be made. Besides serving the purpose of doing CFD, the CAD model would also assist us in manufacturing the physical model of the UAV. The physical replica of the UAV is required as damage is foreseen during the flight-testing phase. It would not be wise to damage the original given model, as it is quite expensive. 4.1 BUILDING OF CAD MODEL Due to the UAV s highly curved shape, it would not be easy to exactly replicate the UAV s shape into a CAD model. To assist us in this process, a 3D laser scanner (Minolta Vivid 900) was used to take 3D scans of the original UAV model. Scans were taken separately for each section of the UAV (i.e. the two wings and the fuselage). For each section, various scans were taken from different angles and pieced together to form a whole 3D picture of each of the sections. Figure 4.1. Taking 3D laser scans of one wing National University of Singapore 9

21 Chapter 4 Building of CAD and Physical Model Even though we could get a 3D model of the UAV using the Laser scanner, the model was not very smooth due to the pre-processing of the UAV before it could be used for scanning. As such, SOLIDWORKS was used to edit the completed 3D model. To model the wings, the shape of the aerofoil, taken from the fuselage end of the wing, was plotted out on paper and the coordinates taken and plotted onto Solidworks. To model the curvature of the wing, coordinates were taken of a few points and joined together to form a guide curve, which was used to loft the aerofoil shape out to the winglet end, assuming that the shape of the aerofoil is constant throughout the wing. This is repeated for the winglet, with the exception that instead of a guide curve in this case, a straight line was used to loft the winglet up to the tip. The elevons were modeled separately and assumed to be flat rectangular pieces. The fuselage was assumed to be a box with a rectangular-base and a top that followed the curvature of the top of the wing closest to the fuselage. The nose was modeled using curves. The final CAD part was converted into a STEP file and meshed using GAMBIT to be used for Computational Fluid Dynamic (CFD) calculations in FLUENT to find the required aerodynamic derivatives. Figure 4.2. CAD model of the UAV Figure 4.3. Mesh of the UAV used in FLUENT National University of Singapore 10

22 4.2 BUILDING OF PHYSICAL MODEL Chapter 4 Building of CAD and Physical Model As flight tests needed to be done on the UAV, there was a possibility that the given UAV would be damaged during these tests. A model needed to be made. Important considerations taken while deciding how to manufacture the model were the methods available to replicate the shape of the given wing as close as possible, as well as to make the aircraft as light as possible. After doing some research on the materials available, it was decided that the model would be made using Glass Fibre Reinforced Plastic. Figure 4.4. Bottom half of wing made of Glass Reinforced Plastic Although balsa wood is normally used for making aircraft models, it was decided otherwise as the UAV has a rather unconventional shape. To tackle this problem, it was decided that plastics could be used. As the model is expected to take rather heavy impacts as it hits the ground during testing, it is possible that the plastics may not be able to withstand the hard knocks. To get around the problem, composites were found to be the solution. Composites have been used in many aircraft applications. As it combines the mechanical properties of both the plastic resin as well as the strengthening fibres, they are light and have good strength properties. By using a mould, an almost exact copy of the given UAV can be made without having to do tedious calculations or technical labour. National University of Singapore 11

23 Chapter 4 Building of CAD and Physical Model A comparison was made between different types of fibres and resins to choose the one suitable in terms of pricing, weight and mechanical strength. Cost Density (Mg/m 3 ) Tensile Strength (GPa) Carbon Mat High Glass Mat Low Kevlar Mat High Table 4.1. Comparison of Fiber properties (Ref: Hull & Clyne) [6] Cost Density (Mg/m 3 ) Tensile Strength (GPa) Polyester Low Epoxy High Table 4.2. Comparison of Resin properties (Ref: Hull & Clyne) [6] From the comparison table above, even though it seems like glass fibres would incur the most weight, we managed to find a glass tissue fibre with less than half the density of the glass mat with similar tensile strength properties. Thus the tissue fibre was chosen as our reinforcement. As for the resin, the choice was simple as judging from the table above, Polyester proved to be both cheap and has similar tensile strength properties as epoxy. Even though epoxy had better thermal resistance and other mechanical properties, it was felt that it did not justify the extra cost. Thus polyester was used. A hand layup method was used to create the two bottom halves of the UAV s wings. Using Solidworks, the side profile of the wing was cut at equal intervals to give the profiles for the ribs to be laid onto the bottom layer of the manufactured wing. The profiles were cut out on balsa wood and pasted onto bottom halves of the wing once they had hardened and dried. This gives the shape of the wings. A thin layer of film is then wrapped onto the balsa wood profiles and the fiberglass bottom to give the complete airfoil shape. The fact that the profiles fit nicely onto the bottom halves proves that the CAD model was rather accurate. National University of Singapore 12

24 Chapter 4 Building of CAD and Physical Model Figure 4.5. Side profiles of the wing to be cut into ribs The fuselage was made using balsa wood and joined to the wings using carbon fibre rods. The completed model is as shown below Figure 4.6. Ribs on the fiberglass layer. Figure 4.7. The completed model National University of Singapore 13

25 Chapter 4 Building of CAD and Physical Model 5. DERIVING AERODYNAMIC COEFFICIENTS To find the gains of a control system, the transfer function of the control system must first be known. In this case, the control system is the reaction of the UAV to various control inputs. For every input to the control surfaces of the aircraft (e.g. turning the elevator upwards), the UAV would have a certain reaction (e.g. pitching upwards). The transfer function required therefore, is the relationship between the deflection of the control surfaces and its respective response. As it was not feasible to find the relationship experimentally, we have to derive the relationship mathematically with the use of force relationships, i.e. the UAV s Equations of Motions. These Equations of motions would be derived based on the X aircraft s Stability axes L N M Y Z Figure 5.1. Definition of the Stability axes To go into how the equations of motions are derived would be beyond the scope of this project, but further details can be found in Cook [7]. To further simplify, it is assumed that whatever disturbance in the lateral plane of the aircraft has negligible effect on the longitudinal movement of the aircraft, and vice versa (e.g. deflection of elevators will not cause the aircraft to go into a roll). As such, the lateral and longitudinal equations of motions are decoupled from each other, giving the equations below: Longitudinal: C C C xu zu mu u + C u + C u + C xα zα α + C α + C mα xq zq α + C q + C q + C mq xδe zδe q + C δe = mgθcosθ + mu& δe = mgθsinθ + mu( & α & θ ) mδe δe = I && θ y National University of Singapore 14

26 Chapter 4 Building of CAD and Physical Model Lateral C C C yβ lβ nβ β + C p + C r + C δa = mu( & β + ψ& ) + mgφ β + C lp β + C yp np lr p + C yr p + C r + C nr lδa r + C yδa δa = I && φ I && ψ nδa x δa = I && ψ I z xz xz && φ With these two sets of equations, we can find the relationship of any one of the variables with respect to another by solving the respective set of the equations. As the values of the mass (m) and moments of inertia ( I, I, I ) are known and have been calculated [15], and the values of velocity (u ), angle of attack ( α ), pitch rate ( q ), elevator deflection ( δ e ); sideslip ( β ), roll rate ( p ), yaw rate ( r ), and aileron deflection ( δ a ) are variables, thus, the objective then would be to determine the value of the other unknowns, i.e. the coefficients of each of these variables: velocity (, C zu, C ), angle of attack ( C, C, C ), sideslip ( C, C, ), elevator mu C x δ e C zδe C mδe xα deflections (,, ), aileron deflections (,, ), pitch ( C C C ), roll ( C C C ) and yaw ( C, C C ). These twenty-four xq zq mq yp lp coefficients are collectively known as the aerodynamic coefficients. np zα mα x yr xz y lr yβ C yδa nr lβ C lδa C n β C nδa C xu As there was no literature on this type of unconventional airfoil shape, it was required to find these coefficients using experimental and computational methods. Computational Fluid Dynamic (CFD) calculations were thus used to find the coefficients due to velocity, angle of attack, sideslip, and control surface deflections. As CFD is not able to model problems due to rate changes, semi-empirical methods were used to derive an estimate of the coefficients due to pitch, roll and yaw rates. National University of Singapore 15

27 Chapter 5 Deriving Aerodynamic Coefficients 5.1 COMPUTATIONAL FLUID DYNAMICS Using Computational Fluid Dynamic (CFD) programs, one is able to determine the values of the forces acting on the aircraft in response to the different variables. In FLUENT, runs were done to estimate how the UAV reacted to changes in velocity, angle of attack, sideslip, and control surface deflections individually while the rest of the variables were kept constant. From these runs, coefficients of forces in the X, Y and Z directions as well as the moments about these axes (L, M and N) with respect to different values of the changes in velocity ( C, C, C ), angle of attack ( C, C, ), sideslip ( C, C, C ), elevator deflections (,, ), and aileron deflections yβ C y δ a lβ C lδa nβ C nδa (,, ) can be found. xu zu mu C x δ e C zδe C mδe xα zα C m α Figure 5.2. Picture of the FLUENT computation display All in all, at least ten different meshes of the UAV with different control deflections were generated and used, and close to about 200 runs involving 6 variables, of over 500 iterations per run were performed. The computations were done in the Supercomputing and Visualisation Unit of the National University of Singapore. National University of Singapore 16

28 Chapter 5 Deriving Aerodynamic Coefficients DETERMINING TYPE OF FLOW Before the calculations can be done, a calculation on the UAV s Reynold s Number (Re) was done to investigate whether the flow over the airfoil would be considered laminar or turbulent. As Re = ρud μ, we would need to estimate the density (ρ) and the viscosity (μ) of the air flowing over the airfoil. The velocity of air over the airfoil (u) is assumed to be constant at 10m/s, which is the cruise airspeed specification given to us by the manufacturer, and the length is assumed to be 0.77m, which is the actual length of the given UAV. Using tables for Air at atmospheric pressure from Holman, the density was found to be kg/m 3 and the viscosity was found to be x 10-5 kg/ms at 300K. Therefore, the Reynold s number is found to be: ρud Re = = μ x 10 x x 10 = 4.91 x 10 5 Even though a flow is considered laminar for a flat plate if the Reynold s number is less than 5x10 5 [7], the laminar flow assumption was found to be inaccurate due to the uneven surfaces inherent on the material used to manufacture the UAV. It was also concluded that the airfoil cannot be considered as a flat plate as it is slightly curved, and the curvature over the small length would thus be significant. Therefore the Reynold s number for transition into turbulent flow would be changed. It is also concluded that since the Reynold s number for a pipe flow (Re = 2000) is lesser than that of a flat plate, the Reynold s number for transition into turbulent flow for this airfoil would be significantly reduced. A turbulent analysis is therefore required. National University of Singapore 17

29 Chapter 5 Deriving Aerodynamic Coefficients AERODYNAMIC COEFFICIENTS ANALYSIS Various methods of solution for modeling turbulence are available in FLUENT. A K- Epsilon (RNG) model was used as the K-Epsilon method [9] has been generally used for turbulence modeling problems. Although another method which requires a lower computational time is available (Spalart-Allmaras model), it is a one-equation model, and thus the results would not be as accurate. The RNG K-Epsilon model [10] was used as it incorporates a formula for lower Reynolds number effects. Along with other features, it serves as a better model than the normal K-Epsilon model. To investigate the effects due to the UAV s velocity, the mesh grid of the UAV with no control surface deflections was used. The mesh was then subjected to air at different speeds. The resultant forces and moments on the UAV was then recorded Figure 5.3. Aiflow to see the effects due to velocity For different angles of attack and sideslip, the airflow was set at various angles to the UAV, at a constant, specified airspeed. As with the case of investigating the effects due to airspeed, the mesh grid of the UAV with no control surface deflections was used. Figure 5.4. Aiflow to see the effects due to angle of attack National University of Singapore 18

30 Chapter 5 Deriving Aerodynamic Coefficients For the control surface deflections, various mesh grids of the UAV with the elevons set at ten different angles were generated and used one by one in the computation. The respective meshes were then subjected an airflow of a constant, specified speed. Figure 5.5. Aiflow to see the effects due to control surfaces The forces and moments on the UAV with respect to the various axes were recorded. Using these values, graphs were plotted to find out each parameter affected the respective forces and moments. The relationships were assumed to be roughly linear, and thus the relevant aerodynamic coefficients were defined as the slope the linear portion of each graph. The graphs can be found in Appendix B. National University of Singapore 19

31 5.2 SEMI-EMPIRICAL DERIVATION Chapter 5 Deriving Aerodynamic Coefficients Unfortunately, the CFD program available was not able to handle dynamic changes of the variables, thus only coefficients of the variables that would not involve dynamic movements of the aircraft can be found. The coefficients due to the pitch ( C C C ), roll ( C C C ) and yaw ( C, C C ) could not be derived. Thus, a xq zq mq yp lp np semi-empirical method of solution as derived in Cook [7] was used for these derivatives. yr lr nr The equations to be used are thus, Longitudinal: C xq zq CD t = Vt α C = V C C mq = C t zq l c Lt t Lateral: C C C yp lp np 1 = Sb 1 = 2Ss = 2Ss 2 H f a c h dh s h h ( a C ) y + 0 s 0 C Ly Dy c y y 2 dy dc D c y y dα y 2 dy C = V C C yr f a lf 1 s H 2 f = CLyc y y dy alfv 2 f Ss 0 b 1 s l 2 f = CDyc y y dy alfv 2 f Ss 0 b lr + nr + To derive these coefficients semi-empirically, certain other values (function of the chord length with respect to Y (C y ), wing area (S), mean aerodynamic chord ( C ) etc.) need to be found. National University of Singapore 20

32 Chapter 5 Deriving Aerodynamic Coefficients To determine C y, three coordinate points were taken from the original wing. The curve was assumed to be quadratic and using the three points, an estimated curve equation was derived. Y Coordinate (mm) y = x x X Coordinate (mm) Figure 5.6. Graph to determine equation of the curvature of the leading edge of the wing By integrating C y, the wing area was derived. 280 C y S = 2 x dx = 2 x x x dx = m 3 The Mean aerodynamic chord is defined by the equation, c = s s s s C C 2 y y dx, where s is dx defined as the one-half of the wingspan. Thus by substituting the values into this equation, the mean aerodynamic chord length was found to be 504.9mm. From the results of these semi-empirical formulations, the coefficients of the pitch, roll and yaw rates were derived National University of Singapore 21

33 Chapter 6 Derivation And Verification Of Transfer Functions 6. DERIVATION AND VERIFICATION OF TRANSFER FUNCTIONS With the derivation of the aerodynamic coefficients, the Equations of motion can be found by substituting these coefficients into the equations. From these equations, the reactions of the aircraft to specific inputs are known. The next step would be to find the required transfer functions of the control loops used in the autopilot control system, so that the gains can be derived. After finding the transfer functions, it is necessary to verify if it is accurate so that accurate calculations can be done to determine the gains in a later stage. 6.1 FINDING TRANSFER FUNCTIONS After substituting the aerodynamic coefficients derived into the equations of motion, these equations are then divided into two separate sets of equations (longitudinal and lateral). By taking Laplace transform on each equation, and using a matrix to separate the coefficients from the disturbance parameters, the equations become: Longitudinal: [ ] e m z x mq y m mu e z e z e zu xq e x xu e e e c c c u s c s I c c )s mu ( c mg sin c s mu c s c mg cos c c ms δ θ α θ θ δ δ δ α α α α = + 2 Lateral: [ ] a n l y nr z np xz n lr xz lp x l yr e yp y e a a a c c c s c s I s c s I c s c s I s c s I c s c mu mg s c c s mu δ ψ φ β δ δ δ β β β = The equations were then divided throughout (using Cramer s Rule) by the effect of the control surfaces, giving the response of the individual parameters due to the control surfaces. National University of Singapore 22

34 Chapter 6 Derivation And Verification Of Transfer Functions Longitudinal: u δ e α δ e θ δ e ms c = czu cmu xu c mu s c e c xα mα zα mg cosθ c mg sinθ ( c I e y s 2 e zα c mq xq + mu s s e )s 1 c c c xδ e zδ e mδ e Lateral : mues c = clβ cnβ β a y cyps mg mue cyrs 1 δ β φ 2 2 I xs clps I xzs clrs δ a ψ 2 2 I xzs cnps I zs cnrs δ a cy cl cn δ δ δ a a a Since we only needed to solve for the gains of the elevator-from-pitch and the aileronfrom-roll loops, only the solutions of θ δ e and φ δ a would be used. The above solutions would thus be the transfer functions of their respective loops. Using MATLAB, the values were input into the above two sets of equations and solved for the transfer functions of θ δ e and φ δ a, giving: θ δ = 1.9 s s s s s s 4 e φ = δ s a s s s s s National University of Singapore 23

35 Chapter 6 Derivation And Verification Of Transfer Functions 6.2 RESPONSE TO CONTROLS From the transfer functions determined previously, it can be seen that both are 4 th order. The denominators of each transfer function should factorise out to give two pairs of roots. These roots will determine how the aircraft will react to control inputs or disturbances during flight. More information on the roots and the stability modes they describe can be found in Appendix C. To determine if the aircraft would be stable longitudinally and laterally, the open loop root locus plots were done for the two transfer functions derived previously to determine how stable the UAV would be. Figure 6.1. Root locus plots for longitudinal (left) and lateral (right) stability From the root locus plots, it can be seen that the aircraft is longitudinally stable as the roots are in the negative real section. As for the lateral stability, the roll and spiral modes appear to be stable as they are within the negative real section of the root locus plot. The Dutch roll mode however seems to be unstable as the roots have positive real parts. As these are open loop root locus plots, it is hoped that using feedback, together with the use of PID gains that a stable solution can be found. National University of Singapore 24

36 Chapter 6 Derivation And Verification Of Transfer Functions 6.3 VERIFICATION OF RESULTS With any calculation, there is always a need for verification as experimental conditions almost always differ from real situations. As most formulas are derived with assumptions to approximate the actual surrounding, there will often be errors between the calculated and actual responses. Experiments involving the physical model of the UAV need to be done to find out if the calculated values were correct. Two tests were used to test the validity of the results of the CFD calculations and semi-empirical derivations. The first was the glide test to verify the calculated Lift to Drag ratio, and the second was an observation to verify the transfer function obtained from all the previous calculations. National University of Singapore 25

37 Chapter 6 Derivation And Verification Of Transfer Functions VERIFY C L AND C D A glide test was used to verify the values of the coefficient of lift, as well as the coefficient of drag obtained from CFD calculations. To prove that these values are accurate, it is necessary to prove that C L / C D = d / h (see Appendix D). Using the values of the coefficients of lift and drag from the CFD calculation, and the values derived from the glide test performed, it was found that the ratios were rather similar. Figure 6.2. UAV in glide test From the CFD calculations, C L / C D = / = From Glide test, d / h = 20m / 3.25m = 6.15 Though the values were not exactly similar, they were rather close. Thus it was concluded that the values of the Coefficients of Lift and Coefficients of Drag derived were rather accurate. This also justified the use of the turbulent model in doing the CFD calculations. National University of Singapore 26

38 Chapter 6 Derivation And Verification Of Transfer Functions VERIFY TRANSFER FUNCTION As seen previously from the root locus plots, it was deduced that the UAV would exhibit oscillations in the Dutch roll mode. To verify that the oscillation would occur, and at the same time verify the frequency of this oscillation, would therefore testify to the accuracy of the transfer function. During the glide test, it was observed that the UAV did indeed show signs of a Dutch roll as it tends to oscillate in response to a slight wind disturbance, even when gliding in a straight line. A comparison is shown below between a sketch of what a Dutch roll looks like [7] and the snapshots of the actual glide test. Figure 6.3. Comparison between a sketch of the Dutch roll mode [7] (left) and the glide test (right) National University of Singapore 27

39 Chapter 6 Derivation And Verification Of Transfer Functions Knowing that the UAV exhibits a Dutch roll is insufficient to prove that the transfer function is correct. By plotting the response of the transfer function to a given step input, it was seen that the UAV would exhibit an oscillation, rolling back and forth with a period of about 1.5 seconds. Figure 6.4. Response of UAV (roll) in response to a unit step input During the glide test, it was also observed that the UAV also exhibited the oscillation with a period of about 1.5 seconds as well. This proved the accuracy of the transfer function, and proved that the UAV indeed had an unstable Dutch roll mode. National University of Singapore 28

40 Chapter 7 Finding PID Gains 7. FINDING PID GAINS The normal aircraft can be simply described as a simple open-loop control system. With the introduction of a control input (e.g. elevator), the aircraft would respond accordingly with the appropriate movement (e.g. pitch). This can be modeled in the block diagram as shown below. Elevator deflection UAV UAV Pitch up Figure 7.1. Open loop control system of the UAV (Longitudinal) For most open loop responses to a given input, the response would not be optimized. Sometimes, it may even be unstable unless a certain feedback is introduced together with use of appropriate gains. By saying that a system is optimized, it means that it is able to reach and maintain at the desired steady state level at the shortest time, as well as having minimum overshoot, i.e. being as close to being critically damped as possible. As the given autopilot control system (Micropilot ), feedback is provided to the control system in the form of pitch, roll and yaw values provided by the gyros found onboard the control card, as well as the GPS receiver which provides position information. Also, the control system is equipped with a PID controller. Elevator deflection + _ PID Controller UAV UAV Pitch up Pitch angle, pitch rate Gyros and GPS Figure 7.2. Closed loop control system of the UAV with PID controller (Longitudinal) National University of Singapore 29

41 Chapter 7 Finding PID Gains The PID controller is able to help to improve the response of a closed loop system by reducing overshoot and oscillations, make the rise time faster, and also help to stabilize an unstable system by appropriate choices of the PID gains. input Control System output input + _ K p + K d S + K i / S Control System output Figure 7.3. Open loop response to step input (left) & Closed loop response to step input (right) It is thus required to find the most appropriate gains to use so as to achieve the desired response as quickly as possible. This would have to be done for both the control inputs for which the transfer functions have been found previously i.e. Elevator vs Pitch, and Aileron vs Roll. To determine the best gains for the system by trial and error would be very tedious as it involves three variables (P, I and D). To do that for two transfer functions would take longer. Thus, a better method has to be used to find the most appropriate gains for the system. By using optimization algorithms, best combinations of the values of the variables can be found as the objective function converges to a desired value. National University of Singapore 30

42 Chapter 7 Finding PID Gains 7.1 OPTIMISATION METHODS There are various methods to perform optimization, each with their various strengths. A simple classification of optimisation methods is shown below (non-exhaustive) Optimisation Methods Derivative Methods Non-Derivative Methods Steepest descent Methods Quasi Newton Methods Sequential Quadratic Programming Simplex/ Complex Methods Genetic algorithms Tabu Search Figure 7.4. Optimisation methods tree The derivative methods make use of gradients to determine the convergence of a function. In some of these methods, second-derivatives of the functions would be investigated as well to get a more accurate answer. However, doing so would incur more computational resources, with a risk of not being able to achieve the global optimum. Also, gradient information is not always available, especially if parts of the objective function are evaluated through simulation of non-linear systems. For non-derivative methods, various ways are used to determine the optimal answer. Genetic algorithms follow genetic or evolutionary patterns to arrive at an optimum. These methods, though efficient, take up a lot of computational resources. Simplex / Complex methods on the other hand, employ a simple search logic, and is able to be used generally to solve most optimization problems. Complex RF [13], a method based on the Neadler-Mead Simplex was thus selected for the use of this project. More information on how Complex RF works can be found on Appendix E. National University of Singapore 31

43 Chapter 7 Finding PID Gains 7.2 OPTIMISATION By using SIMULINK from Matlab, a control system model with a PID controller was created and the respective transfer functions were input into the system. Figure 7.5. Control model set up using SIMULINK To optimize for the gains, the objective function of the above model was optimized to get the minimum sum-squared value of the error with respect to the step input provided using the Complex RF method. The variables used in the optimization were the individual P, I and D gains. The codes can be found in Appendix F. The results of the optimisation are shown below. From these it can be seen that the unstable characteristics of the UAV are corrected to give a stable response to a control input. Gains Elevator vs Pitch Aileron vs Roll P I D Table 7.1. Optimised gains for longitudinal and lateral stability Pitch response to elevator input: Roll response to aileron input: Figure 7.6. Response to a step input before (left) and after optimization (right) National University of Singapore 32

44 Chapter 7 Finding PID Gains To get a more robust value of the gains, a sensitivity study was done (see Appendix G) to see which aerodynamic coefficient had the most significant effect on the stability of the UAV, both longitudinally and laterally. From this sensitivity study, it was found that C had the most effect longitudinal stability, and C had the most mq effect on lateral stability. lp These values were changed by ±10%, the adjusted transfer functions were found, and an optimization was done with the three systems (original, +10%, -10%) together to get the gains that would satisfy all three conditions. Gains Elevator vs Pitch Aileron vs Roll P I D Table 7.2. Optimised gains for longitudinal and lateral stability (3 systems) These two new sets of gains would be the gains used in the Micropilot as they would be more robust and would cater to a larger range of values in the case of errors in calculation of the aerodynamic coefficients Figure 7.7. Optimisation using three systems together National University of Singapore 33

45 Chapter 8 Integration of Autopilot and Flight Tests 8. INTEGRATION OF AUTOPILOT AND FLIGHT TESTS As the computations given above merely serves to derive an estimate of the flying characteristics of the UAV, actual flight test would have to be done to find the correct gains to allow the UAV to fly the way it is supposed to autonomously. 8.1 INTEGRATION OF EQUIPMENT Before flight tests can be done, suitable choices in purchasing a motor, propellers had to be made. Also, the Micropilot card must be integrated into the replicated model previously fabricated. It was also essential to balance the aircraft to move the centre of gravity (CG) to a specific location to ensure that it flies at the correct attitude PROPULSION For the UAV to be able to fly, it has to have a propulsion system capable of producing sufficient thrust to overcome the drag forces that would be incident on the UAV, as well as accelerate fast enough for the UAV to attain the speed at which it would have lift to sustain its flight. After doing some research, it was decided that a brushless motor was required to produce the right amount of thrust for the model UAV as it was rather heavy as compared to the conventional model aircraft. By comparing the characteristics of the motors, and doing appropriate Thrust Calculation and Propeller analysis (see Appendix H), the motor and corresponding propeller was chosen Figure 8.1. Doing Thrust calculation and Propeller analysis National University of Singapore 34

46 Chapter 8 Integration of Autopilot and Flight Tests SYSTEMS To integrate the systems together, connectors have to be made to join each of the components together. The connectors not only have to be able to hold the components together, but also have to be long enough to allow these components to be placed at specific locations within the UAV. This is to ensure that the Centre of Gravity (CG) of the UAV is at the correct position [15] to allow it to fly at the correct attitude. Extra protection was fabricated for the Micropilot card to protect it from shocks and heavy impact during landing and possible mishaps. To prevent unnecessary shifting of the CG during flight, the components were held in place using Velcro or small pieces of balsa wood, strategically placed to restrict their movement. Motor Speed controller Batteries (8.4 &12.6V) RC Receiver Micropilot Card GPS Receiver Servo Controller card Figure 8.2. Connections for the control of the UAV Figure 8.3. Doing CG balancing National University of Singapore 35

47 Chapter 8 Integration of Autopilot and Flight Tests CONTROL To allow the UAV to fly autonomously, a flight plan must be uploaded into the Micropilot card so that the UAV would know where it should be going without having to receive commands from a human pilot. It would thus be the job of the Micropilot to control the UAV to enable it to reach its required destination at a certain airspeed, bearing and altitude. Commands were uploaded to the Micropilot using a laptop installed with the given Ground Control System (GCS) software. From the software, waypoints can be loaded into the control system together with the relevant instructions for the required airspeed and altitude settings for the flight program. Also, the PID gains derived previously are input into the Micropilot through this GCS software to ensure efficient control of the UAV. Figure 8.4. GCS control screen (left) and PID gains screen (right). National University of Singapore 36

48 Chapter 8 Integration of Autopilot and Flight Tests 8.2 FLIGHT TESTS Flight tests were conducted to see the effects of the controller. The PID gains derived from the optimisation were input into the Micropilot using the GCS software. A flight plan consisting of two points, 20 metres apart of each other was also uploaded into the UAV (the onboard GPS receiver has an accuracy of <9m). The UAV is supposed to fly between these two points at an altitude of about two metres above the ground. The UAV would then be observed to see how it reacts during its flight. The gains would be adjusted accordingly until the UAV achieves a stable flight between the two points. A satisfactory set of gains was finally found and the UAV was able to fly autonomously between the two given points. This proved the concept of using calculation and CFD to find the gains. During the flight tests, it was also noticed that wind played an important part in the control of the UAV as strong winds caused it to deviate from its path, sometimes even cause it to turn uncontrollability. This may be due to the fact that the UAV is small and thus very susceptible to the effects of wind, as well as the fact that it has no vertical tail surface, thus it may not be able to damp out the effect of wind that effectively as other aircraft with a rudder. Figure 8.5. Loading commands into the UAV Figure 8.6 UAV in flight National University of Singapore 37

49 Chapter 9 Conclusion 9. CONCLUSION Many things have been learnt from this multi-disciplined project. From the fields of manufacturing, control, fluid dynamics, aerodynamics to optimization, this project has not only proved its industrial relevance, it has also brought about future possibilities for the UAV industry. Through this project, the concept has been proven, and a procedure has thus been established to effectively reverse engineer a given UAV and enable it to fly autonomously. The aerodynamic coefficients have been successfully derived using Computational Fluid Dynamic and semi-empirical calculations, and verified using glide tests. A control model i.e. the transfer functions of the UAV, which describe the its reaction to certain disturbances have been derived from the UAV s equations of motion and verified using glide tests as well. An optimized set of PID gains for both the longitudinal and lateral stability of the UAV was then found using optimization methods. With the use of these gains, the unstable characteristics inherent to the UAV have been corrected. Flight tests have been done and the UAV has proven to be controllable. Figure 9.1 UAV in flight (front view) National University of Singapore 38

50 References REFERENCES 1. David C Aronstein & Albert C. Piccirillo, Have Blue and the F-117A, Evolution of the Stealth Fighter, AIAA Inc Daniel P Raymer, Aircraft Design: A conceptual approach, AIAA Education Series, ISBN /-281-0, Joel M. Grasmeyer and Matthew T. Keennon, Development of the Black Widow Micro Air Vehicle AIAA , 39th Aero. Sci. Meet & Exhibit, Reno, NV, USA, Jan S. Kanowitz, M. C. Nechyba and A. A. Arroyo, "Design and Implementation of a GPS-Based Navigation System for Micro Air Vehicles," 2002 Florida Conference on Recent Advances in Robotics, Miami, May James B.Y. Tsui, Fundamentals of global positioning system receivers: a software approach, New York : John Wiley & Sons, D. Hull and T.W. Clyne, An introduction to composite materials Cambridge; New York : Cambridge University Press, M.V. Cook. Flight Dynamics Principles, John Wiley & Sons Inc., ISBN X, Barnes W. McCormick, Aerodynamics, aeronautics, and flight mechanics, John Wiley & Sons Inc., ISBN , B. E. Launder and D. B. Spalding. Lectures in Mathematical Models of Turbulence. Academic Press, London, England, V. Yakhot and S. A. Orszag. Renormalization Group Analysis of Turbulence: I. Basic Theory. Journal of Scientific Computing, 1(1):1-51, Ogata K., Modern Control Engineering, Prentice Hall International, Second Edition Box M. J., A new method of constraint optimization and a comparison with other methods, Computer Journal, vol. 8, pp , Krus P. and Andersson J., Optimizing Optimization for Design Optimization, in Proceedings of ASME Design Automation Conference, Chicago, USA, September 2-6, Low J.H, Navabalanchandran J. & Gerard Leng, Autopilot Integration For A Flying Wing UAV, RSAF Aerospace Technology Seminar, Navabalanchandran J., Low J.H & Gerard Leng, Reverse Engineering and Aerodynamic Analysis of a Flying Wing UAV, RSAF Aerospace Technology Seminar, 2005 National University of Singapore 39

51 Appendix A GPS Test Results GPS TEST RESULTS Figure A.1. Movement of GPS Receiver 1st posn max E rd posn max E st posn max N rd posn max N st posn min E rd posn min E st posn min N rd posn min N st posn mode E rd posn mode E st posn mode N rd posn mode N Difference Difference nd posn max E th posn max E nd posn max N th posn max N nd posn min E th posn min E nd posn min N th posn min N nd posn mode E th posn mode E nd posn mode N th posn mode N Difference Difference Table A.1. Position recorded by GPS and Difference in location distances National University of Singapore A - 1

52 Appendix B CFD Results CFD RESULTS Longitudinal Forces and Moments Total force (X-Direction) y = x Angle of Attack (rad) Total force (X-Direction) y = x Elevator Deflection (rad) Total force (Z-Direction) y = x Angle of Attack (rad) Total Force (Z-Direction) y = x Elevator Deflection (rad) Total moment (M-Direction) y = x Angle of Attack (rad) Total momen t (M-Direc tion) y = x Elevator Deflection (rad) Figure B.1. Turbulent CFD results (Longitudinal) Coefficients Cx α Cx δe Cz α Cz δe Cm α Cm δe Table B.1. Longitudinal Aerodynamic Coefficients of CFD Calculations National University of Singapore B - 1

53 Appendix B CFD Results CFD RESULTS Lateral Forces and Moments Total force (Y-Direction) y = x Sideslip angle (rad) Total force (Y-Direction) y = x Aileron Deflection (rad) Total Moment (L-Direction) y = x Sideslip angle (rad) Total moment(l-direction) y = x Aileron Deflection (rad) Total Moment (N-Direction) y = x Sideslip angle Total moment (N-Direction) y = x Aileron Deflection (rad) Figure B.2. Turbulent CFD results (Lateral) Coefficients Cy β Cy δa Cl β Cl δa Cn β Cn δa Table B.2. Lateral Aerodynamic Coefficients of CFD Calculations National University of Singapore B - 2

54 ROOTS AND STABILITY MODES Appendix C Roots and Stability Modes For longitudinal stability modes, the denominator should factorise to give two pairs of complex roots in normal aircraft. The lower frequency mode is called the phugoid mode and the higher frequency mode is the short period pitching oscillation mode. Both modes involve oscillation in the pitching motion of the aircraft. For lateral stability modes, the denominator should factorise to give two real roots and a pair of complex roots. The roll mode is a non-oscillatory mode characterized by the restoring reaction of the aircraft to a disturbance in roll, described by one of the real roots. The other real root will describe another non-oscillatory mode called the spiral mode, which involves complex motions in roll, yaw and sideslip. The complex pair would describe an oscillatory mode called the Dutch roll mode, which sees the aircraft oscillating in yaw and roll. For the longitudinal and lateral stability of the aircraft, the roots have to be on the negative portion of the real axis. 1.5 Imaginary Axis Short Period Phugoid Roll Subsidence Spiral Dutch Roll -1.5 Real Axis Figure C.1. Typical Root locus plot for aircraft stability National University of Singapore C - 1

55 Appendix D Glide Test GLIDE TEST In a glide test, the only forces acting on the UAV would be the Lift, Drag and Weight of the UAV. Weight would act vertically downwards towards the earth, Lift would act perpendicular to the airfoil, while Drag would act parallel to the airfoil, as shown in the diagram below. D From the figure, it can be seen that the W θ L forces are related to the glide angle in this equation: Lift / Drag = 1 / tan θ h Since the height and horizontal distance θ glided is also related in the same way, i.e. d Figure D.1. Relationship between glide slope and Forces therefore, d / h = 1 / tan θ, Lift / Drag = d / h Since Lift is equal to ½ C L ρv 2 S, and Drag is equal to ½ C L ρv 2 S, Lift Drag = = ½ C ½ C CL C D L D 2 ρv S 2 ρv S Therefore, C C L D = d h National University of Singapore D - 1

56 Appendix E Complex RF COMPLEX RF In the Complex RF method used, it takes 2n number of random combination of values of P, I and D gains, with n being the number of variables involved (in this case, there are three variables, thus six combinations of values are taken), from the range of values given and finds each of their function evaluations. The combination with the worst evaluation would be set aside and a triangulation would be done for the remaining evaluations to give a centroid. The combination giving the lowest evaluation value would be reflected through the centroid to give another combination. Once again a function evaluation is done for this value and a comparison is done to eliminate the worst function evaluation. The process goes on until the difference between the function evaluations are below a certain prescribed value. The results from this optimization would give the optimal gains for the respective feedback loop to achieve a response with sufficient damping, minimal overshoot and shortest response time. What makes Complex RF method different from the rest of the other Simplex/ Complex methods is that it incorporates a randomization and forgetting factor. The randomization factor introduces random noise during the evaluation of the objective function to prevent it from converging to local optima. The forgetting factor decreases the value of a function each time it is not evaluated. This is important if the objective function varies over time, as older values become increasingly unreliable. The code is given on the next few pages. National University of Singapore E - 1

57 Appendix E Complex RF Complexrf.m % COMPLEXRF Multidimensional constrained nonlinear minimization using the % Complex method. % Reference: Box M. J. (1965), A new method of constraint optimization and a % comparison with other methods, Computer Journal 8:42-52 % % Krus P., Andersson J., Optimizing Optimization for Design Optimization, % in Proceedings of ASME Design Automation Conference, Chicago, USA, % September 2-6, 2003 % Setting the default values for the algorithm MaxEvals = 500; % maximum Number of evaluations Alfa = 1.3; % Reflection distance Rfak = 0.3; % Randomization factor Gamma = 0.3; % Forgetting factor TolFunc = ; % Tolerance for function convergence TolX = ; % Tolerance for parameter convergence IterMax = 30; % Max iterations when creating a new point b=4; % Constant when creating new points % Check that we have sufficient input if nargin < 3, error('complex requires at least three input arguments'); end % Make sure that the we have the right limits on the variables n1=length(x_lower); n2=length(x_upper); if n1 ~= n2, error('upper and lower limits must have the same number of arguments'); end if ~min((max(x_upper,x_lower) == (x_upper))), error ('The upper limits must be larger then the lower limits'); end Nparams = n1; k=2*nparams; % Number of optimization parameters % number of points in the Complex if Nparams == 1, k=3; end fprintf('\n** Complex optimization started **\n') National University of Singapore E - 2

58 Appendix E Complex RF % assign parameter values if they are given in the function call if nargin > 3, [n2 n3] = size(varargin); %nn=floor(n3/2) fprintf('the following parameters have been supplied by the user.\n') for i=1:2:n3, if strcmpi(varargin{i},'alfa'), Alfa=varargin{i+1}; fprintf('alfa = %g \n',varargin{i+1}); end if strcmpi(varargin{i},'gamma'), Gamma=varargin{i+1}; fprintf('gamma = %g \n',varargin{i+1}); end if strcmpi(varargin{i},'maxeval'), MaxEvals=varargin{i+1}; fprintf('maxeval = %g \n',varargin{i+1}); end if strcmpi(varargin{i},'rfak'), Rfak=varargin{i+1}; fprintf('rfak = %g \n',varargin{i+1}); end if strcmpi(varargin{i},'tolx'), TolX=varargin{i+1}; fprintf('tolx = %g \n',varargin{i+1}); end end if strcmpi(varargin{i},'tolfunc'), TolFunc=varargin{i+1}; fprintf('tolfunc = %g \n',varargin{i+1}); end if strcmpi(varargin{i},'k'), k=varargin{i+1}; fprintf('k = %g \n',varargin{i+1}); end end fprintf('\nnumber of evaluations \n'); fprintf('%5.0f ',1); National University of Singapore E - 3

59 Appendix E Complex RF % Start of Complex method exitflag = -1; conv_cond = 0; NoEvals = 0; Iterations = 0; jfmin=1; jfmax=1; % kf implements the forgetting principle. Each objective value will be increased with kf each iteration kf = 1 - (Alfa/2)^(Gamma/k); % Create initial Complex x=ones(k,1)*x_lower + ones(k,1)*(x_upper-x_lower).*rand(k,nparams); % Convert to inline function as needed. Obj_fcn = fcnchk(obj_fcn,length(varargin)); % Evaluate function values for initial complex for i = 1:k, f(i)=feval(obj_fcn,x(i,:)); end allx=x; % Store all x values allf=f'; % Store all f values allx(:,nparams+1) = abs(max((max(x)-min(x))./(x_upper-x_lower))); % store the spread of the initial comlex allf(:,2) = abs(max(f)-min(f)); % store the spread in function values %Do the Complex iteration while NoEvals < MaxEvals, % Check convergence if min (f) == 0, if abs(max(f)-min(f)) <= TolFunc, conv_cond = 1; break; end elseif abs(max(f)-min(f))/abs(min(f)) <= TolFunc, conv_cond = 1; break; end if abs((max((max(x)-min(x))./(x_upper-x_lower)))) <= TolX, conv_cond = 2; break; end National University of Singapore E - 4

60 Appendix E Complex RF % Increase all f-values with a factor kf. This is the forgetting principle. f = f + (max(f)-min(f))*kf; % Identify best and worst point [fmin jfmin] = min(f); [fmax jfmax] = max(f); % Calculate centroid xc = (sum(x) - x(jfmax,:)) / (k-1); %refelct worst point through centroid xnew_1 = xc + (xc-x(jfmax,:)).*alfa; % Add some random noise to the new point xnew_2 = xnew_1 + Rfak.*(x_upper -x_lower)*max((max(x)-min(x))./(x_upperx_lower)).*(rand(1,nparams)-0.5); % Make sure that xnew_2 is within the limits xnew_3=min(xnew_2,x_upper); xnew=max(xnew_3,x_lower); %Replace the worst point by the new one x(jfmax,:) = xnew; % Evaluate the new point f(jfmax)=feval(obj_fcn,x(jfmax,:)); NoEvals = NoEvals + 1; % See if the new point is still the worst. [fmax jfmax_new] = max(f); iter = 0; while (jfmax_new == jfmax) & (iter < IterMax) & (NoEvals < MaxEvals), a1 = 1 -exp(-iter/b); xnew_2 = ((xc*(1-a1) + x(jfmin,:)*a1) + xnew)./2 + Rfak.*(x_upper - x_lower)*max((max(x)-min(x))./(x_upper-x_lower)).*(rand(1,nparams)-0.5); % Make sure that xnew_2 is within the limits xnew_3=min(xnew_2,x_upper); xnew=max(xnew_3,x_lower); %Replace the worst point by the new one x(jfmax,:) = xnew; National University of Singapore E - 5

61 Appendix E Complex RF % Evaluate the new point f(jfmax)=feval(obj_fcn,x(jfmax,:)); % See if the new point is still the worst. [fmax jfmax_new] = max(f); iter = iter + 1; NoEvals = NoEvals + 1; end % Keep on reflecting the worst point until it is not worst any more. Iterations = Iterations + 1; jfmax = jfmax_new; % store the evolution of the optimization in the variables allx and allf allx(iterations+k,1:nparams)=x(jfmax,:); allf(iterations+k,1)=f(jfmax); allx(iterations+k,nparams+1) = abs(max((max(x)-min(x))./(x_upper-x_lower))); allf(iterations+k,2) = abs(max(f)-min(f)); fprintf('\b\b\b\b\b\b'); fprintf('%5.0f ',NoEvals); end % Main Complex loop fprintf('\n\n** Optimization stopped **\n\n') if conv_cond == 2 fprintf('convergence in parameter values\n\n') output.convergence = 'Convergence in parameter values'; elseif conv_cond == 1 fprintf('convergence in function values\n\n') output.convergence = 'Convergence in function values'; else fprintf('max number of evaluation reached\n\n') output.convergence = 'Max number of evaluation reached'; end fprintf('number of evaluations = %g.\n', NoEvals) fprintf('number of iterations = %g.\n', Iterations) fprintf('minimum function value = %g.\n',f(jfmin)) fprintf('maximum function value = %g.\n',f(jfmax)) fprintf('best point found at: \t \t min(x) max(x) \n') for i=1:nparams, fprintf('x(%g) = %8.4g %8.4g \t %8.4g\n',i,x(jfmin,i),min(x(:,i)),max(x(:,i))); end National University of Singapore E - 6

62 Appendix E Complex RF xopt = x(jfmin,:); func = f(jfmin); x_hist=allx; f_hist=allf; output.iterations = Iterations; output.funccount = NoEvals; output.algorithm = 'Complex algorithm direct search'; output.fmin = f(jfmin); output.fmax = f(jfmax); if NoEvals >= MaxEvals, exitflag = 0; elseif (conv_cond == 2) exitflag = 1; elseif (conv_cond == 1), exitflag = 2; end National University of Singapore E - 7

63 Appendix F MATLAB Codes MATLAB CODES LONGITUDINAL Elev_over_simu.m x_up = [1, 1, 1]; % Upper limits of gains x_low = [-1, -1, -1]; % Lower limit of gains [X, F]=complexrf('Elev_ovr_simfcn', x_low, x_up); Elev_over_simfcn.m function ObjVal = Elev_ovr_simfcn(ParamValue); % ParamValue - Vector containing the model parameter values % Output parameters: % ObjVal - a scalar containing the objective function value. % The name of the simulink model MODEL = 'Elevator_ovr'; % Move variables into model parameter names Kp = ParamValue(1); Ki = ParamValue(2); Kd = ParamValue(3); opt = simset('solver','ode45','srcworkspace','current'); [tout,xout,yout] = sim(model,[0 100],opt); % [0 100]= simulation start and stop time yout; % Assign objective function value as the last value in y. ObjVal=sumsqr(yout-1); National University of Singapore F - 1

64 Appendix F MATLAB Codes LATERAL Ail_over_simu.m x_up = [1, 1, 1]; % Upper limits of gains x_low = [-1, -1, -1]; % Lower limit of gains [X, F]=complexrf('Ail_ovr_simfcn', x_low, x_up); Ail_over_simfcn.m function ObjVal = Ail_ovr_simfcn(ParamValue); % ParamValue - Vector containing the model parameter values % Output parameters: % ObjVal - a scalar containing the objective function value. % The name of the simulink model MODEL = 'Aileron_ovr'; % Move variables into model parameter names Kp = ParamValue(1); Ki = ParamValue(2); Kd = ParamValue(3); opt = simset('solver','ode45','srcworkspace','current'); [tout,xout,yout] = sim(model,[0 100],opt); % [0 100]= simulation start and stop time yout; % Assign objective function value as the last value in y. ObjVal=sumsqr(yout-1); National University of Singapore F - 2

65 Appendix G Sensitivity Test SENSITIVITY TEST Longitudinal Coefficients: Response to 10% change in value Sensitivity Cx α 0.23% 3.38% Cx δe 0.00% 0.00% Cx q -0.02% 22.70% Cz α 12.59% -2.82% Cz δe 0.00% 0.00% Cz q 0.11% % Cm α -8.39% 38.57% Cm δe 0.00% 0.00% Cm q -0.88% % Table G.1. Sensitivity of Longitudinal coefficients Lateral Coefficients: Response to 10% change in value Sensitivity Cy β -4.10% 11.57% Cy δa 0.00% 0.00% Cy p 0.01% % Cy r 0.01% 5.34% Cl β 10.12% % Cl δa 0.00% 0.00% Cl p % % Cl r -0.25% % Cn β -0.28% % Cn δa 0.00% 0.00% Cn p 0.60% % Cn r -0.19% % Table G.2. Sensitivity of Lateral coefficients National University of Singapore G - 1

66 MOTOR AND PROPELLER SELECTION MOTOR SELECTION Appendix H Motor and Propeller Selection Himax HA2015 series brushless motors - Speed 280/300 replacement Slotless design for high efficiency. High performance replacement for Speed 280/300 type can motors, For light electric planes under 20oz or 3-D performance under 13oz, Fit most slowflyer/parkflyers _ GWS planes, micro helicopters - Hummingbird & Piccolo, micro cars - mini T & HPI micro, Himax HA2025 series brushless motors - Speed 370/400/480 replacement Slotless design for high efficiency. High performance replacement for Speed 370/400/480 type can motors, For light electric planes under 30oz or 3-D performance under 18oz, Perfect for 3-D aerobatic, pylon race, and any application maximum power/weight ratio is needed, Himax HA2825 series brushless motors Slotless design for high efficiency. High performance replacement for 500/550 type can motor, Gear motors are suitable for small electric trainers, sport fun flyers, Direct drive motors are best for ducted fans & pylon racers where high rpm is a must, For light electric planes under 70oz. CHOSEN *Taken from: Table H.1. Comparison of Brushless Motors National University of Singapore H - 1

67 Appendix H Motor and Propeller Selection THRUST CALCULATION AND PROPELLER SELECTION An experiment was set up to measure the amount of thrust exerted by the motor. This setup uses the Principle of moments to find the amount of thrust exerted. By the Principle of moments, the sum of the moments on the left hand side should be equal to that on the right hand side when the beam is balanced. In this experiment, the equipment were set up as shown below, with the motor at 25cm to the left of the pivot, and the weighing machine on the 50cm mark to the other side. From Newton s 3 rd law, every action has an equal and opposite reaction, the weight shown by the weighing machine should be the force exerted on the weight to cause the moment on the right hand side. Therefore Thrust x 25cm = Weight x 50cm Thrust = Weight x 2 The results of the experiment are shown below 25cm 50cm Thrust Weight Figure H.1. Thrust calculation and propeller selection setup Thrust (N) Motor Prop Idle Mid Max Himax 9x x x x x CHOSEN Table H.2. Results of Thrust Calculation National University of Singapore H - 2