Frequency Domain System Identification of Fixed-Wing Unmanned Aerial Vehicles By KAIWEN XU 8/14/2014 A Thesis Submitted to the Faculty of Graduate Studies of the University of Manitoba In Partial Fulfillment of the Requirements of the Degree of MASTER OF SCIENCE Department of Mechanical Engineering The University of Manitoba Winnipeg, Manitoba Copyright 2014 © Kaiwen Xu Abstract The goal of this thesis is to identify airplanes’ reduced order transfer functions, and aerodynamic derivatives in the longitudinal channel. The outcome of the research will benefit aircraft systems’ controller design, modeling and simulation. Aircraft under study are a conventional fixed wing airplane called Cropcam, and a nonconventional delta-wing aircraft. To identify the system transfer functions and aerodynamic derivatives, direct and indirect frequency domain identification methods are applied. For the direct method, the Equation Error (EE) method is adopted to process the Cropcam’s input-output data pairs and identify the aerodynamic derivatives from the flight data directly. The indirect approach is called the Transfer Function (TF) method. For this method, a commercially available system identification tool kit called CIFER is utilized to identify the longitudinal transfer function of the aircraft first. Then the aerodynamic derivatives are extracted from the identified transfer function. The derivatives identified by the EE method and transfer function method are compared with the ones computed from a Vortex Lattice based program called AVL. The identification results are further verified by comparing computer simulation outputs with flight test responses. Issues such as input excitation design, data gathering, data reliability analysis and result verification are also investigated in this thesis. i | P age Table of Contents Abstract ............................................................................................................................................ i Acknowledgements ......................................................................................................................... v List of Figures ................................................................................................................................ vi List of Tables .................................................................................................................................. x Nomenclature ................................................................................................................................. xi 1 Introduction ............................................................................................................................. 1 1.1 Motivation ........................................................................................................................ 1 1.2 Objectives ......................................................................................................................... 2 1.3 Methodology .................................................................................................................... 3 1.4 Thesis outline ................................................................................................................... 6 2 Background .............................................................................................................................. 9 2.1 Preliminary remark ........................................................................................................... 9 2.2 System identification ...................................................................................................... 10 3 Basic Aerodynamic Theory ................................................................................................... 14 3.1 Coordinate axes .............................................................................................................. 14 3.1.1 Earth axes ................................................................................................................ 14 3.1.2 Aircraft body-fixed axes ......................................................................................... 16 3.2 Definition of variables .................................................................................................... 17 3.3 Axes transformations...................................................................................................... 21 3.3.1 Linear acceleration, velocity and displacement transformation ............................. 21 3.3.2 Angular velocities transformation........................................................................... 26 3.4 Aircraft reference geometry and control surfaces .......................................................... 28 3.4.1 Reference geometry ................................................................................................ 28 3.4.2 Aircraft control surfaces ......................................................................................... 29 3.5 Summary ........................................................................................................................ 30 4 Aircraft Motion Equations ..................................................................................................... 31 4.1 General equations of motion .......................................................................................... 31 4.1.1 Force equations ....................................................................................................... 38 4.1.2 Moment equations ................................................................................................... 40 4.1.3 Kinematics equations .............................................................................................. 41 ii | P age 4.1.4 Navigation equations .............................................................................................. 41 4.2 Aerodynamic terms ........................................................................................................ 43 4.3 Equations of motion for small perturbations .................................................................. 45 4.4 Decoupled motion equations .......................................................................................... 47 4.4.1 Longitudinal motion equations ............................................................................... 47 4.4.2 Lateral motion equations......................................................................................... 51 4.5 Summary ........................................................................................................................ 53 5 Input Excitation ..................................................................................................................... 54 5.1 Input excitation requirements of system identification .................................................. 54 5.1.1 Frequency range ...................................................................................................... 54 5.1.2 Length of input excitation ....................................................................................... 56 5.1.3 Amplitude of input signal ....................................................................................... 57 5.1.4 Flight test conditions ............................................................................................... 59 5.2 Optimal input design ...................................................................................................... 61 5.3 Summary ........................................................................................................................ 69 6 Estimation of Aerodynamic Derivatives ............................................................................... 71 6.1 Transfer function method ............................................................................................... 72 6.1.1 Obtaining transfer function empirically .................................................................. 73 6.1.2 Obtaining transfer function analytically ................................................................. 77 6.1.3 Applying the transfer function method ................................................................... 82 6.2 Equation error method .................................................................................................... 85 6.2.1 Data transformation ................................................................................................ 86 6.2.2 Equation error method in frequency domain .......................................................... 87 6.3 Summary ........................................................................................................................ 91 7 Experimental Results ............................................................................................................. 93 7.1 Identification of Cropcam UAV ..................................................................................... 94 7.1.1 Transfer function method ...................................................................................... 103 7.1.2 Equation error method .......................................................................................... 115 7.2 Identification of a delta-wing UAV ............................................................................. 148 7.3 Summary ...................................................................................................................... 157 8 Conclusions ......................................................................................................................... 159 8.1 Contributions made by this thesis ................................................................................ 159 8.2 Future research ............................................................................................................. 161 iii | P age References ................................................................................................................................... 163 Appendix 1 Derivatives Conversion ........................................................................................... 166 Appendix 2 Simulation Results .................................................................................................. 171 A2.1 Simulation responses .................................................................................................... 171 A2.2 Identification using transfer function based method .................................................... 180 A2.3 Identification using equation error method .................................................................. 189 iv | P age Acknowledgements First, I would like to thank my supervisor Dr. Nariman Sepehri and co-advisor Dr. Hamidreza Bolandhemmat, whose invaluable guidance and support has helped steer my work in the right direction over the past two years. I also thank the staffs from MicroPilot Inc., especially Howard Loewen (president of the company), Asadilan Indrabudi and Adam Toews for providing me technical support and fruitful suggestions. I thank my colleagues from the Fluid Power & Tele- Robotics Research Laboratory, especially Morgan May and Divyarajsinh Raol, for their kind help during this research. I also thank NSERC, MITACS and MicroPilot Inc., for sponsoring the research. Finally, sincere thanks go to Drs. W Kinsner and S Balakrishnan, for the examination of this thesis and their constructive comments. v | P age List of Figures Figure 1.1 Frequency-response system identification procedure ................................................... 6 Figure 3.1 Earth axes (adapted from Cook, 2012) ........................................................................ 15 Figure 3.2 General aircraft coordinate system .............................................................................. 16 Figure 3.3 Aircraft motion variables ............................................................................................. 18 Figure 3.4 Longitudinal channel terms for typical aircraft climbing ............................................ 19 Figure 3.5 Quantities in body axes under symmetric flight (adapted from Cook, 2012) ............. 20 Figure 3.6 Axes transformation (adapted from Cook, 2012) ........................................................ 21 Figure 3.7 Transformation of velocities through incidence and sideslip angles (Cook, 2012) .... 24 Figure 3.8 Transformation of angular velocity (adapted from Cook, 2012) ................................ 26 Figure 3.9 Reference geometry ..................................................................................................... 28 Figure 3.10 Aircraft control surfaces (adapted from Cook, 2012) ............................................... 29 Figure 5.1 Elevator deflection....................................................................................................... 58 Figure 5.2 Body pitch rate ............................................................................................................ 58 Figure 5.3 Doublet input excitation .............................................................................................. 60 Figure 5.4 Body pitch rate ............................................................................................................ 60 Figure 5.5 Typical 3-2-1-1 signal ................................................................................................. 62 Figure 5.6 Frequency sweep input excitation ............................................................................... 64 Figure 5.7 Increasing trend of sweep frequency ........................................................................... 66 Figure 5.8 Typical Chirp input excitation ..................................................................................... 68 Figure 5.9 Finite Fourier transform of input signal ...................................................................... 69 Figure 6.1 Short-period mode flight (adapt from Cook, 2012)..................................................... 78 Figure 6.2 Phugoid mode flight (adapt from Cook, 2012) ........................................................... 78 Figure 6.3 Aircraft responses (w, q, α) to a 1° elevator step input ............................................... 79 Figure 6.4 Aircraft responses (u, θ, ) to a 1° elevator step input ................................................ 80 Figure 7.1 Cropcam aircraft ..........(cid:1)................................................................................................ 94 Figure 7.2 Manually generated Chirp input excitation ................................................................. 96 Figure 7.3 FFT of input signal ...................................................................................................... 97 Figure 7.4 Total axial velocity U .................................................................................................. 98 Figure 7.5 Total lateral velocity V ................................................................................................ 98 vi | P age Figure 7.6 Total normal velocity W .............................................................................................. 99 Figure 7.7 Body roll rate p ............................................................................................................ 99 Figure 7.8 Body pitch rate q ....................................................................................................... 100 Figure 7.9 Body yaw rate r ......................................................................................................... 100 Figure 7.10 Roll angle φ ............................................................................................................. 101 Figure 7.11 Pitch angle θ ............................................................................................................ 101 Figure 7.12 Yaw angle ψ ............................................................................................................ 102 Figure 7.13 Altitude z ................................................................................................................. 102 Figure 7.14 -q data pair ........................................................................................................... 104 Figure 7.15 (cid:2)(cid:3)-w data pair .......................................................................................................... 104 Figure 7.16 B(cid:2)(cid:3)ode plot of -w .................................................................................................... 105 Figure 7.17 Bode plot of (cid:2)(cid:3)-q..................................................................................................... 105 Figure 7.18 Verification o(cid:2)f(cid:3) magnitude of -w .......................................................................... 107 Figure 7.19 Verification of phase of -w(cid:2) ..(cid:3)................................................................................ 107 Figure 7.20 Verification of magnitud(cid:2)e(cid:3) of -q ........................................................................... 108 Figure 7.21 Verification of phase of -q (cid:2)..(cid:3)................................................................................ 108 Figure 7.22 Comparison of simulated(cid:2) (cid:3)w and measured w .......................................................... 109 Figure 7.23 Comparison of simulated q and measured q............................................................ 110 Figure 7.24 Unseen Doublet signal ............................................................................................. 111 Figure 7.25 Comparison of simulated w and measured w with the unseen Doublet signal ........ 112 Figure 7.26 Comparison of simulated q and measured q with the unseen Doublet signal ......... 112 Figure 7.27 Delay removed -q data pair ................................................................................. 116 Figure 7.28 Delay removed (cid:2)(cid:3)-w data pair ................................................................................ 117 Figure 7.29 Delay removed (cid:2)(cid:3)-u data pair ................................................................................. 117 Figure 7.30 Original data sam(cid:2)(cid:3)pling time .................................................................................... 118 Figure 7.31 Verification of EE identified pitching moment related derivatives ......................... 122 Figure 7.32 Verification of AVL estimated pitching moment related derivatives ..................... 123 Figure 7.33 Unseen random signal used for verification ............................................................ 124 Figure 7.34 Verification of EE identified pitching moment related derivatives with unseen random signal .............................................................................................................................. 125 Figure 7.35 Verification of AVL computed pitching moment related derivatives with unseen random signal .............................................................................................................................. 126 Figure 7.36 Unseen Doublet signal ............................................................................................. 127 vii | Page Figure 7.37 Verification of EE identified pitching moment related derivatives with unseen Doublet signal ............................................................................................................................. 128 Figure 7.38 Verification of AVL computed pitching moment related derivatives with unseen Doublet signal ............................................................................................................................. 129 Figure 7.39 Verification of EE identified lift force related derivatives with original signal ...... 131 Figure 7.40 Verification of AVL computed lift force related derivatives with original signal .. 132 Figure 7.41 Verification of EE identified lift force related derivatives with unseen random signal ..................................................................................................................................................... 133 Figure 7.42 Verification of AVL computed lift force related derivatives with unseen random signal ........................................................................................................................................... 134 Figure 7.43 Verification of EE identified lift force related derivatives with unseen Doublet signal ..................................................................................................................................................... 135 Figure 7.44 Verification of AVL computed lift force related derivatives with unseen Doublet signal ........................................................................................................................................... 136 Figure 7.45 Verification of EE identified axial force related derivatives with original signal ... 138 Figure 7.46 Verification of AVL computed axial force related derivatives with original signal 139 Figure 7.47 Verification of EE identified axial force related derivatives with unseen random signal ........................................................................................................................................... 140 Figure 7.48 Verification of AVL computed axial force related derivatives with unseen random signal ........................................................................................................................................... 141 Figure 7.49 Verification of EE identified axial force related derivatives with unseen Doublet signal ........................................................................................................................................... 142 Figure 7.50 Verification of AVL computed axial force related derivatives with unseen Doublet signal ........................................................................................................................................... 143 Figure 7.51 A delta-wing UAV (From uas.trimble.com) ........................................................... 148 Figure 7.52 Input excitation for delta-wing aircraft.................................................................... 149 Figure 7.53 Measured output of the delta-wing aircraft ............................................................. 149 Figure 7.54 Finite Fourier transform plot of input excitation ..................................................... 150 Figure 7.55 Sampling time .......................................................................................................... 151 Figure 7.56 Bode plot of δ -q ...................................................................................................... 152 e Figure 7.57 Comparison of magnitudes ...................................................................................... 153 Figure 7.58 Comparison of phases.............................................................................................. 153 Figure 7.59 Comparison of simulated output and measured output ........................................... 154 Figure 7.60 Unseen input excitation used for transfer function verification .............................. 155 Figure 7.61 Comparison of simulated and measured outputs with unseen input signal ............. 155 Figure A 1 Input excitation ......................................................................................................... 172 viii | Page Figure A 2 Total velocity V ........................................................................................................ 173 Figure A 3 Angle of attack α ...................................................................................................... 173 Figure A 4 Sideslip angle β ......................................................................................................... 174 Figure A 5 Roll rate p ................................................................................................................. 174 Figure A 6 Pitch rate q ................................................................................................................ 175 Figure A 7 Yaw rate r ................................................................................................................. 175 Figure A 8 Roll angle φ .............................................................................................................. 176 Figure A 9 Pitch angle θ ............................................................................................................. 176 Figure A 10 Yaw angle ψ ........................................................................................................... 177 Figure A 11 Axial position x ....................................................................................................... 177 Figure A 12 Lateral position y .................................................................................................... 178 Figure A 13 Altitude z................................................................................................................. 178 Figure A 14 Chirp input excitation ............................................................................................. 180 Figure A 15 FFT of input excitation ........................................................................................... 181 Figure A 16 Bode plot of -w ................................................................................................... 182 Figure A 17 Bode plot of (cid:2)(cid:3)-q .................................................................................................... 183 Figure A 18 Magnitude of(cid:2) (cid:3) -w ................................................................................................. 184 Figure A 19 Phase of -w(cid:2) ..(cid:3)....................................................................................................... 185 Figure A 20 Magnitud(cid:2)e(cid:3) of -q .................................................................................................. 185 Figure A 21 Phase of -q .(cid:2)..(cid:3)....................................................................................................... 186 Figure A 22 Verificati(cid:2)o(cid:3)n of vertical velocity w .......................................................................... 187 Figure A 23 Verification of pitch rate q ...................................................................................... 187 ix | P age