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Simulation and Control Design of a Gliding Autogyro for Precision Airdrop by Joshua F

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Simulation and Control Design of a Gliding Autogyro for Precision Airdrop by Joshua F Simulation and Control Design of a Gliding Autogyro for Precision Airdrop by Joshua F. Torgerson Submitted to the Department of Aeronautics and Astronautics in partial fulfillment of the requirements for the degree of Master of Science in Aeronautics and Astronautics at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2005 @ Joshua F. Torgerson, MMV. All rights reserved. The author hereby grants to MIT permission to reproduce and distribute publicly paper and electronic copies of this thesis document in whole or in part. MASSACHUSETTS INS E OF TECHNOLOGY JUN 2 3 2OO5 Author ...... IBRARIES Department of Aeronautics and Astronautics May 6, 2005 Certified by ........... ... ........ John J. Deyst Professor of Aeronautics and Astronautics Thesis Supervisor C ertified by ........... ................. Sean George Member Technical Staff, Draper Laboratory Thesis Supervisor A ccepted by .......................................... Jaime Peraire Chairman, Department Committee on Graduate Students AERO i [This page intentionally left blank.] Simulation and Control Design of a Gliding Autogyro for Precision Airdrop by Joshua F. Torgerson Submitted to the Department of Aeronautics and Astronautics on May 6, 2005, in partial fulfillment of the requirements for the degree of Master of Science in Aeronautics and Astronautics Abstract Precision airdrop is a technology whose required capabilities have become more exact- ing as combat situations necessitate greater degrees of accuracy. Ballistic and parafoil type delivery vehicles do not have the capacity to consistently deliver a payload on, for example, a particular rooftop in an urban combat situation. A gliding autogyro delivery platform has been investigated as a means of achieving greater airdrop performance. The autogyro has similar gliding characteristics to the parafoil, but has improved wind resilience and control authority. An initial simulation, based on momentum and blade element helicopter theory, has been constructed. A classical controller using a multiple loop closure strategy has been developed that uses a new nonlinear guidance law to follow paths generated by an algorithm considering initial conditions. An extended Kalman filter is used for state estimation. Results from simulations show consistent accuracy of about 5 feet, with the final position error rarely exceeding 10 feet. Thesis Supervisor: John J. Deyst Title: Professor of Aeronautics and Astronautics Thesis Supervisor: Sean George Title: Member Technical Staff, Draper Laboratory 3 [This page intentionally left blank.] Acknowledgments Acknowledgment May 6, 2005 This thesis was prepared at The Charles Stark Draper Laboratory, Inc., under Internal Company Sponsored Research in the Autonomous Vehicles Laboratory, Project Number 5042. Publication of this thesis does not constitute approval by Draper of the findings or conclusions contained herein. It is published for the exchange and stimulation of ideas. Joshua F. Torgerson 5 [This page intentionally left blank.] Contents 1 Introduction 19 2 Simulation Development 23 2.1 Equations of Motion ... .... ... .... ... .... ... ... 23 2.1.1 Quaternions ... ... .... .... .... .... .... .. 26 2.1.2 W ind .. ... .... ... ... .... ... ... .... ... 27 2.2 Forces and Moments ... .... ... ... ... ... ... ... .. 28 2.2.1 Rotor Forces and Moments .. ... .... ... ... ... .. 28 2.2.2 Induced Velocity Near Vertical Descent .. .... ... ... 32 2.2.3 Aerodynamic Forces and Moments . ... ... ... ... .. 33 2.3 Implementation .. .. ... ... ... .. ... ... ... .. ... 35 2.4 Validation . ... .... ... ... ... ... ... ... ... ... 36 3 Guidance Control Design 39 3.1 Linearization .. .... ... ... ... ... ... ... ... ... 39 3.2 Modal Analysis ... .... ... .... ... ... .... ... ... 40 3.3 Control Structure ... ... .. ... ... ... .. ... ... ... 43 3.4 Longitudinal Control Design ... ... ... ... ... ... ... .. 44 3.5 Lateral Control Design . ... .. ... .. ... .. ... .. ... .. 49 3.6 Path Planning .. .. ... .. ... .. ... .. ... .. ... .. .. 54 3.6.1 Steady Wind ... ... ... ... ... ... ... ... ... 57 3.7 Simulation Results . ... ... ... .. ... ... ... .. ... .. 58 7 4 Estimation 61 4.1 Extended Kalman Filter . ...... ...... ..... ...... 61 4.2 Filter Dynamics .............. ................ 61 4.3 Time Propagation. ....... ........ ....... ...... 63 4.4 Measurement Updates ..... ........ ....... ...... 64 4.4.1 G P S . ........... .......... ...... 65 4.4.2 Magnetometer. ..... ....... ....... ....... 65 4.4.3 Range Finder ... ..................... ... 66 4.4.4 Pixel Position ... ....... ....... ....... ... 66 4.5 Estimator Performance .. ....... ........ ........ 69 5 Results 73 5.1 Parameter Setup ........ ........ ....... ...... 73 5.2 Accuracy. ....... ........ ........ ....... ... 74 6 Conclusion 75 A Solutions for Thrust and Induced Velocity 77 A.1 Iterative Solution ....... ........... ........... 77 A.2 Quartic Solution .. ....... ........ ........ .... 79 B Simulation Details 81 B.1 Parameters ........ ....... ........ ........ 81 B.2 Trim Point ... ....... ........ ........ ....... 82 B.3 Linear Model .. ........ ........ ....... ...... 82 C Estimator Definition 85 C.1 Error Covariance Propagation .. .... ............... 85 C.2 Linearized Measurement Equations ........... ........ 88 C.2.1 G PS ....... ...... .............. ..... 89 C.2.2 Magnetometer ....... ..... .............. 89 C.2.3 Range Finder .... .... .......... ......... 89 8 C .2.4 C am era Im age .. .. .. ... .. .. .. .. .. .. .. .. .. 90 9 [This page intentionally left blank.] List of Figures 1-1 GyroBee model aircraft and coordinate system (reprinted with permis- sion from the Autogyro Company of Arizona) .... ........ 20 2-1 Comparison of theory and simulation trim response ...... .... 36 2-2 Trim forward rotor tilt input and glide slope ... ........ ... 37 3-1 Comparison of linear and nonlinear impulse responses ... ...... 40 3-2 Eigenvalues of the linear system on the imaginary plane .... .... 41 3-3 Modal response of linear system .. .... ..... ..... ..... 42 3-4 Nonlinear guidance diagram .. ..... .... ..... ..... .. 44 3-5 Control structure .... ...... ...... ....... ...... 45 3-6 Longitudinal glide slope controller .... ..... ..... ..... 45 3-7 Bode plots of successive longitudinal inner loop transfer functions, from 6f to the controlled variable ........ ........ ....... 46 3-8 Eigenvalues of inner-loop controlled longitudinal system ...... .. 47 3-9 Response of linear model to step in y . ....... ...... .. 48 3-10 Response of linear model to initial altitude offset . ..... ..... 49 3-11 Lateral bank controller ........ ........ ......... 50 3-12 Bode plots of successively closed lateral inner loop transfer functions, from 6, to the controlled variable .... ....... ....... .. 51 3-13 Eigenvalues of inner-loop controlled lateral system ..... ...... 52 3-14 Response of linear model to step in # ... ...... ...... .. 52 3-15 Response of linear model to various commanded paths .. ...... 53 3-16 Trajectory Setup . ..... ...... .... ..... ..... ... 54 11 3-17 Detailed geometry of the first turn in the commanded path .. .. 56 3-18 Path generation with and without wind U, = -2ft/s, W, = 10ft/s . 58 3-19 Controlled simulation result . .. .. .. .. .. .. 59 4-1 Geometry of camera showing vector relations . .. .. .. .. 68 4-2 Camera image details . .. .. .. .. .. .. .. .. .. 68 4-3 Results of baseline estimator simulation . .. .. .. .. .. 70 5-1 Final absolute position error of several simulation runs .. .. .. 74 A-1 Locus of roots of the induced velocity equation ((A.3)) with varying forward flight speeds .. .. .. .. .. .. .. .. .. .. 78 12 List of Tables 3.1 Eigenvalues of the linear system with their interpretations .. ... 43 3.2 Controller parameter summary .. ... ... .. ... .. ... .. 53 4.1 Root mean square values of estimation error during the entire and last ten seconds of the baseline estimator simulation .. .. ... .. .. 69 5.1 Variances of the normally distributed parameters for simulation trials 73 B.1 Simulation parameters ... .. .. ... .. .. ... .. .. ... .. 81 B.2 Simulation trim point . ... .. ... ... .. ... .. ... .. ... 82 13 [This page intentionally left blank.] List of Symbols C ............ direction cosine transformation matrix, body to local frames CDO .. .. .. .. zero lift drag coefficient of blades CT 2 ........... coefficient of thrust pwr,2 t~t tip F ............ force vector [X Y Z]T F ........... equivalent flat plate area of blades G............controller transfer function H ........... angular momentum vector I.............inertia matrix Ib... ......... inertia of rotor blade IX, I yI Izz . inertia KGE ......... ground effect constant L,M N..... torque around x, y, z axes Li.. ......... reference point distance in nonlinear lateral guidance law P, Q,R ...... body rotation rates about x, y, z axes Pi,P ...... .. induced and profile power of rotor T... ......... torque vector [L M N]T T... ......... thrust from rotor U, V, W ...... linear velocities in x, y, z directions Vi .. ......... induced velocity Vh .. ......... induced velocity in hover Vtip . ......... blade tip speed Qr Wb.. ......... aggregate velocity normal to blade W r .......... aggregate velocity normal to rotor disk X, Y, Z ...... forces in x, y, z directions X 0 s, Yv, Z., dimensional drag/lift coefficients Yuv, Zuw.. dimensional lift coefficient due to angle of attack 15 a ............ lift curve slope of rotor blade ao, ai, b1 ..... coning, forward, and side flapping angles of the rotor disk b.............number of blades c.............chord
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