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Trajectory Optimization for Helicopter Unmanned Aerial Vehicles (Uavs)

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Trajectory Optimization for Helicopter Unmanned Aerial Vehicles (Uavs) Calhoun: The NPS Institutional Archive Theses and Dissertations Thesis Collection 2010-06 Trajectory optimization for helicopter Unmanned Aerial Vehicles (UAVs) Gatzke, Benjamin Thomas Monterey, California. Naval Postgraduate School http://hdl.handle.net/10945/5315 NAVAL POSTGRADUATE SCHOOL MONTEREY, CALIFORNIA THESIS TRAJECTORY OPTIMIZATION FOR HELICOPTER UNMANNED AERIAL VEHICLES (UAVs) by Benjamin Thomas Gatzke June 2010 Thesis Advisor: Wei Kang Second Reader: Hong Zhou Approved for public release; distribution is unlimited THIS PAGE INTENTIONALLY LEFT BLANK REPORT DOCUMENTATION PAGE Form Approved OMB No. 0704-0188 Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instruction, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington, VA 22202-4302, and to the Office of Management and Budget, Paperwork Reduction Project (0704-0188) Washington DC 20503. 1. AGENCY USE ONLY (Leave blank) 2. REPORT DATE 3. REPORT TYPE AND DATES COVERED June 2010 Master’s Thesis 4. TITLE AND SUBTITLE 5. FUNDING NUMBERS Trajectory Optimization for Helicopter Unmanned Aerial Vehicles (UAVs) 6. AUTHOR(S) Benjamin Thomas Gatzke 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 8. PERFORMING ORGANIZATION Naval Postgraduate School REPORT NUMBER Monterey, CA 93943-5000 9. SPONSORING /MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSORING/MONITORING N/A AGENCY REPORT NUMBER 11. SUPPLEMENTARY NOTES The views expressed in this thesis are those of the author and do not reflect the official policy or position of the Department of Defense or the U.S. Government. 12a. DISTRIBUTION / AVAILABILITY STATEMENT 12b. DISTRIBUTION CODE Approved for public release; distribution is unlimited 13. ABSTRACT (maximum 200 words) This thesis explores the numerical methods and software development for optimal trajectories of a specific model of Helicopter Unmanned Aerial Vehicle (UAV) in an obstacle-rich environment. This particular model is adopted from the UAV Laboratory of the National University of Singapore who built and simulated flights for an X-Cell 60 small–scale UAV Helicopter. The code, which allowed the team to simulate flights, is a complex system of non-linear differential equations—15 state variables and four control variables—used to maneuver the state trajectories. This non-linear model is incorporated into a separate optimization algorithm code, which allows the user to set initial and final time conditions together with various constraints, and, using the same variable scheme, optimize a trajectory. The optimal trajectory is defined by using a cost function—the performance measure—and the system is subject to a set of constraints (such as mechanical limitations and physical three-dimensional obstacles). Simulations conclude that solutions are readily obtained; however, it is still very difficult to derive trajectories that are truly optimal, and our work calls for more future research in computational programs for optimal trajectory planning. All simulations in this thesis are modeled using the MATLAB program. 14. SUBJECT TERMS 15. NUMBER OF Nonlinear Model, Trajectory Optimization, State and Control Variables, Cost Function PAGES 77 16. PRICE CODE 17. SECURITY 18. SECURITY 19. SECURITY 20. LIMITATION OF CLASSIFICATION OF CLASSIFICATION OF THIS CLASSIFICATION OF ABSTRACT REPORT PAGE ABSTRACT Unclassified Unclassified Unclassified UU NSN 7540-01-280-5500 Standard Form 298 (Rev. 8-98) Prescribed by ANSI Std. Z39.18 i THIS PAGE INTENTIONALLY LEFT BLANK ii Approved for public release; distribution is unlimited TRAJECTORY OPTIMIZATION FOR HELICOPTER UNMANNED AERIAL VEHICLES (UAVs) Benjamin T. Gatzke Major, United States Army B.S., United States Military Academy at West Point, May 1999 Submitted in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE IN APPLIED MATHEMATICS from the NAVAL POSTGRADUATE SCHOOL June 2010 Author: Benjamin Thomas Gatzke Approved by: Wei Kang Thesis Advisor Hong Zhou Second Reader Carlos Borges Chairman, Department of Applied Mathematics iii THIS PAGE INTENTIONALLY LEFT BLANK iv ABSTRACT This thesis explores the numerical methods and software development for optimal trajectories of a specific model of Helicopter Unmanned Aerial Vehicle (UAV) in an obstacle-rich environment. This particular model is adopted from the UAV Laboratory of the National University of Singapore who built and simulated flights for an X-Cell 60 small-scale UAV Helicopter. The code, which allowed the team to simulate flights, is a complex system of non-linear differential equations—15 state variables and four control variables—used to maneuver the state trajectories. This non-linear model is incorporated into a separate optimization algorithm code, which allows the user to set initial and final time conditions together with various constraints, and, using the same variable scheme, optimize a trajectory. The optimal trajectory is defined by using a cost function—the performance measure—and the system is subject to a set of constraints (such as mechanical limitations and physical three-dimensional obstacles). Simulations conclude that solutions are readily obtained; however, it is still very difficult to derive trajectories that are truly optimal, and our work calls for more future research in computational programs for optimal trajectory planning. All simulations in this thesis are modeled using the MATLAB program. v THIS PAGE INTENTIONALLY LEFT BLANK vi TABLE OF CONTENTS I. LITERATURE REVIEW .................................................................................. 1 A. BACKGROUND ................................................................................... 1 B. ARTICLE REVIEWS ............................................................................ 1 C. CONCLUSION ..................................................................................... 6 II. PROBLEM FORMULATION........................................................................... 7 A. INTRODUCTION.................................................................................. 7 B. TRAJECTORY EQUATIONS............................................................... 7 1. Kinematics................................................................................ 8 2. Rigid-Body Dynamics.............................................................. 9 3. Main Rotor Flapping Dynamics ............................................ 11 4. Yaw Rate Gyro Dynamics...................................................... 12 C. CONTROLS ....................................................................................... 12 1. Aileron Servo Input ()δlat .................................................... 13 2. Elevator Servo Input ()δlon .................................................. 13 3. Rudder Servo Input ()δ ped .................................................. 13 4. Collective Pitch Servo Input ()δcol ..................................... 13 D. CONSTRAINT EQUATIONS.............................................................. 14 E. THE PERFORMANCE MEASURE .................................................... 21 III. PSEUDOSPECTRAL METHOD FOR OPTIMAL CONTROL....................... 27 A. INTRODUCTION................................................................................ 27 B. PROBLEM FORMULATION .............................................................. 27 IV. SIMULATIONS ............................................................................................. 35 A. INTRODUCTION................................................................................ 35 B. FLIGHT SIMULATION ....................................................................... 35 C. OPTIMIZATION EXAMPLE ............................................................... 37 D. OPTIMAL CONTROL OF THE HELICOPTER UAV.......................... 39 1. Minimize Time, No Obstacles ............................................... 40 V. CONCLUSIONS AND FUTURE RESEARCH .............................................. 55 A. CONCLUSIONS................................................................................. 55 B. FUTURE RESEARCH........................................................................ 57 LIST OF REFERENCES.......................................................................................... 59 INITIAL DISTRIBUTION LIST ................................................................................. 61 vii THIS PAGE INTENTIONALLY LEFT BLANK viii LIST OF FIGURES Figure 1. A short, block building with center point (0,0,2) and a=4, b=4, c=2, p=20 ................................................................................................... 16 Figure 2. A skyscraper type of building with center point (0,0,15) and a=4, b=6, c=15, p=40 ................................................................................. 17 Figure 3. A flat bridge with two end pillars (plot is constructed using three separate functions; one for each pillar and one for the bridge portion) ............................................................................................... 18 Figure 4. A suspension bridge with two end pillars
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