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An Optimal Control Approach to Mission Planning Problems: a Proof of Concept Calhoun: The NPS Institutional Archive Theses and Dissertations Thesis and Dissertation Collection 2014-12 An optimal control approach to mission planning problems: a proof of concept Greenslade, Joseph Micheal Monterey, California: Naval Postgraduate School http://hdl.handle.net/10945/49611 NAVAL POSTGRADUATE SCHOOL MONTEREY, CALIFORNIA THESIS AN OPTIMAL CONTROL APPROACH TO MISSION PLANNING PROBLEMS: A PROOF OF CONCEPT by Joseph M. Greenslade December 2014 Thesis Co-Advisors: I. M. Ross M. Karpenko 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 December 2014 Master’s Thesis 4. TITLE AND SUBTITLE 5. FUNDING NUMBERS AN OPTIMAL CONTROL APPROACH TO MISSION PLANNING PROBLEMS: A PROOF OF CONCEPT 6. AUTHOR(S) Joseph M. Greenslade 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. IRB protocol number ____N/A____. 12a. DISTRIBUTION / AVAILABILITY STATEMENT 12b. DISTRIBUTION CODE Approved for public release; distribution is unlimited A 13. ABSTRACT (maximum 200 words) This work introduces the use of optimal control methods for simultaneous target sequencing and dynamic trajectory planning of an autonomous vehicle. This is achieved by deriving a control solution that minimizes a cost to maximize target selection. In the case of varying target priorities, the method described here preferentially selects higher priority targets to maximize total benefit in the allowed time horizon. Traditional techniques employ heuristic techniques for target sequencing and then apply a separate trajectory planning process to check the feasibility of the sequence for the given vehicle dynamics. This work uses pseudospectral methods to deterministically solve the problem in a single step. The historic barrier to the application of optimal control solutions to the target sequencing problem has been the requirement for a problem formulation built from continuous constraints. Target points are, by their nature, discrete. The key breakthrough is to transform the discrete problem into a continuous representation by modeling targets as Gaussian distributions. The proposed approach is successfully applied to several benchmark problems. Thus, several variations of the so-called motorized travelling salesman problem, including ones in which it is impossible to acquire all the targets in the given time, are solved using the new approach. 14. SUBJECT TERMS Optimization, Optimal Control, Travelling Salesman Problem, Orienteering 15. NUMBER OF Problem, Mission Planning, Path Selection, Target Selection, Target Priorities, Pseudospectral PAGES Methods, DIDO 107 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. 2–89) Prescribed by ANSI Std. 239–18 i THIS PAGE INTENTIONALLY LEFT BLANK ii Approved for public release; distribution is unlimited AN OPTIMAL CONTROL APPROACH TO MISSION PLANNING PROBLEMS: A PROOF OF CONCEPT Joseph M. Greenslade Lieutenant Commander, United States Navy B.A., Texas A&M University, 1994 B.S., Texas A&M University, 1998 Submitted in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE IN ASTRONAUTICAL ENGINEERING from the NAVAL POSTGRADUATE SCHOOL December 2014 Author: Joseph Micheal Greenslade Approved by: I. Michael Ross Thesis Co-Advisor Mark Karpenko Thesis Co-Advisor Garth Hobson Chair, Department of Mechanical and Aerospace Engineering iii THIS PAGE INTENTIONALLY LEFT BLANK iv ABSTRACT This work introduces the use of optimal control methods for simultaneous target sequencing and dynamic trajectory planning of an autonomous vehicle. This is achieved by deriving a control solution that minimizes a cost to maximize target selection. In the case of varying target priorities, the method described here preferentially selects higher priority targets to maximize total benefit in the allowed time horizon. Traditional techniques employ heuristic techniques for target sequencing and then apply a separate trajectory planning process to check the feasibility of the sequence for the given vehicle dynamics. This work uses pseudospectral methods to deterministically solve the problem in a single step. The historic barrier to the application of optimal control solutions to the target sequencing problem has been the requirement for a problem formulation built from continuous constraints. Target points are, by their nature, discrete. The key breakthrough is to transform the discrete problem into a continuous representation by modeling targets as Gaussian distributions. The proposed approach is successfully applied to several benchmark problems. Thus, several variations of the so-called motorized travelling salesman problem, including ones in which it is impossible to acquire all the targets in the given time, are solved using the new approach. v THIS PAGE INTENTIONALLY LEFT BLANK vi TABLE OF CONTENTS I. INTRODUCTION........................................................................................................1 A. MOTIVATION ................................................................................................1 B. PREVIOUS WORK .........................................................................................3 C. THESIS OBJECTIVE .....................................................................................5 D. THESIS OUTLINE ..........................................................................................7 II. OPTIMAL CONTROL PROCESSES .......................................................................9 A. THE MATHEMATICAL THEORY OF OPTIMAL PROCESSES ........10 B. METHODOLOGY FOR SOLVING OPTIMAL CONTROL PROBLEMS ...................................................................................................15 1. Shooting ..............................................................................................15 2. Collocation ..........................................................................................16 3. Pseudospectral Theory ......................................................................16 C. CODING AN OPTIMAL CONTROL PROBLEM IN DIDO ...................17 III. THE OBSTACLE AVOIDANCE PROBLEM .......................................................19 A. OBSTACLES AS A PATH FUNCTION .....................................................19 B. OBSTACLES AS A COST PENALTY FUNCTION .................................21 IV. THE MISSION PLANNING PROBLEM ...............................................................27 A. VEHICLE DYNAMICS ................................................................................27 B. TARGET DEFINITION ...............................................................................29 C. MISSION PLANNING METHOD: THE STICK AND THE CARROT ........................................................................................................37 V. SPARSE TARGET FIELD .......................................................................................39 A. TARGETS WITH EQUAL PRIORITY ......................................................39 1. Setting Up the Problem .....................................................................40 2. Same Target Priority Mission Planning Problem ...........................45 B. TARGETS WITH VARYING PRIORITIES .............................................50 C. SUMMARY ....................................................................................................55 VI. MISSION PLANNING IN A TARGET RICH FIELD ..........................................57 A. TARGETS WITH EQUAL PRIORITY ......................................................57 B. PRIORITIZED TARGETS...........................................................................62 1. Seeding the Algorithm .......................................................................64 2. Solutions in a Prioritized Target Rich Field ....................................67 C. CONCLUSION ..............................................................................................69 VII. BENCHMARK PROBLEM: MOTORIZED TRAVELLING SALESMAN ......71 A. TWO METHODS FOR SOLVING THE MTSP WITH HYBRID OPTIMAL CONTROL .................................................................................71 1. Methodology .......................................................................................71
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