https://ntrs.nasa.gov/search.jsp?R=19680010999 2020-03-12T08:41:32+00:00Z NASA CR 1000 v.16 e.J- NASA CONTRACTOR NASA CR-l REPORT / GUIDANCE, FLIGHT MECHANICS AND TRAJECTORY OPTIMIZATION Volume XVI - Mission Constraints and Trajectory Interfaces by R. L. Robertson .,.. -. .I- : Prepa red by -I .,; _‘II, Y-:: .., NORTH AMERICAN AVIATION, INC. / I.. c:, .. :. /.‘.(_:_‘-:, I :(; , _,,’ ,..I .;,‘>:, Downey, Calif. .‘I-,.,; .: I.;>/; ..’ . ‘C! r for George C. Marshall Space Flight Center ‘. ,._ *;:’ ,; ., : ‘.s . F h NATIONAL AERONAUTICSAND SPACEADMINISTRATION . WASHINGTON, D. t:,” . APRIL 1968 1; 1 -. I TECH LIBRARY KAFB, NM - ..~ 0060423 NASA CR- 1015 GUIDANCE, FLIGHT MECHANICS AND TRAJECTORY OPTIMIZATION Volume XVI - Mission Constraints and Trajectory Interfaces By R. L. Robertson Distribution of this report is provided in the interest of information exchange. Responsibility for the contents resides in the author or organization that prepared it. Issued by Originator as Report No. SID 66- 1678-8 Prepared under Contract No. NAS 8-11495 by NORTH AMERICAN AVIATION, INC. Downey, Calif. for George C. Marshall Space Flight Center NATIONAL AERONAUTICS AND SPACE ADMINISTRATION For solo by the Clearinghouse for Federal Scientific and Technical Information Springfield, Virginia 22151 - CFSTI price $3.00 FOREWORD This report was prepared under contract NAS 8-11495 and is one of a series intended to illustrate analytical methods used in the fields of Guidance, Flight Mechanics, and Trajectory Optimization. Derivations, mechanizations and recommended procedures are given. Below is a complete list of the reports in the series. Volume I Coordinate Systems and Time Measure Volume II Observation Theory and Sensors Volume III The-Two Body Problem Volume IV The Calculus of Variations and Modern Applications Volume V State Determination and/or Estimation Volume VI The N-Body Problem and Special Perturbation Techniques Volume VII The Pontryagin Maximum Principle Volume VIII Boost Guidance Equations Volume IX General Perturbations Theory Volume X Dynamic Programming Volume XI Guidance Equations for Orbital Operations Volume XII Relative Motion, Guidance Equations for Terminal Rendezvous Volume XIII Numerical Optimization Methods Volume XIV Entry Guidance Equations Volume XV Application of Optimization Techniques Volume XVI Mission Constraints and Trajectory Interfaces Volume XVII Guidance System Performance Analysis The work was conducted under the direction of C. D. Baker, J. W. Winch, and D. P. Chandler, Aero-Astro Dynamics Laboratory, George C. Marshall Space Flight Center., The North American program was conducted under the direction of H. A. McCarty and G. E. Townsend. iii TABLE OF CONTENTS SECTION PAGE 1.0 STATEMENT OF THE PROBLEM . , . , . 1 2.0 STATE OF THE ART .............. 3 2.1 Introduction ............... 3 2.2 Boost Trajectories ............. 3 2.2.1 Introduction .............. 3 2.2.2 Launch Window ......... 3 2.2.2.1 Missions Withoit Specific Ineitiil Orientat& . 4 2.2.2.2 Launch Into Specified Inertial Orientations . 7 2.2.2.3 Orbit Plane Containing a Specific Inertial VeCt& : 8 2.2.3 Range Safety 12 2.2.4 Ascent Trajectoriis......................... 13 2.2.4.1 Equations of Motion ........... 14 2.2.4.2 Velocity Requirements .......... 23 2.3 Rendezvous 25 2.3.1 Introductio; .............................. 25 2.3.2 Direct Ascent ... .......... 25 2.3.3 Rendezvous Compatible OIbi;s ..... 37 2.3.4 Intermediate Orbit Rendezvous'Techniquis ..... 39 2.3.4.1 General ............... 39 2.3.4.2 Concentric Orbits ........... 40 2.3.4.3 Co-Orbital . ...................... 43 2.3.4.4 LOS Delta-V Rendeivous 45 2.3.5 Terminal Maneuver ............ 50 2.4 Spatial ................. 51 2.4.1 Introduction . ... ........ 51 2.4.2 Perturbations and'lifetime ..... 51 2.4.2.1 Variation of Parameters Gene&l Perturbation' Technique ... .. '. 51 2.4.2.2 Perturbativk ALceleration 'Componknts ......... 55 2.4.2.2.1 Asphericity of the Earth ........ 55 2.4.2.2.2 Luni-Solar Perturbations ........ 57 2.4.2.2.3 Solar Radiation Pressure ........ 59 2.4.2.2.4 Atmospheric Drag ........... 60 2.4.3 Earth Trace 67 2.4.4 Earth Coverage' .......................... 68 2.4.5 Tracking ............... 73 2.4.6 Lighting ............. 78 2.4.6.1 1ntroduc;ioh ............. 78 2.4.6.2 Eclipse Geometry. ........... 78 2.4.6.2.1 Penumbra ............. 79 2.4.6.2.2 Umbra .......... 80 2.4.6.2.3 Shadow D'is&im\nait .......... 80 v SECTION PAGE 2.4.6.3 Relation to Orbits . 81 2.4.6.4 Shadow Ellipse Approxiiation . 83 2.4.6.5 Continuous Exposure to Sun . 86 2.4.7 Radiation . 87 2.4.7.1 Van Allen Bells . 87 2.4.7.2 Solar Flare Radiition . , . 90 2.5 Deorbit . 91 2.5.1 Introduction . 91 2.5.2 General Deorbit Mankver . 91 2.5.3 Minimum Energy/Time Deorbit . 94 2.5.4 Deorbit Timing . 97 2.5.5 Deorbit Via Inteimediate &-bit . 101 2.6 Entry Trajectories . 106 2.6.1 Introduction . 106 2.6.2 Entry Dynamics . 106 2.6.2.1 Direct Entry . 108 2.6.2.2 Lifting Entry (E&iibkui Giidi Pithi : : : . 110 2.6.2.3 General Case of Shallow Entry . 113 2.6.2.4 Maneuverability . 11s 2.6.3 Entry Heating . 116 3.0 RECOMMENDEDPROCEDURES ........... 113 3.1 Introduction ............... 119 3.2 Methodology ................ 119 4.0 REFERENCES . 122 vi 1 .O STATEMENT OF THE PROBLEM The subject of this monograph is mission constraints and trajectory interfaces which result from imposing the physical limitations on a trajec- tory or earth orbit. The constraints considered in this document are those which reflect directly or indirectly on the characteristics of the various trajectories which constitute the mission. Each constraint limits, to a certain degree, the choice of the trajectory profile for a given mission phase. Mission constraints affecting trajectory profiles can be thought of as falling into two categories, those which are necessary for accomplishment of mission objectives and those which are due to creb or equipment limitations and capabilities. In the latter category would fall those necessary for crew safety or payload survival. The type of mission constraints required for the accomplishment of mission objectives may be orbital lifetime, earth trace, surveillance pat terns, tracking coverage, stationkeeping, lighting, launch window, launch site, and communications. Mission constraints imposed by crew and equipment limitations may be aerodynamic loading, acceleration/deceleration levels, tracking, recovery sites, lighting, radiation, aerodynamic heating and mission duration. As can be seen, many constraints overlap and conflict with each other in their interface with trajectory capabilities. Trajectories are constrained first and foremost by physical laws governing their motion. Spatial orbit motion for example is described basically by two-body mechanics (except for small perturbations). Any devia- tion of the vehicle’s path to accomplish a particular mission objective must be accounted for in the selection of the orbit or be accomplished by an expen- diture of energy. Similarly, boost and entry trajectories are constrained to paths dictated by the summation of forces acting on the vehicle, whether they be thrust, aerodyanmic or some other. The set of constraints specified for any mission must be compatible with the physics of the trajectories consti- tuting the mission. It is the purpose of this monograph to formulate and examine the more important mission constraints and their relationship to trajectories in order to facilitate the synthesis of a realistic mission. The missions con- sidered here will be earth orbital but will include parking orbits for lunar or planetary missions. The missions will be divided into the five trajectory phases which comprise most earth orbital missions. The phases are: 1 aunch, rendezvous, spatial, deorbit, and entry trajectories. The discussions will be limited primarily to first order detail except where it is necessary for understanding of the problem to include higher order effects. That is, the earth will be assumed either spherical or oblate to the first order in most cases. Maneuvers, except during boost, will generally be assumed to be impul- sive. The atmospheric model will be assumed to be defined by an exponential 1 density gradient. These assumptions are realistic during the preliminary mission design phase where extensive tradeoff analysis is necessary to inte- grate trajectory profiles and mission constraints but must be refined during the detailed design of a particular mission. The constraints will first be discussed individually, in some detail, without regard to overlap areas. Then, the methodology for finding the best trajectory considering all mission constraints will be analyzed. 2.0 STATE-OF-THE-ART 2.1 INTRODUCTION This section presents the formulation of various mission constraints and the relationship of the constraints to the various trajectory phases. The formulations are not intended to precisely define all of the relationships involved for any given constraint. The intent is to provide sufficient detail pertaining to the relationships which are necessary to evaluate mission con- straint and trajectory interfaces. The discussions in this section attempt to identify the parameters which relate a mission constraint to a trajectory profile. Parametric data can be generated from the formulations in this section to aid in the synthesis of the best trajectory complying with a given set of mission constraints. Such analysis will, in many cases, identify mission constraints which are themselves incompatible. Five trajectory phases which constitute