149063 Reports of the Department of Geodetic Science NA'A CR Report No. 156 x t7zKZ OPTIMAL SELENODETIC CONTROL by Haim Benzion Papo 5F~rNOETIC175-73006 -141749) OPTIMAL SELE0DETIC (NASACR (Ohio State Univo ,CONTROL Interim Report Foundation) 273 p unclas ,Research 00/98 12381 Prepared tor Naional Aeronautics and Space Administration Manned Spacecraft Center Houston, Texas Contract No. NAS 9-9695 OSURF Project No. 2841 Interim Report No. 5 The Ohio State University Research Foundation REPRODUCED Y Columbus, Ohio 43212 NATIONAL TECHNICAL INFORMATION SERVICE August, 1971 U S. DEPARTMENT OFCOMERCE SPRINGFIELD, VA. 22161 'K Reports of the Department of Geodetic Science Report No. 156 OPTIMAL SELENODETIC CONTROL by Haim Benzion Papo Prepareci tr National Aeronautics and Space Administration Manned Spacecraft Center Houston, Texas Contract No. NAS 9-9695 OSURF Project No. 2841 Interim Report No. 5 The Ohio State University Research Foundation Columbus, Ohio 43212 August, 1971 1 PREFACE This project is under the supervision of Ivan I. Mueller, Professor of the Department of Geodetic Science at The Ohio State University, and it is under the technical direction of Mr. James L. Dragg, Mapping Sciences Laboratory, NASA/MSC, Houston, Texas. The contract is administered by the Space Sciences Procurement Branch, NASA/MSC, Houston, Texas. A revised version of this report has been submitted to the Graduate School of The Ohio State University in partial fulfillment of the require­ ments for degree Doctor of Philosophy. ii ABSTRACT This study was undertaken to solve the problem of determination of an optimal selenodetic control network on the Moon. Selenodetic control is defined by the coordinates of a network of well identifiable features on the lunar surface with respect to a selenodetic Cartesian coordinate system, __ which is fixed to the lunar crust, is centered at its mass center and is oriented along the three principal axes of inertia of the Moon. The method developed for solving the problem is fully consistent with the theoretical and numerical models for the motion of the Moon in space. For this purpose, the parameters of orientation of the selenodetic system with respect to the mean ecliptic system, identified in this study with the physical [ibrations of the Moon, are made an integral part of the solution for selenodetic control. The solution is based on optical data obtained by photography or by direct angular observations of the Moon taken from the Earth or on board a spacecraft and on range and range-rate data obtained from tracking stations on Earth to a spacecraft orbiting the Moon. In order to achieve orientation of the control network at least part of the optical data- is considered oriented with respect to a certain celestial coordinate system. Scale is introduced to the control solution through the assumption that the lunar ephemeris describes the motion of the center of mass of the Moon, or in this case, the translatory motion of the origin of the selenodetic coordinate system with respect to the geocenter. The lunar ephemeris introduced by JPL under the code-name LE-16 is used for the above purpose. The total observational material modeled in terms of the parameters of the solution is processed by a weighted least squares adjustment procedure and results in estimates for the following parameter groups and their co­ variances: iii a. selenodetic coordinates of a selected number of fundamental control points on the lunar surface, b. parameters of orientation of the Moon (physical libration angles and time ratcs at a standard epoch), c. parameters featuring the low degree terms in the lunar gravitational field. In case the optical data was taken on board a spacecraft, an orbit determination procedure is appled to the range and range-rate data which results in estimates for the selenocentric state vector of the spacecraft and also in estimates for the higher degree terms in the lunar gravitational field. In order to test numerically the mathematical procedure developed in this study, a simulated environment was created which reflects very closely the true world. The Earth, the' Moon and a variety of satellites move and rotate in this simulated envirQmnent strictly according to the laws of Newton and Kepler. The observational material generated is free of unaccounted phenomena and simulates very closely real observations. Numerical tests with the simulated as well as with real data demon­ strated that the solution for the physical librations of the Moon conforms very closely to existing solutions while having many advantages primarily that of being an integral part of the control network solution. A complete selenodetic control solution obtained from simulated Earth-based optical data confirmed the feasibilty of the method developed in this study and introduced estimates for accuracies that could be achieved in the solution for an optimal selenodetic control. iv ACKNOWLEDGEMENTS It gives me a great pleasure to be able to extend my gratitude and appreciatibn to th f-iany persons without whom this work would not have been possible. From my parents I inherited a deep and sincere love for learning and an acute curiosity for the unknown and the unsurveyed. The Technion, Israel's Institute of Technology and The Ohio State University provided me with the best education one could ask for it my field. Working in Room 231, Lord Hall on the OSU campus with other graduate students like myself has been a wonderful experience. There was an atmosphere of enthusiastic learning, of a fruitful exchange of ideas, of sharing ones excitement over finding an answer to a problem, of true friendship of people working together. The OSU Computer Center provided me generously with computer time needed for the study and through the invaluable assistance of Mr. John M. Snowden, I was able to make full use of its excellent facilities. Discussions on many conceptual and practical problems with Doctors Peter Meissl, Dean C. Merchant and Donald H. Eckhardt helped in bringing more clarity and exactness where those were needed. Professors Urho A. Uotila, Richard H. Rapp and Gerald H. Newsom served on the reading committee and provided noncompromising, but con­ structive, criticism which was indispensible in improving many portions of this work. My advisor Professor Ivan I. Mueller played a most vital part in my work. It is only through his guidance, encouragement and unlimited confidence in me that I was able to overcome many difficult moments when it seemed that I was facing an inevitable dead end. The difficult task of typing the text and, in particular, the many v formulae involved was done with exceptional skill and expertise by Mrs. Evelyn Rist, Mrs. Rene Tesfai and Miss Barbara Beer. No graduate study by a married man is possible without the uncon­ ditional support and understanding of his wife and family. To my wife, Viola, and to my children, I shall be indebted forever for their patience and love which sustained me during the years of concentrated study. This study was supported by The Mapping Sciences Laboratory NASA MSC, Houston, Texas under Contract No. NAS 9-9695 and the technical direction of Mr. James L. Dragg. Scholarships were granted me by the Hebrew Technical Institute of New York and also by the Technion, Israel's Institute of Technology. Last, but not least, a word of thanks to The Department of Geodetic Science and The Ohio State University and in particular to the Department Chairman, Professor Urho A. Uotila, for making it possible for me to study and work in this great country. vi TABLE OF CONTENTS page PREFACE ii ABSTRACT iii ACKNOWLEDGEMENTS v LIST OF TABLES x LIST OF FIGURES xi 1. INTRODUCTION 1 1.1 Statement of the Problem 1 1.2 Past Achievements and Problem Areas 3 1.21 Earth Based Methods 5 1222 Satellite Borne Methods 7 1.3 Astronomical Models and Constants 11 1. 31 Lunar Theory and Ephemeris 11 1. 32 Orientation of the Moon in Space 16 1. 33 Fundamental Astronomical Constants 22 1.4 General Plan for the Study 24 2. THEORETICAL SOLUTION OF tHE PROBLEM 28 2.1 Introduction 28 2.2 General Adjustment Model 35 2.21 Condition Equations for an Optical Observation 39 2.22 Condition Equations for Range and Range-Rate 41 2.3 Parameters in the Solution 43 2.4 Linearization 49 2.41 Optical Observations 49 2.42 Range and Range-Rate Observations 55 2.5 Formation and Solution of the Normal Equations 57 2.51 Satellite Borne Optical Observations (Single Arc) 57 vii 2.52 Earth Based Optical Observations 59 2.53 Combined Solution 60 2.6 Programming Considerations 62 3. NUMERICAL INTEGRATION OF THE PHYSICAL LIBRATIONS OF THE MOON 71 3.1 Introduction 71 3.2 Equations of Rotational Motion of the Moon 74 3.21 Euler's Dynamic Equations 74 3.22 Transformation of Euler's Dynamic Equations 80 3.23 Differential Equations of the Physical Libration Angles 87 3.3 Adjustment Theory 91 3.31 Mathematical Model 92 3.32 State Transition and Parameter Sensitivity Matrices 96 3.4 Fitting Numerically Integrated Physical Libration Angles Into Existing Solutions 107 3.41 Spatial Angles Approach 108 3.42 Variational Rotations Method 114 4. SIMULATION OF ENVIRONMENT AND OBSERVATIONS 119 4.1 Introduction 119 4.2 Fundamentals of the Simulation 123 4.3 Constants of the Simulation 127 4.31 The Earth 127 4.32 The Moon 128 4.33 Illuminating Source "Sun" 130 4.34 Initial Values for Position and Orientation of Earth and Moon, Including Linear and Angular Velocities 131 4.35 Motion of the Mass Center of the Moon 132 4.36 Eulerian Angles of the Moon 133 4.37 Eulerian Angles