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Lunar Module Simulator Math Model LUNAR MODULE MOON-REFERENCED EQUATIONS OF MOTION Brian Woycechowsky NASA LUNAR MODULE MOON-REFERENCED EQUATIONS OF MOTION by Brian Woycechowsky Published by the Center for Technology & Innovation, Inc. Binghamton, New York www.ctandi.org © 2021 Contents Foreword ........................................................................................................................................................................ iii About the Authors ....................................................................................................................................................... iv A Perspective from Inside NASA: Memories of the Apollo 13 Mission ................................................... v Preface .......................................................................................................................................................................... viii Introduction ................................................................................................................................................................... 1 Axes Systems Used by the LM’s Equations of Motion (EOM) ...................................................................... 4 Translational Equations of Motion ........................................................................................................................ 6 Rotational Equations of Motion .............................................................................................................................. 8 The M-frame to B-frame Transformation ........................................................................................................ 10 The M-frame to S-frame Transformation Matrix .......................................................................................... 12 Nomenclature ............................................................................................................................................................. 15 References ..................................................................................................................................................................... 18 2021 Supplement ...................................................................................................................................................................... 29 LIST OF FIGURES Figure 1: Lunar Landing Mission Trajectory ..................................................................................................... 1 Figure 2: LM Body Axes – the B-frame................................................................................................................. 2 Figure 3: LEM RCS Thrusters and Body (B-Frame) Geometry ................................................................... 3 Figure 4: Inertial Earth (E-frame) and Moon (M-frame) Geometry ......................................................... 4 Figure 5: Relation between M-Frame and Selenographic S-Frame .......................................................... 5 Figure 6: LM Translational Equations of Motion (EOM) ............................................................................... 7 Figure 7: Rotational Equations of Motion........................................................................................................... 9 Page ii FOREWORD Link’s Apollo team pioneered the first real-time simultaneous simulation of multiple virtual worlds to tr* ain NASA astronauts and Mission Control teams to become proficient in manned spacecraft navigation, operation, and malfunction recovery in order to return crews safely from the previously uncharted territory of earth orbit and the moon. The work of the Link Apollo team, begun in mid-1963, was intensely integrated with the larger Apollo effort, in terms of both spacecraft design and training flight crews and ground control teams. The Link Apollo team reported to three clients – to NASA , to North American Rockwell Corporation for the Apollo Mission Simulator (AMS), and to Grumman Aircraft Engineering Corporation for the Lunar Module Simulator (LMS). Custom real-time capability for digital computation of on-board system status, navigation position, communications, and ground control and monitoring instrumentation was developed to permit the concurrent, interactive training of Mission Control, Command Module, and Lunar Module crews. Computation of the dynamic ephemeris, orbital and cis-lunar dynamics, flight regimes of the spacecraft, visual systems, and simulation of on-board computers were particularly challenging. The Link team responded to the dynamic requirements of evolving spacecraft modifications, telemetry, and malfunction options to deliver mission critical products on target, on time. As an epitome of thinking, designing, fabricating and operating, the Apollo simulators are without parallel – gathering within their steely and angular casings all that was possible (in that largely pre-digital era) to devise systems that would train humans to go to another world. -- Doug Millard, Deputy Keeper, Technologies & Engineering, London Museum of Science During the course of building simulators for the US manned space exploration program, Link's corporate entity shifted from the original Link Aviation Devices. In 1954, the company was sold to General Precision Systems, Inc., becoming* The Link Group, often abbreviated GP-Link. In 1969, General Precision sold The Link Group to The Singer Company, and the group became Singer's Simulation Products Division, often abbreviated Singer-Link. The successor company in early 2021 is L3 Harris Technologies Link Training and Simulation. Page iii ABOUT THE AUTHORS Brian Woycechowsky is a native of Binghamton, NY, who earned a bachelor’s degree in Aeronautical Engineering from Rensselaer Polytechnic Institute (RPI) and a master’s degree from the School of Advanced Technology, Binghamton University. In 1957, Brian started at Link Aviation Devices in the Aerodynamic Section, moving to Systems Engineering and later to Visual Systems. He worked on early passenger jet aircraft simulators, including Boeing’s 707 and Lockheed’s Electra. In the early 1960s, he developed flight dynamics and equations of motion for the full suite of space vehicles: T-20 for Edwards Air Force Base and the Gemini, Apollo, and Space Shuttle programs for NASA. Brian was instrumental in the development of the math models for the Apollo simulator, which trained the astronauts to navigate to/from the Moon and to perform the complex rendezvous maneuvers required. He finished his career at Link in Systems Engineering, responsible for solving general math problems for a range of flight simulators until his retirement in 1990. Frank Hughes started at NASA in 1966 on the day the crates arrived carrying the Link Apollo simulators and retired 33 years later as Chief of Space Flight Training. He was an instructor during the Apollo program, then trained Space Shuttle crews, and went on to lead training efforts for the International Space Station. Currently, he is president of Tietronix Software, Inc., a team of professionals with expertise in software development, mixed realities, process management, medical services, training and simulations, gaming, and information technology. Mr. Hughes joined the Board of Directors of the Center for Technology and Innovation in March, 2019. Frank Cardullo began his career in the simulation industry in 1966. He worked as a Systems Engineer on the Apollo simulators for Link, prior to taking a faculty position at the State University of New York (SUNY) at Binghamton in 1980. Now retired, he remains Emeritus Professor of Mechanical Engineering at SUNY Binghamton and is the technical coordinator for all Watson School simulation short courses. Professor Cardullo is very active in simulation research and serves as a consultant for many aerospace companies and U.S. government agencies. He joined the Board of Directors of the Center for Technology and Innovation in March, 2019. Page iv A PERSPECTIVE FROM INSIDE NASA: MEMORIES OF THE APOLLO 13 MISSION As a simulator instructor at NASA working at the Kennedy Space Center (KSC) in Florida in April 1970, the first 55 hours of the Apollo 13 mission couldn’t have gone better. The vehicle was flying on course to the Moon; not a problem was being tracked in the Mission Control Center (MCC). We were starting to get serious about training the Apollo 14 crew, the next crew to fly. On Monday evening, April 13, the crew did a great 27-minute TV tour of the two spacecraft, both the Command/Service Module (CSM) and the Lunar Module (LM). Then, about 10 minutes later, during a routine procedure of stirring up the oxygen in Tanks 1 and 2, Tank 2 exploded. As we learned later, the force of Tank 2 exploding blew out the entire side of the Service Module. It also took out the plumbing of Tank 1, creating a serious gas escape that started changing the flight path of the entire spacecraft. The crew had lost more than half of their oxygen supply. I had left work after the TV show and did not know about the explosion until I got home, about 30 minutes away. I was called back immediately and so drove back to work and got there about 10 PM. As it turned out, I was to be there until 3 PM on Friday. We were monitoring the activities in the MCC in Houston and were able to listen to their deliberations. We knew that we had to get ready for a lot of simulations to get the vehicle back home, so we sent a team of people across the street to the Operations & Checkout building where the actual
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