# Calculating trajectories for Apollo program

July 10, 2009

As part of EE Times' celebration of the 40th anniversary of Apollo 11 (launched July 16, 1969) and the first moon landing (July 20, 1969), Jack Crenshaw describes his role in studying and creating trajectories for the moon missions.

EE Time's interactive special digital edition available starting July 15th to 20th. The edition can also be accessed on main page of EE Times.
Commentary: Giant Leap has lasted 40 years
Opinion: Applying the lessons of Apollo

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All of my work on Apollo came in a frenetic four-year period, from 1959 through 1963. It was in 1959 that I began work for the Theoretical Mechanics Division at NASA, at Langley Research Center. This was just shortly after NASA was formed. Shortly after I arrived there, a paper came out of the think tank, Rand, Inc., describing a class of lunar trajectories called free-return, circumlunar trajectories--the now-familiar figure-8 paths. It was immediately obvious that this class of trajectories was the only reasonable way to go to the Moon and back. We began studying them intensely, using first a two-dimensional simulation of the restricted three-body problem, and later a 3-D, exact simulation.

In those days, we didn't have spreadsheet programs to draw graphs for us; we had to draw them ourselves. As low man on the TMD totem pole, I got elected to run parametric studies on the computer and plot the results. That task worked in my favor, though, because I gained an understanding of the physics of the problem and the relationship between parameters that I don't think I would have gotten, otherwise. I wasn't content to just make runs and plot curves; I wanted to UNDERSTAND what was going on, and I think that put us ahead of the Rand guys.

 "After lifting off from the lunar surface, the lunar module made its rendezvous with the command module. The Eagle docked with Columbia, and the lunar samples were brought aboard. The LM was left behind in lunar orbit while the three astronauts returned in the command module to the blue planet in the background. (NASA photo ID AS11-44-6642)." -- From Apollo 11 mission, July 20, 1969; photo courtesy of NASA from 30th anniversary site

I pretty much designed the parametric studies. Our group, the Lunar Trajectory Group, was small. Our group leader, Bill Michael, gave me the assignment, and he and I talked daily. But he never had to tell me, Ok, run this trajectory ... now run that one. I was the one making the day-to-day decisions. Bill designed the computer program but neither of us built it. In those days, things were still done "closed shop," and someone from the computer division wrote the code. But I did what would now be called desk-checking, checking the code (in IBM 702 assembler) to make sure it was right.

Later, I did a sensitivity study, plotting the sensitivity of final to initial conditions. Nowadays, we'd call that a state transition matrix, but we didn't know that term, at the time.

My boss and I published a paper in 1961, which was the second paper published on circumlunar trajectories. We also developed quite a number of rules of thumb, approximations, and "patched conic" methods that allowed us to study circumlunar trajectories without spending tons of money for computer time.

Steering
At the time, we weren't thinking of Project Apollo. In fact, I had never heard of it. We had plans to send a "Brownie" camera around the Moon before 1965, using NASA's solid-fuel Scout research vehicle. When Scout's projected payload at the Moon went from a few hundred pounds, through zero and negative, those plans were abandoned. However, the effort left us more than ready for Apollo when it came along.

Next, I began studying the problem of steering the spacecraft, i.e., midcourse corrections. We were all pretty dismayed by the great sensitivity of the circumlunar trajectory to errors in initial conditions, and we knew the accuracy of the Scout boost guidance was orders of magnitude too low. Midcourse guidance would be essential. To my knowledge, my work on that topic was one of the earliest done, though I suspect the fellows from MIT, like Richard Battin, were studying the same problem. In fact, it was Battin's seminal work, seeking a midcourse correction scheme for Apollo, that helped make the Kalman filter practical.

During this time, I also discovered a family of trajectories that were relatively insensitive to injection errors. These trajectories had a nominally vertical reentry. You see, for orbits as highly elliptical as translunar orbit, the perigee is almost totally a function of the angular momentum. The requirement that we return with essentially the same angular momentum as the original orbit implies that the passage past the Moon should not alter that momentum. To achieve this result requires very tight guidance.

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