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Jack Crenshaw's column features algorithms and plug-and-play routines, along with explanations of how they work.
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Posted: 06:15:27 PM, 08/05/2011

Fly me to the Moon (Part 1)

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Fly me to the moon

Let me play among the stars

Let me see what spring is like

On a Jupiter and Mars.

!Bart Howard,?1954 (jazz standard sung by Frank Sinatra, among others)

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We have a new topic this month, and it has absolutely nothing!well, almost nothing!to do with the topics we've been discussing lately!software development, testing, and the like. On the other hand, it certainly has to do with embedded systems and software. Systems that tend to be rather remote. Like 240,000 miles remote.

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If you happened to see my column ("Calculating trajectories for Apollo program") where I reminisced about my days with NASA, you know that I was deeply involved in the planning stages of Project Apollo. During that time, I spent a lot of energy learning neat concepts, making interesting relationships, and picking up skills that make designing lunar missions easier. One of the big disappointments of my young life is that, after Apollo was canceled in 1972, we've never had the curiosity or motivation to go back. So all those neat concepts have pretty much lain fallow.

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What you probably don't know is that now, after more than 40 years, I'm finally getting another chance to go back to the Moon, at least vicariously and in spirit. Since last year, I've been working with one of the groups competing for the Google Lunar X-Prize competition. (www.googlelunarxprize.org/). The competition invites non-governmental teams to soft-land a robotic rover on the Moon, and have it carry out certain tasks, which include relaying back real-time TV images, roving over at least 500m, and surviving the cold, two-week long lunar night. The first two teams to do this will share a $30 million prize.

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I'm working with a group called Part Time Scientists (www.part-time-scientists.com/), which is based in Germany. This effort has been a very exciting one for me. After all these years, I had pretty much abandoned all hope of working on things Lunar again, but here I am doing just that. What I'm doing for the X-Prize effort is pretty much the same as I did before: designing the trajectories. Managing Editor Susan Rambo and I thought you might be interested in how it's done.

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1959 all over again

Years before President Kennedy made his famous "Before the decade is out" speech, lots of people were saying, "Let's go to the Moon." The obvious response, to anyone with the slightest spirit of adventure, was to say, "Yeah, let's!" closely followed by, "How do we do that?" In 1959, we didn't have a very good answer.

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Most of the work I did for NASA was to design the trajectory that gets people to the Moon and back. At the time, space was truly a new frontier for us. We knew virtually nothing about the physics of the problem, we didn't have very good reference books to study, and we didn't have time to go get degrees in astrodynamics. One of the NASA engineers, who had studies astrodynamics, created a handwritten "Cliff Notes" sort of document, and we all studied Xeroxed copies of it.

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My first attempts to get a satisfactory lunar trajectory were pretty pitiful, but I learned. Early on, I decided that I wanted more than just to get a final, satisfactory trajectory. I wanted to understand, at a very deep level, the principles involved.

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In the beginning I had more questions than answers, but from time to time I'd get little glimmers of insight. Whenever I thought about the problem a little more, I'd have one more little glimmer. Now, after more than 40 years of thinking, I believe I finally understand what's going on to that deeper level. I'd like to share that understanding with you.

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It's not easy

The movie Apollo 13 makes the task of getting to the Moon seem pretty easy. You point the nose of the spacecraft at the Moon, light off that big J-2 rocket motor on the Saturn upper stage, and off you go. Simple.

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The reality is not simple at all. You can't just aim at the Moon, because there's the small matter of an Earth and a Moon and their gravitational fields. Before you lit the candle, you were orbiting close to the Earth. Afterwards, the spacecraft still wants to orbit the Earth, so your path is going to be curved in weird and wonderful ways.

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Take a look at figure 1, which depicts the now-familiar "Figure 8" circumlunar trajectory. This trajectory, which goes out, swings around the Moon, and returns to the Earth, attracted a lot of my attention in those early days,1 where the spacecraft goes out to the Moon, swings around it, and returns to the Earth.

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1210embcrenshaw01_circumlunar.jpg

Figure 1: The circumlunar figure-8 trajectory.

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The circumlunar trajectory became central to the Apollo missions, whether they landed on the Moon or not, and for good reason: We didn't just want to send people to the Moon. We wanted to get them back. The nice part about the circumlunar trajectory is that you're coasting in free-fall all the way. Once the booster has done its job, executing what was called the Trans-Lunar Injection (TLI) burn, the spacecraft can coast all the way to the Moon and back. If you've executed that burn perfectly, the spacecraft will coast out to the Moon, swing around it, and return to a safe reentry at Earth, with no more rocket burns required. For obvious reasons, we called it the Free Return trajectory.

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Of course, nobody had any illusions that the Free Return was going to be truly free, because the TLI is not going to be perfect Every time you fire a rocket motor, you risk getting it slightly wrong. and any error at the beginning of the trajectory translates into a bigger error at the end. What's more, the circumlunar trajectory is particularly sensitive to errors in its initial conditions. Even so, using the Free Return trajectory, the astronauts had at least a fighting chance of getting home alive, if something bad happened. And if they could still apply some midcourse corrections, as Jim Lovell and his crew did during Apollo 13, the fighting chance gets a whole lot more optimistic.

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Ironically, the first Apollo mission not to start with a Free Return trajectory was Apollo 13. After the explosion in the liquid oxygen tank, they had to apply a burn, using the LEM motor, to get back onto a Free Return path.

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Since the days of Sputnik, we've become used to seeing orbits depicted as circular or elliptical. Not surprisingly, the physics problem of the motion of two gravitating bodies!Sun and planet, Earth and Moon, or Earth and spacecraft!is called the two-body problem. Way back in 1609, Johannes Kepler conjectured, based on Tycho Brahe's meticulous observations of Mars, that the planets moved in elliptical orbits about the Sun.

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Sixty years later, Isaac Newton solved the two-body problem, and confirmed Kepler's laws. He did it, by the way, by first inventing his three laws of motion, his law of gravity, and calculus. He was 22. So, Isaac, how was your summer vacation?

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Newton found that the general solution of the two-body problem is not always an ellipse. It can be any conic section, which includes the circle, ellipse, parabola, and hyperbola.

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[Continued at Fly me to the Moon (Part 2)]

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[Last update: 06:22:35 PM, 08/05/2011]

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Visitor 02:58:26 AM, 19/04/2013
Comment: My libraries and the ones I used are stnrdaad C' libraries. While they could be used with arduino, I really couldn't tell you how to do it. If you were to transfer the compiled program to an arduino, I would imagine it would most likely run as long as you wired it correctly to the display.
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