INTRODUCTION
When I was a teenager struggling to master algebra, geometry, and
trigonometry in a tiny little high school in the Bluegrass region of
Kentucky, I loved doing mathematical derivations. Those squiggly little
math symbols arranged in such neat geometrical patterns on the printed
pages held endless fascination for me. But never in my wildest dreams,
could I ever have imagined that I might someday be stringing together
long, complicated mathematical derivations that would allow enthusiastic
American astronauts to hop around on the surface of the moon like
gigantic kangaroos.
Nor could I have imagined that someday my Technicolor derivations
would end up saving more money than a typical American production line
worker could earn in a thousand lifetimes of fruitful labor.
I was born and raised in a very poor family. My brother once
characterized us as "gravel driveway poor". At age 18 I had never eaten
in a restaurant. I had never stayed in a hotel. I had never visited a
museum. But, somehow, I managed to work my way through Eastern Kentucky
University, one of the most inexpensive colleges in the state. I
graduated in 1959 with a major in mathematics and physics eighteen months
after the Russians hurled their first Sputnik into outer space. That next
summer I accepted a position with Douglas Aircraft in Santa Monica,
California, and what a wonderful position that turned out to be! At
Douglas Aircraft we were launching one Thor booster rocket into outer
space every other week.
In 1961 after I earned my Master's degree in mathematics at the
University of Kentucky, I was recruited to work on Project Apollo. And I
am convinced that anyone who ever worked on the Apollo Project would tell
you that Apollo was the pinnacle of the rocket maker's art.
At age 18 I had never eaten in a restaurant. I had never stayed in a
hotel. I had never visited a museum. But somehow, by some miracle, six
year later at age 24, I was getting up every day and going to work and
helping to put American astronauts on the moon!
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WHAT IS A MATHEMATICAL DERIVATION?
A mathematical derivation is a series of mathematical and logical
steps that starts with something that every expert can agree is true and
ends up with a useful conclusion, usually one or more mathematical
equations amenable to an easy solution.
Hollywood's version of a mathematical derivation is almost always
carried out on big, long blackboards with no words anywhere. But I did
all of my derivations with words and in Technicolor – using colored
pencils and colored marking pens – on big, oversized quad pads four
times as big as a standard sheet of paper. One day when my daughter,
Donna, was about five years old, she wandered into my den and watched me
struggling over a particularly difficult derivation. "This is
embarrassing," she maintained, "My father colors better than I do."
Most of the derivations my friend, Bob Africano, and I put together in
those exciting days centered around our struggles to enhance the
performance capabilities of the mighty Saturn V moon rocket. The Saturn V
was 365 feet tall. It weighed six million pounds. It generated 7.5
million pounds of thrust and, over nine pulse-pounding Apollo missions,
it carried 24 American astronauts into the vicinity of the moon. Twelve
of those astronauts walked on the moon's surface. The other twelve
circled around it without landing.
Even the simplest mathematical derivation can be difficult,
frustrating work and, over the years, we put together hundreds of pages
of them. For ten years, and more, we worked 48 to 60 hours per week. We
were well paid and treated extremely well and we loved what we were doing
for a living. But we were often teetering on the ragged edge of
exhaustion. One night at a party I observed that doing mathematical
derivations for a living was like "digging ditches with your brain!"
In my career I followed the dictum of the British mathematician
Bertram Russell. "When you're young and vigorous, you do mathematics," he
once wrote. "In middle age you do philosophy. And in your dotage, you
write novels." Sad to say, I just finished my first novel! It is intended
to become a Hollywood motion picture entitled the 51st State. So this
white paper is being composed while I am in my dotage.
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THOSE CHALLENGING DAYS AT ROCKWELL INTERNATIONAL
I joined the staff of Rockwell International at Downey, California, in
1964. Each morning I would jaywalk across Clark Avenue to get to work in
Building 4. I was assigned to a systems engineering group consisting of
about 20 engineers and support personnel led by supervisor, Paul Hayes.
Paul was proficient in several branches of mathematics and he carefully
checked and rechecked the mathematical derivations we were publishing in
internal letters, company reports, and in the technical papers we were
presenting at big conventions around the country and in a few foreign
countries, too.
Most of our time and effort was devoted to figuring out how to operate
the S-II stage (the second stage of the Saturn V moon rocket) with
maximum practical efficiency. We didn't make any modifications to the
hardware; the hardware was already built. Instead, we used the
mathematics and the physics we had learned in school, as effectively as
possible, to maximize the payload of the mighty Saturn V.
Over about ten years on the project, Africano and I – and various
others – developed hundreds of pages of useful mathematical
derivations. Various other engineers scattered around the country were
also trying to figure out how to send more payload to the moon. Joe
Jackson, Scott Perrine, and Wayne Deaton at NASA Huntsville, for
instance, and Carol Powers and Chuck Leer and their colleagues at TRW in
Redondo Beach, California all made significant contributions to this
important work.
During those early days we were taking mathematics and physics courses
at UCLA and UCI (The University of California at Irvine) and teaching
courses of our own at the California Museum of Science and Industry,
Cerritos College, USC, and at Rockwell International in Downey and Seal
Beach, California.
Our supervisor, Paul Hayes, showed remarkable patience and leadership
when the mathematics (or my own stubbornness) led me down blind alleys.
On one occasion, for example, I spent about 3 weeks formulating a more
precise family of guidance equations for our six-degree-of-freedom
trajectory program. Unfortunately, when those equations were finally
finished, checked, and programmed the rocket's trajectory hardly changed
at all. I was rather apologetic, but Paul had an entirely different way
of looking at what we were doing for a living. "It's OK" he told me,
softly. "Try something else."
He exhibited the same magnanimous attitude when I insisted on using
disk storage to replace the nine magnetic tapes we were using for
"scratch-pad" memory. We burned up two weeks or so reprogramming the
routines in an attempt to save computer time (which in those days cost
$700 per hour!). Unfortunately, as our programmer, Louise Henderson, had
predicted, no computer-time saving at all resulted from this tedious and
time-consuming effort.
Paul Hayes realized that we could not make major breakthroughs in the
difficult fields of applied mathematics, orbital mechanics and systems
engineering unless we were willing to risk humiliating failures along the
way! Fortunately, we did, eventually, perfect four powerful mathematical
algorithms that saved an amazing amount of money for the Apollo program.
These algorithms, which required no hardware changes and cost virtually
nothing to implement, involved at least eight difficult branches of
advanced mathematics.
In 1969 Bob Africano and I summarized the salient characteristics of
these four mathematical algorithms in a technical paper we presented at a
meeting of the American Institute of Aeronautics and Astronautics (AIAA)
at the Air Force Academy in Colorado Spring, Colorado. It was entitled
"Schemes for Enhancing the Performance Capabilities of the Saturn V Moon
Rocket." In that paper we showed how those four mathematical algorithms
increased the translunar payload-carrying capabilities of the Saturn V by
about 4700 pounds. Measured in 1969 dollars, each pound of that payload
was worth $2000, or about 5 times its weight in 24-karat gold. NASA ended
up flying nine manned missions around the moon. Consequently, those
mathematical algorithms, liberally laced with physics and astrodynamics,
ended up saving the American space program $2.5 billion valued in
accordance with today's cost of $990 for each pound of gold.
In the paragraphs to follow, I will attempt to summarize the methods
we used to achieve those important payload gains and to describe the
mathematical techniques we employed in accentuating the rocket's
performance.
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PROPELLANT UTILIZATION SYSTEMS
A large liquid-fueled rocket usually includes two separate tanks, one
containing the fuel and the other containing the oxidizer. These two
fluids are pumped or forced under pressure into the combustion chamber
immediately above the exhaust nozzle, where burning of the propellants
takes place.
If we would load 1000 rockets with the required quantities of fuel and
oxidizer, then fly them to their destination orbits, we could expect -
due to random statistical variations along the way – to have a small
amount of fuel left over on 500 of those flights and a small amount of
oxidizer left over on the other 500. Neither the fuel nor the oxidizer
can be burned by itself because burning requires a mixture of the two
fluids.
In order to minimize the average weight of the fuel and oxidizer
residuals on the upper stages of the Saturn V rocket, the designers had
introduced so-called Propellant Utilization Systems. A Propellant
Utilization System employs sensors to monitor the quantities of fuel and
oxidizer remaining throughout the flight. It then makes automatic real-
time adjustments in the burning-mixture-ratio to achieve nearly
simultaneous depletion of the two fluids when the rocket burns out.
For the Saturn V, the necessary measurements were made with
capacitance probes running along the length of the fuel tank and the
oxidizer tank. A capacitance probe is a slender rod encased within a
hollow cylinder. Openings at the bottom of the hollow cylinder allow the
fluid level on the inside of it to duplicate its level on the
outside.
As the fluid level inside the cylinder decreases, the electrical
capacitance of the circuit changes to provide a direct measure of the
amount of fluid remaining in the tank. These continuous fluid-level
measurements are then used in making small real-time adjustments in the
rocket's burning-mixture-ratio to achieve nearly simultaneous depletion
of the two propulsive fluids.
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THE PROGRAMMED MIXTURE RATIO SCHEME
The Propellant Utilization System on the S-II stage increased the
performance of the booster by an extra 1400 `pounds of payload headed
toward the moon. Unfortunately, modeling the behavior of the propellant
utilization systems in flight created a complicated problem for the
mission planning engineers. When we were simulating the translunar
trajectories and the corresponding payload capabilities for the Saturn V,
we found that, if we ran two successive simulations with identical
inputs, each simulation would yield a slightly different payload at
burnout.
These rather unexpected payload variations came about because the
computer program's subroutines automatically simulated slightly different
statistical variations in the Propellant Utilization System during each
flight. In order to circumvent this difficulty, we did what engineers
almost always do – we called a meeting. And at that meeting we
brainstormed various techniques for making those pesky payload variations
go away. Fortunately, no one in attendance that day was able to come up
with a workable solution.
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Sitting in the back of the room was long, lanky propulsion specialist
named Bud Brux. who said almost nothing during the meeting. But, when Bud
Brux got back to his office, he began thinking about the problem we had
encountered. "Hey, wait a minute!" he thought, "The reason we build a
rocket is to put payload into space. If something is causing that payload
to vary, maybe we should try to accentuate the effect, rather than trying
to make it go away."
Bud Brux then wrote us a simple, two-page internal letter suggesting
that we vary the mixture ratio as much as we possible in a few of our
computer simulations to see if we could produce important performance
gains. We were not particularly excited by the letter he wrote; we
received lots of internal letters in those days. But, when those first
few trajectory simulations came back from the computer, our excitement
shot up by a decibel or two. On the best of those simulations, the Saturn
V moon rocket was able to carry nearly 2700 extra pounds of payload to
the moon, each pound of which was worth $2000 – or five times its
weight in 24-karat gold.
Figure 1: The five J-2 engines mounted on the second stage of
the Saturn V moon rocket were originally designed to burn their
propellants at a constant steady-state mixture ration of 5 to 1 (5 pounds
of liquid oxygen for every pound of liquid hydrogen). By working our way
through the proper mathematical derivations, however, we showed that, if
we started out with a mixture ratio of 5.5 to1, then abruptly shifted to
4.5 to 1, the booster rocket could hurl an extra 2700 pounds onto its
translunar trajectory. This so-called Programmed Mixture Ratio Scheme
required no hardware changes. We merely opened 5 existing valves a little
wider in mid flight.
The sketches in Figure 1 highlight some of the salient characteristics
of the Programmed Mixture Ratio Scheme as applied to the second stage of
the Saturn V moon rocket. Early in that rocket's flight, we set the
burning-mixture ratio at 5.5 to 1 (5.5 pounds of oxidizer for every pound
of fuel). But 70 percent of the way through the burn we abruptly shifted
that mixture ratio to a lower value of 4.5 to 1.
As the small graphs in Figure 1 indicate, this shift in the mixture
ratio provided the rocket with high thrust early in its flight at a
slightly lower specific impulse.* Then, following the Programmed Mixture
Ratio shift, it had a lower thrust, but a higher specific impulse.
After studying the computer simulations and putting together several
dozen pages of mathematical derivations, we concluded that the abrupt
Programmed Mixture Ratio shift caused the rocket to leave more of its
exhaust molecules lower and slower as it flew toward the moon. This, in
turn, put less energy into the exhaust molecules and correspondingly more
energy into the payload. The resulting performance gains are not
insignificant. On each of the missions we flew to the moon, the
Programmed Mixture Ratio Scheme allowed us to send 2700 extra pound of
payload onto the rocket's translunar trajectory!
When the last Apollo mission had been completed, I wrote an internal
letter highlighting the clever insights and the important engineering
accomplishments of our illustrious colleague. "If Bud Brux had sent us a
note telling us where five solid gold Cadillacs were buried in the
company parking lot," I concluded, "it would not have been worth as much
as the note he actually wrote!"
In my view, mathematical derivations that involve moving objects such
as a booster rocket or an orbiting satellite can be surprisingly
interesting. Those that center around objects that move along optimal
trajectories are even more interesting. But the most interesting
derivations of all, involve objects that move along optimal trajectories
that are experiencing random statistical variations. The work that we did
on optimal fuel biasing fell into the third category with random
statistical variations superimposed on a booster rocket that was moving
along an optimal trajectory.
__________________ * The specific impulse of a rocket propellant
combination provides us with a measure of the efficiency of the rocket.
It equals the number of seconds a pound of the propellant can produce a
pound of thrust.
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OPTIMAL FUEL BIASING
If we load 1000 identical hydrogen-oxygen rockets with the desired
amounts of fuel and oxidizer in the proper ratio and then fly all 1000 of
them into earth orbit along 1000 statistically varying trajectories,
approximately 500 of them will end up with fuel residuals at burnout, and
the other 500 will end up with oxidizer residuals.
Moreover, on the average, the 500 oxidizer residuals will turn out to
be approximately five times heavier than the 500 fuel residuals because a
typical hydrogen-oxygen rocket carries five pounds of oxidizer for every
pound of fuel. Consequently, if we would add a little extra fuel to each
of those 1000 rockets before lift-off, that extra fuel would reduce the
statistical frequency of the heavier oxidizer residuals. Moreover, the
few remaining oxidizer residuals that do occur will be lighter because of
the fuel bias we have added.
In practice, however, figuring out precisely how much extra fuel to
add to achieve optimal mission performance turned out to be a difficult
and expensive problem in statistics. Our first approach toward
determining the optimal fuel bias is flowcharted in Figure 3. In each of
our simulations we command the computer to choose a fuel bias and then
sample a series of statistically varying values having to do with the
variation of the rocket's thrust, its flow rate, its specific impulse,
its mixture ratio, and so on. The computer then substituted each of these
statistical values into our optimal trajectory simulation program, and at
burnout, it recorded the type of residual (fuel or oxidizer) and its
corresponding weight.
This so-called "Monte Carlo" simulation procedure was repeated
hundreds or thousands of times to allow the computer to construct an
accurate statistical "snapshot" similar to the one sketched at the bottom
of Figure 2. Repetitions of those computerized procedures executed with
different fuel-bias levels allowed us to determine the fuel bias that
provided the optimum rocket performance.
This technique worked as advertised, but it turned out to be extremely
costly, in the days when computer simulation time was so incredibly
expensive. However, after several hours of mind-bending mathematical
manipulations, I managed to reduce the essence of the optimization
problem we faced to a single mathematical equation. It was an integral
equation from calculus with variable limits of integration based on the
normal distribution functions from the statistics courses I had been
attending at UCLA.
Figure 2: In the 1960's this Monte Carlo sampling procedure
provided our analysis team with a simple and convenient method for
finding the optimum amount of fuel bias to add to the S-II Stage to
minimize its "3-sigma" fuel and oxidizer residuals. Although this
procedure was conceptually simple and easy to implement, finding the
optimum fuel bias turned out to be extremely costly in an era when a
rather primitive IBM 7094 computer rented for $700 per hour. On a typical
Apollo mission we were burning though $95,000 worth of computer time to
find the optimum bias level. Practical alternatives were mathematically
elusive, but eventually we developed a far more economical approach based
on Leibniz' rule for the differentiation of integral
equations.
That equation, though simple in appearance, could not be integrated to
get a simple answer in closed form. Fortunately, that summer I had been
studying a powerful branch of mathematics called the calculus of
variations pioneered, in part by my hero, Isaac Newton.
Isaac Newton, Christmas present to the world, was born on December 25,
1642. In that era, if a talented mathematician would solve a difficult
mathematical problem, he would sometimes pose the problem to various
other famous mathematicians before publishing the solution.
Such a problem had been posed by the Bernoulli brothers, two famous
Swiss mathematicians. It centered around the optimal shape for a wire on
which a small bead would slide in minimum time from one point to another
under the influence of gravity. The Bernoulli brothers had posed this
problem to Newton's rival Gottfried Wilhelm von Leibniz who had not been
able to solve it within the three months they had allotted. So he
requested six more months in which to devise a solution. The Bernoulli
brothers granted his request, but they also included Newton in their new
challenge.*
That day Newton came home from a tiring day of working in the British
mint, read his mail, and began working on the problem. By the time he
fell into bed that night, he had devised a brilliant solution which he
published anonymously. On seeing the solution, John Bernoulli is said to
have remarked, "I recognize the lion by his paw!" In his view, no other
living mathematician was clever enough to have devised the published
solution.
As luck would have it, one of the key relationships in the calculus of
variations turns out to be Leibniz's rule for the differentiation of
integral equations with variable limits of integration! I had never seen
Leibniz's rule applied to a statistics problem, but it turned out to be
the key to obtaining the solution to the optimal fuel biasing problem we
were
______________ * Egged on by British and continental mathematicians
and scientists, Newton and Leibniz engaged in a lifetime rivalry. At one
point, however, Leibniz paid Isaac Newton a supreme compliment: "Of all
the mathematics developed up until the time of Isaac Newton," he wrote,
"Newton's was, by far, the better half." seeking. By using Leibniz's
rule, some well-known identities from statistics, a back-handed
interpretation of "standard deviation", and a closed-form version of the
rocket equation as derived in 1903 by that lonely Russian school teacher,
Konstantin Tsioikovsky, I finally managed to develop a simple closed-form
solution to our optimal fuel-biasing problem!
For Rockwell International's hydrogen-fueled S-II stage, our Monte
Carlo approach had typically required 10,000 computer simulations
executed at a total cost of $95,000 per flight. The new closed-form
approach, based on Leibniz's rule, required only 13 computer simulations
at a cost of around $3000.
My supervisor, Paul Hayes, again demonstrated his leadership when he
secretly submitted a company suggestion in my name indicating that I had
managed to develop a derivation that saved the Saturn S-II Program over
$700,000 based on nine manned missions flown into the vicinity of the
moon. Paul was sorely disappointed when the reply came back from the
suggestion group: No award was to be forthcoming because, as they pointed
out: "That's what he does for a living."
The parametric curves at the bottom of Figure 3, which were
constructed using the closed-form equations I derived, were used to
determine the optimum fuel-bias level. For a typical Apollo mission, the
optimum amount of fuel to add turned out to be about 600 pounds, assuming
that we wanted the smallest residual propellant remaining at the "3
sigma" probability level (99.87 percent).
Bob Africano and I later published a technical paper in which we
discussed the fact that biasing to minimize residuals is not the same as
biasing to maximize payload. We reasoned that these two bias levels must
be slightly different because, when we add fuel bias to minimize the
residuals, the fuel bias itself represents a dead weight that the rocket
must carry into space. However, we soon discovered that no matter how
many times we manipulated the relevant mathematical symbols, we could not
discover the desired relationship. Several years later, however, John
Wolfe, a superb space shuttle engineer, read our paper and figured out
how to bias to maximize payload. John Wolfe was such a generous soul, he
even claimed, in print, that Bob Africano and I had solved the problem on
our own. Actually, all we had done was to formulate the problem. John
Wolfe, himself, provided the solution!
Figure 3: A clever mathematical algorithm based on Leibniz'
rule for the differentiation of integral equations with variable limits
of integration allowed us to find the fuel bias that would minimize the
"3-sigma" fuel and oxidizer residuals remaining at burnout of the Saturn
S-II stage. This new approach saved $92,000 per flight while achieving
essentially identical results. Later a highly creative space shuttle
engineer, John Wolfe, figured out how to modify our procedure to maximize
the payload of the reusable space shuttle.
It was not a difficult derivation; we understood it immediately. But
finding it did required a rather unusual mathematical approach that had
eluded us throughout several dozen oversized pages of Technicolor
derivations.
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POSTFLIGHT TRAJECTORY RECONSTRUCTION
On January 1, 1801, the first minor planet, Ceres, was spotted by
alert telescope-equipped astronomers as it hooked around the sun. Ceres,
which we now call an asteroid, was a new type of object never seen by
anyone on Earth up until that time. Unfortunately, after Ceres had been
in view for only 41 days, it traveled so close to the harsh rays of the
sun it was lost from view. The astronomers who were tracking it were
afraid that it might never be found again.
However, as Figure 4 indicates, the famous German mathematician Carl
Frederich Gauss accepted the challenge of trying to reconstruct the
trajectory of Ceres from the small number of closely spaced astronomical
observations available to him. Under his brilliant direction, Ceres was
located again on the other side of the Sun on the last day of 1801,
almost exactly one year after it had first been discovered.*
More than 160 years later in 1962, we adapted the mathematical methods
Gauss had used in reconstructing the orbit of Ceres to determine the
performance of the Saturn V moon rocket on a typical mission. When we
were executing a preflight trajectory simulation, we would feed the
thrust and flow-rate profiles into the program together with the initial
weight of the vehicle, its guidance angle histories, and the like, and
then we would simulate the resulting trajectory of the rocket. In a
postflight trajectory simulation, we did exactly the opposite. We would
feed the program the trajectory of the
______________ * When Gauss was in elementary school in Germany, one
of his teachers asked her students to "add up all the values of the 100
integers ranging from 1 to 100." While his classmates were struggling to
obtain the solution, the young Gauss wrote down the answer immediately.
He had noticed that there were 50 pairs of numbers – each of which
totaled 101; they were 1 + 100, 99 + 2, 98 + 3 . . and so the desired
total was equal to 50 (101) = 5050. rocket – as ascertained by the
tracking and telemetry measurements – and then we would use the
computer to determine the thrust and flow-rate profiles and the guidance
angles the booster must have had in order to have traveled along the
observed trajectory.
Figure 4: In 1801 the brilliant German mathematician Carl
Frederich Gauss devised a marvelously efficient mathematical algorithm
that allowed the astronomers of his day to relocate the asteroid Ceres
– a tiny pinpoint of light – as it emerged from the harsh rays of the
sun. Approximately 160 years later our analysis team adapted this so-
called iterative least squares hunting procedure to help us reconstruct
the postflight trajectories of the various stages of the mighty Saturn V.
Over time these mathematical techniques increased the rocket's translunar
payload by 800 pounds.
Years later in a television interview on the ABC television network,
my host asked me what a trajectory expert does for a living. "We predict
where the rocket will go before the flight," I replied. "Then, after the
flight, we try to explain why it didn't go there."
Those of us who worked as trajectory experts on the Saturn V moon
rocket developed one of the most sophisticated postflight trajectory
reconstruction programs ever formulated up until that time. It included
more than 10,000 lines of computer code (five boxes of IBM cards!) and it
required 300 inputs per simulation, all of which had to be correct if the
program was to produce the desired results. Unfortunately, 75 percent of
our simulations blew up due to incorrect inputs. A small percent of the
others blew up because we made various mistakes when we made
modifications to the program.
In a typical postflight reconstruction, we simulated a 400-second
segment of the rocket's trajectory which required about 2.5 hours of
computer time on an IBM 7094 mainframe computer at a cost of about $700
per hour. Our six-degree-of-freedom iterative least squares hunting
procedure was structured so we could, on any given simulation, choose up
to nine independent variables, such as vehicle attitude, slant range,
inertial velocity, and the like. We could choose up to nine dependent
variables, such as the rocket's thrust profile, flow-rate history, the
initial weight of the rocket, and so on.
We initially formulated the six-degree-of-freedom trajectory program
so that all the search variables were added to or multiplied by the prime
variables (e.g .the thrust profile or the weight history of the rocket
stage). Later we figured out how to include additive or multiplicative
polynomials with variable coefficients that were determined automatically
by the computer. We also figured out how to "segment" (chop up) the
relevant polynomials with automatic computer-based determination of the
polynomial coefficients in each of the segments being determined
independently. The independent variables were measured during the flight
with tracking devices located on the ground and telemetry devices carried
onboard the rocket. On a typical Saturn V trajectory reconstruction, the
computer calculated about 30 partial derivatives at each of the 400 time
points spaced one second apart. The resulting partial derivatives –
around 12,000 of them – were arranged sequentially in a special matrix
format and recorded on as many as nine magnetic tapes.
On a typical Apollo flight, the average deviation between the
predicted preflight trajectory and the actual postflight trajectory was
about one mile. However, after 2.5 hours of simulation time on an IBM
7094 computer, the iterative least squares hunting procedure typically
reduced this average error to only about one foot!
After running a series of computer simulations of this type, we were
able to get a much better handle on the statistical variations in the
dependent variables such as the rocket's thrust and it's specific
impulse. This new knowledge, in turn, allowed us to increase the
performance capabilities of the rocket by several hundred pounds of
payload headed for the moon.
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THE LEGACY
Today virtually every large liquid rocket that flies into space takes
advantage of the performance-enhancement techniques we pioneered in
conjunction with the Apollo moon flights. NASA's reusable space shuttle,
for example, employs modern versions of optimal fuel biasing and
postflight trajectory reconstruction. However, more of the critical steps
are accomplished automatically by the computer.
Russia's huge tripropellant rocket, which was designed to burn
kerosene-oxygen early in its flight, the switch to hydrogen-oxygen for
the last part, yields important performance gains for precisely the same
reason the Programmed Mixture Ratio scheme did. In short, the fundamental
ideas we pioneered are still providing a rich legacy for today's
mathematicians and rocket scientists most of whom have no idea how it all
crystallized more that 40 years ago.
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THE CONCLUSION
Figure 5 summarizes the performance gains and a sampling of the
mathematical procedures we used in figuring out how to send 4700 extra
pounds of payload to the moon on each of the manned Apollo missions. We
achieved these performance gains by using a number of advanced
mathematical techniques, nine of which are listed on the chart. No costly
hardware changes were necessary. We did it all with pure mathematics!
In those days each pound of payload was estimated to be worth five
times its weight in 24-karat gold. As the calculations in the box in the
lower right-hand corner of Figure 5 indicate, the total saving per
mission amounted to $280 million, measured in 2009 dollars. And, since we
flew nine manned missions from the earth to the moon, the total savings
amounted to $2.5 billion in today's purchasing power!
We achieved these savings by using advanced calculus, partial
differential equations, numerical analysis, Newtonian mechanics,
probability and statistics, the calculus of variations, non linear least
squares hunting procedures, and matrix algebra. These were the same
branches of mathematics that had confused us, separately and together,
only a few years earlier at Eastern Kentucky University, the University
of Kentucky, UCLA, and USC.
I was born and raised in a very poor family. At age 18 I had never
eaten in a restaurant. I had never stayed in a hotel. I had never visited
a museum. But somehow, by some miracle, six years later, at age 24, I was
getting up every day and going to work and helping to put American
astronauts on the moon!
Even as a teenager I loved doing mathematical derivations. Those
squiggly little math symbols arranged in such neat geometrical patterns
were endlessly fascinating to me. But never in my wildest dreams, could I
ever have imagined that someday I might be stringing together long,
complicated mathematical derivations that would allow enthusiastic
American astronauts to hop around on the surface on the moon like
gigantic kangaroos!
Nor could I have ever imagined that someday my Technicolor derivations
would end up saving more money than a typical American production line
worker could earn in a thousand lifetimes of fruitful labor!
Figure 5: Over a period of two years or so a small team of
rocket scientists and mathematics used at least nine branches of advanced
mathematics to increase the performance capabilities of the Saturn V moon
rocket by more than 4700 pounds of translunar payload. As the
calculations in the lower right-hand corner of this figure indicate, the
net overall savings associated with the nine manned missions we flew to
the moon totaled $2,500,000,000 in today's purchasing power. These
impressive performance gains were achieved with pure mathematical
manipulations. No hardware modifications at all were required.
Read more here
http://www.articlesbase.com/gps-articles/schemes-for-enhancing-the-
saturn-v-moon-rockets-translunar-payload-capability-4191020.html