So in the fall of 2002 that’s what I did, and before long I was writing NRAs (National Research Announcements) to solicit universities and corporations to bid on technology work for tethers. We put the first NRA out for tethers and got responses and had a meeting where a committee picked the winners in March 2003. After that things started getting serious. We had real money for the first time to do momentum-exchange tether work, and there were still so many unanswered questions that needed to be solved.

Sometimes fate or luck or serendipity drops things in your lap. In the summer of 2002, I met one of the most clever and hard-working people I’ve ever had the good fortune to meet–Dr. Stephen Canfield of Tennessee Technological University. The next summer he was down at MSFC and I was in the middle of trying to figure out the answer to a very thorny problem: if you have a tether that’s spinning, how do you keep the solar panels pointed at the Sun? My friend Kyle Frame and I would sit in my cubicle for long stretches of time with pieces of paper pretending to be solar panels and pencils and sticks standing in for the tether, trying to figure out some way to do it that wasn’t totally foolish.

One day Steve Canfield stopped in and asked us what we were up to. We described the problem and he asked a simple question:

“Do you care what orientation your solar panel is in so long as it is pointed at the Sun?”

I said no, we didn’t care, and then he showed me something he’d been working on since he was a grad student. It looked like this:

He called it a “Trio-Tristar Carpal Wrist Joint.” I thought that sounded like a real mouthful so I just called it “Canfield’s joint” and eventually everyone (except Canfield) began to call it a Canfield joint. It was kind of a crazy looking thing that you couldn’t figure out what to do with it unless you held it in your hands and started playing with it. Unfortunately, in a blog post I can’t reach out of your screen and hand you your own Canfield joint to play with, because if I could you’d figure out in a few seconds what I’m talking about, but the real magic of the Canfield joint is that you can point the joint anywhere in a hemisphere without winding up anything.

The joint has several parts. There’s the “base plate” which stays attached to whatever the joint is mounted to, like your spacecraft, and then there’s the “distal plate”, which points to whatever it is that you want to point at. There are six legs on the joint, in three units. The joint is called a “parallel structure” because there’s more than one load path for the loads to follow, and this is what gives it its potential strength. Where the legs mount to the plates is a simple revolute joint. I didn’t know what that meant so I asked Canfield and he said that it just meant that it was a simple, one-degree-of-freedom (one way to move) joint or hinge. Where the two legs come together you could have a spheric joint (a ball and socket with two degrees-of-freedom) or you could have three revolute joints in series. That’s what we usually do.

I asked Canfield what the joint was for. He said that he originally wanted to use it to replace the CV joints in cars, since if it had all revolute-joints then it wouldn’t need a boot. If I hadn’t had to replace the boot on the CV joint in my car when I was in college and dirt-poor, I wouldn’t have had any idea what he was talking about, but the loss of money was still burned in my mind, so I appreciated that application.

Well, to make a too-long story shorter, I learned how the Canfield joint worked and figured out how to solve my little problem on the tether. Tell me if you like the result:
Canfield Joint on MXER Tether
Medium View of Canfield Joint
Closeup of Canfield Joint

The expression uses the Gaussian error function, erf(x), which is not typically available in a spreadsheet or scientific calculator. erf(x) also cannot be calculated in closed-form–typically the expressions used to calculate it are iterative. Since the tapered tether mass ratio is such a useful design tool to have, I derived a recursive algorithm that computes the ratio in a simple loop, given only the velocity ratio of the tether (tip velocity/characteristic velocity).

Here is the recursion:

and here is psuedocode for the recursion:

subroutine getRatio(double VR, int k) {
double VR2 = VR*VR;
double sum = 0.0;
for (int i = k; i >= 1; i--) {
sum = (VR2/(double)i)*(1.0/(double)(2*i+1) - sum);
}
return 2.0*exp(VR2)*VR2*(1.0 - sum);
}

With about 8 recursions, the results are extremely accurate. The recursion is unstable when the velocity ratio is greater than about 3, but no one should be building tethers with velocity ratios greater than 3! and you can just use Moravec’s expression with erf(x) = 1.0, which is a pretty safe assumption for x > 2.0 or so.

The first idea of concept of a tether dates back to the imagination of Konstantin Tsiolkovsky, the Russian schoolteacher who first developed our modern concepts of rocketry and first derived the rocket equation. In the late 1800s, Tsiolkovsky visited Paris and saw the Eiffel Tower. He was so impressed by the sight that he imagined a tower reaching up far into space. He calculated the height at which such a tower would have to be before the centrifugal force from the Earth’s rotation balanced the pull of gravity (inadvertently calculating the altitude of geosynchronous orbit).

Tsiolkovsky, of course, could not conceive of any material that could withstand the compressive forces of such a structure, but sixty years later, a Russian engineer named Yuri Artsutanov picked up the thread of Tsiolkovsky’s work and first worked out the engineering principles of what is now called a “space elevator”, a long tether hanging all the way from geosynchronous orbit to the surface of the Earth. The space elevator required materials with specific tensile strength far in excess of any known material, and still does. Further conceptual engineering work on the space elevator concept was done in the early 1970’s by American engineer Jerome Pearson.

The space elevator was a hanging tether, and payloads were required to traverse its length in order to achieve orbit. The beginnings of rotating momentum-exchange tethers date to the late 1970s, when Hans Moravec, a robotics researcher at Stanford University (now at Carnegie-Mellon) was intrigued by a suggestion of his friend John McCarthy of a satellite that “rolled like a wheel” around the Earth. Moravec began a scientific investigation of the concept, which he first called a “non-synchronous orbital skyhook” and later a “Rotovator”. Like the space elevator, it reached all the way to the surface of the Earth, but unlike the elevator, it rotated about its axis a number of times per orbit. A payload would be picked up by the tip at the surface of the Earth and then thrown half a rotation later into an interplanetary trajectory. The Rotovator was a good deal shorter than the space elevator (~4200 km vs. 40, 000 – 100,000 km) but was not much better in terms of materials required. Moravec published a paper on the subject in the Journal of Astronautical Sciences where he speculated on advanced forms of matter that might have the strength needed to build the Rotovator.

About a year after the JAS paper was published, Dupont’s development of Kevlar excited Moravec to the possibilities of Rotovators built with conventional materials. He wrote a short paper called on the subject which was never published. The paper showed that Kevlar skyhooks were not feasible around the Earth but could be reasonably built around the Moon and Mars. In an appendix to this unpublished paper, Moravec speculated on the possibility of skyhooks built in interplanetary space that would assist spacecraft traveling between the Earth and Mars. To the great benefit of future tether researchers, his equations for the cross-section of a tether, in the absence of a gravitational field, could be integrated in closed-form. Thus, the Moravec “tether equation” was first derived.



Moravec was able to derive analytical expressions for the area of the tether as a function of its distance from the rotational center. He then numerically integrated the area expression along the length of the tether to calculate volume and mass. As an aside, in an appendix, he considered the case of a tether spinning in free space. When the tension on the tether was only due to centrifugal forces, the area expression could be analytically integrated to a closed-form solution. Thus the Moravec mass ratio was derived.

The equation could be simplified by realizing that fundamentally, the mass ratio is a function only of the velocity ratio, which itself is the ratio of the tip velocity of the tether and the characteristic velocity of the tether material.


Further insight into the value of the equation was gained by comparing it to the rocket equation and noting the similarities and differences.

Moravec wrote a few articles on the subject for space-themed publications, but basically returned to his robotics work. Nevertheless, Moravec’s equation still serves as a foundation to all momentum-exchange tether work to this day.

The rendezvous required for a rotating tether and its payload is far more dramatic. The whole point of the operation is to have the tether and the payload in different orbits, so that the rendezvous can lead to an exchange in angular momentum and orbital energy between the two, resulting in a payload boosted to a higher energy orbit (or dropped to a lower energy one).

Thus, you can’t match orbits like you do in conventional rendezvous. The best that you can do is to instantaneously match position and velocity (but not acceleration). So you need an approach to rendezvous that is pretty tolerant of error.

So we threw out the book when it came to trying to think of how to do rendezvous, and came up with something totally different and designed to meet the specific needs of the mission. And I was pretty proud of the result, and still am. Because, you see, this is a bit of an anniversary for tether rendezvous technology. It was five years ago (February 2005) that we successfully demonstrated that the rendezvous technology we had postulated could work, at least at the lab scale.

We took advantage of the fact that the tether was under rotation and experiencing centrifugal acceleration, and that the payload was in free-fall. We simulated this (quite accurately) by hanging the tether’s “catch mechanism” from the ceiling of a racquetball court at Tennessee Tech, and then we “shot” our simulated payload up to the catch mechanism, with its boom positioned to be captured by the catch mechanism when it penetrated the aperture of the catch mechanism. Then the catch mechanism would release and close around the boom, quite quickly, allowing the simulated payload to be caught.

It all worked out a lot better than I thought it would–take a look at our results:

First Catch Mechanism Test

Second Catch Mechanism Test

More Testing with Animation

And here was the press release that came out months later announcing the accomplishment. Our video footage of successful testing got on NASA TV…once.

NASA Engineers, Tennessee College Students Successfully Demonstrate Catch Mechanism for Future Space Tether

So we might want to try to figure out what kind of size of tether it would take to do these nifty tricks. The first place to start is to ask how big the tether needs to be–is it huge or tiny relative to our payload? To answer that question we need to know two things: what the velocity of the spinning tether is at its tip, and what it’s made out of.

There is a simple relationship between the mass of an untapered tether and the payload it is meant to catch and throw. Actually, it’s a ratio, much like a mass ratio in the rocket equation:

MR = 2(VR)^2/(1 – (VR)^2)

MR is the mass ratio, or the mass of the tether divided by the mass at the tip.
VR is the velocity ratio, which is the tip velocity divided by the characteristic velocity of the material.
The characteristic velocity of the material is the square root of two times the tenacity divided by the product of the safety factor and the material density.

VR = Vtip/Vc

Vc = sqrt(2*tenacity/(safety-factor*density))

An example might make this more clear:

Let’s say we wanted to throw a payload from LEO to a geosynchronous transfer orbit (GTO). That’s about a 2400 m/s delta-V beyond LEO, applied impulsively at LEO altitude. Assuming that the tether is in an orbit intermediate between GTO and LEO, and assuming that the tether will give half of the delta-V at catch and the other half at throw, let’s run the numbers:

Let’s assume we’re using a material with a characteristic velocity of 1600 m/s. To get the 2400 m/s of DV we need a tip velocity of 1200 m/s. This gives us a velocity ratio of (1200 m/s)/(1600 m/s) = 0.75. Plugging VR=0.75 into the equation gives 2*(0.75)*(0.75)/(1 – (0.75)*(0.75)) = 1.125/0.4375 ~2.57.

So the untapered tether will have a mass 2.57 times that of its payload, in this scenario. This very simple analysis also assumes that the tether is connected to a counterweight that is, for all intents and purposes, infinite, so as to save us the step of computing the mass of the tether on the other side of the center-of-mass. So assuming we had a payload of, say, 1000 kg, the mass of the tether would be 2570 kg.

The equation I’ve just described shows that the characteristic velocity of a material has a physical meaning. If you look at the denominator of the equation, you can see that when the velocity ratio goes to one, the denominator goes to zero and the mass ratio goes to infinity. So the characteristic velocity is the maximum tip velocity of an untapered tether (of any length) with a safety factor of 1, at which speed it will break under its own tension.

In my next post, this simplified approach to tether modeling will get more complicated by describing TAPERED tethers whose velocity ratio can be greater than one. But the mathematics will be more complicated.

After working on the effort for a few weeks, I summoned up the courage to show my ideas to my coworkers. One of them read through my work, and said to me “that’s it? can’t you come up with something better than that?”

I was really crushed and I asked him what he meant. He said, “Don’t you think that after 30 years that we can come up with a better way to go to the Moon than big throwaway rockets?”

Like someone who had just been told that their baby was ugly, I went back to the drawing-board, so to speak, humbled by my colleague’s response to my idea.

I started poking around the internet and found a company called Tethers Unlimited, owned by Robert Hoyt and Robert Forward, the late science fiction writer. This company (TUI) was talking about a kind of space tether I had never heard of before. It spun round and round and caught and threw a payload from one orbit to another. It sounds fantastic (as in fantasy) and I sat there and thought about, fresh from my orbital mechanics course a year earlier. After a few hours I had convinced myself that it didn’t violate the laws of physics or orbital mechanics, and that, in theory, it should be possible to do what they claimed. But I thought it was hopelessly complicated from an engineering perspective. The “catch maneuver” in particular seemed nigh-unto impossible.

Little did I know what I would be getting myself into–some of the most professionally rewarding and frustrating years were to follow that little discovery…

…as a preview, enjoy this animation, which a team of animators and I spent the better part of a year (2004) working on, so that we might make momentum-exchange tether principles easier to understand.