Spudis writes:
Sub-orbital “hops” (ballistic flights from point-to-point) are possible, but come at fairly high cost—it takes nearly as much energy to fly hundreds of kilometers on the Moon in a ballistic hop as it does to go into orbit and then descend elsewhere.This is incorrect. Here I look at suborbital hops on airless worlds. A minimum energy ellipse going from point A to B would have a focus on the midpoint of the chord connecting A & B:
The other focus would be at the moon's center, of course.
The vis viva equation tells us
v=sqrt(GM(2/r - 1/a)
In this case GM is the moon's gravitational parameter, r is the moon's radius and a is the semi major axis of the ellipse.
Let's say A is 300 km from B. That'd be about 9.9 degrees separation. Here's a pic:
.67 km/s to hop and another .67 km/s for a soft landing. For low lunar orbit that would be 1.68 km/s to take off and another 1.68 km/s to soft land. Energy goes with square of speed. (.67/1.68)2=.16. The energies differ by more than a factor of 6! How on earth did Spudis conclude these are nearly the same?
Here is my Lunar Hopper spreadsheet. There's a tinted cell user can input distance between point A and B. This is the first document I've uploaded to Google docs, hope it works.
Spudis suggests spherical pit bots for lunar drones. These bots use micro thrusters to hop and hover. Whether the hop is 5 meters or 500 kilometers, the most efficient hop is the minimum energy ellipse described above. On the moon a ten minute hover costs about one km/s delta V. Spudis justifiably grouses about the tyranny of the rocket equation. But these pit bots rely on reaction mass to move. They don't circumvent said tyranny.
Pogo Hoppers
Various folks talk about lunar drones at Spudis' forum. Someone who goes by the name finkh mentioned pogo sticks. An interesting notion, in my opinion.
When I was a kid, my pogo stick used a spring. Solar cells might provide energy over time to compress a spring, thus avoiding the use of reaction mass. No more nasty rocket equation! On landing the spring absorbs the impact. The compression on impact might be a way to recover some energy.
I moved over a variety of terrains with my pogo stick. I could move forward, backward, left or right. It seems feasible to develop a robot with similar abilities.
But as I recall, getting from point A to B on a pogo stick was more strenuous that walking. So I'm not sure compressing a spring on impact is a great way to regain energy. Looking at existing robots like Big Dog, it looks like powerful engines are needed to power the device. Once again, the need for a better Alpha rears its ugly head. Elon Musk seems to be working on improved solar panels and energy storage. Hopefully Tesla Motor's R&D will have applications in space exploration.
How much impact can a pogo stick take? The 300 km hop pictured above hits the ground at .67 km/s or about 1500 miles per hour. No, I wouldn't want to be on that pogo stick.
This list of pogo stick records says Biff Hutchison jumped nearly 3 meters high. By my arithmetic he hit the ground at about 7.5 meters/sec or about 17 miles per hour.
Assuming 17 miles per hour is maximum jumping and landing velocity, a lunar pogo could jump about 36 meters (assuming the hop was a minimum energy ellipse from point A to B). This hop would be about 9 meters high.
A jump 36 meters long and 9 meters high isn't spectacular but such a device might have uses. And I like the image of a pogo stick on the moon.
9 comments:
Your sheet is not allowing public access...
Something to consider...
At least one Lunar space elevator proposal includes tramways running to the poles. These are meant to be placed every 1-3km depending on terrain. A suborbital hop of that length is only 40-70 m/s, potentially within range of an airbag and anchor (or maybe even a pogo spring) for landing. During the construction phase payloads could travel along the tramway to the last tower, fire a short engine burst and land without using further fuel. 15 seconds of flight time vs. 18 minutes of drive time (10km/hr) might not justify it unless the terrain is nasty or the mass of the drive mechanisms are much larger than the mass of fuel, engine and landing gear.
The pogo stick record you mentioned is about 3 meters / 7.5m/s for an adult male, maybe 60 kg? That would be 1688 joules of kinetic energy. With that much force a 10-kg pit bot could hit over 18 m/s, more than 200 meters on the moon. I have doubts about handling that much energy in a 1.5kg subsystem of a 3kg bot, but I bet you could top 36 meters.
If it is a pressurized gas system you could absorb (some of) the landing force with a ratchet mechanism. A gas generator or onboard pressurized nitrogen could be used to re-pressurize the chamber; there would be some gas loss since no system is perfect, but at least it would be unchained from the rocket equation.
Attitude control and mechanical power storage could be handled by a set of flywheels (arranged to avoid gyroscopic problems). Primary mobility would be by rolling along the ground using the angular momentum of the flywheels. Secondary mobility using the pogo would be assisted using flywheels to ensure a good contact angle for energy recovery.
I don't pretend to know if that could be engineered into a 3kg exploration bot, but it seems such a system could offer better energy density and energy efficiency than expendable rocket fuel. It would still use gas of some kind as a pressurant, but at a much lower rate.
Another option would be a conductive tether that unreels from inside the exploration bot; that would give you fiber-optic data uplink plus robust electrical power. Tethers unlimited has some prototypes, but I'd guess you could stuff a few kilometers of tether inside the 3kg bot. With electric motors and mechanical springs there would be no consumable gas and no need to use flywheels for energy storage (though they could still be used for attitude control). This would most likely be lighter than the reference design and would avoid the snags problem. It does mean that either the bot or the tether is expendable if it snags on the way back and there is no hover capability.
If your bot is a mobile mini-lab like Curiosity then putting using pogo stick locomotion might be a bad idea. Your instruments would be rattled every time the bot moved. I've been thinking on the logistics of doing a science mission on the Moon with bots. A limited-budget mission might be similar the Mars rover missions with a slow-moving, RTG-powered bot sending data back to Earth. A more ambitious mission might involve the following elements:
*A human-rated lab in lunar orbit.
*A reusable lander with a laser spectrometer, drill, ground penetrating radar, deployable seismometers and sample return to lunar space.
*Delta-v might be further minimized by using a hanging tether or rotating tether in a lunar frozen orbit.
While a reusable lander could take samples from all over the Moon's surface the environment near each landing site would be disturbed upon landing. Some secondary means of non-disruptive locomotion might be desirable. By returning to a the orbital lab the lander would not have to be built to survive the lunar night.
What about a trampoline? :-D
Wouldn't the lower surface gravity on the Moon lower the efficiency of a pogo stick proportionally? You'll jump high, but you'll land softly and not charge the spring very much.
Dmitry, a trampoline on the moon would be a hoot! If they could get a hab with a high enough ceiling, I'd bet that be a big attraction.
Anonymous, please give yourself a label so it's clear what comment I'm replying to. On a level surface, speed coming down is the same as the speed going up. This would be true regardless if the gravity's .1 g or 10 g's.
Have you seen the latest idea from JPL? Using a tether and a hook to decelerate at comets or KBOs:
http://www.centauri-dreams.org/?p=33943
Thought this might be right up your alley. :-) And they propose to use that system to accelerate again, in order to visit multiple targets.
Tony, I saw that Centauri dreams suggesting harpooning comets. Something falling from the Kuiper Belt or even Jupiter height is moving pretty fast when in our neighborhood. Rendezvous by harpoon would be extremely difficult (IMHO). So I wasn't super enthusiastic.
However I was happy to see JPL guys thinking of tethers. As you probably know, I'm obsessed with tethers.
John Hare did a recent Selenian Boondocks post on using tethers on asteroids http://selenianboondocks.com/2015/04/asteroids-as-transportation-hubs/ Asteroids typically have high angular velocity and a shallow gravity well. So elevators are a whole lot easier than the deep wells of planets or big moons. If we make it to the Main Belt, I expect tethers will be used a lot. Vesta and Ceres are perfect for elevators.
I know it's late, but I checked out the maths for one mile (1.60934 km). Spreadsheet claims 114 mph per impact; then, of course, you slow down until aposelene (r = a x (1+e)) and speed back to 114 mph on return.
The "ToF" at cell D27 has the flight taking 0.185726903 minutes overall. Where one mile / minute is 60 mph (famously), we've traveled that mile at 5.384 times 60 mph. A golf cart leaving and meeting the pogo would be travelling an average 323 mph. Almost three times the pogo takeoff=impact speed.
Are cells D16 and D17 right? I wonder about pogo semimajor D16=(1+SIN(D14))*D10 / 2. The halving could be removed; D17 might shift that =SQRT($B$4*D9*(2/D10-2/D16)). Then both the pogo velocity and the overall mph approximate 114 mph.
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