Sibling Rivalries, Barstools, and Kebral Space Program

[Alshain]

My little brother an I got into a debate of engineering prinicples this evening (yeah we are strange that way). He put forward that a barstool with only 3 legs is more stable than one with 4. Of course I denied it could be possible, more legs is always more stable, right? So we first took to Google where we found a lot of sketchy information about how 3 points make a plane (the dimensional kind, not the kind with wings) and how the extra leg destabilizes the plane. But we couldn't find anything really concrete that said for certain.

So I had the great idea, let's try this in practice. Let's land on the Mun! We then spent several hours designing and testing at least 6 different mun landers and landing them on different terrain. We had tall skinny landers, wide landers, we landed on hills and flat terrain, each with a 3 leg and 4 leg variant. On a flat landing surface it seemed no different, but then we moved on to steep slopes. The wide landers performed well in either case, but to my surprise, I could not get the 4 legged tall lander down after about 10 tries, it always tipped over. Once I switched to the 3 leg variant, it landed on the first try. Not only that, but it just felt easier. The trick was to move 2 legs up hill with the 3rd downhill just before touchdown.

So my brother was right (and I'm not too happy about that) but at least now I know a better way to build a Mun lander. Leave the 4th leg back at mission control.

[DarkGravity]

Cameras and telescopes come with tripods. Never quadrapods. In geometry, 3 points define a plane. When you sit at a wobbly 4-legged table, you might fold a piece of paper and wedge it under the one leg that doesn't reach the ground. Something supported by a tripod will never wobble, cuz all three legs will always be in contact with the ground.

Also triangles are inherently strong building shapes. Take a drinking straw and fold it into a triangle, and tuck the two ends into one another. Do the same to make a square. Try to wiggle them. The triangle won't wiggle. The square will wiggle a lot. Triangles rock.

[Dres]

My theory on this is that it depends on the number of "tipping points" an certain layout of legs has. For example, an object with more legs can have more directions it can fall more easily, one in each area between two legs. However, larger tipping points relative to circumference of the object can also lead to instability, as it is more likely for a slope to be toward a tipping point. For most slopes, fewer tipping points would be better because it minimizes the chance that the force down is close to the center of any said tipping point. The reason the third leg down-slope method worked is because it kept the pull down the slope on a corner, a direction which an object is difficult to tip. At least, that is my theory, it is probably wrong, so just ignore me

Edited May 18, 2015 by Dres

[Temeter]

Interesting. I might have to try a 3 legged lander. Btw, since I was curious and googled it, here a short version of the explanation:

It's just a triangles rigidity-thing: 3 legs are perfectly limiting all dimensions of movement without any extra. Every leg will push the lander towards a situation where all legs are on the ground. A 4 legged lander on the other hand adds imperfect complexity and will easier create situations where solving the direction of the wobble is done in multiple ways.

Most notable, when 4 legged lander is tilting toward the site, then it's really just behaving like a two legged lander, and there are two logical directions 2 legs can tilt. Compared, a three legged lander has a stronger bias to tilt inwards.

edit: dem ninjas

Edited May 18, 2015 by Temeter

[adinfinitum]

Did you try it so that the four legged lander had a leg pointing downslope, one upslope, and two to the sides?

[DarkGravity]

Tripods are useful because the three feet will always find a plane to touch down in. However, four-legged tables do have their uses too. Stability occurs when the center of mass is over a place that is supported. What is useful is not the number of tipping zones, but the distance between the feet that are supporting the object, or the size of the tipping zone. If you put a tripod and a four-legged table of the same height indoors on a level floor, the tripod is easier to knock over because it is easier to move the center of mass of the object over a spot in the middle of one of the tipping zones, away from the supports. In this setting, the table is more stable, because its tipping zones are shorter, making it harder to move the center of mass to an unsupported location. However, on rugged terrain, the tripod will probably win out because the table will be very unlikely to touch down with more than three feet, and the distance between the feet that are in contact with the ground is now larger, because you have 3 legs of a table in contact with the ground (leaving one whole quadrant unsupported), rather than three evenly-spaced legs of a tripod. So that makes one tipping zone extremely large for the table.

[SpacePi]

So that makes multiples of 3 the best? So six-legged landers are better, or just redundant?

[Ival70]

When the Apollo Lunar Lander was being designed, several iterations were considered with 3, 4 and 5 legs. The 4 legged design was chosen as a compromise between stability, redundancy and weight. I believe essentially, a lander with more legs has more angles of rotation which they are optimally stable.

For example, imagine a weight with a triangle base, and try to push it over such that one of the corners is directly opposes your push. Visualize how far you have to push it before it tips over. Now do that same except with a side of the triangle on the opposite side of your push. You will notice that you don't have to tip it as far before it tumbles over. This is because the side of the triangle is much closer to the center. Now do the same thing with a square base, then a pentagon, hexagon, and so on. You will notice that the stability when a corner of the polygon opposes your push is the same for all. The extra stability for polygon bases with more corners comes from in-optimal positions, in which a side is opposing your push, not a corner.

To put it into other terms: the more sides a regular polygon has, the greater its apothem (the distance from the center to the middle of an edge). That greater distance means that the center of mass can lean farther out in any direction without tipping over.

Of course theory is quite a different subject than practice. One that note, I would like to ask if you tried landing the 4 legged lander on the hill, in which 1 leg is facing downhill, 1 is uphill, and the other two are sideways. I believe the result should replicate what you saw with your three legged design. Also try it with a lander with many many legs, 8+ and see if it will stay upright on the hill regardless of whether a leg is facing directly downhill. I am curious what the results will be, so please let us know.

[amorymeltzer]

So that makes multiples of 3 the best? So six-legged landers are better, or just redundant?

My kind of thinking!

[katateochi]

I tend to build most of my small Landers with 3 legs, partly to keep weight down but also it just seemed right... and anyway; "'cos three is the magic number" - De La Soul.

[problemecium]

No, multiples of 3 are not the best. The mathematical argument here claims that exactly 3 is the best number.

If all four (or more) legs are the exact same length, and the terrain is flat within the space occupied by the lander (it can be tilted, but must not be uneven, curved, lumpy, etc.), then the number of legs does not matter.

If, however, the terrain curves, extra legs may or may not contact the ground. Only the first three to touch down will do so, thus defining a triangle. If this triangle is very skinny, and the terrain is tilted, then the lander will be likely to tip.

However, due to the legs' suspension and the fact that any terrain a player is likely to consider "good enough for landing" will be relatively smooth, having four or more legs rarely makes a big difference. In some cases it actually helps if the lander has an uneven mass distribution or lands on a hillside.

[Alshain]

Did you try it so that the four legged lander had a leg pointing downslope, one upslope, and two to the sides?

Indeed I did, inevitably it wanted to tip one way or the other because of the difficulty of landing at an angle without altering lateral velocity. With the 3 legs, even when it tipped, it just fell on the 3rd leg and stopped. As long as you can touchdown with two legs, you are good. If you position one leg below either the hatch or the window then you can position your camera perfectly behind the hatch and the controls will match up with the legs so if it does start to become too unstable, you can save it with SAS a lot easier. Just hit the W or S in the direction of that 3rd leg.

So that makes multiples of 3 the best? So six-legged landers are better, or just redundant?

Well, I would think if leg suspension on 3 were an issue, prime numbers are best. 2 obviously won't work, but 3, 5 and 7 should be the most stable due to the polygons that would be created, with 3 being the absolute most stable. Unfortunately, placing 5 and 7 legs in KSP would be difficult and would have to be done manually.

FYI, my computer chair has 5 casters. Coincidence?

Edited May 18, 2015 by Alshain

[adinfinitum]

Ah yep, that makes sense, sounds like the best option for both landing stability and static stability would be to have six legs, with only 3 extended for landing, and the other 3 extending after you land. That way you get the landing benefits of 3 legs, and its harder to tip over once settled.

[Alshain]

Ah yep, that makes sense, sounds like the best option for both landing stability and static stability would be to have six legs, with only 3 extended for landing, and the other 3 extending after you land. That way you get the landing benefits of 3 legs, and its harder to tip over once settled.

I found it was pretty hard to tip over the landed craft on 3 as well. Even flying into it full force with a Kerbal on EVA. I think the 6 is just unnecessary mass. If you think about the 3, what angle would you hit it from to knock it over? If you hit it between two legs, the leg on the other side is going to be very strong resistance. Likewise, hitting it just above a leg means you have to push it on the full force of two supporting legs.

[adinfinitum]

Well I mean in most cases you wouldn't need max static stability, but if for example you have a tall light tower, and you're flying a kerbal back to base on Minmus you might want that extra stability, for any unintended collisions.

[Dilli]

I think it's the leg suspension that makes the 3 legged lander more stable on a slope. With the four leg lander the up-slope leg tends to have the least force on it causing it to extend further and tilt the craft.

It comes to mind as for low gravity landings I often have to lock the suspension of the down-slope leg(s) or raising the up-slope leg(s) to counter the slope.

[Hotblack Desiato]

I usually use that as lander.

http://i.imgur.com/H7MIHCB.png

4 legs, stable as hell.

so, I think it doesn't depend so much on the number of the legs, but how far they are apart from each other. and a lander-can + 4 FL-T400 tanks, with the legs on the outer edge of the tanks, that gives a very nice base, and it can't tip over.

and that was my ten-wheel rover and my 6 leg kerbal home.

http://i.imgur.com/kdNW6y7.png

[tomf]

Years ago I read a prrof that on continuous ground (with no bottomless holes in it or vertical cliffs) it is always possible to rotate a four legged table so that all the legs are in contact with the ground. Thye table won't be level and could still fall over though.

[Sky_walker]

I build few test landers and in general I found that for me it's easiest to land with 4 or more legged lander.

3 legged is too easy to tip-over when you don't come down perfectly vertical down.

[Raptor9]

Forgive me for coming at this from a lowest denominator perspective, but a lander's leg geometry and spacing should be designed for your target landing environment. There is no one size fits all. Further, the pilot will play a big part in the success or failure of the landing. If he lands on a slope when there is a relatively flat gradient nearby, or lands with lateral velocity, I don't care how stable his craft is the pilot is the factor increasing the chances of tipping over.

[Skorpychan]

I, my computer chair has 5 casters. Coincidence?

THAT is actually because of redundancy. If one breaks, you still have four legs for it to be resting on, so it PROBABLY won't tip over.

Also, I suggest people read this: http://www.projectrho.com/public_html/rocket/landing.php

Scroll down to 'landing legs' for the relevant bit.

Ideally, three is always the most stable for bumpy terrain, or hills with no lateral movement on touchdown.

If you didn't get the landing burns quite right, or didn't have time to rotate, three is tippy as hell. Try leaning over on a three-legged stool sometime. I generally use four legs, because there's that little bit of extra stability on landing when you've got lateral velocity to deal with.

Also because I use boosters in fours, and it's just plain easier than messing about with trying to make different levels of symmetry work on the same rocket.

It's not, however, perfect. My Minimus landing tipped over the other day on touchdown, and I had to right it with pod torque.

[zarakon]

A lander with 3 legs won't have wobble, which is why 3 legs is a good choice for camera tripods and barstools.

However, I think more legs would be more reliable.

When landing on a hill, the 3-legged lander works great as long as you have one leg pointed down the hill. But that same strategy should be equally effective no matter how many legs you have, assuming that in each case the feet are the same radial distance away from the center of mass. The problem with 3 legs is that if you don't get that leg pointing straight down the hill, you're going to be in a lot more trouble than if you had more legs.

As for why you had an easier time landing with 3 legs on a slope? I'm guessing that the idea of pointing a leg downhill might have just been more obvious with 3 legs, and you didn't think about doing that with 4. Or maybe all the practice you did with 4 legs just happened to finally pay off when you tried with 3

Really, I don't think the number of legs makes a huge difference. I sometimes use 3, sometimes 4, occasionally more, it just depends on the rest of the lander design - usually how many radial fuel tanks I have. The more important things for making a lander reliable are to get a low center of mass and a wide stance. A tall single-core lander with legs right off the core will tip over much more easily than a short lander with radial tanks and legs connected to those. Unfortunately the wide lander is also significantly harder to launch now, thanks to aerodynamics.

[Temeter]

I usually use that as lander.

http://i.imgur.com/H7MIHCB.png

4 legs, stable as hell.

so, I think it doesn't depend so much on the number of the legs, but how far they are apart from each other. and a lander-can + 4 FL-T400 tanks, with the legs on the outer edge of the tanks, that gives a very nice base, and it can't tip over.

and that was my ten-wheel rover and my 6 leg kerbal home.

http://i.imgur.com/kdNW6y7.png

There is of course more to stable legs. OPs solution was good for a lander which is inherently instable, namely a tower with high CoM and closely packed landing gear. Your lander on the other hand has a CoM that's practically on the same level as the landing gear and is very wide. In that case 3 legs might indeed become more of a hindrance.

Compared, the real lunar lander did land on a pre-determined, optimal landing site, not the accidental side of a hill like you'd do in KPS, so the counter-wobble of 3 legs wasn't needed and they went for the advantages of having 4 legs.

[zarakon]

Thought about this some more

When tipping over, a lander will tend to tip on two legs at once, pivoting on what is essentially an edge. A 3-legged lander can be thought of as having 3 edges like a triangle. A 4-legged lander would be a square.

The distance from the center to an edge determines how far the lander can tilt before tipping over (when the CoM goes beyond the edge).

Suppose that the feet are at a distance R from the center of the lander. In an equilateral triangle, the distance to an edge is half the distance to a corner, so R/2. In a square, that distance is R/sqrt(2) (or R/1.414). As you add more legs, the shape approaches that of a circle, so the distance to an edge keeps increasing, approaching R.

So on a flat surface, a lander with more legs can tilt farther before falling over because it effectively has a wider stance.

On a slope though, there's more going on. If you put an edge downhill, then obviously the 3-legged lander will do the worst and a many-legged lander would do best. However, what happens when you put a leg downhill is that you change the angle that it wants to tip. A 4-legged lander will tip 45 degrees away from straight downhill, and a 3-legged lander will tilt 60 degrees away. This effect helps to counteract the shorter distance to edge of the 3-legged lander, though I'm not certain enough of the math to say which one actually comes out advantageous.

[moronwrocket]

One other thing I've found with 4vs3 is that with 4, that leg pointing uphill will hit the slope first much earlier than a 3 legged design, knocking you off course, or requiring MECO sooner than a 3 legged design. The extra speed tends to cause frantic panic as I try to stop tipping over. Here's a thought - how about putting on 4 legs, but retracting the upslope leg during descent? Should give the best of all worlds (Except for the extra leg mass). Find some 20* slopes to test please!