Optimal Rocket Landing

[FlyingDutchman_]

Hi everyone,

I'm trying to figure out what is the most efficient way, fuel-wise, of landing on a celestial body. What are the equations that I must plug my values in? What is the T value that I should start retrograde burning so that just when I'm going to touch base I'll have a velocity of 1 m/s or so?

 

How to figure out this physics problem?

[Reactordrone]

The usual quick and dirty way to estimate your landing burn time is to set a manoeuvre node where your trajectory touches the surface and pull the retrograde handle until you get to a vertical trajectory. You can afford to delay the burn a little from the predicted time since your TWR will increase as you burn fuel.

[Guest]

What you're attempting is called a "suicide burn."

I'll leave the maths to someone else (it'll be a function of your orbital velocity, altitude, thrust, dry mass, and propellant mass), but I'll point out that in addition to Reactordrone's suggestion, there's Kerbal Engineer Redux. It will provide you with a running estimate of distance and time to suicide burn. Note that it only takes into account vertical speed, i.e. if you're in a low-thrust vehicle with a lot of horizontal speed and follow it, you will end up a a crater.

 

[Delay]

14 minutes ago, Brikoleur said:

I'll leave the maths to someone else (it'll be a function of your orbital velocity, altitude, thrust, dry mass, and propellant mass)

I'll do that, then.

Generally I like to do suicide burn calculations as distance, rather than time. This might be because you've got an altimeter on screen and my brain thinks "I can use that!", when it actually shows altitude  as above SL. Though there are mods (I think at least) that set the altimeter to ground level, so the following is applicable.

Anyway, the distance you need to start burning at until the ground is given by this equation:

d=v²/2(F/m-g)

v is your current velocity (in an atmosphere that would probably be terminal velocity), F is the thrust provided by the engines, m is the current mass of the vehicle and g is the acceleration due to gravity.

Edited February 22, 2018 by Delay

[Flavio hc16]

2 hours ago, Delay said:

I'll do that, then.

Generally I like to do suicide burn calculations as distance, rather than time. This might be because you've got an altimeter on screen and my brain thinks "I can use that!", when it actually shows altitude  as above SL. Though there are mods (I think at least) that set the altimeter to ground level, so the following is applicable.

Anyway, the distance you need to start burning at until the ground is given by this equation:

d=v²/2(F/m-g)

v is your current velocity (in an atmosphere that would probably be terminal velocity), F is the thrust provided by the engines, m is the current mass of the vehicle and g is the acceleration due to gravity.

you will end up in a crater if you do your mats 99% of the time....why? because the game altimeter shows your altitude over the lowest part of the body you are landing, and even if you do your math using a mod that shows you the altitude to the ground, as soon as you move just 100 meters you can have a heigh discrepancy that will end up blowing you up. And even if you calculate the terrain around you, you have to consider the horizontal speed.

[Delay]

2 hours ago, Flavio hc16 said:

you will end up in a crater if you do your mats 99% of the time...

Strange, my suicide burns rely on this equation and so far all have worked. The real problem is that without atmosphere, your velocity is constantly rising, so it might be better to add 50 or even 100 m/s to your actual velocity.

 

2 hours ago, Flavio hc16 said:

because the game altimeter shows your altitude over the lowest part of the body you are landing, and even if you do your math using a mod that shows you the altitude to the ground, as soon as you move just 100 meters you can have a heigh discrepancy that will end up blowing you up.

I already wrote that this method most likely needs a mod that changes the ways the ingame altimeter works. That, or you land in IVA if possible. Also, I don't understand what you mean by "as soon as you move just 100 meters". Do you mean "if you start your burn 100m too late"? In that case, the problem hardly lies in the equation.

In theory, using distance instead of time should be even better as you can get the burn start right down to a meter if you do it correctly. Compare that to time, where the distance you overshoot by is a function of velocity and time instead of being absolute: 10 meters is always 10 meters, 1 second can, however, equate to hundreds of meters, perhaps even kilometers if you take the very direct route.

Also, consider that this equation assumes constant acceleration. But TWR will increase as the engines drain fuel, so you're going to stop a few meters above the ground.

3 hours ago, Flavio hc16 said:

And even if you calculate the terrain around you, you have to consider the horizontal speed.

If your horizontal velocity is this great you probably shouldn't do a suicide burn.

[linuxgurugamer]

You can do a suicide burn that takes horizontal velocity into account as well, but it's harder

[Starman4308]

What I prefer to do is not a true suicide burn, but rather a constant-altitude descent, which has been shown to be more efficient, particularly for low-TWR landers.

It also makes for a relatively simple simulation: just set the pitch high enough to maintain 0 vertical velocity at each timestep, and run until horizontal velocity is zero.

In practice, because airless bodies tend not to be perfectly smooth spheres, I run an approximately-constant-vertical-velocity descent, managing pitch to maintain a velocity setpoint, adjusting if I think I'm descending too fast or too slow.

[Delay]

One thing you can also do to nail your burn times: If you land at a point you were at previously, like the KSC, you plant a flag there to judge your distance to it by targeting the flag. Generally speaking, targeting allows accurate distance readouts.

I also target asteroids to suicide burn there. Otherwise I'd often overshoot because I tend to overestimate TWR.

[Zhetaan]

It's not too difficult to solve for a suicide burn that accounts for horizontal velocity; the trick is to take your trajectory and decompose it into horizontal and vertical components.  Then you can apply the gravitational acceleration only to the vertical component, solve for how much acceleration is needed in each direction to reach zero velocity at the same time, and compose the answers into your thrust vector, which is necessarily going to change direction as you descend because the vertical part has constant acceleration, whereas the horizontal part does not.

It's certainly not something that you'll do on the fly, but it is doable with not-too-complex maths and so is well within the capability of a good autopilot if you want to install or write one.  And you will need to install something; to do this calculation, you'll need to know the gravitational parameter of the body you're orbiting, your descent angle relative to down, and your exact distance to both the ground and the gravitational centre of the body--all this is information which is available or calculable, but none of it is given in the standard KSP HUD.

One of the more common new-player approaches involves reducing the horizontal component to near-zero while still high up and focusing on the vertical component only on the way down.  Because the horizontal velocity is zero, it's an easier calculation, but it's also horrifyingly inefficient--any net gain from the suicide burn is lost in the amount of burn required to cancel the horizontal velocity to zero in the first place and again in cancelling the gravitational vertical velocity acquired on the descent.

On the other hand, you can employ @Starman4308's trick of using what's called a trivial solution to solve the problem low enough to stop you near the ground and quickly enough to leave a bit for pilot reaction time:  drop your periapsis to just above ground level and while there, thrust to hold the vertical velocity (and acceleration) at zero (the readouts are accurate for that) and use the rest of your engine's thrust, without bothering to see how much it is, to reduce the horizontal velocity only.  You still need to constantly change the thrust angle relative to gravity, but with a simple readout to tell you whether you're holding zero vertical velocity, it's easy to see how to correct.  Once the horizontal velocity is zero, then you can shift focus to the vertical and manage a controlled descent.  It takes longer than a straight suicide burn, but so long as you don't go down you're not going to hit anything, and if you set your orbit so that you perform the manoeuvre while close to the ground, it leaves you with only a few m/s to cancel out when you make your final landing.  The real beauty of it is that it allows you to make a very efficient landing without requiring a physics degree and a bank of computers.

@Delay:  I think that what @Flavio hc16 meant by saying 'as soon as you move just 100 meters' is that even with an actual-distance-to-ground altimeter, the value given is for the distance to the ground directly underneath the spacecraft.  Since terrain is sloped and quite variable, drifting by 100 metres in any sideways direction would skew that figure and leave you at the end of the burn ten metres too high or, worse, too low.  I agree with your simple expedient of adding extra velocity in order to ensure an above-ground stop, but there is a point to the concern.

[bewing]

24 minutes ago, Zhetaan said:

but it's also horrifyingly inefficient

Nah, it's not that bad. But the bit you lose in fuel efficiency, you gain in targeting accuracy.

[GoSlash27]

Posting this since Photobucket killed the original tutorial.
 This is the method I use for all powered landings on airless bodies. The video shows it being executed from a lowered periapsis, but I do it straight from low orbit.

Best,
-Slashy

[Zeiss Ikon]

I'm surprised no one has mentioned Better Burn Time.  It's a tiny mod, which puts virtually no load on your system (a tiny display overriding the stock burn time display, and a little background math,which CPUs are really good at and which can, in theory, run on a different core/thread than the physics engine, though I don't know if it's coded that way), and when you're within 120 seconds of impact, displays a running count of time to impact, estimated burn time to zero velocity (total vector velocity, not just horizontal or vertical) and a very intuitive countdown to start the burn.  It also does the same for maneuver nodes, rendezvous, and reentry/atmosphere exit when your orbit intersects atmosphere.

For suicide burns, the only real flaw in BBT is that it doesn't take into account your deceleration in calculating time to impact -- which means, if you start your burn when the countdown reaches zero, you'll stop well above surface.  This is a design decision; the alternative, to assume you'll start burning at countdown time and calculate from max thrust, is too sensitive to variations in terrain (for which, as far as I can tell, BBT uses vertical altitude from the vessel, essentially what you'd get from a radar altimeter -- and if you have significant horizontal velocity, though the velocity vector is correctly accounted for, changes in terrain height can't be predicted).

Now, for an optimum landing burn, you want to land SpaceX style: reach a non-zero vertical velocity and non-zero lander tilt, with non-zero horizontal velocity, both within parameters for the landing legs (not to collapse, not to tip the lander), and run out of fuel just as you'd otherwise cut off for touchdown.  This has been coined as a "hoverslam" -- that is, if your legs can take, say, 14 m/s impact you'd like to land at 13 m/s, +- 1 m/s; if your lander can handle 2 m/s horizontal, you want to land as close to that figure as possible -- and then have your fuel calculated so you land with none.  Perfection of this computer-controlled operation (well, near-perfection, as demonstrated by the Falcon Heavy central core running short of ignition hypergols to relight three engines for landing burn) is what allows SpaceX to launch commercial/government payloads and recover their  boosters -- it allows minimizing the fuel that must be reserved for boost back, reentry, and landing burns.  Every kilo of fuel left in the tanks at landing burn shutdown requires more kilos to lift, reverse, reenter, and land it, so zero is the ideal number -- and the higher your landing velocity (short of breaking stuff) the less fuel you need to brake to that figure.