Pushing the limits of the hoverslam landing

[PreferToLurk]

Warning: Wall of text incoming!   

tl;dr:  Looks like SpaceX could save fuel/weight by landing more aggressively.  Could they? Will they?

I've tried to do a bit of calculating based on the wonderful video analysis that Hrissan provided of the OG2 landing. I was trying to figure out just how efficient the landing was in terms of propellant used to decelerate the stage vs cancelling gravity, and then throw some hypothetical aggressive landing scenarios and see how those would look comparatively.  Anyway, heres some preliminary results followed by some questions I have for the community. 

Assumptions and observations: 
145m/s terminal velocity (Hrissan stated "about" 150, but my eyeballs say a little less, plus it fits the deceleration estimates/times better)
Initial deceleration regime of 4m/s/s for 10s
Final deceleration regime of 7.5m/s/s for 14s
These two regimes are about 20% apart in impulse, which lines up well with an assumption of 100%-62% throttle range and 10% throttle buffer on both ends.
Assuming no major sources of aerodynamic drag during the landing burns.
Assuming constant thrust AND deceleration for each regime.  (which we all know can't be true, hoping it wont throw off the results too much)
Assuming constant engine ISP through different throttle settings such that a given % drop of thrust will produce the same % drop in Mdot.
I used 282s ISP and 273kg/s Mdot.

From this I came up with an actual landing that used 5415kg of propellant to produce 380m/s of total impulse while slowing the stage by 145m/s for a total efficiency of 38.1% (62% of the thrusting merely cancelled out gravity)

I created a few different hypothetical scenarios some of which may be unrealistic or even fully impossible. 

MB1 (Make Believe 1):
1 landing engine, 1 thrust regime (90% throttle)
Burn would last 19.3 seconds @ 7.5m/s/s deceleration, using 4748kg of fuel while providing 334m/s of total impulse for total efficiency of 43.4% and a fuel savings of 667kg from the baseline.

MB2:
3 landing engines 1 thrust regime (72% throttle)
Burn would last 4.6 seconds @ 31.6m/s/s deceleration, using 2718kg of fuel while providing 190m/s of total impulse for total efficiency of 76.3% and a fuel savings of 2697kg (!) from the baseline.

MB3:
3 landing engines 2 thrust regimes and 2 engine shutdown @ stage velocity of 40m/s (72% throttle on all 3 initially with final landing on 90% single engine)
First burn would last 3.3 seconds @ 31.6m/s/s using 1962kg of fuel while providing 138 m/s of total impulse.
Second Burn would last 5.3 seconds @ 7.5m/s/s using 1304kg of fuel while providing 92 m/s of total impulse.
All together 3266kg of Fuel, 230m/s of impulse, efficiency of 63% and a fuel savings of 2149kg from the baseline.

MB2 is obviously the most crazy. stable landing control with three thrusting engines and a final T/W of over 4.2 is far from assured.  Of course one could imagine a scenario with three engines firing at 90% till landing, but that one seemed too crazy for me to even bother modeling.

MB3 is still very aggressive even with a final T/W no worse than already demonstrated -- Still only 3.3 seconds (including startup and shutdown transients) for the three engine burn and only 5.3 seconds to iron out a final landing solution. 

MB1 is very tame merely doing away with the initial low throttle period, but still managing to save over 600kg of fuel. 

So, first off can anyone point out any obvious math errors? (i know "total impulse" for breaking + Gravity losses is probably the wrong terminology)  Beyond the feasibility of my scenarios, I would like to know if I didn't even calc the fuel use to within an order of magnitude.

But more interestingly, how useful is a fuel savings of about 2000kg? or even just 600kg?  Is there any reason to expect that SpaceX will attempt to land more aggressively and really push the limits? 

I don't want to say that I was disappointed by the landing, but it wasn't exactly as "brown pants" of a maneuver as I was expecting. But maybe my perception betrays just how difficult of landing it already was?

Anyway, after doing all these calcs I figured that with how much use I get out this forum, that I should try to give back and hope someone else finds this interesting as well.  It is now way too late for me to still be awake please forgive typos. 

[macpacheco]

I think the entry burn is far more critical to save fuel. That's a three engine burn already.
But a higher acceleration landing is desirable as it saves fuel but also reduces the stage exposure to winds.
I don't think it will ever mean using two engines though.
Running the center engine at 100% of original thrust (around 85% of upgraded thrust) is a heck of a lot of thrust. The stage is very light. It also leaves the question of any variability in restart times. If you push too hard, even a fraction of a second = not enough time to land.
This will likely be a gradual process where SpaceX will slowly push the envelope until they fail (or some metric is achieved).

SpaceX will likely push this on ASDS landings. Where the most they have to loose is the stage itself (ASDS cheaper than incoming stage), and in missions where fuel is more likely to be short.

[IRobot]

I think they will go conservative on this because:

1) savings on 1st stage do not translate directly on payload increase
2) increases risk considerably
3) decision between expendable or reusable will come from the payload mass and target orbit and probably the decision will not be different if there is a bit more upmass capability.

[Jcc]

I wonder what's really preventing them from deploying the landing legs earlier. That would save fuel in theory, but maybe it's a question of aerodynamic stability even assuming the legs would stay rigid, but maybe there is a risk that the legs would bend and oscillate.

[macpacheco]

I wonder what's really preventing them from deploying the landing legs earlier. That would save fuel in theory, but maybe it's a question of aerodynamic stability even assuming the legs would stay rigid, but maybe there is a risk that the legs would bend and oscillate.

I think the legs could burn with rocket exhaust if exposed too soon.

[guckyfan]

I think the legs could burn with rocket exhaust if exposed too soon.

If the deploy mechanism would be changed to allow this they could do a partial deploy first. Make it look like an arrowhead. It would provide drag and keep the legs away from the flames. It should cause less stability issues too.

[Kabloona]

I wonder what's really preventing them from deploying the landing legs earlier. That would save fuel in theory, but maybe it's a question of aerodynamic stability even assuming the legs would stay rigid, but maybe there is a risk that the legs would bend and oscillate.

I think the legs could burn with rocket exhaust if exposed too soon.

Yes, we saw how badly the metal Grashopper legs got smoked. SpaceX obviously wants to reuse these F9 composite legs which will be more easily damaged by heat than the metal GH legs, and the obvious way to preserve them is to minimize time on the barbecue grill.

[abaddon]

I think a likely required part of the landing burn is correction of targeting errors from using just the grid fins.  Unless the grid fins are pinpoint accurate, you want to start the burn early so the rocket has time to adjust for the final targeted location.  Even then, upper-layer winds might push the stage off-target and the grid fins might not be able to compensate as much as the engine can.  This might be a limiting factor in starting the burn later, however many engines are used.

[acsawdey]

I wonder what's really preventing them from deploying the landing legs earlier. That would save fuel in theory, but maybe it's a question of aerodynamic stability even assuming the legs would stay rigid, but maybe there is a risk that the legs would bend and oscillate.

I thought we understood the legs required deceleration force to deploy and thus couldn't deploy until the landing burn was in progress. Elon mentioned a redesign to allow the legs to act as airbrakes, have we heard anything about that since this 11/2014 tweet?

https://twitter.com/elonmusk/status/536268250620125185

[macpacheco]

I think a likely required part of the landing burn is correction of targeting errors from using just the grid fins.  Unless the grid fins are pinpoint accurate, you want to start the burn early so the rocket has time to adjust for the final targeted location.  Even then, upper-layer winds might push the stage off-target and the grid fins might not be able to compensate as much as the engine can.  This might be a limiting factor in starting the burn later, however many engines are used.

That's an unintuitive aspect of landing.
The faster the vertical speed, the less winds will affect you (less time exposed to them).
The sensors and computers on the stage detect wind effects instantly (1/10th of a second).
The faster the vertical speed, the more control authority the grid fins have.

Finally you're assuming sideways control margins are tight. That might be right or very wrong.
And cold gas thrusters add more control authority.

The fact SpaceX allows for 50mph winds for land landings suggest there are plenty of margins.

[AS-503]

I think the legs could burn with rocket exhaust if exposed too soon.

If the deploy mechanism would be changed to allow this they could do a partial deploy first. Make it look like an arrowhead. It would provide drag and keep the legs away from the flames. It should cause less stability issues too.

How would that partially deployed leg position effect the air flow in/around the grid fins?
Would it cancel out your proposed stability enhancement?

[Llian Rhydderch]

Yes, we saw how badly the metal Grashopper legs got smoked. SpaceX obviously wants to reuse these F9 composite legs which will be more easily damaged by heat than the metal GH legs, and the obvious way to preserve them is to minimize time on the barbecue grill.

I recall that some of those later Grasshopper tests had the metal legs covered with a black material that the earlier flights did not.  I always assumed it was SpaceX engineers covering the legs with some/all of the sorts of materials and/or coatings they were planning to use in the deployable legs design on F9 v1.1.  I believe there was some discussion of this in other threads.

In other words, I suspect that a good bit of that extended smoking we observed was materials testing.

Now, back to the original idea of this thread by PreferToLurk:  what about using different thrust and fuel usage regimes for saving propellant on hoverslam landings?

[Kabloona]

I thought we understood the legs required deceleration force to deploy...

That was someone's speculation which I don't believe is correct. There's a pneumatic piston inside the telescoping cylinders that forces the leg open, like the nitrogen strut on a car's hatchback.

[abaddon]

I think a likely required part of the landing burn is correction of targeting errors from using just the grid fins.  Unless the grid fins are pinpoint accurate, you want to start the burn early so the rocket has time to adjust for the final targeted location.  Even then, upper-layer winds might push the stage off-target and the grid fins might not be able to compensate as much as the engine can.  This might be a limiting factor in starting the burn later, however many engines are used.

That's an unintuitive aspect of landing.
The faster the vertical speed, the less winds will affect you (less time exposed to them).

I'm talking about upper-layer winds, following the entry interface burn, and before the current powered landing envelope.  This is the terminal velocity of the stage, so velocity is constant in this regime.  It is true that the grid fins have more control authority at a higher speed, but we don't know that they are capable of a pinpoint landing without any engine burn.  In other words, I am asserting the engine burn is not just to bring vertical velocity to zero, it is also to target the final landing point.  I think this would be difficult to argue against.  I'm further arguing that it is likely that the burn starts higher up, to allow the engine to target the pad with terminal guidance that the other systems are unlikely to be able to provide.  This seems reasonable to me, but is clearly not proven.

Finally you're assuming sideways control margins are tight. That might be right or very wrong.

I don't think they have to be "tight" to prevent a last-second burn from being practical.

And cold gas thrusters add more control authority.

Cold gas thrusters are irrelevant for this purpose, they aren't very powerful and are on the wrong end of the stage to provide any real lateral displacement that late in the game.

The fact SpaceX allows for 50mph winds for land landings suggest there are plenty of margins.

With the engine burn, starting where it does, yes.

Finally, one purpose of the final burn appears to be a "divert" from an offshore-based trajectory to one that targets the pad.  This divert did seem to happen in the successful landing.  You'd probably have to eliminate the divert maneuver before considering reducing the final burn by any significant amount of time.

(As an aside, I have no issue with the hypothesis that SpaceX is being conservative where they are starting the engine burn, and that they might be able to reduce it over time.  I think it is unlikely we will ever see a true "hoverslam" though).

[LouScheffer]

But more interestingly, how useful is a fuel savings of about 2000kg? or even just 600kg?  Is there any reason to expect that SpaceX will attempt to land more aggressively and really push the limits? 

Here's a back-of-the-envelope calculation of how much this helps.  I assume a GTO mission since all LEO missions so far have big margins anyway.

If you are super aggressive, you might save 2500 kg of fuel.  Each engine uses 270 kg/sec, so that's 1 more second you can burn all 9 engines on ascent.  Since you are accelerating at roughly 5 Gs at cutoff, that's 50 m/s more you might get from the first stage.  So for the same payload mass, you could get 50 m/s more final delta-v.  Compared to the 300 m/s differences between different GTO orbits, that's mildly helpful but not game-changing.

Alternatively, you could get a bigger payload to the original orbit.  For GTO, the second stage needs about 8210 m/s ( https://forum.nasaspaceflight.com/index.php?topic=34077.msg1463298#msg1463298 ), so if it starts at 121 t, it masses 10,897 kg at cutoff.  If it only needs 8160 m/s, then it masses 11,057 kg at cutoff, so the payload goes up by 160 kg.  (These mass number have way too many significant digits, but the difference should be pretty close to the real value.)  This only makes a difference for those few payloads right on the very edge of F9 performance, and even there the difference is not big.

If the fuel saved is only 600 kg, the improvements will be about 4x smaller.  In either case, it's hard to see making a big change in landing strategy.  A more aggressive throttle schedule with the current number engines?  Perhaps.  A three-engine landing burn to a screeching halt?  Probably not worth the risk.

On the third hand, suppose you had an engine failure or some other under-performance on the way up.   You make up for it by burning more fuel to fight more gravity losses.  After achieving the correct velocity for the start of the second stage, you don't have enough fuel for a "conventional" landing.   Now might be the right time to try a super-aggressive landing sequence - shorter than normal re-entry burn, followed by a minimal-hover maximum-slam landing.  But it's hard to imagine SpaceX devoting the engineering resource needed to attempt a landing for such an unlikely case, plus you'd probably need extra inspections/refurbishment after a novel landing sequence even if it succeeded.  So overall the screeching halt landings seem unlikely.

[macpacheco]

Cold gas thrusters and grid fins job is to incline the stage so it acts like some highly inefficient wings.

Considering the stage flew for about 7 minutes after MECO on OG2, I wager less than 3 minutes through the normal windy atmosphere range (below 60k ft).

Say we have 200 mph winds = 90 m/s
Three minutes exposure = 16Km lateral deviation.

16Km is a LOT, but the stage will be compensating for that all the way through the descent, and I assume winds aloft will be uploaded prior to launch and the stage will be pre positioned to drift through that.

Weather forecasts and weather balloons pre launch are excellent at characterizing aloft winds.
I also think those can be input into the descent profile.

[LouScheffer]

It is true that the grid fins have more control authority at a higher speed, but we don't know that they are capable of a pinpoint landing without any engine burn.  In other words, I am asserting the engine burn is not just to bring vertical velocity to zero, it is also to target the final landing point.  I think this would be difficult to argue against. 

Actually, it's extremely easy to argue against this.  Smart bombs routinely achieve meter-level accuracy using only fins, and have for decades.  It's well proven technology.

[whitelancer64]

I just want to point out that the Falcon 9 first stage carries approximately 286,400 kg in LOX and 123,100 in RP-1, approximate total of 409,500 kg.

Source: http://spaceflight101.com/spacerockets/falcon-9-ft/

A savings of 2,000 kg of propellants is about 0.5% of the Falcon 9's total fuel capacity.

So more aggressive landings would only help in the most marginal of return scenarios, where the Falcon 9 has already depleted almost all of its available fuel and oxidizer.

[Nomadd]

It is true that the grid fins have more control authority at a higher speed, but we don't know that they are capable of a pinpoint landing without any engine burn.  In other words, I am asserting the engine burn is not just to bring vertical velocity to zero, it is also to target the final landing point.  I think this would be difficult to argue against. 

Actually, it's extremely easy to argue against this.  Smart bombs routinely achieve meter-level accuracy using only fins, and have for decades.  It's well proven technology.

Is it known that the fins are for guidance and not just stability?

[macpacheco]

It is true that the grid fins have more control authority at a higher speed, but we don't know that they are capable of a pinpoint landing without any engine burn.  In other words, I am asserting the engine burn is not just to bring vertical velocity to zero, it is also to target the final landing point.  I think this would be difficult to argue against. 

Actually, it's extremely easy to argue against this.  Smart bombs routinely achieve meter-level accuracy using only fins, and have for decades.  It's well proven technology.

Smart bombs have far more control authority and glide capability than a F9R stage.
The state of the art in smart bomb is the SDB, which have been demonstrated to glide for tens of miles with high subsonic drop.
In the cast of SDB, you have actual crude wings.
Even a 500lb JDAM can glide much better than a F9R stage.

But as long as the stage can receive pre launch winds aloft programming, it can seek an estimated aim point at end of entry burn that lateral inertia + winds aloft will significantly reduce grid fin efforts in lateral navigation.

This is a well known technique in aircrafts in general. Easy peasy for the SpaceX software guys to program (compared to the other nightmares they have to handle in the whole RTLS/ASDS thing).

The altitudes where winds are the strongest (above 25k ft) are exactly where terminal speed is still significant (but subsonic), hence grid fins will have maximum effect.

[abaddon]

Actually, it's extremely easy to argue against this.  Smart bombs routinely achieve meter-level accuracy using only fins, and have for decades.  It's well proven technology.

I see, so you are asserting that the F9 first stage is identical in size/shape/density/mass distribution to a smart bomb?  I think poking a hole in your "easy" argument is even easier...

[abaddon]

Is it known that the fins are for guidance and not just stability?

Yes, pretty much.  Earlier flights without fins were only able to achieve a very gross landing area.  Cold gas thrusters were able to maintain attitude successfully (once they added more gas) but that was not sufficient with targeting.  It was the fins that enabled the stage to get in the vicinity of the barge.

[rpapo]

The altitudes where winds are the strongest (above 25k ft) are exactly where terminal speed is still significant (but subsonic), hence grid fins will have maximum effect.

Which brings up the question: At which point in the descent (how high up is it still) does the rocket go subsonic?  The fact that the sonic booms were so loud in the December landing indicates (to me, at least) that the rocket remained supersonic until rather late.  Keep in mind that "terminal velocity" is a function of air density (and therefore altitude), so that throughout the descent the rocket is going at some speed above terminal velocity because terminal velocity itself is decreasing.

[mme]

It is true that the grid fins have more control authority at a higher speed, but we don't know that they are capable of a pinpoint landing without any engine burn.  In other words, I am asserting the engine burn is not just to bring vertical velocity to zero, it is also to target the final landing point.  I think this would be difficult to argue against. 

Actually, it's extremely easy to argue against this.  Smart bombs routinely achieve meter-level accuracy using only fins, and have for decades.  It's well proven technology.

Is it known that the fins are for guidance and not just stability?

I can't say it's known, but they water landed stages without grid fins and those landings where within 10 km rather than 10m accuracy.  I'm pretty sure that is mostly do to the grid fins, but I don't have a specific source saying that is the only change.  Also, running out of hydraulic fluid on the first attempt "missed the mark," requiring a huge divert.

As too the rest of the argument, I'm a bit lost.  Clearly the engine is required to avoid lawn darting, and slowing the stage will reduce the effectiveness of the grid fins.  So the engine becomes more important to control in the last few seconds.  That's for an incredibly small portion of the landing and it better be really close.

I'm in the "grid fins are pretty darn accurate AND SpaceX chooses to use engines to divert the IIP while not lawn darting" camp.

[LouScheffer]

It is true that the grid fins have more control authority at a higher speed, but we don't know that they are capable of a pinpoint landing without any engine burn.  In other words, I am asserting the engine burn is not just to bring vertical velocity to zero, it is also to target the final landing point.  I think this would be difficult to argue against. 

Actually, it's extremely easy to argue against this.  Smart bombs routinely achieve meter-level accuracy using only fins, and have for decades.  It's well proven technology.

Smart bombs have far more control authority and glide capability than a F9R stage.

Of course they have more control authority.  You can't reschedule a war for when the winds are calmer, the launching aircraft may have other things to optimize for (like surviving, avoiding missiles), rather than lining up straight on target, the enemy is unlikely to broadcast ground-level winds for your convenience, etc.

I would assume the SpaceX landing requirements will guarantee they would not run out of control authority.  Then the accuracy will be determined by the magnitude of the disturbances and the combined loop gain of the correcting algorithm, the fins, and the aerodynamics.  Both have similar sideways disturbance accelerations (the GBU-38 has a mass of 250 KG and a surface area of about 1 m^2, the Falcon 9 about 30t and 120 m^2).  The Falcon is about 10x longer so you would expect the control to be about 10x slower (the actuators are 10x further from the center, but the moment of inertia goes like the square, and is 100x more).  So you'd expect the response to slow gusts should be identical (both systems can compensate) but fast gusts could be up to 10x worse than the JDAM.  Overall it should be somewhere between 1x and 10x worse accuracy than JDAM, depending on the temporal structure of wind gusts.  10 meters of accuracy using only the fins seems pretty likely.

[Stan-1967]

Warning: Wall of text incoming!   

tl;dr:  Looks like SpaceX could save fuel/weight by landing more aggressively.  Could they? Will they?

......

So, first off can anyone point out any obvious math errors? (i know "total impulse" for breaking + Gravity losses is probably the wrong terminology)  Beyond the feasibility of my scenarios, I would like to know if I didn't even calc the fuel use to within an order of magnitude.

But more interestingly, how useful is a fuel savings of about 2000kg? or even just 600kg?  Is there any reason to expect that SpaceX will attempt to land more aggressively and really push the limits? 

I don't want to say that I was disappointed by the landing, but it wasn't exactly as "brown pants" of a maneuver as I was expecting. But maybe my perception betrays just how difficult of landing it already was?

Anyway, after doing all these calcs I figured that with how much use I get out this forum, that I should try to give back and hope someone else finds this interesting as well.  It is now way too late for me to still be awake please forgive typos. 

It was a long read, I appreciated it & it was worthwhile.   Thanks for contributing to the discussion. 

I think what you have done is mapped out a "window" where it is mathematically possible to land the stage.   There are infinite mathematical solutions to landing the stage, but functionally, the physical limitations of the hardware and fidelity of control responses narrow down what is possible.  ( i.e grid fins, wind, valve response time etc.)  When I have thought through the calculations you were going through, in my mind I draw out a flight regime on a graph:   

The left vertical axis has height above the landing zone where the landing burn begins, and it ends at zero elevation.

The bottom horizontal axis has thrust level,  say 70-100% is the range of interest.

The top horizontal axis has probability of failure figures throughout the flight regime  that assigns increasing risk as higher thrust levels progress rightward on the horizontal axis.  For arguments sake, maybe failure probability ranges from 5% to 90% as throttle goes from 70-100%.

As you start the burn lower everything has to happen much faster and all the sensor and mechanical responses will overlay their uncertainty & variability into greater risk of failure.  Some factors may favor higher speed, like wind response, but overall I think the risk increases with low elevation/high throttle landing.

In the end, I think you get a landing window for a given amount of propellant available, and then you try to optimize your landing probability based on your characterization of your electrical/mechanical/software systems.  You could theoretically do a "brown pants" landing every time, but the real goal is a rocket standing on the pad undamaged. 

 I recall E. Musk citing a 10% increased probability of successful landing when he explained delaying the launch by a day.   He cited "Monte Carlo" simulations as justification of this.  That statement tells me they are modeling each landing based on all variables they have characterized that are inputs to the landing algorithms and process.


Again, thank for posting your ideas!

[cambrianera]

Nobody realized that saving 2000 kg of propellant in the landing burn decreases the mass during reentry burn (and boostback burn, if done).
Therefore saving could be easily double that, because propellant for reentry burn (and boostback burn) can be less.

[Kabloona]

Nobody realized that saving 2000 kg of propellant in the landing burn decreases the mass during reentry burn (and boostback burn, if done).
Therefore saving could be easily double that, because propellant for reentry burn (and boostback burn) can be less.

Not really, because the dominant factor is the 26,000 kg dry mass on the first stage, so carrying 2,000 kg less propellant through boostback and re-entry reduces vehicle mass by less than 10%, and thus gives you less than 10% additional propellant savings for a given deltaV.

[PreferToLurk]

Thanks for all the great replies.  40m/s to payload certainly doesn't seem worth the risk.   I wonder though if they could use the extra propellant for more boost back burn? 3 more seconds and around 150m/s extra boost back might enable rtls for more missions, or enable a less lofted trajectory. Certainly lots of variables at play here, and a safe/reliable landing is priority one. Leads me to believe we may still see testing at Spaceport America. Even if they only squeeze out another 1000kg of saved fuel, you can triple that for FH.

[rpapo]

...or enable a less lofted trajectory.

Careful what you wish for: More loft is more air time, which means less velocity back to landing site required, which means less boostback required.  On the other hand, it isn't worth it to loft the stage yet more, as that would require propellant too.

[cambrianera]

Nobody realized that saving 2000 kg of propellant in the landing burn decreases the mass during reentry burn (and boostback burn, if done).
Therefore saving could be easily double that, because propellant for reentry burn (and boostback burn) can be less.

Not really, because the dominant factor is the 26,000 kg dry mass on the first stage, so carrying 2,000 kg less propellant through boostback and re-entry reduces vehicle mass by less than 10%, and thus gives you less than 10% additional propellant savings for a given deltaV.

Propellant consumed in boostback and reentry is more than 25,000 kg, so saving less than 10% doubles the saving in braking burn.

[PreferToLurk]

...or enable a less lofted trajectory.

More loft is more air time, which means less velocity back to landing site required, which means less boostback required.

Right, but the more lofted the trajectory, the less the first stage contributes to the required horizontal orbital velocity. And we just saved 1000-2000kg of fuel for the extra boostback.

In the OG2 mission, there was over 250 seconds of free flight from the first stage between boostback and reentry burns. lets say less lofted trajectory cuts off 50 seconds of ballistic free flight, and adds 100m/s (horizontal) to the second stage. 150m/s of extra boostback burn completely cancels the extra horizontal velocity while adding 50m/s in the opposite direction. So the final question is will 10km of extra boostback distance be enough to compensate for the extra distance downrange and the loss of 50 seconds of free return?

We would need to know just how much dV the original boostback burn provided (likely proprietary information) to really know.   They don't announce boostback completion on the webcast, but from the cheers it seems like it is about 15-20 seconds long. At a minimum the boostback burn is 50% cancelling out downrange velocity (probably more like 60%-75%, but just for arguments sake...) so 8-10 seconds of building up rtls velocity at most. if you add 2-3 seconds to that burn you get at least 20% more rtls velocity with 20% reduction in "hang time". 
At the very worst it would appear to still get you back to your landing point while providing more dv to the payload than simply delaying MECO1 by one second. 


Anyway, like I said, TONS of variables at play here.  But I've got to think that SpaceX can find a good use for an extra 1000kg of fuel. Also, there are a lot of good reasons for SpaceX being conservative in this landing.  Sticky valve from the last attempt not being the least of them.  As they gain confidence I would expect to see incrementally more aggressive landings. Getting the center core of a FH even back to a barge is going to be a lot more difficult, and they may not be able to afford such a conservative landing.

[whitelancer64]

Time for a smaller analogy.

Let's say you have a car that gets 40 mpg. This car has a 10 gallon tank, so for each tank you can go 400 miles. Put another way, to go one mile your car burns about 3.2 ounces of gas.

Let's say you think you might save gas by coasting in downhill sections of roadway rather than keeping your foot on the gas. Let's say your calculations show that by doing this you will save 0.5% of your car's tank of fuel, which is 6.4 ounces of gas. This enables you to drive approximately 2 additional miles.

Given the car normally goes 400 miles on a tank anyway, is this extra 2 miles going to be worth it?

[CJ]

Time for a smaller analogy.

Let's say you have a car that gets 40 mpg. This car has a 10 gallon tank, so for each tank you can go 400 miles. Put another way, to go one mile your car burns about 3.2 ounces of gas.

Let's say you think you might save gas by coasting in downhill sections of roadway rather than keeping your foot on the gas. Let's say your calculations show that by doing this you will save 0.5% of your car's tank of fuel, which is 6.4 ounces of gas. This enables you to drive approximately 2 additional miles.

Given the car normally goes 400 miles on a tank anyway, is this extra 2 miles going to be worth it?

It is absolutely worth it *if* the next gas station is 402 miles away.

IMHO, a likely scenario in future is that available margin will dictate the landing profile. If it's really tight on margin due to payload requirements, they'll fly a very aggressive (fuel-minimal but higher risk) profile plus land on the ASDS. They'd do this because if they don't, the lose the stage for sure. If there's a bit more margin, they'll use it. If there's plenty of margin, they'll fly a very conservative profile. In a nutshell, my guess is they'll adapt the profile to the specific mission as needed.   

[MrHollifield]

As mentioned upthread, this looks like a perfect case for testing in NM.

Each case can be simulated then the most promising flight tested without risking a customer's payload and a new, unflown F9FT. The ones that work can be tried out on LEO launches then made a standard return method if they improve payload to orbit or margin for landing error. If the recovered stage in NM (F9-19 is my guess) lawn darts, then they'll know not to try that landing sequence on a future flight.

[Norm38]

If we're talking about reusable vehicles, then I don't get the need. FedEx doesn't pack extra cargo onto a 737 by coasting it into the airport on fumes. They use a 757.

If the F9 needs another 200kg of payload mass, then SpaceX will build a bigger rocket. Which it's doing.
Landings should be boring, safe, routine, with plenty of margin.

[guckyfan]

Landings should be boring, safe, routine, with plenty of margin.

I agree with boring, safe, routine. We need to get there. But I disagree with plenty of margin. Nothing in space is done with plenty of margin, except for the first tries. They aim for just enough margin. Of course many flights will have plenty, it's about the largest payloads possible without going to the next more capable and expensive means. They will home in on that and risk some failures on the way.

[Llian Rhydderch]

I wonder if we might get this thread back on topic.  The OP said:

"Looks like SpaceX could save fuel/weight by landing more aggressively.  Could they? Will they? "

This is the sort of rocket equation/propellant mass flow/trajectory stuff that is typically catnip for NSFers.  Let's talk about that.

[PreferToLurk]

I'll Cross quote from the SES-9 thread.   Dante80 put together a very interesting analysis of upcoming launches, masses, and RTLS/Barging implications. As GuckyFan just stated, this is likely to be of most importance when looking at squeezing an expendable flight into a barge flight, or a barge flight into a RTLS flight. The absolute performance gain is not as important as moving the break-points in your favor. 

To re-iterate a little more about the projected capability for FT.

1. F9 v1.1 has "demonstrated" an expendable capability of ~ 4,850kg to GTO-1800. Two sets of data compound to that assertion.

a. The biggest GTO payload was the 4,707kg TurkmenAlem52E. It was placed in a roughly GTO-1765 orbit (180x36600x25.5)
b. Thaicom 6 (a 3,016kg sat) got to a GTO-1500 equivalent orbit (295x90000x22.5). This stretched S2 fuel reserves to almost complete depletion (according to USAF, which evaluated this flight as part of the SpaceX EELV certification procedure).

2. F9 FT as a whole is reported to be around 30% more capable than F1 v1.1

3. DPL (barging) costs about 15% payload.

This means that F9 FT has a theoretical capability of about 6,300kg to GTO-1800
Barging moves it to 5,355kg.
SES9 is 5330 kg.

Its looking very close. Depending on whether the rocket goes to DPL or not, as well as what the end orbit is, we are going to get a lot of info about the current Falcon variant capabilities. 

If this pans out, and RTLS removes another 15% of performance, F9 FT would be able to RTLS after sending a 4410kg payload to GTO-1800. This number is interesting for some of the following missions (quoting from here):

Thaicom 8                      3100kg   GTO    Yes
ABS 2A, Eutelsat 117 West B   ~4000kg?  GTO    Possibly (based on ABS-3A, Eutelsat 115 West B mass)
JCSAT-14                      ~3400kg?  GTO    Probably (based on JCSAT-15 mass)
BulgariaSat-1                 ~3400kg?  GTO    Probably (based on JCSAT-15 mass, same SSL-1300 bus)
JCSAT-16                      ~3400kg?  GTO    Probably (based on JCSAT-15 mass)
KoreaSat-5                     4465kg   GTO    Possibly
Es'hail-2                     ~3000kg   GTO    Probably

If F9 FT performance upgrade over v1.1 I listed above is correct (some say its more, like 33%), then almost all the missions above would be eligible for RTLS, OR a better orbit than GTO-1800 + DPL.

For Barging especially, a few extra seconds of breaking burn paired with an aggressive landing barging might be the difference between throwing away a stage and getting it back.  Or also likely, the difference between SpaceX bidding on a launch worth millions and not bidding at all. Ariane 5  is still winning a lot of launches, at least in part because there are plenty of large comsats that SpaceX can't/wont bid on.  Being able to win even one or two of those launches seems like a very large reason for SpaceX to at least test out aggressive landings.

Although unless someone can figure out the most efficient use of 1-2k kg's of fuel (i still think boostback/breaking burn paired with a lower loft trajectory, but i can't back that up) I think this discussion may have already gone about as far as it can. 

[Joffan]

I think the legs could burn with rocket exhaust if exposed too soon.

If the deploy mechanism would be changed to allow this they could do a partial deploy first. Make it look like an arrowhead. It would provide drag and keep the legs away from the flames. It should cause less stability issues too.

They could do this really early, at the start of the re-entry burn. One difficulty is that the force of the burn is acting to deploy the legs, but you want them to stop at this intermediate position. So as well as the partial-deploy latch (which holds the legs open), there would need to be a deploy-limit mechanism which would then need to be disabled for the full leg deploy during the landing burn.

Talking of the burn acting to deploy the legs - I wonder how well the legs fared in the SES-9 deploy at high speed/high thrust?

[gospacex]

For RTLS, they can try optimizing boostback burn initiation delay after MECO. Thus far it was quite long, some 30-40 seconds. During this time, S1 distance from LS nearly doubled!

[Robotbeat]

Has anyone figured out the T/W ratio of the RTLS F9FT landing through video analysis?

[cmcqueen]

For RTLS, they can try optimizing boostback burn initiation delay after MECO. Thus far it was quite long, some 30-40 seconds. During this time, S1 distance from LS nearly doubled!

Does it really matter? If it's mostly above the atmosphere, then isn't it just delta-V that matters?

[sewebster]

For RTLS, they can try optimizing boostback burn initiation delay after MECO. Thus far it was quite long, some 30-40 seconds. During this time, S1 distance from LS nearly doubled!

Does it really matter? If it's mostly above the atmosphere, then isn't it just delta-V that matters?

After cancelling out the horizontal velocity, it needs to add some back in the opposite direction to get back to the launch site... more if it is further away, right?

[PreferToLurk]

Has anyone figured out the T/W ratio of the RTLS F9FT landing through video analysis?

Sorry, haven't  been paying attention to this section for a few days.  But to answer your question, yes.  Hrissan did a pretty good analysis of the landing and it is what I used as the basis for all my tinkering in the OP.   

[Kabloona]

For RTLS, they can try optimizing boostback burn initiation delay after MECO. Thus far it was quite long, some 30-40 seconds. During this time, S1 distance from LS nearly doubled!

The boostback burn has to be delayed that long because that's how long it takes for the relatively weak GN2 thrusters to reorient the stage and settle propellant for the burn.

[Comga]

Has anyone figured out the T/W ratio of the RTLS F9FT landing through video analysis?

Sorry, haven't  been paying attention to this section for a few days.  But to answer your question, yes.  Hrissan did a pretty good analysis of the landing and it is what I used as the basis for all my tinkering in the OP.   

I agree that Hrissan's post is terrific work and a great place to start.

He finds terminal velocity is 150 m/sec
Baseline deceleration is 7.5 m/sec^2  or ~3/4g, which means the Thrust to Weight ration is ~1.75. (ignoring drag)

A simple model (constant mass, constant thrust, instant start, no air drag, * ) says the landing burn would take 20 seconds (=150/7.5) and start at 1500 m altitude (0.5*7.5*20^2).  Gravity loss would be 20 sec *~10 m/sec^2 = 200 m/sec which is more than the terminal velocity.  Total velocity killed would be 350 m/sec

If this landing used three engines at the same thrust, T:W~ 5.25 and a~42.5 m/sec^2.
The simple model  says that this would take only ~3.5 sec (=150/42.5)  and start at ~265 m altitude.  (Talk about BPL!)

Gravity loss would be only 3.5 sec *10 m/sec^2 = 35 m/sec, or about a seventh of that for a single engine landing.

Total velocity killed would be ~185 m/sec, which is 53% or just over half, of the single engine landing, so it would need little more than half the fuel.  Hence the motivation to try.

As for the thread title, "pushing the limits" imagine the absurd case of using all nine engines.  T:W~16.  a~150 m/s^2. The landing burn takes ~1 second, and starts around 75 meters!  Gravity loss is 10 m/sec, so total deceleration is 160 m/sec.  This is 86% of the three engine case and 43% of the one engine case.  That's only an additional 7% savings,  definitely diminished returns.

* I was going to add "a spherical cow in a vacuum" but that's sort of an in joke among people who took, or worse yet majored in, physics in college.   It implies gross over-simplification.

edit: corrected error in 9 engine calculation.  Now shows even less benefit

[guckyfan]

A simple model (constant mass, constant thrust, instant start, no air drag, * ) says the landing burn would take 20 seconds (=150/7.5) and start at 1500 m altitude (0.5*7.5*20^2).  Gravity loss would be 20 sec *~10 m/sec^2 = 200 m/sec which is more than the terminal velocity.  Total velocity killed would be 350 m/sec

If this landing used three engines at the same thrust, T:W~ 5.25 and a~42.5 m/sec^2.
The simple model  says that this would take only ~3.5 sec (=150/42.5)  and start at ~265 m altitude.  (Talk about BPL!)

I had no idea a 3 engine burn would save that much. Thanks.

[the_other_Doug]

A simple model (constant mass, constant thrust, instant start, no air drag, * ) says the landing burn would take 20 seconds (=150/7.5) and start at 1500 m altitude (0.5*7.5*20^2).  Gravity loss would be 20 sec *~10 m/sec^2 = 200 m/sec which is more than the terminal velocity.  Total velocity killed would be 350 m/sec

If this landing used three engines at the same thrust, T:W~ 5.25 and a~42.5 m/sec^2.
The simple model  says that this would take only ~3.5 sec (=150/42.5)  and start at ~265 m altitude.  (Talk about BPL!)

I had no idea a 3 engine burn would save that much. Thanks.

Yes, thanks!  I was trying to get a concept of when the landing burn would begin if it was a 3-engine burn, and I was estimating between 500 and 750 meters in altitude.  A landing burn starting at 265 meters altitude means that, if the engines don't light up properly,  you can be hitting the barge less than two seconds later.

On SES-9, we obviously had an ignition attempt (the bright light coming down off-center in the barge video), but if the engines never came up to thrust, we would have had an impact within just a few seconds of ignition.  If the hole in the barge is any indication, the engines couldn't have come up to thrust and left the stage with enough kinetic energy to punch that hole...

[Okie_Steve]

if the engines don't light up properly,  you can be hitting the barge less than two seconds later.

Which is about the time from ignition to release at launch after they verify engine thrust etc.
I'm sure they don't hold longer than necessary. Insane suicide burn indeed.

[LouScheffer]

Here's an analysis of the accuracy needed, from the SES-9 discussion thread:

Let's say it's a = 50m/s^2 (with respect to the surface, not freefall), and the stage has to be within v=2m/s of zero in order to land safely.  How accurate do you have to be within the z-direction?

a = v^2/(2*d) becomes: 2*d*a = v^2 becomes d = v^2/(2*a) = (2m/s)^2/(2*50m/s^2) = 4 centimeters (!)

You have to be within 4 centimeters in the z-direction in order to stay within your landing velocity constraint when you're hoverslamming with 3 engines. If something doesn't throttle up fast enough or you start too early or late, you're toast. This isn't impossible, but it's DANG challenging.

I think this analysis is too pessimistic.  It's OK for the ends of the legs to hit the ground faster, provided the body of the rocket reaches 0 vertical speed before the legs run out of travel (or the engine bell hits the ground, whichever comes first).  Assuming the legs can absorb one meter of bend before breaking, then you need the lower vertex of the parabola to be between the deck and a point one meter below. 

Is this practical?  With 3 engines, 30 tonnes mass, your acceleration varies from 3.8G at 70% throttle to 5.5G at 100%.  Assume you plan your burn for 4.5Gs so you have leeway in both directions.  If you are falling at 250 m/s (about what you'd guess from the one engine landings) you'd want the engine to start at 82% throttle at 5.5 seconds before impact, at a height of 694 meters.  You get about a 1/2 second of slop since as long as you start before 568 m you can still stop at full thrust.

Once (if) your engines start you are in good shape.   On this time scale the radar altimeter and calculations should be instantaneous, so you immediately know the desired acceleration to place the  vertex 50 cm below the landing pad (or whatever your target).  You don't know the exact mass of the stage, nor the actual thrust for a commanded amount, but measuring the achieved acceleration tells you the proportionality constant.   Now you start adjusting the commanded thrust to get the acceleration right.

At 1 second before landing at 4.5 Gs , you are 22.5 meters up.  A 1% acceleration error will move the vertex +- 22 cm.  That's about all you can afford, since it's already half your error budget.  So you need to have the acceleration right to the 1% level by 1 second to go.  You get 4.5 seconds of correction to do this.  If the initial error is 20% (say 10% for throttle and 10% for mass) the you need to reduce the error by a factor of 20.  Assuming a linear system, this level of correction requires 3 time constants (e^3 = 20) so if your time constant for throttle response is 1.5 seconds or less, it should be possible.  Given that the engine can get to (nearly) steady state during either a static fire or the short time before liftoff, such a time constant seems possible.

Now this analysis assumes you are coming straight down with the rocket vertical, no attempt to steer horizontally, no errors in the radar altimeter or IMU, etc.  But even given these errors, it seems possible to make this work.

The potential fuel savings are similar:   A single engine landing burn might have 25 sec at full thrust followed by 5 sec at 70%, so 28.5 engine-sec.  For three engines, 5.5 sec x 82% x 3 = 13.5 engine-sec.  So 15 engine-sec savings means 1.66 more seconds of first stage burn.  At about 5G, that's 80 m/sec more to the payload.  Well worth it if you can make it work.

[Comga]

Simplistic model in table form

Engines	T/W	Accel	t	h	g-loss	ratio	fuel
		(m/s^2)	(sec)	(m)	(m/s)
1	1.75	7.5	20	1500	200	2.33	100%
2	3.5	25.0	6	450	60	1.40	60%
3	5.3	42.5	3.5	265	35	1.24	53%
4	7.0	60.0	2.5	188	25	1.17	50%
5	8.8	77.5	1.9	145	19	1.13	48%
6	10.5	95.0	1.6	118	16	1.11	47%
7	12.3	112.5	1.3	100	13	1.09	47%
8	14.0	130.0	1.2	87	12	1.08	46%
9	15.8	147.5	1.0	76	10	1.07	46%

Conclusion: This shows quickly diminishing returns.  Most of the fuel savings happens in the first step.
It seems obvious why the SpaceX went from 1 to 3 engines.  Three engines are rigged to restart for boostback and entry burns.
But there is 85% of the fuel savings, with almost twice the time for adjustments, if it uses 2 engines instead of 3. 
(A symmetric configuration would be the outer 2 of the 3 restarting engines.)
What would be the issues with a 2 engine landing burn?                                          

[The_Ronin]

What would be the issues with a 2 engine landing burn?

The outer engines do not have the same gimble range as the center.  Make make corrections difficult.

[PreferToLurk]

Simplistic model in table form

Engines	T/W	Accel	t	h	g-loss	ratio	fuel
		(m/s^2)	(sec)	(m)	(m/s)
1	1.75	7.5	20	1500	200	2.33	100%
2	3.5	25.0	6	450	60	1.40	60%
3	5.3	42.5	3.5	265	35	1.24	53%
4	7.0	60.0	2.5	188	25	1.17	50%
5	8.8	77.5	1.9	145	19	1.13	48%
6	10.5	95.0	1.6	118	16	1.11	47%
7	12.3	112.5	1.3	100	13	1.09	47%
8	14.0	130.0	1.2	87	12	1.08	46%
9	15.8	147.5	1.0	76	10	1.07	46%

Conclusion: This shows quickly diminishing returns.  Most of the fuel savings happens in the first step.
It seems obvious why the SpaceX went from 1 to 3 engines.  Three engines are rigged to restart for boostback and entry burns.
But there is 85% of the fuel savings, with almost twice the time for adjustments, if it uses 2 engines instead of 3. 
(A symmetric configuration would be the outer 2 of the 3 restarting engines.)
What would be the issues with a 2 engine landing burn?

Another solution for 3 engine landing is to throttle lower.  I think I originally calculated that the 1.75 T/W regime of the OG2 landing was with a throttle setting around 90% (though it might have been as high as 95% with slightly different assumptions).  We know for a fact that the center engine can throttle at least to 80% of the final thrust setting (based on the first landing burn regime of about 4 m/s^2  deceleration). 

If all three engines can throttle to the same extent (and there is some consideration that they cannot, or maybe just not as precisely), then you should be able to get a 3 engine landing burn with 31 m/s^2 deceleration.  This would net you at least another second of burn time without having to sacrifice much in terms of control authority or fuel.

Complicating this is the fact that with a dry mass of around 30000kg, coming in for a landing with 4-6k kg less fuel (assuming saved fuel is burned before meco and assuming a fuel reserve/ballast on OG2 that would also be mostly burned prior to meco) affects the landing T/W in non trivial terms.  At the first landing burn of OG2 I assumed 2500kg of fuel reserve plus 5500kg of fuel used in the burns. A 38000kg stage with a 54kgf (about 72% of 75kgf full thrust) will nicely create a 1.4 T/W, but if you pull 3000kg of fuel off the stage, then your 90% throttled engine will produce about 1.93 T/W, not 1.75 which is going to speed up all of your landing assumptions.

Anyway, just some food for thought.  Personally, I think they could start all three engines at absolute minimum thrust (no throttle down margin), and if they accidentally lit the engines too high up and need to throttle down further just cut either the center engine (after zeroing enough lateral error to let the outer engines gimble range take over) or the outer two if needed.  This would allow for the greatest throttle up margin (which apparently is what doomed this last landing) while giving as much time as possible to resolve ignition transients and compute a landing solution.  IMHO

[CraigLieb]

Simplistic model in table form

Engines	T/W	Accel	t	h	g-loss	ratio	fuel
		(m/s^2)	(sec)	(m)	(m/s)
1	1.75	7.5	20	1500	200	2.33	100%
2	3.5	25.0	6	450	60	1.40	60%
3	5.3	42.5	3.5	265	35	1.24	53%
4	7.0	60.0	2.5	188	25	1.17	50%
5	8.8	77.5	1.9	145	19	1.13	48%
6	10.5	95.0	1.6	118	16	1.11	47%
7	12.3	112.5	1.3	100	13	1.09	47%
8	14.0	130.0	1.2	87	12	1.08	46%
9	15.8	147.5	1.0	76	10	1.07	46%

Conclusion: This shows quickly diminishing returns.  Most of the fuel savings happens in the first step.
It seems obvious why the SpaceX went from 1 to 3 engines.  Three engines are rigged to restart for boostback and entry burns.
But there is 85% of the fuel savings, with almost twice the time for adjustments, if it uses 2 engines instead of 3. 
(A symmetric configuration would be the outer 2 of the 3 restarting engines.)
What would be the issues with a 2 engine landing burn?

Another solution for 3 engine landing is to throttle lower.  I think I originally calculated that the 1.75 T/W regime of the OG2 landing was with a throttle setting around 90% (though it might have been as high as 95% with slightly different assumptions).  We know for a fact that the center engine can throttle at least to 80% of the final thrust setting (based on the first landing burn regime of about 4 m/s^2  deceleration). 

If all three engines can throttle to the same extent (and there is some consideration that they cannot, or maybe just not as precisely), then you should be able to get a 3 engine landing burn with 31 m/s^2 deceleration.  This would net you at least another second of burn time without having to sacrifice much in terms of control authority or fuel.

Complicating this is the fact that with a dry mass of around 30000kg, coming in for a landing with 4-6k kg less fuel (assuming saved fuel is burned before meco and assuming a fuel reserve/ballast on OG2 that would also be mostly burned prior to meco) affects the landing T/W in non trivial terms.  At the first landing burn of OG2 I assumed 2500kg of fuel reserve plus 5500kg of fuel used in the burns. A 38000kg stage with a 54kgf (about 72% of 75kgf full thrust) will nicely create a 1.4 T/W, but if you pull 3000kg of fuel off the stage, then your 90% throttled engine will produce about 1.93 T/W, not 1.75 which is going to speed up all of your landing assumptions.

Anyway, just some food for thought.  Personally, I think they could start all three engines at absolute minimum thrust (no throttle down margin), and if they accidentally lit the engines too high up and need to throttle down further just cut either the center engine (after zeroing enough lateral error to let the outer engines gimble range take over) or the outer two if needed.  This would allow for the greatest throttle up margin (which apparently is what doomed this last landing) while giving as much time as possible to resolve ignition transients and compute a landing solution.  IMHO

It seems with only a 3 second burn, thrust variability during start-up and shut-down could play a much larger role in reduced reliability of the landing scenario.   

[drzerg]

gimbal range wich is limited due to other engines is irrelevant because they could gimbal outwards just before shutdown

[Comga]

The_Ronin:  Two engines with limited gimbal range looks like it is still more force than one with full range
PerferToLurk: This model is woefully insufficient to use for such subtleties, but that misses the point.  There is little reason to use more than two engines.  The is not much more to gain in terms of fuel savings.
CraigLeib: Precisely, even at 3+1 seconds
drzerg:  That doesn't work because it counts on cosine losses, and for the small gimbal angles Cos~1.

[meekGee]

I like the idea of a planned nominal one engine burn, but have all three engines spun up, so that if the center engine fails, you can still switch to the other two in time to land.

On more demanding missions, go for a planned 2 engine burn, with three engines as a fall back...

[the_other_Doug]

My only concern is that, with the gimbal limitations of the outer engines, you just don't have the required control authority to actually hit the target unless you use the center engine.  We're talking about the end of the descent, when the grid fins are losing their control effectiveness and the control is shifting more and more to the engine gimbals, until right at the end it is entirely controlled by engine gimballing.

As a caveat, though, only SpaceX knows the control authority allowed by using two opposing outer engines for a hoverslam.  Without this knowledge, anything we say about it is nothing more than gross speculation.

[Kabloona]

On the subject of the hoverslam, there has been some debate in other threads about whether the stage reached terminal velocity before the landing burn started, and what that velocity might be.

For reference, two NSF members did earlier independent video analyses of the OG2 landing and derived "terminal" velocities before landing burn of 150 m/sec and 180 m/sec on that flight.

In the following SES-9 simulation, landing burn starts at a velocity of about 170 m/sec, and a relative acceleration of 1.18 g, so the stage is decelerating relatively slowly, therefore almost at terminal velocity (which would be 1.00 g relative acceleration).

Also, the entry burn starts at 65 km altitude at about 2,300 m/sec and decelerates to about 1,300 m/sec. From that point on, aero drag takes over and decelerates the stage continuously as it falls towards the ASDS.

The simulation was done by someone (Zach) who says (in the comments section) that he compared the output of his simulation against telemetry data scraped from the webcast in order tweak his aero model. I don't know how accurate the landing portion of the trajectory is, but it seems credible and consistent with what we saw on OG2.

Update: In response to my question about how he modeled the trajectory after entry burn, Zach replied with this comment:

"There is no public information or telemetry on the first stage after MECO. The only hints we get from the webcast are the callouts for when the burns start and sometimes when the stage goes transonic. My simulation currently doesn't take lift into account so I would say the trajectory is only accurate until it starts getting deep into the atmosphere."

"The only thing we know for sure is that the landing burn used three engines this time. My simulation shows that burning 3 engines on 60% thrust will generate 4.2G. I'm assuming the dry mass of the stage is ~25,000kg and the engines generate 756kN of thrust at sea level (known value). If the burn lasted 4-5 seconds then the kinematic equations will show it had to be moving around 170 m/s."

[Robotbeat]

If you're going to be a stickler about it, the stage would never reach terminal velocity even with greater drag. Terminal velocity is something you can only approach. But it likely gets within 10-20% of terminal velocity, so close enough for practical purposes.

[Okie_Steve]

So successful 3 engine landing. Presumably they were running somewhere close to full throttle. I've been surprised that the airframe of an an almost empty stage could take the G loading, but obviously it can - at least once. It's diminishing return but I wonder if it could handle  a 'we who are about to die' 5 engine burn. Re-plumbing  for 5 vs 3 engine TEA/TEB would allow changing both the boost back and landing burns. Did we ever establish for sure that only 1 engine is used for the entry burn 'plasma shield'?

IIRC Elon commented recently that small percentage changes add up in the rocket business.

[Lars-J]

So successful 3 engine landing. Presumably they were running somewhere close to full throttle. I've been surprised that the airframe of an an almost empty stage could take the G loading, but obviously it can - at least once. It's diminishing return but I wonder if it could handle  a 'we who are about to die' 5 engine burn. Re-plumbing  for 5 vs 3 engine TEA/TEB would allow changing both the boost back and landing burns. Did we ever establish for sure that only 1 engine is used for the entry burn 'plasma shield'?

No. And if it was just a single engine burn, it wouldn't be for just a 'plasma shield'. It is done to seriously brake the stage.

[Saabstory88]

Now, I'm not a structural engineer, but I would imagine that a ~4G landing burn with little propellant remaining would be less stressful on the airframe than the ~3.5G at the top of the ascent with ~125 tons on top of the partially loaded vehicle.

I would imagine that the streses in the legs would be of the greatest concern in the 3 engine scenario.

[acsawdey]

So successful 3 engine landing. Presumably they were running somewhere close to full throttle. I've been surprised that the airframe of an an almost empty stage could take the G loading, but obviously it can - at least once. It's diminishing return but I wonder if it could handle  a 'we who are about to die' 5 engine burn. Re-plumbing  for 5 vs 3 engine TEA/TEB would allow changing both the boost back and landing burns. Did we ever establish for sure that only 1 engine is used for the entry burn 'plasma shield'?

IIRC Elon commented recently that small percentage changes add up in the rocket business.

Regarding the G loading ... from the F9 user's manual v 2.0, max axial acceleration loading is 8.5g. But, from the point of view of the first stage, that is with a ~120 ton weight sitting on top of it. That's equivalent to the first stage sitting on the ground with a 1000 ton weight sitting on top of it. So, the tanks by themselves can probably take a lot more than 8.5g with nothing on top. Full throttle with 3 engines and completely empty stage would be about 10g -- 3*167000/(~50000).

On another thread somebody did fuel savings calculations for different numbers of engines for landing. By the time you get to 3 engines you have got most of the benefit and increasing the number of engines beyond that doesn't improve it by very much.

[Okie_Steve]

Simplistic model in table form

Engines	T/W	Accel	t	h	g-loss	ratio	fuel
		(m/s^2)	(sec)	(m)	(m/s)
1	1.75	7.5	20	1500	200	2.33	100%
2	3.5	25.0	6	450	60	1.40	60%
3	5.3	42.5	3.5	265	35	1.24	53%
4	7.0	60.0	2.5	188	25	1.17	50%
5	8.8	77.5	1.9	145	19	1.13	48%
6	10.5	95.0	1.6	118	16	1.11	47%
7	12.3	112.5	1.3	100	13	1.09	47%
8	14.0	130.0	1.2	87	12	1.08	46%
9	15.8	147.5	1.0	76	10	1.07	46%

This is up thread. *IF* it could be made to work 5 vs 3 would save a bit more fuel and little bits add up with rocket. But, if it's already pulling ~10Gs with 3 engines then 5 would push that up to ~16 which is likely beyond the structural limit. Although, the published G limit may be more about the payload than the rocket. I guess if they ever try 5 we can surmise that it's higher than we thought. 

As I understand it, if the the stage was made 'stationary' relative to the surface the terminal velocity it would achieve would be a relatively benign event. The figure ' about 15 seconds' was mentioned in the web cast for the entry burn. I'm sure that produces significant slowing of the stage but comes nowhere close to 'stationary' so the rest of the kinetic energy has to be bled off as drag and the resulting heat. The 'plasma shield' I was referring to is the engine(s) exhaust diverting much of that heating around the stage and keeping it from cooking and ablating the exterior surface. That being the case there might be an advantage of using a single engine since you could run it longer to create an increased protection duration. If that's not true, then there would be no benefit to waiting to initiate the burn once you were out of proximity  to S2 and could effectively combine the boost back and entry burns.

[savuporo]

But, if it's already pulling ~10Gs with 3 engines then 5 would push that up to ~16 which is likely beyond the structural limit.

"Part of our post-landing checkout on the rocket is measuring it with a yardstick, need to make sure it aint a wee bit shorter than it used to be"

[CJ]

I have to admit that when I first saw this thread, I dismissed it as  hyperbole, and unworkable. I mean, the very idea that SpaceX would try, especially during it's early landings, something as preposterous and absurdly difficult as a 3-engine landing burn - it was just too absurd...

Well, rather clearly, I was wrong (and I'm quite glad I kept my mouth shut on the issue until now).

Thank you, Prefer to Lurk, for starting this thread and crunching the numbers.

I hope better video of yesterday's 3 engine landing is released at some point - I'd love to see if they shut down the outer engines a second or two before the center engine.

[2552]

Simplistic model in table form

Engines	T/W	Accel	t	h	g-loss	ratio	fuel
		(m/s^2)	(sec)	(m)	(m/s)
1	1.75	7.5	20	1500	200	2.33	100%
2	3.5	25.0	6	450	60	1.40	60%
3	5.3	42.5	3.5	265	35	1.24	53%
4	7.0	60.0	2.5	188	25	1.17	50%
5	8.8	77.5	1.9	145	19	1.13	48%
6	10.5	95.0	1.6	118	16	1.11	47%
7	12.3	112.5	1.3	100	13	1.09	47%
8	14.0	130.0	1.2	87	12	1.08	46%
9	15.8	147.5	1.0	76	10	1.07	46%

This is up thread. *IF* it could be made to work 5 vs 3 would save a bit more fuel and little bits add up with rocket. But, if it's already pulling ~10Gs with 3 engines then 5 would push that up to ~16 which is likely beyond the structural limit. Although, the published G limit may be more about the payload than the rocket. I guess if they ever try 5 we can surmise that it's higher than we thought. 

As I understand it, if the the stage was made 'stationary' relative to the surface the terminal velocity it would achieve would be a relatively benign event. The figure ' about 15 seconds' was mentioned in the web cast for the entry burn. I'm sure that produces significant slowing of the stage but comes nowhere close to 'stationary' so the rest of the kinetic energy has to be bled off as drag and the resulting heat. The 'plasma shield' I was referring to is the engine(s) exhaust diverting much of that heating around the stage and keeping it from cooking and ablating the exterior surface. That being the case there might be an advantage of using a single engine since you could run it longer to create an increased protection duration. If that's not true, then there would be no benefit to waiting to initiate the burn once you were out of proximity  to S2 and could effectively combine the boost back and entry burns.

Given the diminishing returns of 5 engines vs 3 for landing, I think it's more likely SpaceX tries a 5 (or more) engine reentry burn instead. There's a lot more fuel mass onboard (15-20 tons?) when it starts, so a 5 engine reentry burn should give more of a benefit to T/W without pulling too many Gs. The reentry burn for JCSAT-14 was about 15 seconds with 3 engines, so would a 5 engine burn cut that to under 10 seconds? What kind of fuel savings would that give?

[Herb Schaltegger]

Nice to see some confirmation of what some folks had guesstimated earlier:

Per Elon's tweet (quoted in the post below), the landing burn starts with 3 engines, ends with one. I'm sure the SpecialSauce™ is in the timing and details.

http://forum.nasaspaceflight.com/index.php?topic=39843.msg1530435#msg1530435

[Okie_Steve]

Simplistic model in table form

Engines	T/W	Accel	t	h	g-loss	ratio	fuel
		(m/s^2)	(sec)	(m)	(m/s)
1	1.75	7.5	20	1500	200	2.33	100%
2	3.5	25.0	6	450	60	1.40	60%
3	5.3	42.5	3.5	265	35	1.24	53%
4	7.0	60.0	2.5	188	25	1.17	50%
5	8.8	77.5	1.9	145	19	1.13	48%
6	10.5	95.0	1.6	118	16	1.11	47%
7	12.3	112.5	1.3	100	13	1.09	47%
8	14.0	130.0	1.2	87	12	1.08	46%
9	15.8	147.5	1.0	76	10	1.07	46%

This is up thread. *IF* it could be made to work 5 vs 3 would save a bit more fuel and little bits add up with rocket. But, if it's already pulling ~10Gs with 3 engines then 5 would push that up to ~16 which is likely beyond the structural limit. Although, the published G limit may be more about the payload than the rocket. I guess if they ever try 5 we can surmise that it's higher than we thought. 

As I understand it, if the the stage was made 'stationary' relative to the surface the terminal velocity it would achieve would be a relatively benign event. The figure ' about 15 seconds' was mentioned in the web cast for the entry burn. I'm sure that produces significant slowing of the stage but comes nowhere close to 'stationary' so the rest of the kinetic energy has to be bled off as drag and the resulting heat. The 'plasma shield' I was referring to is the engine(s) exhaust diverting much of that heating around the stage and keeping it from cooking and ablating the exterior surface. That being the case there might be an advantage of using a single engine since you could run it longer to create an increased protection duration. If that's not true, then there would be no benefit to waiting to initiate the burn once you were out of proximity  to S2 and could effectively combine the boost back and entry burns.

Given the diminishing returns of 5 engines vs 3 for landing, I think it's more likely SpaceX tries a 5 (or more) engine reentry burn instead. There's a lot more fuel mass onboard (15-20 tons?) when it starts, so a 5 engine reentry burn should give more of a benefit to T/W without pulling too many Gs. The reentry burn for JCSAT-14 was about 15 seconds with 3 engines, so would a 5 engine burn cut that to under 10 seconds? What kind of fuel savings would that give?

Given the tweet above I think I agree.

[Kabloona]

The reentry burn for JCSAT-14 was about 15 seconds with 3 engines...

Just curious, what was your source for that? I didn't see the whole webcast. On the technical webcast, the entry burn callouts were 25 seconds apart, apparently not very accurately timed...

By comparison, the entry burn callouts for SES-9 were spaced 17 seconds apart. And this time they may have shortened the entry burn slightly in order to reserve more propellant for landing.

So 15 seconds makes sense, but I'd like to find the source.

[2552]

It was said on the hosted webcast at about 27:34.

[schaban]

Merlin takes about 3 sec to start so whatever fuel savings gained in reduced gravity losses using 5 engines might get lost in 6 sec of wasting fuel of extra 2 engines start up


Sent from my iPhone using Tapatalk

[ChrisWilson68]

Merlin takes about 3 sec to start so whatever fuel savings gained in reduced gravity losses using 5 engines might get lost in 6 sec of wasting fuel of extra 2 engines start up

Huh?  In what way do you think fuel is "wasted" during engine start-up?

[rocx]

Huh?  In what way do you think fuel is "wasted" during engine start-up?

The fuel/gases that come out during spin-up are nowhere near the exit velocity of a fully running engine, so they only give little thrust for their mass.

[ChrisWilson68]

Huh?  In what way do you think fuel is "wasted" during engine start-up?

The fuel/gases that come out during spin-up are nowhere near the exit velocity of a fully running engine, so they only give little thrust for their mass.

You cut out the context of my post, and that makes all the difference.  I wasn't asking in a general sense about whether there was any Isp loss from engine start-up.  I was asking specifically how that poster I was responding to meant it.  In particular, that poster was quoting "six seconds" of wasted engine running time and drawing the conclusion from that that adding two more engines during the landing burn couldn't have any beneficial effect.

The point of the question was to elicit whether the poster thought some significant portion of those "six seconds" was simultaneously using a significant portion of the normal engine prop flow rate and at a significantly lower-than-average Isp.

[gin455res]

a) What kind of upper stages might one design for a Falcon-3R, or a Falcon-5R?

(Is a Falcon-3R with parachute light enough to be caught by a helicopter?) 
     
 

[Kabloona]

a) What kind of upper stages might one design for a Falcon-3R, or a Falcon-5R?

(Is a Falcon-3R with parachute light enough to be caught be a helicopter?) 
     

That's a question for a different thread, like this one:

http://forum.nasaspaceflight.com/index.php?topic=39314.0

[gin455res]

A Falcon-5R would be equivalent to landing on 1.8 engines. How much fuel would that save?

[envy887]

The numbers are up the thread... 2 engines saves about 40% and three engines about 47% of landing fuel, but sacrifices a lot of control.

According to Musk's tweet, the landing thrust goes down to 40% of a single Merlin. That would be about 16% of a single Raptor.

That's either a super sporty landing, or an incredibly throttleable engine. But S1 Raptor will probably be overexpanded at sea level. Over expanded engines and super deep throttling don't mix well.

Landing a F9 S1 on a Raptor would be quite challenging indeed.

[gin455res]

The numbers are up the thread... 2 engines saves about 40% and three engines about 47% of landing fuel, but sacrifices a lot of control.

According to Musk's tweet, the landing thrust goes down to 40% of a single Merlin. That would be about 16% of a single Raptor.

That's either a super sporty landing, or an incredibly throttleable engine. But S1 Raptor will probably be overexpanded at sea level. Over expanded engines and super deep throttling don't mix well.

Landing a F9 S1 on a Raptor would be quite challenging indeed.

Not sure if you are responding to my question about a falcon-5R? 

As you first mention landing on 2 engines which is approximately 1.8 (the ratio of the  T/W of landing a Falcon-5 on one engine with : the T/W of landing a F9 on one engine), but then mention landing on a raptor. The falcon-5R I'm imagining is a 5 merlin falcon rocket.  The 'R'stands for reusable not Raptor (It would only be able to throttle down to 72% {F9 equivalent} at the end of the landing burn.)

[CraigLieb]

and if you need a distraction from how pushing the limits of hoverslam landings effect inventory...

New poll how many stages will they have to store by Dec 31st 2016!
http://forum.nasaspaceflight.com/index.php?topic=40267.0

(a bit of shameless self promotion again)

[envy887]

The numbers are up the thread... 2 engines saves about 40% and three engines about 47% of landing fuel, but sacrifices a lot of control.

According to Musk's tweet, the landing thrust goes down to 40% of a single Merlin. That would be about 16% of a single Raptor.

That's either a super sporty landing, or an incredibly throttleable engine. But S1 Raptor will probably be overexpanded at sea level. Over expanded engines and super deep throttling don't mix well.

Landing a F9 S1 on a Raptor would be quite challenging indeed.

Not sure if you are responding to my question about a falcon-5R? 

As you first mention landing on 2 engines which is approximately 1.8 (the ratio of the  T/W of landing a Falcon-5 on one engine with : the T/W of landing a F9 on one engine), but then mention landing on a raptor. The falcon-5R I'm imagining is a 5 merlin falcon rocket.  The 'R'stands for reusable not Raptor (It would only be able to throttle down to 72% {F9 equivalent} at the end of the landing burn.)

I did think you were referring to a Raptor...

Is there any basis to think the Falcon-5 is ever coming back? My understanding was that all F1 and F5 payloads were permanently moved to F9 launches. F5 LEO payloads maxed at 4000 kg, which can fly in Dragon's trunk. There's no real economic case to invest in a smaller launcher.

Economics aside, it is probably feasible to land the F5 on it's center Merlin, although that would still be a pretty sporty landing. Definitely easier than landing a 5-Raptor Falcon but still certainly pushing the limits of a hoverslam.

[speedevil]

According to Musk's tweet, the landing thrust goes down to 40% of a single Merlin. That would be about 16% of a single Raptor.

I don't think he quite said that.
That's 260kN or so, and with a stage weighing 20 tons, a residual accelleration after gravity of only 3m/s.
The rocket would go the last rocket-height in ~6 seconds.
That seems rather too slow.
I'd think that they'd be up near the top of the thrust range - 80% perhaps, not 40%.
If you're at 80%, you can throttle up 20 and down 40%. If you're at 40%, you can't throttle down at all.

40% might be desirable to get a human-scale timeline for landing - but it is actively bad to slow down that far IMO.
The problem is the slower you are decelerating, the more time you have for wind to blow the stage sideways with limited time to correct, and no authority from the grid fins at all.

[envy887]

According to Musk's tweet, the landing thrust goes down to 40% of a single Merlin. That would be about 16% of a single Raptor.

I don't think he quite said that.
That's 260kN or so, and with a stage weighing 20 tons, a residual accelleration after gravity of only 3m/s.
The rocket would go the last rocket-height in ~6 seconds.
That seems rather too slow.
I'd think that they'd be up near the top of the thrust range - 80% perhaps, not 40%.
If you're at 80%, you can throttle up 20 and down 40%. If you're at 40%, you can't throttle down at all.

40% might be desirable to get a human-scale timeline for landing - but it is actively bad to slow down that far IMO.
The problem is the slower you are decelerating, the more time you have for wind to blow the stage sideways with limited time to correct, and no authority from the grid fins at all.

It seems pretty clear that he is actually saying that: https://twitter.com/elonmusk/status/728753234811060224 Context before and after the 40% part is obviously referencing a 3-engine landing burn.

How long it runs at 40% is not obvious; it may only be throttled that low for >1 second before touchdown.

[sanman]

The numbers are up the thread... 2 engines saves about 40% and three engines about 47% of landing fuel, but sacrifices a lot of control.

How do they do 2 engines? Both outer engines and no central gimbaled engine? Or is it the gimbaled central engine plus one of the outer ones? (sounds asymmetric)

For 3 engines, is it the central gimbaled one and 2 outer engines? Or are all 3 outer engines?

[spacenut]

I'm not sure, but I think all the engines are gimbaled.  At the very least 4 outer engines would be. 

[sevenperforce]

The numbers are up the thread... 2 engines saves about 40% and three engines about 47% of landing fuel, but sacrifices a lot of control.

How do they do 2 engines? Both outer engines and no central gimbaled engine? Or is it the gimbaled central engine plus one of the outer ones? (sounds asymmetric)

For 3 engines, is it the central gimbaled one and 2 outer engines? Or are all 3 outer engines?

Only three engines have plumbing for TEA/TEB reignition, and all three receive a reignition pulse regardless of which ones are lit. They are the central engine and two opposite outer engines.

IIRC all the engines are gimbaled, though the center might have more freedom than the outer ones.

[mvpel]

You can see the TVC pistons in the two engines in the forefront of this picture, and a hint of the top of one in the rightmost engine, which is indicative that all engines are vectorable.

[Kabloona]

It seems pretty clear that he is actually saying that: https://twitter.com/elonmusk/status/728753234811060224 Context before and after the 40% part is obviously referencing a 3-engine landing burn.

How long it runs at 40% is not obvious; it may only be throttled that low for >1 second before touchdown.

Here is one notional scheme for a 3-engine landing burn.

The final single-engine throttle-down sequence can be a pre-programmed curve with known total impulse based on engine testing. It's important that the thrust-vs-time curve during throttle-down be well-characterized by engine testing on a thrust stand because there's no room for error that close to touchdown. So that curve can be characterized/optimized by experiment on the test stand and then programmed into the autopilot for repeatability.

With the throttle-down curve pre-programmed, the critical variable becomes the time lag between shutdown of the 2 outer engines and initiation of the center engine throttle-down program. This delta-t can be calculated in real time based on altitude, descent speed, and acceleration with the center engine running at 100% throttle (from which stage mass can be derived). Those three variables should be sufficient for the autopilot to determine when to intitiate the throttle-down sequence for a soft landing.

[dorkmo]

wonder how much fuel reserve is taken into account by the computer. would the most fuel efficient option be to jump directly from 3 engines at 100% to 1 engine at 40%?

[sevenperforce]

It seems pretty clear that he is actually saying that: https://twitter.com/elonmusk/status/728753234811060224 Context before and after the 40% part is obviously referencing a 3-engine landing burn.

How long it runs at 40% is not obvious; it may only be throttled that low for >1 second before touchdown.

Here is one notional scheme for a 3-engine landing burn.

The final single-engine throttle-down sequence can be a pre-programmed curve with known total impulse based on engine testing. It's important that the thrust-vs-time curve during throttle-down be well-characterized by engine testing on a thrust stand because there's no room for error that close to touchdown. So that curve can be characterized/optimized by experiment on the test stand and then programmed into the autopilot for repeatability.

With the throttle-down curve pre-programmed, the critical variable becomes the time lag between shutdown of the 2 outer engines and initiation of the center engine throttle-down program. This delta-t can be calculated in real time based on altitude, descent speed, and acceleration with the center engine running at 100% throttle (from which stage mass can be derived). Those three variables should be sufficient for the autopilot to determine when to intitiate the throttle-down sequence for a soft landing.

One other issue to keep in mind is that the shutoff itself is also a gradual throttledown in thrust, though a more rapid one....

[Kabloona]

wonder how much fuel reserve is taken into account by the computer. would the most fuel efficient option be to jump directly from 3 engines at 100% to 1 engine at 40%?

It's more fuel efficient to burn at high thrust as long as possible, so if you're going to shut down 2 engines, it's better from an efficiency POV to run the center engine at 100% as long as possible before throttling down to 40%.

And the computer doesn't need to know how much propellant remains. It's probably pre-programmed to do an entry burn of specific time duration, and the landing burn is probably pre-programmed to start at a certain altitude/velocity, based on pre-launch Monte Carlo simulations that give them a good idea of how much propellant will be used during the burns. And they would do Monte Carlo runs to calculate the propellant reserves at MECO and verify that the expected reserves at MECO exceed the expected consumption during entry and landing.

For SES-9 the results of those two Monte Carlo simulations were probably quite close, resulting in their public prediction that successful landing was unlikely.

But the flight computer probably just flies the mission assuming it has enough propellant to land safely. And you wouldn't want it to know if it didn't. 

One other issue to keep in mind is that the shutoff itself is also a gradual throttledown in thrust, though a more rapid one....

Yup, my drawing left out a few details...

[llanitedave]

wonder how much fuel reserve is taken into account by the computer. would the most fuel efficient option be to jump directly from 3 engines at 100% to 1 engine at 40%?

It's more fuel efficient to burn at high thrust as long as possible, so if you're going to shut down 2 engines, it's better from an efficiency POV to run the center engine at 100% as long as possible before throttling down to 40%.

And the computer doesn't need to know how much propellant remains. It's probably pre-programmed to do an entry burn of specific time duration, and the landing burn is probably pre-programmed to start at a certain altitude/velocity, based on pre-launch Monte Carlo simulations that give them a good idea of how much propellant will be used during the burns. And they would do Monte Carlo runs to calculate the propellant reserves at MECO and verify that the expected reserves at MECO exceed the expected consumption during entry and landing.

For SES-9 the results of those two Monte Carlo simulations were probably quite close, resulting in their public prediction that successful landing was unlikely.

But the flight computer probably just flies the mission assuming it has enough propellant to land safely. And you wouldn't want it to know if it didn't. 

One other issue to keep in mind is that the shutoff itself is also a gradual throttledown in thrust, though a more rapid one....

Yup, my drawing left out a few details...

Shouldn't really have to assume anything.  The system knows how many engines it's running and how they are throttled.  It knows what the turbopump speed and pressure is.  It knows how the engine responds.  It should be able to calculate to a large degree of accuracy what the thrust being produced by those engines is at any time slice.


At the same time, it knows what acceleration the stage is experiencing, and in what directions.  It knows the empty mass of the stage and it knows how much fuel and oxygen has been loaded.  Therefore, it's a straightforward calculation to derive the current mass of the stage, and therefore how much propellant is currently remaining.


How it would use that information, what kind of autonomous decision-making ability it has, I wouldn't know.  At the very least, it can be comparing its actual performance with that of the pre-flight simulations to help refine future plans.  At most, it might be able to make adjustments on the way down to try to optimize its trajectory and touchdown velocity.

[Kabloona]

How it would use that information, what kind of autonomous decision-making ability it has, I wouldn't know.  At the very least, it can be comparing its actual performance with that of the pre-flight simulations to help refine future plans.  At most, it might be able to make adjustments on the way down to try to optimize its trajectory and touchdown velocity.

My point was, the autoilot doesn't need to care or know how much propellant is left. The boostback/entry/landing burn parameters are probably pre-programmed in such a way that the propellant usage is pretty much fixed by the flight profile (subject to the usual dispersions of course) which has been run on Monte Carlo simulations, so they know ahead of time how much propellant will be needed at MECO.

All the flight computer is doing is running the boostback/entry/landing burn sequences based on given parameters, and controlling the landing burn throttle-down timing to achieve zero-zero landing. None of that requires knowledge of propellant remaining. Naturally the computer is adjusting throttle based on accelaration and velocity, but that also is a function of stage total mass at any given time and doesn't require derivation of propellant mass remaining.

So I was being flip about the flight computer "assuming" it has enough propellant, but it really doesn't need to know anything about remaining reserves. It simply runs its boostback/entry/landing burns until it either touches down or runs out of propellant. Either way, knowledge of propellant quantity remaining wouldn't change how the autpoilot operates, IMO, so there's no need to try to derive it in real time.

[llanitedave]

How it would use that information, what kind of autonomous decision-making ability it has, I wouldn't know.  At the very least, it can be comparing its actual performance with that of the pre-flight simulations to help refine future plans.  At most, it might be able to make adjustments on the way down to try to optimize its trajectory and touchdown velocity.

My point was, the autoilot doesn't need to care or know how much propellant is left. The boostback/entry/landing burn parameters are probably pre-programmed in such a way that the propellant usage is pretty much fixed by the flight profile (subject to the usual dispersions of course) which has been run on Monte Carlo simulations, so they know ahead of time how much propellant will be needed at MECO.

All the flight computer is doing is running the boostback/entry/landing burn sequences based on given parameters, and controlling the landing burn throttle-down timing to achieve zero-zero landing. None of that requires knowledge of propellant remaining. Naturally the computer is adjusting throttle based on accelaration and velocity, but that also is a function of stage total mass at any given time and doesn't require derivation of propellant mass remaining.

So I was being flip about the flight computer "assuming" it has enough propellant, but it really doesn't need to know anything about remaining reserves. It simply runs its boostback/entry/landing burns until it either touches down or runs out of propellant. Either way, knowledge of propellant reserves wouldn't change its performance, IMHO.

It may be minor, but the total amount of mass remaining would affect its acceleration at a given thrust level.  So either it needs to be able to adjust it parameters on the fly, or the preflight predictions need to be very accurate.  I'm sure they are.

[Kabloona]

It may be minor, but the total amount of mass remaining would affect its acceleration at a given thrust level.  So either it needs to be able to adjust it parameters on the fly, or the preflight predictions need to be very accurate.  I'm sure they are.

True, but again the instantaneous acceleration at any given time is based on thrust/total mass. It already knows its acceleration during the 100% thrust portion of the landing burn, from which instantaneous total mass can be derived, and it knows how fast it's burning propellant to subtract from that total mass. So it knows how fast total mass is/will be decreasing based on throttle level.

Again, you *could* derive remaining propellant mass from that, but it's unnecessary.

[CameronD]

It may be minor, but the total amount of mass remaining would affect its acceleration at a given thrust level.  So either it needs to be able to adjust it parameters on the fly, or the preflight predictions need to be very accurate.  I'm sure they are.

True, but again the instantaneous acceleration at any given time is based on thrust/total mass. It already knows its acceleration during the 100% thrust portion of the landing burn, from which instantaneous total mass can be derived, and it knows how fast it's burning propellant to subtract from that total mass. So it knows how fast total mass is/will be decreasing based on throttle level.

Again, you *could* derive remaining propellant mass from that, but it's unnecessary.

That all sounds great in theory, but does anyone know precisely what they'd be basing those mass figures on?  To know how fast it's burning propellant, does it use mass flowmeters - or something else?  If they use mighty-accurate mass flowmeters, can anyone hazard a guess at a make/model??

Reason I ask is that the results of calculations of this nature are only going to be as accurate as the accuracy/response time of their sensors (input data), so it'd be nice to know what that was to know whether or not it's possible to do it..

[Mark K]

That all sounds great in theory, but does anyone know precisely what they'd be basing those mass figures on?  To know how fast it's burning propellant, does it use mass flowmeters - or something else?  If they use mighty-accurate mass flowmeters, can anyone hazard a guess at a make/model??

The point being made is that the acceleration measurement, plus the known thrust of engine at the measured throttle allow the system to derive the mass of the booster at any point. Subtract the known non-propellent mass and you know the mass of the remaining fuel plus oxidiser. Whether the system cares or uses that is another issue.

[Gliderflyer]

Only three engines have plumbing for TEA/TEB reignition, and all three receive a reignition pulse regardless of which ones are lit.

Was this ever confirmed? I remember some speculation on it, but I don't recall if there was ever a source for it. In the Orbcomm 2 landing video there is only one green flash during the landing burn startup. If the igniter was firing in three engines, I would think the green flash would be visible in the other 2 engines as well as the center one.
https://www.youtube.com/watch?v=ANv5UfZsvZQ#t=2m35s

[Kabloona]

Only three engines have plumbing for TEA/TEB reignition, and all three receive a reignition pulse regardless of which ones are lit.

Was this ever confirmed? I remember some speculation on it, but I don't recall if there was ever a source for it. In the Orbcomm 2 landing video there is only one green flash during the landing burn startup. If the igniter was firing in three engines, I would think the green flash would be visible in the other 2 engines as well as the center one.
https://www.youtube.com/watch?v=ANv5UfZsvZQ#t=2m35s

That was stated as fact on reddit by someone who seemed to know (u/EchoLogic). And I haven't looked again at the OG2 landing video, but maybe all 3 engines were on line-of-sight to the camera, so the viewer would see only one flash.

As it happens, the OG2 stage is the one where we noticed the white TEA/TEB residue on the 2 outer engines and made the conjecture about how the TEA/TEB was plumbed. So it's virtually impossible that there *weren't* three flashes on OG2. Later I noticed what seemed to be authoritative confirmation of the plumbing configuration on reddit by EchoLogic. Hopefully he wasn't just repeating what he read here. 

[abaddon]

That was stated as fact on reddit by someone who seemed to know (u/EchoLogic). And I haven't looked again at the OG2 landing video, but maybe all 3 engines were on line-of-sight to the camera, so the viewer would see only one flash.

As it happens, the OG2 stage is the one where we noticed the white TEA/TEB residue on the 2 outer engines and made the conjecture about how the TEA/TEB was plumbed. So it's virtually impossible that there *weren't* three flashes on OG2. Later I noticed what seemed to be authoritative confirmation of the plumbing configuration on reddit by EchoLogic. Hopefully he wasn't just repeating what he read here. 

EchoLogic is the (an?) r/SpaceX moderator.  So any information should be considered secondary in nature (not saying it is bad).

ISTR there was a poster who seemed to be likely to be from SpaceX directly posting here around the time, who was indicating the same.

In general, we have a lot of sources pointing to the same thing, and nothing I am aware of contradicting it, so it seems like we should consider it as factual until otherwise proven different.

[launchwatcher]

That all sounds great in theory, but does anyone know precisely what they'd be basing those mass figures on?  To know how fast it's burning propellant, does it use mass flowmeters - or something else?  If they use mighty-accurate mass flowmeters, can anyone hazard a guess at a make/model??

I'd imagine that some combination of turbopump RPMs plus temperature and pressure readings at various points would give you a good idea about mass flow.

[John Alan]

I think you all are overthinking how this works... 

I will quote myself from a different thread yesterday...

It's likely a carefully crafted sequence developed thru many simulations then just played back and run thru...
There is likely code to adjust some things in real time... but mostly just to adjust the timeline playing out...

Example... compares timeline expected radar altitude to actual reading...
Jumps back or forward in timeline to make equal and continues to monitor...

Example... I'm falling faster then I should be at this point on timeline...
Applies small plus offset to thrust commands in timeline to attempt to compensate...

Point is with a good timeline sequence laid out and then played out... it works...
Just has enough wiggle room built in to adjust for the actual deck height before it gets there...

The above is very simplified and just my opinion... 

To add to the above... my opinion... 
SpaceX runs their simulations preflight and builds a launch (and landing) program sequence for the stage S1 main controller...
While it likely has built in correction calculation loops to make small adjustments to output values on the fly...
...It's very committed to assuming reality matches the assumed values made when programming the simulation...

It also (and this is key) logs in real time what it signaled out and what all input values were... and saves it.
Every stage landed and log file downloaded helps correct for modeling errors in the simulator...

In industry, this is how you automate high speed machinery...
You break it down to sequences and correction factors...
A) Sequences assume the laws of physics will not change and the machine is in working order...
B) Correction factors allow for things that can and will vary, to be allowed and adjusted for in real time...

In the end... the S1 controller just 'drives'... (like you drive a car)
It does not have to do heavy math... calculate all these things in real time and keep up...

Just my opinion on topic... 

[envy887]

Can someone with insight comment on how SpaceX is actually measuring or calculating thrust in flight?

[John Alan]

Can someone with insight comment on how SpaceX is actually measuring or calculating thrust in flight?

Likely off chamber pressure sensors and assuming the engine is in working order...
Would not surprise me if each engine controller is not reporting back a value on the controller network for all controllers to see...
If chamber pressure is X and altitude is Y then thrust equals Z...
Uses a lookup table... real common way of doing such a thing in the embedded controller world...
Turbine speed is indirectly an indication of power output... Chamber pressure is more direct...
BUT... that is all just my opinion... 

[Herb Schaltegger]

Can someone with insight comment on how SpaceX is actually measuring or calculating thrust in flight?

They don't need to. They can infer it from stage acceleration at any given time, which they can adjust within given parameters to modulate.

And that said, I think a lot of people posting should do some reading on closed-loop control system logic and design. You're making things way more complicated than they need to be from a "vehicle smarts" standpoint. 

[Kabloona]

Can someone with insight comment on how SpaceX is actually measuring or calculating thrust in flight?

Not claiming insight into SpaceX specifically, but other LV's I have worked on (Pegasus, Taurus, which admittedly are different beasts as solids) don't measure or calculate thrust per se. The IMU just measures vehicle acceleration, etc.

For liquid vehicles, the only difference is throttleability, and the thrust characteristics of the engines at given throttle settings are determined in static test firings. Once you have the engine thrust vs. throttle setting curves established through testing, you can program the autopilot accordingly. It can then command a given throttle setting expecting a certain thrust level, and the result is a given acceleration sensed by the IMU.

If fine control of acceleration (and therefore thrust) is required, eg in the case of F9 landing burn, the IMU can adjust throttle up or down as needed to obtain the desired acceleration, because of course the vehicle mass is always decreasing. So the IMU ends up "closing the loop" on desired acceleration by adjusting throttle accordingly. But it doesn't need to measure (or calculate) thrust per se, only acceleration.

[the_other_Doug]

Can someone with insight comment on how SpaceX is actually measuring or calculating thrust in flight?

Not claiming insight into SpaceX specifically, but other LV's I have worked on (Pegasus, Taurus, which admittedly are different beasts as solids) don't measure or calculate thrust per se. The IMU just measures vehicle acceleration, etc.

For liquid vehicles, the only difference is throttleability, and the thrust characteristics of the engines at given throttle settings are determined in static test firings. Once you have the engine thrust vs. throttle setting curves established through testing, you can program the autopilot accordingly. It can then command a given throttle setting expecting a certain thrust level, and the result is a given acceleration sensed by the IMU.

If fine control of acceleration (and therefore thrust) is required, eg in the case of F9 landing burn, the IMU can adjust throttle up or down as needed to obtain the desired acceleration, because of course the vehicle mass is always decreasing. So the IMU ends up "closing the loop" on desired acceleration by adjusting throttle accordingly. But it doesn't need to measure (or calculate) thrust per se, only acceleration.

I'm pretty certain that the Falcon stage 1 control and guidance system knows exactly where it should be, and how fast it should be going, at every point along its pre-defined trajectory.  It uses its controls (grid fins, engine gimbals and engine throttling) to maintain itself as close as possible against the pre-defined trajectory.

It doesn't need to know exact thrust or exact weight of the stage at any given time, it just knows what kinds of inputs to put into its controls to keep it on the right path.  If adding a little downrange velocity is called for, the system will do so until it is on profile.  It doesn't need to know it's fighting winds, what the wind speed is, etc.  It just needs to know where it's supposed to be and where it is.  It will then apply control inputs to get there.

This keeps the variables it needs to solve for down to a bare minimum, and based on the successful targeting we've seen, it seems to work pretty darned well...

[Robotbeat]

Indeed. There is some value to knowing those things to some extent, but a control loop can be pretty dumb and still work just fine getting you to where you need to go.

[envy887]

My question about thrust was more about the data sent home than anything used real-time for control.

Isn't engine thrust a metric examined in post mission analysis?

[the_other_Doug]

My question about thrust was more about the data sent home than anything used real-time for control.

Isn't engine thrust a metric examined in post mission analysis?

Not really.  "Thrust" is an ever-changing number and has as much to do with the air pressure as it does with the engine.  What is measured are things chamber pressure and acceleration.  From these kinds of measurements, one can compute thrust at a given moment.

In the old line about how, for example, the Saturn V would sit during engine start-up and not ask to be released until the engines had "come up to full thrust," the measurements that were actually used to determine the thrust level of the engines were chamber pressure measurements.  There was nothing that measured the actual thrust being produced, just what the chamber pressure was.  Per the physics of the engine design, at a given chamber pressure, the engine must be generating a given amount of thrust...

[Herb Schaltegger]

My question about thrust was more about the data sent home than anything used real-time for control.

Isn't engine thrust a metric examined in post mission analysis?

As I posted earlier, "thrust" can be determined by measuring acceleration of the vehicle in response to control inputs. During post-mission analysis, the mass of the stage at any given time can be determined to a reasonable degree of certainty by integrating along the flight profile. SpaceX knows the empty mass of the vehicle; they know the mass of the payload; the know the mass of propellants loaded; they know the mass flow-rate curves for the engine along the throttle points for the M1D engine.  From those starting values, they know all they need to.

EDIT: Yeah, what the_other_Doug said.

[MP99]

My question about thrust was more about the data sent home than anything used real-time for control.

Isn't engine thrust a metric examined in post mission analysis?

As I posted earlier, "thrust" can be determined by measuring acceleration of the vehicle in response to control inputs. During post-mission analysis, the mass of the stage at any given time can be determined to a reasonable degree of certainty by integrating along the flight profile. SpaceX knows the empty mass of the vehicle; they know the mass of the payload; the know the mass of propellants loaded; they know the mass flow-rate curves for the engine along the throttle points for the M1D engine.  From those starting values, they know all they need to.

EDIT: Yeah, what the_other_Doug said.

Plus, also deriving air resistance from altitude and descent rate?

Edit: also attack angle and grid fin settings?

Cheers, Martin

[MP99]

But the flight computer probably just flies the mission assuming it has enough propellant to land safely. And you wouldn't want it to know if it didn't. 

I would not expect MECO to occur until S1 has met its performance targets.

If there is insufficient prop left for a landing, I'm not sure if it would go ahead with a doomed landing attempt, or switch to expendable mode and abandon recovery?

One last tweak - if prop is tighter than expected, perhaps a single engine landing would be switched to three engines? (But not sure if single engine landings will ever be planned again.)

Cheers, Martin

Sent from my Nexus 6 using Tapatalk

[Saabstory88]

(But not sure if single engine landings will ever be planned again.)

Wont that depend on the state #24? If the first thing they did upon reaching the droneship was to pull off the engine covers, it may be a little more harsh of a technique than is good for reusability.

[gadgetmind]

Isn't it the short re-entry burn that's harsh rather than the three engine landing burn?

[llanitedave]

Indeed. There is some value to knowing those things to some extent, but a control loop can be pretty dumb and still work just fine getting you to where you need to go.

I think the question wasn't so much about staying on track, but in cases where fuel reserves are at a bare minimum, knowing whether or not there will be enough fuel remaining to complete the landing.

[Herb Schaltegger]

Indeed. There is some value to knowing those things to some extent, but a control loop can be pretty dumb and still work just fine getting you to where you need to go.

I think the question wasn't so much about staying on track, but in cases where fuel reserves are at a bare minimum, knowing whether or not there will be enough fuel remaining to complete the landing.

Why does the stage guidance system need to know that? Either there is, and the landing presumably succeeds, or there isn't, and it fails one way or another. Telemetry lets the ground later figure out why it failed (e.g., due to lack of fuel) and they adjust things to hopefully improve odds of success the next time. Or decide landing isn't feasible with that payload and they don't even try further for a payload/mission of that type in the future.

[matthewkantar]

Indeed. There is some value to knowing those things to some extent, but a control loop can be pretty dumb and still work just fine getting you to where you need to go.

I think the question wasn't so much about staying on track, but in cases where fuel reserves are at a bare minimum, knowing whether or not there will be enough fuel remaining to complete the landing.

If some set of data,such as remaining fuel, is moot, or would not in any way change the control inputs, I don't think it would be included in the landing program. Finding out the stage is running out of propellants is a bummer, but unless there is something that can be done with that information it is useless. No filling stations on the way down.

There is one action the stage could take based on such data, but to me it seems unlikely it would switch to progressively more aggressive burns to save on propellant.

Matthew

[llanitedave]

Indeed. There is some value to knowing those things to some extent, but a control loop can be pretty dumb and still work just fine getting you to where you need to go.

I think the question wasn't so much about staying on track, but in cases where fuel reserves are at a bare minimum, knowing whether or not there will be enough fuel remaining to complete the landing.

Why does the stage guidance system need to know that? Either there is, and the landing presumably succeeds, or there isn't, and it fails one way or another. Telemetry lets the ground later figure out why it failed (e.g., due to lack of fuel) and they adjust things to hopefully improve odds of success the next time. Or decide landing isn't feasible with that payload and they don't even try further for a payload/mission of that type in the future.

You could save yourself the expense of patching holes in the barge.

[Herb Schaltegger]

Indeed. There is some value to knowing those things to some extent, but a control loop can be pretty dumb and still work just fine getting you to where you need to go.

I think the question wasn't so much about staying on track, but in cases where fuel reserves are at a bare minimum, knowing whether or not there will be enough fuel remaining to complete the landing.

Why does the stage guidance system need to know that? Either there is, and the landing presumably succeeds, or there isn't, and it fails one way or another. Telemetry lets the ground later figure out why it failed (e.g., due to lack of fuel) and they adjust things to hopefully improve odds of success the next time. Or decide landing isn't feasible with that payload and they don't even try further for a payload/mission of that type in the future.

You could save yourself the expense of patching holes in the barge.

They've hit the barge a few times, and holing the deck has happened once. Pretty obviously, at some point once they've reached the point where they think they've learned all they can, SpaceX won't continue to rocket-punch their barge if they don't think recovery is going to work. They'll go into some semblance of an "operational mode" - recovery will either be routine and expected to succeed most of the time, or it won't and they'll stop bothering for at least some heavy/high-energy payloads.

[malu5531]

And the computer doesn't need to know how much propellant remains. It's probably pre-programmed to do an entry burn of specific time duration, and the landing burn is probably pre-programmed to start at a certain altitude/velocity, based on pre-launch Monte Carlo simulations that give them a good idea of how much propellant will be used during the burns.

My impression is that they're not pre-programming control timings such as when to start engine burns, rather they tweak constraint parameters for an onboard algorithm for real-time optimisation of landing trajectory. I.e., the rocket decides in flight when to start / stop or throttle engines.

Attached is a nice paper by Lars Blackmore, the person responsible for F9 EDL at SpaceX. It describes an onboard "Powered Descent Guidance" algorithm, which optimise landing trajectory for minimal landing error and fuel use, with given limits set on throttle, speed, position, etc.

Continuous onboard optimization during EDL is needed since initial conditions at staging are not known beforehand AND conditions change during flight such as wind gusts, high altitude jet-streams, engine performance, control accuracy, etc.

I believe SpaceX is currently in a phase of iteratively adjusting constraints. If SpaceX is very aggressive in finding the constraint envelope, it may be that there's quite a bit more fuel left in the rocket after landing. I.e. the warning that they might crash stage may not be due to "landing on fumes", but rather due to constraint experiments in order to expand the envelope and evaluate the overall performance.

[Jcc]

You would think propellant load could be calculated based on knowing the thrust of the engines and the deceleration it produces. More deceleration for the same thrust means less mass of the stage, subtract the dry mass and you get the fuel. Maybe that can't be measured with enough precision, if so, upgrade the sensors.

[Bob Shaw]

You would think propellant load could be calculated based on knowing the thrust of the engines and the deceleration it produces. More deceleration for the same thrust means less mass of the stage, subtract the dry mass and you get the fuel. Maybe that can't be measured with enough precision, if so, upgrade the sensors.

No. Rocket fuel is notoriously difficult to measure in-flight, especially when in zero-G. It also sloshes, gurgles and bounces around when under thrust, changes density, and will generally seek the 'lowest' point at the end of the flight which may well be different from the 'lowest' point during launch, as the stage may be canted into the wind. You have to have reserves; the saving grace for SpaceX is that the thrust/weight ratio on the F9 is absurd when the tanks are almost empty and there's no second stage/payload on top, so they can get away with murder, eg the infamous hoverslam.

[philw1776]

And the computer doesn't need to know how much propellant remains. It's probably pre-programmed to do an entry burn of specific time duration, and the landing burn is probably pre-programmed to start at a certain altitude/velocity, based on pre-launch Monte Carlo simulations that give them a good idea of how much propellant will be used during the burns.

My impression is that they're not pre-programming control timings such as when to start engine burns, rather they tweak constraint parameters for an onboard algorithm for real-time optimisation of landing trajectory. I.e., the rocket decides in flight when to start / stop or throttle engines.

Attached is a nice paper by Lars Blackmore, the person responsible for F9 EDL at SpaceX. It describes an onboard "Powered Descent Guidance" algorithm, which optimise landing trajectory for minimal landing error and fuel use, with given limits set on throttle, speed, position, etc.

Continuous onboard optimization during EDL is needed since initial conditions at staging are not known beforehand AND conditions change during flight such as wind gusts, high altitude jet-streams, engine performance, control accuracy, etc.

I believe SpaceX is currently in a phase of iteratively adjusting constraints. If SpaceX is very aggressive in finding the constraint envelope, it may be that there's quite a bit more fuel left in the rocket after landing. I.e. the warning that they might crash stage may not be due to "landing on fumes", but rather due to constraint experiments in order to expand the envelope and evaluate the overall performance.

Looking at Dr Blackmore's impressive list of publications I notice a couple on balloons in Titan's atmosphere.
A future FH launched Dragon 2 destination in the 20s?

[Zed_Noir]

You would think propellant load could be calculated based on knowing the thrust of the engines and the deceleration it produces. More deceleration for the same thrust means less mass of the stage, subtract the dry mass and you get the fuel. Maybe that can't be measured with enough precision, if so, upgrade the sensors.

No. Rocket fuel is notoriously difficult to measure in-flight, especially when in zero-G. It also sloshes, gurgles and bounces around when under thrust, changes density, and will generally seek the 'lowest' point at the end of the flight which may well be different from the 'lowest' point during launch, as the stage may be canted into the wind. You have to have reserves; the saving grace for SpaceX is that the thrust/weight ratio on the F9 is absurd when the tanks are almost empty and there's no second stage/payload on top, so they can get away with murder, eg the infamous hoverslam.

Recent videos from propellant tank interiors show that visual markings inside the tank can be use for gauging the volume of remaining propellants. At least while the vehicle is under acceleration.

[Bob Shaw]

Oh, certainly - things will get smarter. But not all at once. The big thing SpaceX has going for it is, simply, that empty tankage is light. Everything else pales into invisibility compared to that.

[hkultala]

I just want to point out that the Falcon 9 first stage carries approximately 286,400 kg in LOX and 123,100 in RP-1, approximate total of 409,500 kg.

Source: http://spaceflight101.com/spacerockets/falcon-9-ft/

A savings of 2,000 kg of propellants is about 0.5% of the Falcon 9's total fuel capacity.

So more aggressive landings would only help in the most marginal of return scenarios, where the Falcon 9 has already depleted almost all of its available fuel and oxidizer.

2000 kg of fuel saved in the landing phase does not mean only either
1) Having 2000 kg more fuel to spent in the ascent and boostback phase (0.5 % here is not much)
2) Having 2000 kg less weight to first accelerate to staging velocity(~1.5%)
3) Having 2000 kg less weight to boost back. (>5% for this)

It means the combination of  ALL of these.

And because rocket equation is a highly nonlinear equation, the benefits of these together is much higher than the sum of either done separately.

[speedevil]

And because rocket equation is a highly nonlinear equation, the benefits of these together is much higher than the sum of either done separately.

It's only highly nonlinear if you are at the portion where the mass is changing rapidly.
At the state where the tank is nearly empty, and the fuel is no longer overwhelmingly dominant, it's gets close to linear.

[AncientU]

There seem to be a few factors that optimize fuel use (and thus delivered payload) by conducting a three engine landing burn:
1.  Extra fuel saved for landing is fuel that could have been used for boost during the most productive final few seconds of the burn -- 5g burn with almost empty tankage.  This is the fuel you most want to conserve -- it provides much more than 0.5% of acceleration (I think).
2.  Landing with minimal fuel also improves the ballistic coefficient, allowing the atmosphere to slow the booster more instead of the landing burn doing that deceleration, and
3.  Waiting till last seconds allows more deceleration in the thickest portions of the atmosphere. 
These last two each reduce the amount of fuel needed for landing.

[John Alan]

I have sometimes wondered the trades... of adding some sort of speed brake to be deployed when it goes subsonic...
A)drogue chutes (then cut all loose when speed <~20 knots)
B)form fitted 'speed brakes' that deploy in between the grid fins (likely NFG on a mass trade study)

Obviously these all need thick air to work in...
...and they have to sort out the 3 engine super slam first, to maximize that free resource to work with it... 

[MikeAtkinson]

There seem to be a few factors that optimize fuel use (and thus delivered payload) by conducting a three engine landing burn:
1.  Extra fuel saved for landing is fuel that could have been used for boost during the most productive final few seconds of the burn -- 5g burn with almost empty tankage.  This is the fuel you most want to conserve -- it provides much more than 0.5% of acceleration (I think).
2.  Landing with minimal fuel also improves the ballistic coefficient, allowing the atmosphere to slow the booster more instead of the landing burn doing that deceleration, and
3.  Waiting till last seconds allows more deceleration in the thickest portions of the atmosphere. 
These last two each reduce the amount of fuel needed for landing.

Agree, but there is likely to be a larger reserve of un-burned fuel on landing to ensure that none of the three engines run out of fuel.

[deruch]

There seem to be a few factors that optimize fuel use (and thus delivered payload) by conducting a three engine landing burn:
1.  Extra fuel saved for landing is fuel that could have been used for boost during the most productive final few seconds of the burn -- 5g burn with almost empty tankage.  This is the fuel you most want to conserve -- it provides much more than 0.5% of acceleration (I think).
2.  Landing with minimal fuel also improves the ballistic coefficient, allowing the atmosphere to slow the booster more instead of the landing burn doing that deceleration, and
3.  Waiting till last seconds allows more deceleration in the thickest portions of the atmosphere. 
These last two each reduce the amount of fuel needed for landing.

Though, it may not be a 1:1 transfer from savings on the landing to extra boost, as increasing the boost may also require that they add some to the entry burn in order to remain survivable/maintain reusable condition, etc.  But, any addition can be worth it.  The point is that optimizing the landing burn as much as possible either expands the envelope for what is considered a recoverable mission, allows them to improve the condition of recovered boosters by protecting them more on reentry, or improves the performance on what orbit a given payload can be delivered to.