* https://en.wikipedia.org/wiki/LGM-25C_Titan_II#Mishaps
* https://en.wikipedia.org/wiki/1965_Searcy_missile_silo_fire
* https://en.wikipedia.org/wiki/1980_Damascus_Titan_missile_ex...
I do see military vehicles traveling to and from the one that I am close to semi regularly, perhaps a month or so on average.
As far as fire response, they likely have their equipment for that at the control centers as well.
Sounds like a serious weak point
Digging into one of the cables is going to get you a prompt and unpleasant visit from base security.
https://en.m.wikipedia.org/wiki/LGM-30_Minuteman#/media/File...
There are alternate ways to launch the missiles (e.g. radio from a plane) in the event of an attack.
Actual change in procedure? Or just extra cheek-clinching?
Around 2000 land owners are affected, and anecdoctally it seems to add a few days to a 811 response (~a week instead of 2-3 days).
This sounds like it couldn't possibly work (surely all the little errors compound?) but apparently it's how Apollo navigated
https://wehackthemoon.com/tech/inertial-measurement-unit-mec...
But honestly only about half as crazy as GPS would sound if you tried to put it in similar terms. And that's before considering the signal itself is way below the noise floor.
As much as I love these technical writeups, I wish more people who understand the slightly expanded bigger picture/ implications of the missile technologies would write what they know before the generation goes extinct.
There's so many dots that are easily connected from articles like these... but I suspect some level of classification prevents those in the know from being able to publish.
The bulletin of atomic scientists is the only non-fringe source that ever comes close. (Their article on the new generation of warhead fuses is a great resource for those wanting to go down the rabbit hole. Even it now seems to have been scrubbed from their site. [0])
[0] https://web.archive.org/web/20170307194641/https://thebullet...
Eta working bulletin link: [1]https://thebulletin.org/2017/03/how-us-nuclear-force-moderni...
"To target a Minuteman I missile, the missile had to be physically rotated in the silo to be aligned with the target, an angle called the launch azimuth. This angle had to be extremely precise, since even a tiny angle error will be greatly magnified over the missile's journey. " ... "The guidance platform was completely redesigned for Minuteman II and III, eliminating the time-consuming alignment that Minuteman I required. The new platform had an alignment block with rotating mirrors. Instead of rotating the missile, the autocollimator remained fixed in the East position and the mirror (and thus the stable platform) was rotated to the desired launch azimuth. "
Most of the tech for the Minuteman I was developed in the mid-1950s.
With that level of processing, would you rather solve a 2d problem by precisely orienting the missile before launch? Or a 3d one by requiring it to orient during flight?
Keep in mind: any equipment to self-orient in-flight also needs to be carried on the missile itself, while being tolerant of launch, acceleration, and reentry forces.
Any precision machinery at the launch site has no such requirements.
Had never heard of gyrocompasses before. I worked on a small robot in the past and remember having to calibrate the magnetic compass, which was not very accurate (similar to smartphone compasses). I never thought about how they’d get super precise headings for ICBMs.
The Encyclopedia Britannica article on gyrocompasses is really good. Here it explains why you can’t use a gyrocompass on a vehicle on fast aircraft (and I guess small robots that are jostled around a lot):
> A major contribution by Schuler was the discovery that, when the period of oscillation is 2π√(Earth radius/gravity), the heading precession of the gyroscope spin-axis due to acceleration is exactly the rate of change of the angle between the apparent and true meridians seen on a moving vehicle. The gyrocompass will then read true north at all times if its indicating reference is offset by the angle between these two meridians. The angle, at ship speeds, is a direct function of the north-south speed and is easily set into the system. The need for accurate speed measurement for this offset is the main reason why a gyrocompass is not practical for use in aircraft.
https://www.britannica.com/technology/gyrocompass
Love this article!
I really wonder what the failure rate would be if they were all actually launched today. And I mean failure, from not lifting off, to failure in flight, to misguided warheads etc.
„Aligning the missile was a tedious process that used the North Star*t* to determine North.„
Minuteman D-17b: The Desktop Computer Was Born in an ICBM
https://www.youtube.com/watch?v=MJPnZzZtswc (Alexander the ok)
Check him out and give him slack for his vocal fry. It's mildly annoying but the content is just so good.
Obtaining position & veocity: I think it even more interesting when one compares the difficulties of getting these fundamental navigation data in an aerial, ground and undersea platforms.
https://aviation.stackexchange.com/questions/49961/why-are-a...
Minuteman III used a smarter targeting algorithm that only needed 70 words of data per target, so the missile could support something like 8 targets at once, selected by a knob on the launch console. (The launch officers didn't know what the targets were; they were just told to use target #3 for example.) The targeting data was read off punched tape for Minuteman II and a magnetic tape cartridge for Minuteman III.
Simplified launch orders via the football, etc.
One group scenario might have been silo coordinates for an offensive first-strike. One group city coordinates for launch on warning strategic counter-strike, etc.
Each missile got a target from each scenario list programmed into a 'memory slot' with some overlap.
The organization/ optimization is mind-boggling.
But few understand that this is WHY the wargames and strikes had to be pre-planned ahead of time. It wasn't political hubris, but a technical requirement due to memory allocation.
The future so bright, I gotta wear shades.
The scary thing is that, when left alone for a long time, and these rockets have been, "the plates" keeping the chemicals from meeting each other ahead of schedule, corrodes, just a tiny bit at a time. each time raising the possibility of premature ejaculation just a tiny little fraction.
Which to his point would be even more scary, but just isn't the actual real world risk with the way the things were designed.
Plus hypergolics are usually toxic on their own, even without mixing and/or booming, in a quieter, more-deadly-to-technicians way.
Spills and defueling and meeting well-intentioned but bad safety guidelines that require abundant fiddling were the real source of danger. More fiddling == bad.
Iirc, the fuels/oxidizers/reagents/ whatever-liquids mainly behaved like aluminum oxidizing, such that reaction with the tanks components actually created an increased buffer layer of oxidation/ protection.
Tank corrosion wasn't high on the list of risks after it was figured out on a per-chemical basis.
I think it's one of the aspects covered fairly well in (the great, often posted) Ignition! [0]
> O SON OF SPIRIT! The best beloved of all things in My sight is Justice; turn not away therefrom if thou desirest Me, and neglect it not that I may confide in thee. By its aid thou shalt see with thine own eyes and not through the eyes of others, and shalt know of thine own knowledge and not through the knowledge of thy neighbor. Ponder this in thy heart; how it behooveth thee to be. Verily justice is My gift to thee and the sign of My loving-kindness. Set it then before thine eyes. > > ~ Baha'i Teaching
> Be united, O kings of the earth, for thereby will the tempest of discord be stilled amongst you, and your peoples find rest, if ye be of them that comprehend. Should any one among you take up arms against another, rise ye all against him, for this is naught but manifest justice.
> (“Gleanings from the Writings of Bahá’u’lláh”, pp. 253-254)
Examples of that would include ego, greed, petty revenge, etc...
But I guess the way I said it did come across that way, so yeah, my bad
I'm just saying that if we could evolve past such petty ego-based sentiments in the world, then wouldn't such pressure to develop weaponry in such massive amounts, and hence we could focus on actually making a functioning society
GPS can be jammed (see Russia-UKraine war), so inertial systems are still very important for rockets, for example some HIMARS rockets start with GPS and then rely only on inertial while getting close to target.
Sidewinders are another example. Both developed at China Lake.
I remember taping two together back to back and integrating acceleration across them. That's when I learned Kalman filters. It was accurate enough so I could throw it across my desk and measure the desk length :)
https://en.wikipedia.org/wiki/Kalman_filter#History
https://github.com/chrislgarry/Apollo-11/blob/master/Luminar...
I think if I kept messing with it, it'd get a lot better, but I sorta lost interest. This was more of a fun weekend toy.
I think all phones have them, and they might be reachable through chrome/safari. And it is kinda fun to play with, but you'll probably hit sampling rate errors pretty quick. you gotta guess the shape of the curve between datapoints.
Besides inertial navigation, they had a transponder that would echo back a continuous pseudorandom bit stream, and the delay gave a precise measurement of distance.
Or you can add an external correcting factor, such the Trident's astronav system that takes star-shots to recalibrate the INS.
But it's based on the same idea, only getting position as a derivative of velocity. (And some borderline-magic statistics applied.)
And that's before taking into account the absurdity of how low power the broadcast signal is.
Maybe this has changed or is ineffective now that smartphone/quadcopter IMUs have caught up.
They did not caught up. There are two kind of IMUs: one where you have to account for the rotation of the Earth during signal processing and one where there is no point because it will be lost in the noise anyway. The smartphone/quadcoptee IMUs are the second kind. The first kind is still export controlled.
Random errors (i.e. noise) cancel out in the long run thanks to integration. You're then only left with systematic offset errors which can presumably be calibrated out to a large extent.
It's also mentioned that the computer uses one of the first integrated circuits for miniaturization. Do you know if this can be definitely traced to advances in industrial/consumer products? It's a common trope that military research trickles down - so it's a "good" thing. It's not clear if this actually happens or if progress would have been made eventually without the need for these machines.
I've been doing a lot of research on the impact of Minuteman and Apollo on the IC industry (which led to the current post). The Air Force likes to take credit for the IC industry, as does NASA, but the actual influence is debatable. My take is that both projects had a large impact on the IC industry, more from Minuteman. However, even in the absence of both projects, there was a lot of interest and demand for ICs. If I had to take a quantitative guess, I'd say that those projects advanced ICs by maybe a year, but the basic trajectory would have remained the same.
Chromates are effective corrosion inhibitors for aluminum alloys and some other metals. Here's a brief article about how they work with aluminum:
"Inhibition of Aluminum Alloy Corrosion by Chromates"
https://www.electrochem.org/dl/interface/wtr/wtr01/IF12-01-P...
When the Wikipedia entry's "Safety" section says that sodium chromate is corrosive, in context it means "destructive to human tissue by contact." That is, like sodium hydroxide (lye) and many other chemicals, in concentrated form it can destroy skin and eyes.
Or maybe early RADAR systems of 1942 and later?
(been reading about WW2 Pacific naval/air wars, and I am curious about these new-at-the-time technologies)
The machinery is ironically beautiful to look at!
How did this compare to its approximate contemporary in the USSR ?
Note that you need to start with the lowest bit with a serial computer, which explains why x86 is little-endian. It goes back to the Datapoint 2200, a desktop computer made from TTL chips and running serially. The Intel 8008 processor was a copy of the Datapoint 2200 (as was the Texas Instruments TMX 1795). Although the 8008 was parallel, it copied the little-endian architecture of the Datapoint 2200.
(As labeled in https://static.righto.com/images/minuteman-mmiii/guidance-la...)
> Also note the window in the side of the missile to allow the light beam from the autocollimator to reflect off the guidance platform for alignment.
https://www.spaceforce.mil/News/Article-Display/Article/3796...
> I really wonder what the failure rate would be if they were all actually launched today
I hope we will never find out. But certainly there would be many duds. But this is calculated into the effectiveness of the system by simply having more missiles. That is how it achieves its goal of dettering a would be attacker. (Not even talking about how there are two other totaly separate legs of the nuclear triad with dissimilar personel and technical solutions.)
Jeff! Did you remember to take out the warheads? Jeff?!
https://www.youtube.com/watch?v=zUg7x1zo7D0
They can test the missiles and reentry vehicles, everything except the nuclear warhead. The closest those come to a test is a supercomputer simulation, since those tests are forbidden by treaty.
Minuteman III has an excellent but not perfect [0] failure rate. Some other older systems, like the UK's submarine launched Trident missiles... not so much.
[0] https://www.airandspaceforces.com/icbm-test-failure-nuclear-...
https://en.wikipedia.org/wiki/UGM-133_Trident_II
However, the warheads on the UK missiles are designed and manufactured by the UK.
Obviously you can't factor in unknown problems but that's what drills and test flights are for.
Opinion: I don't think the US would if Russia or China didn't first. China likely won't for the same reason the US doesn't need to: they have super-computers and the sims line up with the data from prior tests. Russia might however if only to saber rattle, although they likely don't need to either. Russia however is likely not in any hurry to have a test failure right now. So while testing could resume, I wouldn't put money on it.
Note too, that mil spec silicon is different in that it is resistant to CMOS latch-up, redundant CRC protected self-correcting consensus register ops, and large gate sizes less sensitive to Gamma radiation.
It was an interesting time, and a few people still think living under the Sword of Damocles builds character. =3
Each stage needs to function in the presence of nearby nuclear detonations, resulting from both adversary and friendly weapons.
These detonations are expected to cause severe shock, thermal, radiation, and electromagnetic transients.
In the case of the most important targets, it is guaranteed that numerous detonations near the target, from ABM systems and friendly impacts, will occur, and these systems have been engineered and are expected to perform reliably under such conditions.
I don’t understand why it would be “political hubris”.
Proper targeting is hard work. You need to map your enemy territory to make optimal choices. Not just in a geographical “what are the coordinates” sense, but also in a “what are the important nodes to get the coordinates for” sense. At the same time your enemy doesn’t want to be mapped and resists your efforts.
Doing this properly takes time. On the order of months. But once you are under attack you don’t have that time. So you have to select your targets ahead of time.
It is not because the missiles have limited memory. If they would have needed more memory they would have added more memory. It is because the President doesn’t have time once under attack to name each enemy railway depot one by one and decide which ones are important, and which ones are better left unharmed. Instead what they have is a menu of options. Something like option 1 destroy all red military ports, military airports and military bases; option 2 destroy major military installations plus main industrial centers; option 3 destroy main population centers.
Thinking that the memory allocation is why it is the way it is is super tech centered and quite frankly putting the cart before the horse.
Multiple search engines still pointed to the stale link.
When there's as much secrecy by obscurity as I've seen re: the topic, my tinfoil hat always assumes the worst.
The minuteman missiles are solid fueled. There are no liquids and no hypergolics involved in the stages which loft it towards the enemy. Structurally it is more similar to a candle than a fuel tank. There are no spills or defueling with this system. This is a fact. In this system you won’t find a separate oxidiser/fuel. The two components are mixed together and they form a kind of rubber like cylinder with a hole in the middle. The hole is shaped appropriately so the rocket engine burns at the right rates.
There are hypergolic fuels at the very end of the rocket in the payload. They are used for deorbiting and to control the return vehicles. But it is a much smaller part of the whole missile. (Both by mass, and by encapsulated energy.)
Should have prefaced that response with a big IFF/WHEN old liquid rockets.
"Optical navigation subsystem sightings of celestial bodies and landmarks on the Moon and Earth are used by the computer subsystem to determine the spacecraft's position and velocity and to establish proper alignment of the stable platform."
And Wikipedia (https://en.m.wikipedia.org/wiki/Apollo_PGNCS):
"The CM optical unit had a precision sextant (SXT) fixed to the IMU frame that could measure angles between stars and Earth or Moon landmarks or the horizon. It had two lines of sight, 28× magnification and a 1.8° field of view. The optical unit also included a low-magnification wide field of view (60°) scanning telescope (SCT) for star sightings. The optical unit could be used to determine CM position and orientation in space."
That would be nice, but it seems that would make people no longer human. Happily following a petty ego-based Dear Leader's orders into warfare seems to be the norm for much of the human population, looking at history, so this tendency appears to be deeply-rooted into the human psyche.
as the quote above says:
thou shalt see with thine own eyes and not through the eyes of others, and shalt know of thine own knowledge and not through the knowledge of thy neighbor
it all comes down to better education, specifically moral education, and learning to critically evaluate any information we get.
if people blindly follow a leader, then this is exactly what they are missing.
Everything that goes wrong later can be traced to that inequality. Things that go right can usually be traced back to it as well. Any faith or philosophy that doesn't begin with an understanding of that aspect of human nature is a waste of time at best. Thank you for coming to my TED talk.
https://en.m.wikipedia.org/wiki/Gombe_Chimpanzee_War
Cue the opening scene of 2001 A Space Odyssey.
It's labeled in another image towards the end:
https://en.wikipedia.org/wiki/2007_United_States_Air_Force_n...
If that was an accident, that means they didn't properly know which warheads are where. That's.... upsetting.
I will definitely be taking a look at "Minuteman: A Technical History". Books dealing at least in part with the history of IMUs are few and far between.
Then they got a lot more accurate than .1%
The bit serial CPU in `bit.vhd` is actually customizable, you can make a 32-bit, a 14-bit, or a 27-bit CPU if you want from that VHDL quite easily.
https://thebulletin.org/2017/03/how-us-nuclear-force-moderni...
So even though Russia withdrew from START II almost immediately, the US continued to unilaterally remove the MIRV capability from its ICBM fleet and stick to single warhead Minuteman IIIs.
true, but perhaps we do have capacity to learn to think well enough to pick leaders who are better people than the most of us overall, e.g. learning to not mistake charisma for pure intention or intellectual insight
with some self awareness, it takes less effort overall to know who is better than us or our friends in terms of ethics and intellect, than it is to be right on every issue
and thats probably more important than the specifics of each issue, especially since the lay person doesn't have time to delve into the finer details of everything
now if we can get in a cycle of electing leaders like that then that would set us on a trajectory of constant societal growth, but to get to that our system probably also needs to change so as to actually present people with realistic options
that is probably a lot harder, and the two are certainly locked into a feedback loop
and I don't have answers on that one, but I believe if its possible to get to where we are from pond scum, then perhaps we can get to that point in the future as well
The random component I assume to be gaussian (thermal noise, for example) and therefore symmetrical around the real value. It's obvious we can remove this type of noise through averaging (of which the core operation is integration).
The non-random component I assume to be a skew that can be calibrated out.
With these two assumptions in mind you can see that yes, it's indeed a random walk, but a very well behaved one.
As far as calibrating out the skew, of course you can do that to some extent, but it's not a magic bullet. The Minuteman periodically measures skew and even applies equations for the change in skew with acceleration. The problem is that skew is not constant; it changes with time, changes with temperature, changes with position, and changes randomly, so you can't just calibrate it out. That's one reason why missiles use strategic-grade IMUs for a million dollars rather than a commercial-grade gyro for $10: you're getting drift of .0001º/hour instead of .1º/second.
Short-term random effects (as in, the part of the gyro's random walk error significantly higher in frequency than the inverse of the integration period) will get cancelled out by integration, assuming they're Gaussian.
Long-term random effects (mainly from time and temperature like you mentioned) will instead tend accumulate with integration aka worsening with time.
P.S. great fan of your many ventures into retro tech, keep them coming!
This is how the Russians have been throwing double digit percentages of launches off course.
F.ex. the 90s Tomahawk used terrain contour matching to orient itself
For more details see https://apps.dtic.mil/sti/tr/pdf/ADA315439.pdf (US translation of a mid-90s Chinese survey of the guidance space, but it covers the material and is publicly available)
Afaik, most modern systems use infrared target matching for final course correction. (Initially developed to allow anti-shipping missiles to autonomously prioritize targets, but now advanced enough to use in land scenarios as well)
It wouldn't make much sense to me, as most ATACMS warheads are area based, not point target based, so they wouldn't expect to aim at a single target. Also these systems are relatively cheap compared to things that DO have such guidance
I would guess you must solve the 3D problem at least to some degree.
Vs something like a Polaris SLBM that has a much more variable guidance problem
It'd be curious to see how early ICBM and SLBM guidance systems differ.
Submarines can determine their position accurately enough, but their orientation data can be improved upon using stars.
The MIRV bus takes in the angular fix just before it starts giving the warheads their individual nudges.
There's a lot of math that phrase is hiding. It's not a magic black box. It just often seems that way.
Your point stands, though... the way it does work is pretty much indistinguishable from magic, from a 1970s perspective. Those guys were wizards.
It's all summing dx/dt + dy/dt + dz/dt, for i paths between satellites and ground stations (or more receivers for differential or rtk or vrs style). [2]
Which reduces most of the time to summing DELTA-Xi + DELTA-Yi + DELTA-Zi + delta-t(timeerrors). For i paths between each sat and ground receiver.
Which you should recognize the transformation if you've ever taken calculus. Even if you don't integrate every time you get a fix.
Part of what I describe as math 'magic' is that you can cancel out most of the unknowns and most of the unsolved calculus if you add a second fixed receiver.
Google and Apple location services 'cheat' and do this via subbing a nearby wifi MAC with known coordinates, which for them is good-enough. But augmented gps from FAA or DOT or coastguard etc work the same way, but with real gps receivers on the ground in realtime. Obviously without having to substitute anything.
Either way- the extra known variable greatly simplifies the math via canceling-out terms.
Plus there are both closed and open form solutions developed since initial GPS deployment that allow solving without direct integration.
Chapter 12 of [0] Surveying gets into the math, including transformations, if you want to see the math details.
Or [1] GPS by van Sickle for a good overview of the various methods/ technologies. (Also survey-centric).
[0]https://books.google.com/books/about/Surveying_theory_and_pr...
[1]https://books.google.com/books?id=J0fLBQAAQBAJ&pg=PA63&sourc...
[2] despite wgs84 and lat/lon being associated as default 'GPS coordinates', the 'raw' gps system data is xyz Cartesian in feet, then transformed to lat lon or whatever else.
The delivery vehicle and the reentry/payload vehicle have entirely different life cycles and deployment concerns.
https://en.wikipedia.org/wiki/Fractional_Orbital_Bombardment...
It's main advantage being that it could attack the US from the South (or presumably any other direction).
I think anything developed before ~2005 that wasn't explicitly anti-ship doesn't have terminal guidance. Cruise missiles maybe/not.
Things after (e.g. SBD2 / StormBreaker) started to, because components were finally cheap and mature enough during the development cycle.
https://m.youtube.com/watch?v=h5KejRbD5s0&t=34s
That's a lot more dynamic of a launch orientation. Which way is it rotated? Is it inclined off vertical?
https://en.wikipedia.org/wiki/Wafer_backgrinding
Best regards =3
Edit: this article seems to support my view that they don't start with velocities: https://insidegnss.com/wp-content/uploads/2018/01/marapr15-S... "How does a GNSS receiver estimate velocity?"
The doppler mostly comes into play with the small delta-t errors, but again, more math magic cancels most of it out in most cases, or what remains is negligible.
It's more of a signals/sync thing that gets into antenna design and (to simplify) getting all signal cycles from the various satellites working within a single aligned synced cycle, if that makes sense.
One reason the old gps units needed a long time to get an initial fix was waiting to download the broadcast, in ~bits/sec. This can now be downloaded much quicker via internet or other methods.
And there are dozens of other similar shortcuts possible depending on receiver capabilities/ connectivity/ observervation methods.
Which is to say that there's no one 'right' way to get a fix- and the 'most' correct original design was the ~hour long broadcast download. And no one does that anymore.
But just about every method (I'm aware of) is derived one way or another from the general eqns I gave above.
(But my exposure is almost entirely geodesy, engineering, and surveying, and my military (encrypted) knowledge comes from my PLS instructor being ex army intelligence, not hands on. But which is also why I am at least aware of so much of the missile tie-in issues.)
And there are signals processing and CS tricks also, which I only barely grasp.
But if something says it starts with baseline (propagating signal path) lengths to get position, it's skipping the step of how it measures/ estimates those initial baseline lengths.
This is a terrible assumption to make if you’re trying to deter nuclear war.
Unless any random outage or terrorist/conventional strike against one’s early-warning radars, or errant satellite launch by a low-grade nuclear power, is automatic grounds for a universal MAD offensive.
They are, that's what Stanislav Petrov is famous for helping prevent. A single early warning satellite had an anomalous reading and everyone wanted to start nuclear Armageddon, only prevented because he didn't believe the US would send only a few warheads, to the point he was willing to bet his country on it.
Famously the US plans early on didn't really have any distinction as to who fired, and the only retaliatory option available was to go full send with everything on Soviet cities.
Specifically about satellite launches, yeah, that's a genuine concern, which is why they are talked about publicly, even when it's a classified spy satellite going up, why you are very loud and public about testing ballistic missiles anywhere you do so, and why both the US and USSR worked very very hard on making computers to quickly estimate the landing point of a ballistic launch.
Meanwhile the entire point of nuclear missile submarines is that ground based missiles are an explicit target of known enemies, and thus not likely to survive a first strike. They are not intended or planned to survive such a first strike, which is why we had SAC flying B52s 24/7 for like 40 years. Any missiles that aren't out of their tubes before enemy detonation are mostly assumed lost.
In fact, nuclear submarines have gone a long, long way to improve the situation, because now you don't have to rely on those ground based or air based warheads as much, so you can be more conservative in your judgement. If you're wrong and the soviets really did initiate nuclear war, oh well, Tridents will show them the error of their ways.
The biggest "eh, we should wait before we launch" cause is simply the lack of tension between world powers. Despite Russia's blustering, in official capacities they have not signaled that they are looking to launch nukes. They have not significantly increased their readiness, towards a large scale anti-NATO war, to the point that they are literally removing defenses from the border with Finland in order to divert those resources to invasion.
There has only been one "Oh shit oh god" moment that I know of; when an S300 missile (which I believe later turned out to be Ukrainian) landed in Poland and killed a farmer. There was a crisis meeting of NATO members.
An ideal integrator has a response of 1/s. That's just a 1st order low-pass filter with the pole at 0. Therefore, it will filter out high frequency noise.
> Take a step to the left for heads and a step to the right for tails. Most of the time you won't end up where you started, i.e. you have residual error.
I wrote a quick simulation based on your suggestion [1]. Started by generating 1e6 random points and then applied a high-pass filter. Calculated the cumulative sum on both the original and the filtered version. TL;DR: filtered version has small and very fast variations but doesn't feature the much larger amplitude swings seen in the original.
Integration indeed does not help for those large slow swings (I'd call it drift in case of a gyroscope), but that's what I was trying to get at when I started to distinguish between short and long term random effects. What I was trying to get across originally is that "all the little errors" (which I read to mean tiny fast variations, forgetting that drift is a much bigger issue in gyroscopes) which OP mentioned get filtered/canceled out. I totally missed to explain that this will vary with frequency, which was my bad.
> if you buy a commercial-grade gyroscope for [us]$10, it will have a random walk error of several º/√h. So after summing the errors for an hour, you're left with several degrees of random error, which is bad. If you spend [us]$100,000 on a navigation-grade gyroscope, you'll get a random walk error < 0.002º/√h, which is much better.
if the slope was anything else, the unit of °/√h wouldn't make sense; it would have to be °/h or °/∛h or something. similarly for noise figures given in nanovolts/√Hz
