Why 99 Percent Coverage Costs 4.6 Times the Floor

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Because it cannot tell, and the reason it cannot tell is a piece of probability rather than a broken sensor. If each patch a machine sweeps lands more or less independently of the ones before it, then the fraction of the floor covered after sweeping a total area S over a room of area A is one minus the exponential of minus S over A. That curve has no knee, no completion signal and no end, and it produces a result worth carrying around: reaching 95 percent coverage takes three passes over the whole floor, and 99 percent takes 4.6.

Then one line of calculus answers the question as asked. The chance that the next patch swept is new ground is exactly the fraction still missed. At 95 percent coverage, nineteen of every twenty square metres the machine does from then on is re-cleaning. At 99 percent it is ninety nine out of a hundred. From the inside, those two situations are indistinguishable, which is the whole of why a robot stops looking finished and starts being finished at no particular moment.

The one table worth remembering

Invert the law and the sweep needed is the negative natural logarithm of the fraction you are still missing. Six entries cover every argument anyone has about this:

Half the floor costs 0.69 passes. Ninety percent costs 2.30. Ninety five costs 3.00. Ninety nine costs 4.61. Ninety nine point nine costs 6.91.

Two things fall straight out of that. Every further factor of ten off the missed fraction costs a flat 2.30 passes over the entire floor, because that is the natural logarithm of ten, and the increments never fall, no matter how far you go. And the step from 90 percent to 99 percent is exactly double the work for nine points of coverage, because the logarithm of a hundred is twice the logarithm of ten. There is no diminishing-returns curve to find the sweet spot on. There is a constant price per decade, charged forever.

A listing that publishes enough to test its own claim

This catalogue carries no robot vacuum at all, which is worth saying plainly before using anything as a stand-in. The only machine we stock that covers an area on its own is a Neptunova Pro cordless robotic pool cleaner, and it happens to publish the three figures the arithmetic needs together, which is unusual enough to be the reason it is here: a rated area of up to 2,150 square feet, a runtime of 150 to 180 minutes, and AI sonar navigation rated by the maker for up to 99 percent coverage. It adds a 180 watt brushless motor, a stated 45,000 pascals of suction, tank style tracks that climb walls and scrub the waterline, three modes, auto parking at the pool edge and a 2 litre filter basket.

So test it. 2,150 square feet is 199.74 square metres and 180 minutes is 10,800 seconds, so an overlap free route has to sweep 0.0185 square metres a second. At a representative 28 centimetre track that is 6.6 centimetres a second, a slow and entirely plausible crawl.

A random route reaching the same 99 percent has to sweep 4.61 times as much, which is 0.0852 square metres a second, or 30.4 centimetres a second on the same track.

Now read it the other way, because that is where the runtime comes in. A random route travelling at the overlap free speed would need 4.61 times 180 minutes, which is 13.8 hours, against a published battery of three. So the claim to plan an efficient route rather than, in the listing's own words, "bumping around randomly", and the claim to a three hour runtime, are the same claim stated twice. The factor between them is not a marketing number. It is the 4.61 that 99 percent costs. Full record in our review of the Neptunova Pro, whose runtime is written as "up to 150 to 180 minutes", a range inside an "up to", so the honest reading of it is 150.

Why a map changes the shape of the error, not just the size

Here is the part that decides what you should actually do, and it is a different kind of mathematics rather than a bigger number.

A machine that navigates randomly leaves a missed area that decays exponentially with how long it runs. Give it a second pass and you halve what it left, roughly, and a third and you halve it again. Time genuinely buys coverage.

A machine that builds a map leaves a missed area that is a constant. It misses what its map excludes: the strip under a sofa its bumper cannot enter, the triangle behind a chair leg, the room past a threshold it will not climb. None of that shrinks with running time at all, because the machine is not failing to reach it by chance, it is declining to reach it by plan.

So a second pass halves what a random machine left and changes nothing on a mapped one. Which reframes a question several people in the vault are asking from the other direction: when a robot cannot get over a doorway strip, the fix is the strip. Running the machine longer, buying a longer battery, or scheduling a second cycle addresses an exponential that is not the problem. The geometry is.

And it says something about the 99 percent on the listing above. A mapping machine's 99 percent is a statement about its map, not about a sweep, and the two are tested completely differently. One is falsified by a stopwatch. The other is falsified by looking under the furniture.

Two of our records lead with the number that is not the constraint

The same habit shows up on the machines a person pushes, and here two published pairs settle it without any assumption at all.

A Qimedo Cordless Electric Mop M3 publishes three speeds at 400, 600 and 800 revolutions a minute, two 21 volt batteries, a stated 420 minutes of continuous low speed running, a 2.5 hour charge, a removable 450 millilitre tank, a 50 inch telescopic pole, an LED head light and a display that prompts a recharge below 25 percent.

Divide the two published numbers. 450 millilitres over 420 minutes is 1.07 millilitres per minute of running. A press to spray pump of the kind its own record describes delivers something in the range of one to five millilitres a second while pressed, so the tank is between a minute and a half and seven and a half minutes of actual spraying inside seven hours of spinning. The runtime is not the thing that stops you. The tank is, and it is not on the front of the listing. More in our review of the Qimedo M3 and the category view in how to choose a cordless electric spin mop.

The same division on a different machine. A GARYOG cordless window vacuum squeegee publishes 2000 pascals of suction, a 280 millimetre soft rubber blade, a 2,200 milliamp hour battery giving up to 40 minutes, a removable transparent 150 millilitre dirty water tank, a 31 to 40 inch extendable pole and 1.1 pounds.

150 millilitres over 40 minutes is 3.75 millilitres per minute. At a representative 15 millilitres recovered per square metre of sprayed glass, that tank holds 10 square metres of glass, which a 280 millimetre blade moving at a third of a metre a second clears in about two minutes. The battery is sized for twenty tanks. Its own record is honest about the tank being "modest for very big jobs"; the arithmetic says modest by a factor of twenty against the number printed beside it. See our review of the GARYOG window vacuum.

What a person actually sweeps, for scale

Worth doing, because the robot arithmetic above is easy to mistake for a statement about speed. A 24 inch flat floor mop with a 360 degree swivel head and four microfiber pads publishes a screwless aluminium frame, plastic covered edges, a pole extending from 80 to 155 centimetres and four pads for wet or dry work.

Twenty four inches is 0.6096 metres. At an unhurried 0.8 metres a second that is 0.488 square metres a second, which is 26.4 times the sweep rate the pool cleaner needs to finish its own rated area in its own rated time. A person is not slower than a robot. A person is twenty six times faster and unwilling, and that is the entire value proposition of the category, stated honestly.

Its own drawbacks name the limit on that figure: "Exact head width beyond the 24 inch name is unstated." So the 0.6096 metres is a product name rather than a measurement, and the real swept width is whatever the pad is minus the frame. More in our guide to choosing a flat floor mop, and the robot side of the same job is in best robot mops.

The one product here that addresses knowing rather than doing

An EICEM green vacuum dust detection light attachment clips to a vacuum head and rakes a low green beam across the floor so fine debris casts a shadow. Our page on why a bag is the filter and a cyclone is not already uses this product for a different argument, the head against the filter, and quotes the four word drawback that makes it useful: "Adds visibility, not suction."

It earns a third mention here for a reason neither of those pages needs. The coverage law gives you the total area a random machine has missed and tells you absolutely nothing about where. There is no term in it for position. So the only instrument that can answer the question the owner is actually asking, which patch did it miss, is one that looks at the floor, and the simplest such instrument in this catalogue is a raking light that takes nothing from the machine and adds no suction. Its own second caveat is the right one for a device that works by shadow: "Contrast fades in bright rooms." Details in our review of the EICEM detection light.

The honest limit, and what it leaves out

What is exact here is the law and every ratio inside it. The 2.3026 passes per decade, the factor of two between 90 and 99 percent, the 4.6052, and the identity that the new ground fraction equals the missed fraction are all consequences of the exponential and do not depend on any number being right. The 1.07 and 3.75 millilitres per minute are exact, because each is one published volume divided by one published time on the same listing. The 199.74 square metres and the 10,800 seconds are published conversions.

What is representative rather than published is a short list and one item on it matters more than the rest. The 28 centimetre track width is an estimate, because no pool cleaner in this catalogue publishes one, and every speed in the third panel scales inversely with it: assume 40 centimetres instead and both speeds fall by 30 percent while the 4.61 between them does not move. The 0.8 metres a second walking sweep, the 15 millilitres a square metre of recovered water and the one to five millilitres a second spray rate are all typical values. And the model itself assumes each newly swept patch lands independently of the ones before, which no real machine does. A bump and turn robot is strongly correlated over short distances, which makes it locally better than this law and globally no better at all, because the correlation decays long before the room does.

Three things this page deliberately does not compute. It does not compute the size or shape of the patches a random machine leaves behind, because the law has no position in it and the distribution is precisely what the owner sees. That is the gap, and naming it is more useful than guessing at it. It does not compute coverage for any robot vacuum, because this catalogue does not carry one and borrowing a figure from a pool cleaner for a carpet would be a different claim wearing the same arithmetic. And it does not evaluate the sonar navigation itself, because that is a claim about a method rather than a published quantity, and the only thing the numbers can say about it is what this page said: it is worth a factor of 4.61, and the battery beside it is the proof that the machine is relying on having it.

Three numbers to carry. Random coverage costs 2.30 passes over the whole floor for every factor of ten you want off the missed fraction, so 95 percent is three passes and 99 is 4.6. The chance the next patch is new ground equals the fraction still missed, which is why no machine can feel the difference between nearly done and done. And a mapped robot's misses are a constant rather than an exponential, so when yours leaves the same strip every time, the thing to change is the strip.