The computer then takes the time differential between any two of those received times, and generates a mathematical hyperbola, on a map of the yard. It does this for each pair of received times and creates more hyperbolic curves. At only one location do all the hyperbolas intersect, and THAT describes the exact current location of the lawn mower on a map of the yard.
If three transducers are used, then there are three intersecting hyperbolas that identify the exact location, to better than the nearest tenth of an inch, both E-W and N-S. If four transducers are used in the yard, there are six hyperbolas that intersect. If five transducers are used, there are ten mathematical hyperbolas that intersect. The main advantage of using more than three transducers is for when some object causes an echo of the ultrasonic sound that might delay its time of reception. In the position processing, the computer can discard an outlier hyperbola that does not intersect at the same location of the others. Position location is still only accurate to about the nearest tenth of an inch.
During the night while you are sleeping, a valuable meteorite crashes SOMEWHERE in the Grand Canyon. You don't know WHEN it crashed or WHERE. But using your simple tape recorder, and some math, the next morning you will be able to walk EXACTLY to where the valuable meteorite is lying on the ground! I will slightly approximate here and say that the speed of sound is 1000 feet per second (it is actually slightly faster than that). The meteorite did NOT contain any advanced electronics! We aren't even sure if it was going 40,000 mph when it crashed. None of that is important. It IS important that it made a SINGLE crash sound when it hit, and that sound propagated all over the Canyon. We also know that that crash sound hit the echo wall and was then reflected toward your tape recorder. The result is that your tape recorder recorded TWO crash sounds, one directly from wherever the crash site was, and the second a little later, when the echo sound got to the recorder.
You listen to the tape and discover that the two sounds arrived exactly 0.500 second apart on the tape. What that means is that the total echo path was 500 feet longer than the direct sound path. Your maps had already told you exactly how far you were sleeping (or rather, how far the recorder was) from the echo wall. All you needed was the DISTANCE and not even the SPACE DIRECTION. Now, some relatively simple math establishes that, wherever the meteorite crashed, it HAD TO BE somewhere along a mathematical hyperbola. This alone does not tell you exactly where the meteorite now is lying.
But there was also another flat 'echo wall' and you had also used your maps to know exactly where it is. So your tape recorder actually recorded THREE crash sounds, first the direct path sound, and then the two echoes, slightly later. So you do some more calculating regarding the first sound and the echo off the second wall. Some more relatively simple math establishes that, wherever the meteorite crashed, it HAD TO BE somewhere along a second mathematical hyperbola. This is a DIFFERENT hyperbola, and one that intersects with the first hyperbola, somewhere in the Canyon. You plot these two hyperbolas on a map of the Canyon floor, and the intersection point of the two hyperbolic lines IS WHERE THE VALUABLE METEORITE IS WAITING FOR YOU!
Essentially, NO advanced technology beyond a simple tape recorder is involved. And you never learned WHEN the meteorite hit during the night. But you now know EXACTLY where it is now laying in the Canyon. Depending on how well you can TIME THE INTERVAL between the two crash sounds on your tape recording, maybe to the nearest thousandth of a second, you should know its position, TO WITHIN ONE FOOT ACCURACY.
IF you like to do math, you could also create a hyperbola based on the difference of time of the two echo receptions. You will find that it also EXACTLY intersects at the SAME intersection point of the first two hyperbolas on your map. This can confirm that you did not make any math errors and that the meteorite WAS waiting for you.
Do you see how amazing this hyperbolic approach is? You were SLEEPING in a large Canyon, and at SOME INSTANT during the night, a meteorite (with no transmitter or other electronics on board) crashed, which caused three crash sounds to be recorded on a cheap tape recorder. And you wake up, do some math, and walk EXACTLY to where the valuable meteorite is waiting for you! Both the (audio) robot lawn mower and the (microwave) PSST Rape Prevention System are very precise at locating a target, and FAST (as any computer can do the hyperbola math nearly instantly) and neither requires the target to have ANY SOPHISTICATED ELECTRONICS, except for the crudest 'clicker' which emits a brief single pulse click, which the computer uses to identify the exact moment when a click was emitted. The computer then does the hyperbola thing to calculate WHERE the pulse had been emitted.
This is WITHOUT ever caring exactly WHEN the pulse was emitted (as a GPS system has to precisely know and which causes GPS to never be very precise. Satellite-based actual GPS is only accurate to about 330 feet in any direction). The GPS system DOES now include GROUND-BASED transmitters, called DGPS, which are both much faster and more accurate (about 10 feet). But even DGPS cannot match the precision of this hyperbola-based approach, nor the speed of analysis. Actual satellite GPS requires about two minutes to capture four different satellites' signals and then do a LOT of time checking regarding how many nano-seconds the clocks in all those satellites have changed as well as how different the GPS device clock has gotten. This hyperbola-based system does not need to know what time it actually is anywhere. GPS has a Carrier Wave improvement, to better than one inch accuracy, but that actually is all ground-based and really has nothing to do with any GPS satellites.
Each lady in the city can wear a very simple necklace, bracelet, or anklet that normally does not do anything at all. If the lady feels she has a personal danger, there are two tiny pushbuttons on the small charm, which then sends out a 'click' once per second. Similar to the mower system, three or more receiver microwave receivers around the city receive the click and record the exact time of that reception. A simple number is then sent to a city Police Station to compare the three (or more) times, where mathematical hyperbolas are then generated and the intersection point of the hyperbolas is calculated. This is all done in a tiny fraction of a second, and to a position accuracy of better than one-foot precision.
Better still, if the lady is being taken in a vehicle, the Police Station computer display SHOWS her precise updated position every second, so it is easy to see what street the vehicle is on, what lane it is in, and exactly how fast and in what direction the vehicle is moving. Saving the lady should be rapid and reliable.
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The transducers are standard items, around $10 each. Telephone Bell wire can send the signals to the PC, or a more sophisticated version can use something like BlueTooth or a Cordless Mouse technology to send the signals to the PC's input ports. Modification to the mower to add a battery and motors, and the R/C equipment is the greatest expense in doing this.
The lawn mower system works like a charm!
The PSST versions of this require microwave electronics circuitry, which I have never personally developed.
The mower system CAN get confused if only three transducers are used and there are objects in or near the yard which can cause audio echoes, such as a parked car. Four or five transducers seems to eliminate any such confusion, nearly always. I never had the mower go wander off down the street!
The PSST presentation was first placed on the Internet in May 1997. Rape Prevention Personal Security System - PSST
This lawn mower presentation was first placed on the Internet in June 2015.
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