## The P and C/A Codes

A cell phone constantly emits roaming radio signals that may be picked up by three or more cell towers enabling the triangulation to work. Trilateration calculations estimate the coordinates of a mobile device using the coordinates longitude,latitude of nearby cell towers as well as the estimated distance of the device from the cell towers e.

In this challenge we will investigate the math equations used in trilateration calculations.

On the diagram above, each circle represents all the possible locations of a mobile phone at a given distance radius of a cell tower. The aim of a trilateration algorithm is to calculate the x,y coordinates of the intersection point of the three circles. Each circle is defined by the coordinates of its center e. Step 1 The three equations for the three circles are as follows:. Step 2: We can expand out the squares in each of these three equations:.

Likewise, we can now subtract the third equation from the second:. This would result in the following system of 2 equations:. Step 5: The solution of this system is:. Your task is to complete the code of the trackPhone function that we have started for you:. Get ready for the new computing curriculum. Find new computing challenges to boost your programming skills or spice up your teaching of computer science.

This is done by measuring the time delay that a signal takes to be sent from the satellite to the GPS satnav, and converting this time delay into a distance, Using more advanced multilateration formulas based on a 3D model rather than a 2D model. Other challenges you may enjoy Search for:.

Recent Posts. View more recent posts View all our challenges Take a Quiz Our Latest Book. View all books. Follow this blog. Email Address.Easton, S. Cohen, F. Mobile Robots VII: Madsen, C. Optimal landmark selection for triangulation of robot position. Robotics and Autonomous Systems23 4 : McGillem, T. Casanova, S. Quijada, J. Mobile Platform Self-Localization.

Information, Decision and Control : Font-Llagunes, J. Consistent triangulation for mobile robot localization using discontinuous angular measurements. Robotics and Autonomous Systems57 9 : New method that solves the three-point resection problem using straight lines intersection. Journal of Surveying Engineering2 : Porta, F. Journal of Surveying Engineering4 : Esteves, A. Carvalho, C. Journal of Surveying Engineering3 : Three point resection problem.

Mobile Robot Localization from Landmark Bearings. World Congress on Fundamental and Applied Metrology : Pierlot, M. Van Droogenbroeck, M.

Case 450c partsA new three object triangulation algorithm based on the power center of three circles. Van Droogenbroeck. Keywords: 2D positioning, triangulation, mobile robot positioning, algorithms, benchmarking, software, C source code, documentationToTal algorithm. Triangulation setup in the 2D plane.

R denotes a device such as a robot.

Pile spacing ukOur triangulation algorithm computes the robot position and orientation based on these angles. We provide the C source code, programs, documentationas well as the instructions to reproduce all the results given in the paper.

The last column of the following table provides the command and arguments to obtain the execution times of the different algorithms:.

Follow this link to get the programs and C source code [ triangulation. The package contains programs that you can directly use:. To run the program, copy the appropriate program depending on your platform into your current directory and type:. These images are the ones shown in Fig.By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service.

Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up. I post this question in stackoverflow here and was advised it was best suited for here. Use the distance equation. I've got information from 3 cell towers about the exact time when they received a signal, down to the nanosecond.

But I don't know when the message was sent.

Minecraft xp to levels calculatorSo my problem is to look for 3 intersecting circles, that have a common unknown offset added to their radius. At light speed, every nanosecond allows a radio wave to advance 0.

I'm approximating the map of Holland, where all this is happening, as a flat rectangle where every degree longitude counts for 66 kilometers, while a degree latitude equals kilometers.

The online graphing with desmos. Thanks MatMc. The drawback is that this makes units of both axis different in size, so every real-life circle on the map gets drawn on the mathematical grid as an ellipse.

Now what really bothers me, is the appearance of 2 perfect intersections, upon picking the proper time offsets. Earlier on, I drew some quick circles by hand, and I concluded that the northernmost intersection was the location.

A triangulation van was dispatched, and thankfully they found the signal at that location. Now I realize that I just got lucky.

I could just as well have bumped into the southernmost intersection, by choosing a different common offset, and then the guys would have found nothing there. I have a solution that works quite well. It involves finding the centroid which will help for scenarios where the circles don't intersect perfectly. And suppose the the distance from each receiver is known; therefore a circle can be drawn around each receiver with a radius as follows:.

### Henrik Mühe

Note that the 'y' term will cancel out and only leave an 'x' value as follows, which draws a line between them:. Try it out on a website like www. Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Asked 5 years, 8 months ago. Active 1 year, 6 months ago. Viewed 20k times.I happened upon such a dataset and wanted to try rediscover the original location through trilateration.

Given the aforementioned setup with absolutely accurate values and under the assumption that the earth is flat, L can be found by finding the intersection of the three circles around AB and C with their respective diameters. Unfortunately, neither is the earth flat nor du we commonly have perfect values for the diameters and potentially the coordinates of AB and C. With reasonably good data however, we can compute L as the intersection of the spheres around AB and C with their respective diameters.

In this example, the three known points are Moscow, Madrid and London and the diameters are the distance from Munich, Germany calculated using the Haversine formula.

We need to map this to a coordinate system where we can actually determine the intersection somewhat easily.

Wow macro castsequence skip if cooldownA good candidate is ECEF. Ideally, a library should be used as this gets iffy very quickly.

**Everything about GPS: What is GPS?Trilateration-C/A PRN code Generation(Seminar Version)**

Forward mapping from geo coordinates to ECEF is still reasonably straight forward, the back-mapping is an iterative algorithm and not trivial. A great library for this with bindings to many languages is PROJ. Determining the intersection can be done as described in this stackexchange post.

The reason I am investigating this is that I happened upon a website which allows me to search for things sorting them by distance from my location. It does not reveal the location of the search results. I figured I may be able to search for them from 3 different locations and do the trilateration, which, when tried visually by drawing circles on Google Maps, looked very promising. I first tried the approach posted here.

## Mika Tuupola

This does in fact yield points which are in the right vicinity but not quite where the location actually is. I initially only looked at the distance between the trilateration results and the actual location. Plotting the results on a map is a lot more helpful in understanding what the problem is:.

The actual location is Haimhausen. It looks like we are a little short of the target in the east-west axis. Then, much worse, the points seem quite spread out on the north-south axis. This led me to suspect that the conversion from geo to ECEF listed on stack overflow is not quite accurate enough for what I wanted. In fact, the code for doing the proper ECEF conversion is fantastically more complicated.

A good example for validating ones implementation and for looking at the code used in their implementation can be found here. Low and behold, using an ellipsoid:. This is vastly better than before. North-south as well as west-east axis seem to be about equally skewed now. I was however still convinced that this could be improved.

My suspicion was that the distance metric used for calculating the geo-coordiante distance pairs on the server differed from Haversine or Vincenty. I played around with the numbers a little bit but eventually tried multiplying all distances with a small factor.

This yields a much better result and makes the trilaterated points cluster a lot more densely. I tried this with many known points from the dataset and this constant helps all trilaterations about equally. It looks like internally, the source of the data set maps all geo coordiantes on a flat x-y plane and calculates distances on that plane. Result with the correction factor:. I mentioned this to a friend who suggested thinking about it as an optimization problem.GPS can be compared to trilateration.

Both techniques rely exclusively on the measurement of distances to fix positions. One of the differences between them, however, is that the distances, called ranges in GPS, are not measured to control points on the surface of the earth.

Instead, they are measured to satellites orbiting in nearly circular orbits at a nominal altitude of about 20, km above the earth. GPS is often compared to triangulation, which is actually not entirely correct. More correct would be trilateration. Trilateration is based upon distances rather than the intersection of lines based on angles. Now, in a terrestrial survey as indicated in this image here, there would probably be a minimum of three control stations, and from them would emanate three intersecting distances, i.

This is very similar to what's done with GPS, except instead of the control points being on the surface of the Earth, they are orbiting the Earth.

The GPS satellites are the control points orbiting about 20, kilometers above the Earth. There's another difference; instead of there being three lines intersecting at the unknown point, there are four. Four are needed because there are four unknowns - X, Y, Z, and time - that need to be resolved. There are also some similarities between this image of terrestrial surveying and the GPS solution. The distances need to be paired with their correct control points in both cases.

Another is that the distances are measured electronically based upon the speed of light the speed of electromagnetic radiation and the amount of time that the signal takes to go from the control point to the unknown point, and back in some cases.

Please note that in GPS that trip is one way. We'll talk more about that. There are other similarities too, but these ideas of distances being used, several simultaneous distances, being used to find the position of an unknown point is one of the fundamental ideas behind the functioning of GPS. Skip to main content. Trilateration Print Trilateration. Click here to see a text description.In fact, they are known as pseudorandom noise, or PRN, codes.

Actually, they are carefully designed. They have to be. They must be capable of repetition and replication. They carry the raw data from which GPS receivers derive their time and distance measurements. The P code is called the Precise code. It is a particular series of ones and zeroes generated at a rate of It is carried on both L1 and L2 and it is very long, 37 weeks 2x10 14 bits in code. Each GPS satellite is assigned a part of the P code all its own and then repeats its portion every 7 days.

For example, if a satellite is broadcasting the fourteenth week of the P code, it must be Space Vehicle 14 SV The encrypted P code is called the P Y code. There is a flag in subframe 4 of the NAV message that tells a receiver when the P code is encrypted into the P Y code.

This security system has been activated by the Control Segment since January of It is done to prevent spoofing from working. Spoofing is generation of false transmissions masquerading as the Precise Code. Therefore, most have developed proprietary techniques both for carrier wave and pseudorange measurements on L2 indirectly. They're carried on the carrier waves. The way these codes are modulated onto the carrier is important.

For example, when you listen to a radio in your car and the announcer says you're listening to, let's say, megahertz If you could hear it, it would be a constant buzz. What you hear is the modulation of speech and music onto the megahertz carrier. The same idea is used in GPS. But with a radio, of course, the modulation is typically a frequency modulation or an amplitude modulation for FM and AM, respectively.

In GPS, the modulation is done differently. Phase modulation is used.This page provides instructions and printable materials that teachers can use to introduce the concept of GPS trilateration to students. Be sure to print the map without scaling; otherwise, the calculated measurements of the strings will not work. Get software. The concepts behind GPS positioning are very simple, but the application and implementation require amazing precision.

GPS positioning works on two basic mathematical concepts. The first is called trilateration, which literally means positioning from three distances.

The second concept is the relationship between distance traveled, rate speed of travel and amount of time spent traveling, or:. The first concept, trilateration, is the focus of this activity.

It centers around finding your position on the Earth by knowing the location of orbiting GPS satellites and the distance from those satellites to your location on the planet. However, there is no way to actually take a yardstick, tape measure, etc.

So how can we use trilateration if we can't physically measure the distances? The answer lies in the second concept, relating distance, rate and time.

The trick lies in the fact that GPS satellites are always sending out radio signals. In GPS positioning the rate is how fast the radio signal travels, which is equal to the speed of lightmeters per second. With a known rate and a known time we can solve for the distance between satellite and receiver. Once we have the distance from at least 3 satellites, we can determine a 3 dimensional position on the surface of the earth.

To teach you the basic concept of how GPS works, we will conduct an exercise to demonstrate trilateration.

Trilateration is determining a position by knowing your distance from at least 3 known points. In GPS those known points are the satellites themselves. It is important to understand that this is a simple exercise in trilateration itself, and not an exact representation of how the process of GPS positioning works. We will be using a flat map and string, when in reality the earth is round and the satellites are in the sky, not on the ground.

Also, one often can "see" many more than three GPS satellites in the sky at any time, so we are going to use four points instead of just 3, to exemplify some of the issues surrounding extra satellites. This exercise should give you and your students a good example of how GPS positioning works. In this exercise we are going to simulate GPS positioning using 4 satellites.

You are going to pretend to be a GPS receiver somewhere on the map and will figure out where you are based on the 4 "signals" you receive.

3dmigoto doaxvvBut for you and for a GPS receiver all those signals tell you is where the satellite was when it sent the signal, and how long it took for the signal to get from the satellite to you. In other words, you have the time elapsed from when the signal left the satellite to when it arrived at your location. You also know where the satellite was when it sent you the signal, since the positions of the satellites are shown on the map.

You need to determine where you could be, based on that amount of time elapsed. Since we know the speed of the signal Rand the elapsed time Twe can figure out the distance D. In true 3 dimensional GPS positioning, the signals from the satellites are represented by spheres.

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