Introduction to Positioning Fundamentals

TL;DR

Positioning is about figuring out where something is in space, relative to a known reference. You'll learn about different ways to define position, like coordinates, and the types of measurements involved. Understanding these basics is key to using positioning in real-world applications.

1. The Mental Model

Imagine you're trying to tell a friend exactly where to meet you. You wouldn't just say "over there." You'd use landmarks, street names, or even your phone's GPS coordinates. Positioning is just that: precisely locating something in a structured way.

2. The Core Material

Positioning is the science of determining the precise location of an object or person. This involves using various techniques and technologies to pinpoint coordinates in a specific reference frame.

How Do We Define a Position?

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Generally, we define a position using coordinates. These are a set of numbers that specify a point's location. The most common systems are:

• Cartesian Coordinates (X, Y, Z): Think of a graph. X, Y, and sometimes Z (for height) tell you how far along each axis you are from an origin point.
• Geographic Coordinates (Latitude, Longitude, Altitude): This is what GPS uses. Latitude measures distance north or south of the equator, longitude measures east or west of the Prime Meridian, and altitude is height above a reference surface (like sea level).
• Polar Coordinates (r, θ): These use a distance (r) from a central point and an angle (θ) from a reference direction.

Types of Measurements

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To get these coordinates, we rely on different kinds of measurements:

• Range (Distance): How far away an object is from a known point. This can be measured using time-of-flight (like radar or sonar), signal strength, or direct measurement.
• Bearing/Angle: The direction from a known point to the object. Think of a compass reading or an angle measured with a protractor.
• Relative Position: Describing an object's location in relation to another moving or unknown object, rather than a fixed reference.

The Positioning Process

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No matter the technique, positioning generally follows a similar flow:

graph LR
    A["Know Reference Points/Stations"] --> B["Measure from Reference Points (Range/Angle)"];
    B --> C["Collect Multiple Measurements"];
    C --> D["Apply Geometric/Algorithmic Calculation"];
    D --> E["Determine Object's Position (Coordinates)"];

• Reference Points: You need at least one, often several, known locations. These could be GPS satellites, Wi-Fi access points, or survey markers.
• Measurements: You take readings (distance, angle, signal strength) from your object to these reference points.
• Calculation: You use geometry, trigonometry, or more complex algorithms to combine these measurements and pinpoint the object's location within your chosen coordinate system.

3. Worked Example

Let's say you're trying to locate a drone using two ground stations with known positions.

• Ground Station 1 (GS1): Located at (0, 0) on a 2D Cartesian grid.
• Ground Station 2 (GS2): Located at (100m, 0m).
• Drone: At an unknown position (X, Y).

You measure the distance (range) from each ground station to the drone:

• Range from GS1 to Drone: 80 meters
• Range from GS2 to Drone: 60 meters

We can use the distance formula (which is just the Pythagorean theorem) to set up two equations:

1. From GS1: $(X - 0)^2 + (Y - 0)^2 = 80^2$
   $X^2 + Y^2 = 6400$
2. From GS2: $(X - 100)^2 + (Y - 0)^2 = 60^2$
   $(X - 100)^2 + Y^2 = 3600$

Now, let's solve for X and Y. Expand the second equation:
$X^2 - 200X + 10000 + Y^2 = 3600$

Substitute $X^2 + Y^2 = 6400$ from the first equation into the expanded second equation:
$6400 - 200X + 10000 = 3600$
$16400 - 200X = 3600$
$200X = 16400 - 3600$
$200X = 12800$
$X = 12800 / 200$
$X = 64$

Now substitute X = 64 back into the first equation to find Y:
$64^2 + Y^2 = 6400$
$4096 + Y^2 = 6400$
$Y^2 = 6400 - 4096$
$Y^2 = 2304$
$Y = \pm\sqrt{2304}$
$Y = \pm 48$

So, the drone is at either (64m, 48m) or (64m, -48m). To resolve this ambiguity, you'd typically need a third measurement or additional context (e.g., you know the drone isn't underground). This process of using distances from multiple known points is called trilateration (or triangulation if using angles).

4. Key Takeaways

• Positioning determines an object's location relative to a known reference frame.
• Coordinates (Cartesian, Geographic, Polar) are the common ways to express a position.
• Range (distance) and bearing (angle) are fundamental types of measurements used.
• Multiple measurements from known reference points are usually required to pinpoint a location.
• Geometric calculations, like those based on the Pythagorean theorem, translate measurements into coordinates.
• The choice of coordinate system depends on the application and environment.

Common mistakes to avoid:
- Confusing 2D and 3D: Forgetting to account for altitude (Z-axis) when working in a 3D space.
- Inadequate Reference Points: Trying to solve for a position with too few or poorly placed reference points.
- Ignoring Measurement Errors: All measurements have some error; assuming perfect readings will lead to inaccuracies.
- Mixing Coordinate Systems: Ensure all your data is in the same coordinate system before performing calculations.

5. Now Try It

Imagine you're standing somewhere in a large park. You have a map with two distinct landmarks marked: a statue at (50m East, 0m North) and a large tree at (0m East, 70m North). Using your phone's "measure distance" app, you find you are 60 meters from the statue and 50 meters from the tree.

Task: On a piece of graph paper, or by setting up equations similar to the worked example, determine your approximate position (East, North) in the park.

Success looks like: You should be able to narrow down your position to one or two possible (X, Y) coordinates, demonstrating an understanding of how ranges from known points help locate you.