Absolute vs. relative accuracy

Here is a very typical response to our system:
“You did tests outside and they seem to be very precise.
But how do you know that locations from beacons are exact (+- 2cm)?

I mean what reference coordinate did you use to compare?”

The questions touch several critically important aspects of any positioning system and even more of those for indoor positioning system:

  • What coordinate system is used?
  • How do you define “the ground truth”?
  • How do you define accuracy at all?
  • How do you measure it?
  • What is the difference between absolute and relative accuracy?

There are many questions like this. They are related to each other and very important to understand in detail how positioning system work and how to assess them. Let’s discuss the points one by one.

What coordinate system is used

Greenwich meridian or GPS/GLONASS coordinates are not really relevant when we talk about indoor positioning system (IPS), because your premises are indoor. The premises or the IPS can be anywhere – in Nairobi, underground in Canada, or on Mars – and if the IPS doesn’t have a link to external coordinate system (a sort of georeferencing), the system cannot calculate which part of the world it is even in theory. It can be anywhere.Coordinate system

Imagine, you are in a closed room without windows. Your robot is precisely driving indoor using, for example, Marvelmind Indoor Positioning System. But you have no way to measure which part of Earth you and your robot are – the windows are closed. Maybe, while you were driving inside, the aliens stole you from the Earth and transferred to Alpha Centaurus. If you haven’t experienced acceleration, you can’t say whether you stayed or moved or is being moving at that moment and with what speed.

Something similar is already happening to us in real life – not in a hypothetical case with aliens – but in real life and every second. We believe we measure our very accurate coordinates against the Earth, but the Earth is flying several km/s in space against the Sun and it is flying with several tens of km/s against the center of the galaxy, etc. What are my coordinates? Against what?

Thus, we establish that coordinates are always relative.

What is Marvelmind indoor positioning system uses as a relative point?

  • When you build the Marvelmind indoor positioning system, one of the stationary beacons is simply chosen by the system to have (0,0,0) coordinates
  • You can assign any beacon to have (0,0,0) coordinate. Even more, you can assign any point on the map to be (0,0,0)
  • Since in some cases the indoor system is linked with outdoor, it is quite easy to make georeferencing: you link the IPS’s (0,0,0) point with external coordinates, for example, GPS coordinates, and turn the map so that it would match the angular direction between the two coordinate systems as well. Effectively, you align and lock not one, but two points on the indoor map and two points on the external map. See more about georeferencing in the Operating Manual
 
Marvelmind Indoor Positioning system is using internal (0,0,0) point as a reference and builds a coordinate system against this point.

Driving from point A to point B

Our system is designed, first of all, to guide an autonomous robot from point A to point B. Of course, absolutely the same applicable for autonomous drones, forklifts or even people tracking. All these case are the same for this discussion. But the sake of simplicity let’s discuss autonomous robots.Autonomous forklift

With this approach in mind (driving from point A to point B), it does not matter what the absolute distance between point A and point B measured in meters is. Even when we say “absolute distance” – it is not really absolute. It is a distance related to some arbitrary chosen distance unit – meter. It could be in meters, it could be lengths of the robot, it could be in some units of our indoor positioning system.

For precise autonomous drive or for precise indoor positioning it is important that the system can very precisely and very accurately measure coordinates of mobile beacons. Coordinates in the coordinate system of stationary beacons. Not in meters. Not in some GPS coordinates, but in the internal coordinate system of the stationary beacons. Because we know the coordinate of the point A in internal coordinates, coordinates of the point B in internal coordinates, and the robot’s current location in internal coordinates. So, internal distances, coordinates and dimensions matter – not some “external meters” or external coordinates.

Here how it works in practice on an example of a mobile robot:

  • You drive or push your robot to a point A and record/set/assign a point on the map of stationary beacons (IPS map/RTLS map) corresponding to the point in the real space
  • Then you drive/push your robot to a point B in space and assign a point B on the map of stationary beacons
  • You repeat the same for whatever number of points you wish, for example, to all points in the between of points A and B, if the robot turns along the way and has to drive on a complex curve
  • Robot’s controller measures the difference between the robot’s current location – in internal coordinates of the indoor positioning system – and adjust driving (change power on the motors, turns, if needed, etc.). Then, it repeats the same with some logic until the coordinates are close enough to make sure where are on the right path or reach the right point. All this is done, measured in internal coordinates of the indoor positioning system

Absolute distance

When you already have an indoor positioning system up and running, your robot is ready to drive, because the robot has coordinates of the point A, the point B and its own current coordinates. The robot’s autopilot will compare the expected location (the next desired waypoint) with the robot’s current location and drive the robot accordingly.

What is important, the robot doesn’t really need to know the distance in meters. Everything is measured in some internal units – inside the coordinate system that is inside the indoor positioning system. Not in meters.Meter

Why do we focus on this so much? Why do we stress that the absolute distances do not really matter and even harm sometimes, when mixed with internal measurement units/distances:

  • Our system is using time of flight of ultrasound. This is why Marvelmind IPS is so precise. But this is why it is important to remember that the speed of sound in the air is not an absolute constant. It depends on many parameters and, first of all, the air temperature. Likely, the air temperature indoor is typically the stable. However, even that is not always the case. Besides, the temperature can be stable, but the absolute air temperature may differ drastically – from -40 to +60C, for example. And the speed of sound in the air would be notably different in those conditions. Thus, the temperature must be taken into account, right? Yes and now.
  • Imaging an air balloon. If you have more pressure inside, the balloon is slightly larger in size, if you have lower pressure inside – slightly smaller. Draw a triangle on the surface of the balloon. It will have three pivoting points – your driving waypoints. BalloonsThey are relative coordinates inside the coordinate system of the balloon. The relative coordinates of the waypoints wouldn’t change whether you have a slightly larger or slightly smaller balloon. The relative coordinates remain the same. The ratio of distances remain the same.
  • You can always precisely drive from one point to another. But if you start measuring sizes of each line in “external meters” you will easily find that the distances are different (obviously, because the diameter of the balloon changed). And if you measured coordinates of the those points in “ground truth”, for example, against something external and using, for example, a laser distance meter, one may conclude that the “absolute” coordinates are measured incorrectly. But the conclusion is not correct, because: a) there is no “absolute” coordinates, b) there is confusion between the indoor positioning system and some external positioning system. The robot driving on the surface wouldn’t notice any problem with the dimensions. Maybe, only indirectly, like: increase energy consumption, because the size of the balloon increased in terms of the size of wheels of the robot and the robot must do more revolutions in order to cover the same distance

Therefore, it is very important to clearly distinguish between relative distance and absolute distance. It is very important to understand that “absolute” distance is not absolute either – it is relative distance measured in something else – platinum meters or revolution of the wheels, etc.

Summary and key points:

  • Marvelmind indoor positioning system is designed for driving from a point A to a point B. It is not “a laser distance meter”, at least, without a calibration
  • We define accuracy against the points on the map of stationary beacons. It is relative accuracy. And this accuracy is typically better than ±2cm

Relative accuracy vs. absolute accuracy

Remember, Marvelmind Indoor Positioning System is, first of all, designed to drive a robot from point A to point B. In order to be very accurate in doing that, we need to have as little dispersion of the measurements of the location of point A, point B and the robot’s current location as possible.Measurement dispersionOption 1 has higher dispersion, but lower average shift against (0,0) coordinates. Whereas, Option is relatively more accurate, but has higher absolute shift against the (0,0) point.

Once again, an absolute shift from an expected point measured in meters or some other measurement units outside of our own coordinate system doesn’t really matter. We must make very precise measurements inside the system: for the point A, point B and the robot’s current location. Then, we will be able to drive from point A to B very accurately and return back very accurately and it doesn’t really matter whether it takes us 10.000m or 10.200m to drive from A to B. Therefore, when we drive, we simply bring the robot to the point A and the Option 2 measures it very accurately. Then, we move the robot to a point B. It can be shifted against some external coordinate system, against and external grid – it doesn’t matter. After we moved the robot, we once again record the measured location of the point B in our indoor positioning system. And now we want to drive back to the original point A as precisely as possible. The correct choice would be the Option 2, because it relatively more accurate and it doesn’t matter that it is shifted in the external coordinate system.

Summary:

  • What option is better: Option 1 or Option 2? – It depends on the needs towards the indoor positioning system. Marvelmind IPS focuses on the Option 2. When we refer to ±2cm accuracy, we refer to the Option 2. And the Option 2 is much better for the real-life applications, when you need to precisely drive from a point A to a point B
  • What is the absolute accuracy for Marvelmind IPS? – a shift from an externally defined (0,0) coordinate – Without a calibration, typically – 1-2%. Not ±2cm, but 1-2%, because the measured distance depends on the speed of sound in the air; and the speed of sound, first of all, depends on temperature. Notice, that 1-2% is significantly more than ±2cm on typical sizes for our system. For example, for a 10m size, it is 10-20cm absolute error. Thus, the measured result using Marvelmind Indoor GPS will have a very low dispersion – typical sigma of ±2cm – but it can easily be shifted by 1-2%, i.e perhaps 10-20 times more than the diameter of the measurement spot

Therefore, if you want to use very Marvelmind RTLS for autonomous driving or flying, we are very good. If you want to use us as an alternative to a laser distance meter – only with calibration and with clear understanding of the limitations (air temperature, first of all).

Real measurements and results

If anything is unclear, contact us via info@marvelmind.com