My Algorithm to Calculate Position of Smartphone – GPS and Sensors

As some of you mentioned you got the equations wrong but that is just a part of the error.

1. Newton – D’Alembert physics for non relativistic speeds dictates this:

   // init values
   double ax=0.0,ay=0.0,az=0.0; // acceleration [m/s^2]
   double vx=0.0,vy=0.0,vz=0.0; // velocity [m/s]
   double  x=0.0, y=0.0, z=0.0; // position [m]
   
   // iteration inside some timer (dt [seconds] period) ...
   ax,ay,az = accelerometer values
   vx+=ax*dt; // update speed via integration of acceleration
   vy+=ay*dt;
   vz+=az*dt;
    x+=vx*dt; // update position via integration of velocity
    y+=vy*dt;
    z+=vz*dt;
   

2. the sensor can rotate so the direction must be applied:

   // init values
   double gx=0.0,gy=-9.81,gz=0.0; // [edit1] background gravity in map coordinate system [m/s^2]
   double ax=0.0,ay=0.0,az=0.0; // acceleration [m/s^2]
   double vx=0.0,vy=0.0,vz=0.0; // velocity [m/s]
   double  x=0.0, y=0.0, z=0.0; // position [m]
   double dev[9]; // actual device transform matrix ... local coordinate system
   (x,y,z) <- GPS position;
   
   // iteration inside some timer (dt [seconds] period) ...
   dev <- compass direction
   ax,ay,az = accelerometer values (measured in device space)
   (ax,ay,az) = dev*(ax,ay,az);  // transform acceleration from device space to global map space without any translation to preserve vector magnitude
   ax-=gx;    // [edit1] remove background gravity (in map coordinate system)
   ay-=gy;
   az-=gz;
   vx+=ax*dt; // update speed (in map coordinate system)
   vy+=ay*dt;
   vz+=az*dt;
    x+=vx*dt; // update position (in map coordinate system)
    y+=vy*dt;
    z+=vz*dt;
   

   • gx,gy,gz is the global gravity vector (~9.81 m/s^2 on Earth)
   • in code my global Y axis points up so the gy=-9.81 and the rest are 0.0

3. measure timings are critical

   Accelerometer must be checked as often as possible (second is a very long time). I recommend not to use timer period bigger than 10 ms to preserve accuracy also time to time you should override calculated position with GPS value. Compass direction can be checked less often but with proper filtration

4. compass is not correct all the time

   Compass values should be filtered for some peak values. Sometimes it read bad values and also can be off by electro-magnetic polution or metal enviroment. In that case the direction can be checked by GPS during movement and some corrections can be made. For example chech GPS every minute and compare GPS direction with compass and if it is constantly of by some angle then add it or substract it.

5. why do simple computations on server ???

   Hate on-line waste of traffic. Yes you can log data on server (but still i think file on device will be better) but why to heck limit position functionality by internet connection ??? not to mention the delays …

[Edit 1] additional notes

Edited the code above a little. The orientation must be as precise as it can be to minimize cumulative errors.

Gyros would be better than compass (or even better use them both). Acceleration should be filtered. Some low pass filtering should be OK. After gravity removal I would limit ax,ay,az to usable values and throw away too small values. If near low speed also do full stop (if it is not a train or motion in vacuum). That should lower the drift but increase other errors so an compromise has to be found between them.

Add calibration on the fly. When filtered acceleration = 9.81 or very close to it then the device is probably stand still (unless its a flying machine). Orientation/direction can be corrected by actual gravity direction.