Calculation of acceleration, speed and distance from GPS
A common question is “if your GPS system has a positional error of say 1m, and you are calculating speed at 5Hz, then that would give you a speed error of 1m x 5Hz = 5m/s or about 10mph. How can you claim 0.1mph?”
The answer is that speed is not calculated by simply differentiating the GPS positional data. The GPS position is firstly calculated using the method outlined about and then speed is calculated separately based on either the Doppler shift of the satellites, or the carrier phase of the GPS signal.
It is commonly documented that GPS speed is always calculated from the Doppler shift in the satellites’ signal – however this is not strictly true. When available, it is more accurate to use the carrier information to calculate speed. For a typical quality GPS receiver the carrier noise is normally around 1mm, so when sampling at 20Hz the speed error is approximately 1mm x 20Hz = 20mm/s or 0.05mph. If the carrier signal is not available then the Doppler shift of the radio signal can be used, but with an associated reduction in accuracy. In practice, GPS receivers do not continuously use one method or the other; the output is a weighted average depending on the relative quality of the signals and other conditions.
This highlights a problem with GPS speed measurements – the higher the sample frequency of the GPS system, the higher the noise on the speed signal. If we assume that only the carrier information is used to calculate the speed then:
At 5Hz, with 1mm carrier noise we get about 5mm/s or 0.01mph error.
At 20Hz, with 1mm carrier noise we get about 20mm/s or 0.04mph error.
At 100Hz, with 1mm carrier noise we get about 100mm/s or 0.2mph error.
Clearly, these are very much “order of magnitude” figures, but they illustrate that as sampling frequency increases, so the error/noise on the speed measurements also increases – so there is a direct trade-off between speed accuracy and the measurement update rate. Road cars have a resonant frequency of approximately 3Hz, race cars are lighter and more stiffly sprung so they have a higher resonant frequency; 6Hz would be a reasonable estimate. It is normal to want to sample at least twice the resonant frequency, so in this case 20Hz is a good engineering compromise.
Once speed is calculated, distance and acceleration can be very simply derived. Distance is calculated by integrating the speed information; acceleration is calculated by differentiating the speed information.
Distance can either be calculated from the “3D” speed, which includes the effects of going up and down hills, or the so called “2D” speed which can be thought of as the distance measured from a map. The advantage of only using 2D speed is that it is less “noisy” because vertical errors are approximately twice the size of horizontal errors. However, on a steep gradient the error in 2D speed measurements could be as much as 3%. For this reason all Race Technology products use 3D speed, for higher dependable accuracy under all conditions at the expenses of a small increase in measurement noise.
GPS based acceleration can be calculated by differentiating the speed signal – however differentiating any signal increases its noise a great deal, consequently GPS-only based accelerations are often too noisy to be used directly. Therefore, all Race Technology equipment measures acceleration directly using multi axis digital accelerometers.