GPS速度/加速度/距離計算方式

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?”
如果GPS計算是採用位置變化計算得來，則1米位置誤差的GPS系統,接收率為5Hz,其速度誤差應為10mphè不合理!!

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.
GPS速度的計算方式不是採用位置的變化計算所得，而是利用衛星傳送訊號至接收天線之都普勒效應所計算得來。

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 0.045mph. 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.
採用GPS都普勒效應計算速度,一般接收器的雜訊為1mm，如果取樣率為20Hz，則速度雜訊大約為20mm/s=0.072kph=0.045mph

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.

5Hz的衛星訊號，速度雜訊約為0.01mph

At 20Hz, with 1mm carrier noise we get about 20mm/s or 0.045mph error.

20Hz的衛星訊號，速度雜訊約為0.045mph

At 100Hz, with 1mm carrier noise we get about 100mm/s or 0.2mph error.

100Hz的衛星訊號，速度雜訊約為0.2mph

è取樣率越高,雜訊越大,一般改善精度方式如下有3種

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.

Methods of improving on GPS positional accuracy

The basis of the positional calculations are outlined above, whereby the receiver uses the measured ranges from the satellites to fix its position in 3D space. Using a GPS receiver the standalone accuracy by this method is about 3m. However, there are several methods that can be used to improve on this:
DGPS有1.分廣域參考站(WAAS)與2.局部參考站 Local Basestation

台灣目前並沒有廣域參考站

1. Wide area differential or “WAAS” corrections. This method utilises corrections either transmitted from beacons or from the GPS satellites to remove some of the error from the satellite orbit, clock, and atmospheric effects. In practice the improvements are quite small. In common with all good quality GPS systems, all Race Technology products have the ability to use WAAS corrections when appropriate.

2. Local differential corrections (single differencing). This uses a local static GPS receiver to correct the received ranges of the moving GPS receiver. In a low multipath environment, single differencing typically reduces errors to the sub meter range.

若要達到好的速度精度,須架構局部參考站(Local Basestation),否則儘管有100Hz的更新率,精度會更差。

3. Real Time Kinematic (RTK) calculations. Again this method requires a trackside reference receiver and works by tracking the number of carrier waves between the satellite and the receiver. Because the carrier phase can be tracked far more accurately than the raw range information (about 1mm compared to around 20cm) the result is a positional accuracy of just a few centimetres. The problem with this method is that it is not particularly robust, as both receivers need to have a clear view of all satellites for several minutes before any positional information is available and even trees by the trackside can cause the solution to fail. In practical terms this is satisfactory for surveying applications; however it is not readily suited to automotive testing.

即時動態測量（Real-Time Kinematic）：簡稱RTK，為結合無線電數據通訊設備，可以立即解算求得點位坐標，適用於空曠地區移動物體之軌跡定位，道路中心線測量，水道測量及地籍測量。

Note that none of these methods directly impact the accuracy of the speed measurements; there is very little that can be done to improve these using differential corrections.

GPS and its application to automotive testing

GPS is now widely used for automotive testing with regard to position, speed, distance measurements, and is generally accepted as the preferred technology due to its combination of very high accuracy, flexibility, and convenience.

“Whilst there is no theoretical upper limit to how fast you can sample the GPS signal, the faster you sample, the more error you get due to noise”. So you need to trade off the measurement bandwidth you require, against the measurement noise that is acceptable.

A typical road car has a resonance of around 3Hz, so sampling at 20Hz seems like a good engineering compromise to make sure you are “not missing anything”. An analogy to this would be sampling a 3mm thermocouple with a response time of five seconds: you might sample the voltage twice a second to make sure you don’t miss any information, but there is absolutely no point in sampling it any faster.

取樣率高可以確保資料不遺失，但對於增加過高的取樣率是沒有意義的＝＝＞對精度是沒有幫助的。

Finally, please be cautious of directly comparing specifications of GPS systems from different manufacturers – there is a huge difference between “best case” GPS accuracy and “real world” GPS accuracy. The only way to compare the accuracy of two systems is to test them on the same day, under the same conditions, at the same time.