## Table of Contents

## Principle

A Pitot-Darcy tube is an air speed sensor commonly used in aviation. It consists of a tube pointing in the forward direction. When the sensor moves forward a stop pressure $P_t$ appears at its tip. The differential pressure $P_{diff}$ is measured between the tip of the tube $P_t$ and the ambient pressure $P_s$. One variant named from its inventor Prandtl tube has static air ports directly on the side of the tube.

The pressure measured is the square of the air-speed : $P_t = P_s + \frac{1}{2}\rho v^2$ where $ P_t $ and $P_s$ are measured in Pascal unit (Pa). $\rho$ is the air density constant typically in $[1.14 \ 1.34]$ depending on temperature & altitude. $v$ is the air-speed in $m.s^{-1}$. The differential pressure measured is $P_{diff} = P_t - P_s = \frac{1}{2}\rho v^2$

## Prandtl tube

Prandtl tube is made of an inner tube placed within an outer tube. At their tip, both tubes are centered. Space between the two tubes is filled-in with an epoxy adhesive (Araldite or equivalent) on few mm near the tip. The side of the outer tube is drilled to sense the static pressure. At the bottom, the inner tube which is longer act as a connector for the dynamic pressure sensor and third short tube is added to create a connector to the static pressure area which lies in the empty space between the two tube.

The static pressure holes must be drilled at a minimum distance from the tube tip where airflow perturbations are reduced. A distance 4 times the diameter (d=4mm)of the outer tube is retains here (16mm). Four 1mm holes were drilled with a dremel hand tool. The three tubes are presented on the figure below.

First test was done using brass tubes. Inner tube width is 2/1mm (outside/inside) and outer tube is 4/3mm.

References:

- report by W.Gracey on the impact of the various shape on the measurement.
- report (french) from ANSTJ with general information on Pitot sensor.
- website (French) from Rémi Bourgin with experimental Pitot tube for RC plane

## Electronics

### Sensitivity requirements and Pressure sensor

The differential pressure measured vary with the square of the speed. RC plane speed can be quite low. Challenge is to be sensitive enough to obtain good measurement even in the low speed range of few meters per seconds.

The MP3V5004dp from NXP is a sensitive differential pressure sensor. Its differential measurement range goes from 0 to 3.92 kPa. The analog output is ratio-metric and swing from 0.6V to 3V ($\Delta V =2.4V$). For reference, 1 kPa is approximately the pressure of 10cm of water.

We are interested in speed range varying from $0$ to $25 m.s^{-1}$ ($90km/h$) which correspond to a maximum differential pressure of 3.75 cm of water, or 0.375 kPa. The MP3V5004DP sensor is used in 10% of its nominal range.

Table below provide theoretical differential pressure expected for various speed using $P_{diff} = \frac{1}{2}\rho v^2$ with $\rho=1.15$.

v (m/s) | v (Km/h) | $P_{diff}$ (Pa) | ~ $H_2O$ (cm) | MP3V5004dp & MCP3428 (0.2041Pa / LSB^{1}) |
---|---|---|---|---|

1 | 3.6 | 0.6 | 0.006 | 3 |

2 | 7.2 | 2.4 | 0.024 | 12 |

4 | 14.4 | 9.6 | 0.096 | 47 |

7 | 25.2 | 29.4 | 0.294 | 144 |

10 | 36 | 60 | 0.6 | 294 |

15 | 54 | 135 | 1.3 | 661 |

20 | 72 | 240 | 2.4 | 1176 |

25 | 90 | 375 | 3.75 | 1837 |

30 | 108 | 540 | 5.4 | 2646 |

40 | 144 | 960 | 9.6 | 4704 (4095 sat) |

### Signal conditioning and conversion

The pressure sensor is placed close (~10cm) to the Pitot tube on the wing. The analog to digital conversion is also done on site to avoid noise pollution of analog signal. The MCP3428 from Microchip is a high resolution sigma-delta converter with four differential inputs and a programmable gain factor from 1 to 8. It has a digital I2C interface to connect to the dsPIC.

The MP3V5004dp output analog signal is connected to the MCP3428 Sigma-Delta ADC through a first order RC low pass filter with a cut off frequency at $28Hz$ ($R=5.6 kOhm$,$C=1\mu F$). The 2nd differential input is connected on a voltage divisor with 1.2k and 560 ohms to the 3.3V ref and GND. $10\mu F$ decoupling capacitor are used on power supply. The I2C bus wires are pulled up with 10kOhm and are connected to a 10pf capacitor protecting from glitches.

The ADC Sigma-Delta is configured for

- 12 bits
- x8 Gain
- 240 Samples Per Seconds (SPS)

The overall resolution is:

- (1) MP3V5004dp Analog output sensitivity is 1633 Pa/V
- (2) MCP3428 12 bits with a gain of 8 : 0.125mV/LSB
^{1} - with (1) and (2), we obtain $1633 * 0.125e^{-3} $ =
*0.2041 Pa / LSB*^{1}

### Sensor static tests

A 90 minutes lasting measurement is performed indoor without movements. Sensor output were logged at 100Hz. The figure below shows a slow drift. The sensor standard deviation measured on the overall measurement is 0.54 Pa (including the drift).

The sensor is sensitive to its own orientation. Moving the sensor up-side down create an offset of (100 LSB^{1}; to be checked).

## Flight setup

The sensor is mounted on the wing of a FirStar 1600 RC plane.

A 2mm flexible tube, originally for protecting fish lines, connect the sensor to the Pitot tube.

One magnet is glued on the Pitot tube. A 2nd magnet integrated in the wing allows to fix the Pitot tube. Both magnet are in contact. The tube stays firmly in place during flights and ejects in case of hard landing reducing damages. It is a flexible solution for testing other Pitot tube design ; It’s also practical for storage and transportation.

## Experimental results

The MCP3428 samples $P_{diff}$ at 250Hz during flights. $V_{Pitot}=\sqrt{\frac{2}{\rho}*P_{diff}}$ with $\rho = 1.15$.

No calibration were required. Sensor theoretical output scaling were used with the formula to compute the speed from the pressure. $\rho$ value was tune from a default value 1.2 to 1.16 which minimized the mean error however the improvement was not a big deal. The only value to adjust is the raw pressure measurement offet (zero). The zero value for the sensor useed is -1800.

Matlab script converting 250Hz raw MCP3428 to Pressure (Pa) and Speed (m/s):

```
% Pitot Calibration. P_pitot is a vector with raw MCP3428 output read from I2C bus.
P_pitot_cal = 0.2041* (double(P_pitot) + 1800); % to Pa unit
% Compute Speed (m/s) from diff pressure (in Pa).
V_pitot = sqrt(max(0,(2/1.15) * P_pitot_cal));
```

### Comparison with GPS ground speed

In calm condition, the Ground Speed $V_{GPS}$ and Air Speed $V_{Pitot}$ are equal.

In windy condition, wind average direction and strength are estimated combining $V_{GPS}$, $V_{Pitot}$ and the plane yaw $\Theta_{heading}$ direction (using GPS COG^{2}). This wind estimation is described below.

The airplane ground speed (GPS) is estimated with the difference of the air speed (Pitot) with the projected wind to the aircraft forward direction:

$V_{GPS} \approx V_{Pitot} - V_{wind}*cos(\Theta_{heading} + \Theta_{wind}) $

The figure below presents the air speed in dashed blue. The reconstructed ground speed (black) matches accurately with the GPS velocity (red) which prove the correctness of the air-speed measurement as well as the wind strength and direction. The onshore wind is laminar with limited turbulences. The air-speed measurement presents a high sensitivity even at low speed.

The error is defined with $error = V_{gps} - \left( V_{Pitot} - V_{wind}*cos(\Theta_{heading} + \Theta_{wind}) \right) $. For the 200s of the flight shown on the figure, the error measured is presented in the table:

error | m/s | km/h |
---|---|---|

mean | 0.017 | 0.06 |

standard deviation | 0.74 | 2.6 |

Further figures are available in slides.

RMS error is relatively low regarding the measured speed value. Part of the error is also due to wind gust and GPS limited accuracy particularly at estimating fast change of vertical speed.

### Wind estimation

Wind model considered is constant and characterized by

- its constant direction $\Theta_{wind}$
- its constant strength $V_{wind}$

#### Wind => $V_{Pitot} - V_{gps}$

The curve below shows the difference between the Ground speed (GPS) with the air speed (Pitot) in function of the plane flight direction. This error (blue dashed points) fit with a sine wave. This sine wave is the speed offset added for each plane direction in $[0 \ 2\pi]$ by a wind with a constant strength and direction. Sine wave amplitude and phase is the wind strength and direction.

This wind estimation is used to compensate the air-speed when comparing the GPS ground speed $V_{gps}$ with the Pitot air-speed $V_{Pitot}$ above.

#### Script

Matlab script to estimate wind off-line:

```
% V_gps and V_pitot are two vector with all data measured.
% V_pitot was under-sampled (averaging) by a factor 5 to fit the 50Hz log from the GPS.
% GPS chip update frequency is 10Hz, but it is logged at 50Hz.
V_err = V_pitot - V_gps; % Ground and Air speed difference (i.e. wind)
M = [cos(COG); sin(COG) ]'; % COG is the direction (Theta in rad) vector data measured from the GPS
y = V_err';
x = M\y; % Solve wind strength and direction using linear algebra (MMSE)
Theta_Wind = -atan2(x(2),x(1)); % Wind (go to) direction. Azimuth direction is opposite to trigo
V_Wind = sqrt(sum(x(1:2).^2)); % Wind strength (m/s)
plot(COG,V_err','.'); hold on; % plot Error blue dots
plot([0:.01:(2*pi)],V_Wind*(cos([0:.01:(2*pi)]+ Theta_Wind)),'-k','linewidth',3); % plot wind
```

#### Discussion

Using the GPS COG^{2} field is not exactly the plane yaw direction $\Theta_{heading}$ but the plane flight direction.
Thus the COG is a biased plane yaw $\Theta_{heading}$ direction. It would be best to use plane orientation from the IMU sensor. It is not done here to reduce the number of sensors for this demonstration. The COG bias is small enough if we assume the wind speed to be small compared to the airplane air speed. It might be possible with a more sophisticated script to compensate this bias.

W. Premerlani propose a wind estimation relying exclusively on GPS data. (To be tested with the same data set to compare…)

#### Other flight data

Plane was equipped with two GPS:

- one uBlox M8N GPS and
- one MTK3339.

The uBlox trace presented on the map below is better than the MTK trace. The uBlox chip was used exclusively for the curves above. The MTK trace can be shown on the map (top left icon). The KML file can be opened with Google Earth which provide a 3D view of the trace showing the height of the plane.