How are these parameters derived from readings of sensors in an aircraft: angle of attack, slip angle, and airspeed?

Question

From what I've learned, the reading comes from

1. an angle of attack sensor and a slip angle sensor. Let's call the readings $\hat\alpha$ and $\hat\beta$ respectively. These are of either the weathervane type or the differential pressure type.
2. airspeed sensor, i.e. a pitot tube. This probe measures the total and static pressure $\hat p_\text{total}$ and $\hat p_\text{static}$, and the airspeed $\hat u$ is obtained from them.

Here's the catch: in real flight, the measured angles $\hat\alpha$ and $\hat\beta$ may depend on not only the true values $\alpha$ and $\beta$ but also other factors which include true airspeed $u$, and the measured airspeed $\hat u$ may depend on $\alpha$ and $\beta$ too. How is this problem solved? How are these sensors calibrated? Are they tested in a wind tunnel where every possible combination of $(\alpha,\beta,u)$ is given to produce $(\hat\alpha,\hat\beta,\hat u)$ and form a lookup table?

Answer 1

Here's a fun fact. Nothing you measure is what you intend to measure. The corrections of these measurements to satisfy the objectives form a good chunk of development flight test by the airplane manufacturer. Even after the airplane type is certified and enters into service, the corrections may continue!

Now specifically for AOA ($\alpha$) and AOS ($\beta$). In development flight test, an accurate measurement of AOA and AOS is typically done with a nose-boom that places it outside of the aircraft boundary layer (as reasonably can). But there's an error even with that, due to Mach, manufacturing tolerance, etc. Flight path reconstruction could be performed to further decrease the errors, if needed. Usually, the boom AOA and AOS measurements are pretty close to the actual AOA defined relative to the body axis and used in wind tunnel.

In service, AOA is measured with vanes located at the nose of the aircraft. Even at low speed where Mach effect is small, these measurements are not the aircraft AOA. They are local AOA due to upwash and boundary layer. For non FBW aircraft, AOA measurement is used primarily for stall warning and stall protection (SPS). There are lookup tables in the SPS that register the stick shaker and pusher (if required) angles, as a function of the local AOAs. An extra mapping, as a function of Mach, configuration, whatever, is required to transform the local AOAs to aircraft AOA, should the design need it.

Development flight test airspeed is usually measured by the nose-boom, often augmented by an additional trailing cone. The trailing cone is extended behind the aircraft to measure the far field static pressure. Total pressure is measured by the boom. The resulting impact pressure needs to be transformed to dynamic pressure to output airspeed.

In service airspeed is measured most often by pitot-static tubes at the nose. Due to boundary layer, static source error correction (SSEC) would be required. These lookup tables are stored in the air data system (ADS). There may also be a mixing of the left/right static ports to decrease the effect of asymmetry.

Answer 2

They use test flights.

They attach more instruments to the plane and move them outside the influence of the planes own turbulence.

From the known good values of these test instruments they can calibrate the actual instruments.

Answer 3

When the manufacturer builds a plane, they make a table with correction of readings according to altitude and position of the sensor