What equipment or method is used to obtain measurements of the pitch angle (θ) and pitch rate (q) during flight?
Edit: I am asking about how the aircraft systems sense these quantities
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The pitch of an aircraft relative to the horizon is observed on the Attitude Indicator which also displays roll angle. These are required for IFR flight but not VFR flight here in the US. As such they are found on most but not strictly all aircraft. These are typically gyroscopically based.
Here is a basic cutaway of the interior showing where the gyro is mounted. They are often vacuum (or positive pressure) driven but can be optical or electrical as well.
Angle to the relative wing (angle of attack) is shown on an angle of attack indicator. While not required these are becoming more popular. These are controlled by a sensor on the outside of the aircraft.
The sensor that measures this on the exterior looks like this:
As far as I know there is no pitch rate info on your standard panel.
There are three principle competing electronic technologies replacing vacuum and electric powered gyros.
Higher end systems use ring laser gyros (RLG), and rely on interference pattern changes to detect motion in the axis of the ring laser. Similar, but slightly lower cost are fiber optic gyros (FOG). Lower end systems (including $20 hobby drones) are using MEMS devices. MEMS devices can be extremely low cost and make effective accelerators.
From a current systems design standpoint, MEMS devices tend to have less bandwidth and higher noise, but for general aviation and INS/GNSS systems for general aviation, it is expected that MEMS will continue to grow market share. They are inexpensive, environmentally tolerant, and provide adequate performance even today.
Traditionally, gyros (rather than electronic gyros and accelerometers) have been the mainstay of aircraft attitude and rate information sensing.
Expect that many systems today, and future systems will employ primarily accelerometers, and attitudinal information will be inferred.
If you have a starting reference, and not allot of system noise, then a positional and acceleration reference can be established by just accelerometers. This means that the need for a "gyro" where positional or attitudinal reference isn't needed and can be derived from the accelerometer data alone. (Most likely your smartphone only has accelerometers, and derives attitude.)
Also, higher end MEMS systems may use 6 sensors, to provide finer granularity in small accelerations, compensate for some system noise, and provide higher slew rates.
Integrated MEMS IMUs with integrated GPS/GNSS receivers are available at the chip level now. In the US, they are available in non-ITAR configurations, which eases export of integrated products. There are also ITAR configurations which are popular in non-military applications such as survey, autonomous vehicles and aircraft. The ITAR units require Dept of State approval for export, and are harder to integrate with non-US teammates.
Prices will continue to drop, and capabilities will continue to grow. I believe that the steam gauge style instruments in aircraft will be virtually obsolete within the next 20 years. They will be replaced by integrated systems, and replacement instruments for backward compatibility. The cost to manufacture MEMS instruments is rapidly falling below the cost to manufacture vacuum and electric powered gyros.
ALSO, with MEMS based IMUs it is standard that the unit will provide attitude information, and rate of change information, strictly from accelerometers. This statement may be redundant, but I wanted to make clear that as an example, pitch, roll and yaw angle (position or attitude) and rate are readily available, as well as Lat/Long type positional information.
And just one more tidbit. When you have an integrated GPS receiver, the filtering on the GPS processing can provide very accurate velocity information (more accurate and responsive than by interpolating positional fixes). That is a nice feature about GPS and receiver signal processing.
The predominant technology today is AHRS, pronounced A-hars, for Attitude Heading Reference System. A typical Garmin unit provides:
The existing answers cover aircraft with full instrumentation, so I add my 2 cent for simpler ones.
In aircraft that have only basic instrumentation, the pilot can use the cockpit view to judge the pitch attitude. Especially in gliders, where the forward window sill is at some distance from the pilot's eyes, the height of the horizon relative to the window sill gives a very good indication of the pitch attitude.
For the pitch rate the vestibular system of the pilot is used in combination with visual cues. This works very well in VFR flight. As @Koyovis points out, very small angular accelerations are below the threshold of the vestibular system, so without visual cues spatial disorientation is very likely.