I've been learning how to do dead-reckoning using just an ordinary IMU chip (a 3-axis accelerometer and a 3-axis gyroscope). In a number of textbooks I've read on aircraft navigation, the motion of a navigating object is often summed up in four equations. Aside from the last equation - (3.17) rotational dynamics - where we need to separately calculate the inertial matrix in CAD, the other three equations requires only the input data registered by the accelerometers or gyroscopes, or integration of the values.
I have so far no problem understanding the derivation for each equation or knowing what they physically mean. The trouble I had is I couldn't relate the rotational dynamics equation (3.17) of an object to its navigation.
Navigation is a matter of tracking the position and orientation of an object and perhaps also its acceleration and velocity at a specific point in space. The first three equations (3.14) (3.15) (3.16) will suffice:
- Given an initial condition and gyro data, we can numerically update the Euler angles in (3.16) and determine the object's orientation
- With data measured by accelerometers and gyros in the body-frame, and if we know our externally applied force, we can find out the object's velocity in (3.15)
- Since we already know our orientation and body's velocity, we can calculate our inertial velocity in (3.14). Integrating that, we get the object's inertial position
Up to this point I would have everything I need for navigation and the last equation (3.17) seems out of place here. I've already had my orientation per unit time - what's the point of knowing the angular acceleration around the bodily axis? Calculating all the different moments of an object doesn't seem to have any meaning to navigation of an object either. So why does that equation (3.17) matter in navigation?
I start to suspect it could be my entirely lack of basic knowledge in aeronautics or aviations that gives rise to this question. I don't see any mathematical difference in mounting an IMU on a banana or on an actual aircraft. The IMU provides me with accelerations and angular rates that I can calculate where the object navigates in what orientation. Navigation care less for the inertia or moments of the object. Even if I need angular acceleration for whatever reason, wouldn't it be much easier (if not more accurate) to differentiate the gyros data directly?
The only situation I can think of that makes inertia and rotational acceleration matter in navigation is when the object collides into something else where a transfer of momentum occurs. But that's just my wild imagination and am unsure if that makes any sense.