How to sensor Jerk=$d^3{\bf r}/dt^3$, or higher derivatives (4th, 5th, 6th order) when applied in the equation of motion of a ballistic missile? Given a 6th order differential equation of motion as usually used in ballastic missile dynamic models. What kind of sensors are usually used to measure Jerk, (or higher order derivatives in kinematics)?
 A: From when I worked among missile engineers, accelerometers were used, along with gyroscopes (mechanical or laser).
I don't know of 6th order differential equations.
I do know of 3rd order, namely in the steering by swiveling the engine nozzle.
Specifically, the engine nozzle angle is off-center by a certain amount, causing
an angular acceleration (2nd derivative) of the airframe pointing direction.
The motor that swivels the nozzle has a rate, and that's the 3rd derivative.
Keep in mind that rockets are not always pointed in the desired direction of travel.
If they are solid-fueled they have a certain fixed amount of delta-V which may be more than needed.
If so, they spend a certain amount of time blasting sideways, to use up the excess fuel.
Submarine depth control is a similar situation.
A motor drives the angle of the bow planes at a certain rate.
The angle of the bow planes (and speed) controls the pitch rate of the vehicle.
The pitch of the vehicle (and speed) controls the rate of depth change.
