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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)?

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  • $\begingroup$ Just out of curiosity, could you link to this 6th order differential equation? $\endgroup$ – Mikael Kuisma Jan 21 '16 at 23:10
  • $\begingroup$ What's wrong with using a bog standard accelerometer and measuring its 1st to 4th derivatives? $\endgroup$ – Duke of Sam Jan 22 '16 at 17:38
  • $\begingroup$ The 6th order ode is just what you'd expect from a Taylor expansion for position wrt time. i.e. x0 + vt + at^2/2 + ... $\endgroup$ – Duke of Sam Jan 22 '16 at 17:40
  • $\begingroup$ @DukeofSam: It's difficult to take derivatives of physical signals because the noise is amplified. The reason accelerometers are so useful is that you can integrate them, once to get velocity, and twice to get position. They do need to be calibrated to offset constant error, and they do drift, as gyroscopes do, so they occasionally need correction. But there's no good way to take numerical derivatives without heavy smoothing. $\endgroup$ – Mike Dunlavey Jan 22 '16 at 17:53
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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.

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