# 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)?

• Just out of curiosity, could you link to this 6th order differential equation? – Mikael Kuisma Jan 21 '16 at 23:10
• What's wrong with using a bog standard accelerometer and measuring its 1st to 4th derivatives? – Duke of Sam Jan 22 '16 at 17:38
• 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 + ... – Duke of Sam Jan 22 '16 at 17:40
• @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. – Mike Dunlavey Jan 22 '16 at 17:53