Peter,
Now for the parting questions to make you think. Why bother? Why would one need to calculate acceleration? Why not use an accelerometer?
Well what you are also trying to control is "jerk" or the rate of change of the aceleration in order to minimise the peak force applied to the system.
Torque = Inertia * Acceleration
Rate of Change of Torque = Inertia * (Rate of Change of Acceleration)
Actually I struggle with this a bit, because "Rate of Change of Torque" seems a bit meaningless to me. Given that Torque is produced by Current in a servo motor, or servo valve position in a hydraulic system, if both of these can be manipulated at an infinite rate, then "Jerk" would have no practical consequence.
However in all real systems the servo motor torque or hydraulic valve bandwidth (ability to respond to a change in command) is limited. Therefore if the system commands the actuator to generate torque faster than it can produce it, then a transient error is created.
To correct this error the loop then tries to compensate by commanding MORE torque...so you can see that very quickly the torque producing loop can saturate at 100% output for a very brief period until the position loop catches up. It is the limited bandwidth of the torque producing loop that causes this effect.
Therefore if you are trying to control the Rate of Change of Acceleration, ie Jerk, you need to calculate it. As the original example suggests if:
Therefore the distance resolution is 0.001"
The velocity resolution will be 0.001"/0.001 sec = 1"/sec
And the acceleration resolution will be 0.001"/(0.001^2)sec = 1000"/sec^2
Then:
Jerk resolution will be 0.001"/(0.001^3) = 1,000,000"/sec^3
Which is very poor indeed. Will an accelerometer help? My instinct is to say yes. After all we could control the entire system with just an accelerometer, by integrating it to get velocity and again to get position, but then we introduce another kind of error...absolute position is impossible to measure directly, ie we cannot find the integration constants.
Without the benefit of much math, I am guessing that combining both an position and aceleration measuring sensors into the system, it should be possible to get a result that gives the best from both, while nullifying the weaknesses of each. I suppose what I am saying is that:
Differentiating creates noise errors. (or resolution errors).
Integrating creates drift errors (or absolute errors).
If I can measure BOTH position and aceleration then:
Differentiating position gives me velocity, and integrating aceleration also gives me velocity, so I can correlate the two sensors for validity if nothing else.
Differentiating acceleration gives me jerk, and given that I have only had to differentiate once (not three times as I would have if I derived jerk from position) then in principle I may be much better off.
Where all this MAY come unstuck is that I am not very familiar with accelerometers and I am not sure of their inherent behaviour in terms of bandwidth and resolutiuon.