Quite an Unusual one

[EddyCurrent]

Just thinking very quickly about control, it may not be good enough for your requirement though. (trying to keep it cheap and simple)
Forget software.

The base would be a simple motor and gearbox
The screwjack would be simple motor operated.

The base would rotate at a constant speed.
At each full turn it will trigger a switch that will move the screwjack a small amount using a timer to run the motor.
2 limit switches will indicate when the hammer is at each end of it's travel.

OR

There will be one motor that drives everything.
The base and screwjack are connected by belts and pulleys to provide the correct speeds.

As the base rotates the hammer will slowly spiral into the centre.

OR

A VFD will control the screwjack
A simple motor will rotate the base

A tacho generator will be connected to the base so that it's output will be proportional to the rotation speed.
The output from the tacho will feed the reference (0 to 10v) of the VFD.
As the base rotates the hammer will move a small amount to spiral into the centre and if the base speed changes so will the screwjack speed.