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What type of a mechanical device can I use and program to apply a controllable force within the range of what a human can to an area about the size of a fingertip? It is desirable for the applied force to be precise and known.

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I agree the compressed air is the way to go. There are many vendors that sell small cylinders and even have bumpers available for the end of the actuator rod that could approximate a fingertip.

For control, you don't mention any requirements, but if they're simply manually controlled on/off application of force, a regulator with some simple manual valving should do the trick, just make sure you have a way to vent the air in the cylinder when you're trying to release the force. Keep in mind that unless you control the flow into the cylinder, the force will be applied quite suddenly unless you use a flow control valve or orifice as well.

What I've done for applications like this is to use an I/P transducer, controlled by computer which would lets me precisely control the pressure applied to the cylinder by varying the current flowing in a loop, typically 4-20mA. (There are also voltage controlled versions, but I have no experience with them) This way, you could ramp the pressure at any rate you wish and the process is much more repeatable.

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There are many possible options, but possibly this simplest would be a cylinder driven by compressed air. Since compressed air pressure is easy to measure, and your cylinder area will be constant, it is easy to know how much pressure is being applied. Compressed air is also nice because the amount of pressure can be easily and quickly controlled.

If you are looking for something that already exists, there are a number of manipulators for robotic automation that are readily available, if relatively expensive.

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