mbed action lib kinetic

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If there is no simple_action_server.h, how do you implement the tutorial on an mbed?

Unfortunately, you don't.

The tutorial you link to targets the roscpp variant of ROS 1, the things you're trying to use target rosserial. Those are two very different client libraries, and without someone porting actionlib to be compatible with rosserial, you cannot use it directly

The tutorial-specific FibonacciAction.h is in the correct directory. However, there does not appear to be "actionlib/server/simple_action_server.h" as listed in the includes of the tutorial.

FibonacciAction.h is a header which most likely contains the goal/message definition. That's part of what's needed to use actionlib, but not enough.

There are various related Q&As here on ROS Answers about using actionlib with rosserial. I'm not aware of what the state-of-the-art is here, so you might want to try and find those (tip: use Google and append site:answers.ros.org to your query).

The various rosserial issue trackers (Github) may also have information.

Edit: there is a way to implement an Action server without actionlib, as in essence, it's "just" a set of topics and a state machine. I'm not aware of someone having implemented that in a generically reusable way for rosserial though.