I used the method in the attached file without a skeleton. This can be done in several ways but the idea is the same.
Open the mechanism app and run the analysis.
The links are bound to each other with an axis bound to a fixed plane; the 1st link is bound to the floor plane with a point, and final link is bound to the "cylinder" disk with a point. The connection is a cam just to provide animation. This could be done in many ways including a rack and pinion gear set.
The assembly is free to float until the final link is connected. If over-driven, it can double on itself since there are 2 valid positions. An additional angle limited constraint set would solve this as well but not necessary if your range is limited.
This is the only way I know that doesn't require regeneration in animation.
Creo 2.0 full version attached.
A curved spring can be constrained to a curve with a point, rather than constraining the axis to a plane. The idea is that each semicircle part has a path to follow.
No problem. Love a challenge 
Here is a slightly different technique mostly to aid assembly of so many elements.
And if both ends are on the same "arc", you could drive it from what is here the inside radius. You will notice in the video (#5) that the fixed end floats.
This is because the center is bound as it is in my model. If you bind the inside diameter, and both ends are tied to the inside diameter, the ends won't move. It will require some additional constraints to make sure the angles between coils remain consistent.
Again, a servo motor is driving the motion. Open mechanism and run the analysis. For some reason, it doesn't get added to the zip automatically.
Creo 2.0 full version attached.
