I didn't specify, but it is almost impossible to obtain linear movement as you said earlier. Obviously, I have to go for a sinusoidal movement.
I really like your idea to optimize weight. Thank you so much ! I'll think about that
You do not say much about the support structure for you project, but if it were mine, I would use a cable loop and a windless powered by a geared DC motor. The motor and windless would be mounted on the structure. The cable attached to your carriage.
Linear with fast start and stop is far easier than sinusoidal.
Momentary start/stop is impossible according to classical physics. So take that into account...
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