Petroleum engineer want Electrical engineer help?
I would suggest a single motor and a mechanical arrangement utilising some kind of 'cam' would work far better and be more reproducible and reliable than a rope or other based mechanism.

If different ranges of motion are required or the range is critical, to avoid having to design everything mechanically I would, depending upon load (including plate) utilizing a stepper motor or servo.

Is this for mixing by any chance? 
If you did not need to tilt and the samples were not effected by magnetic fields, you could use two electromagnets per vessel and 'drag' the ball across by alternately switching on the magnets, negating the need for mechanical design and simplifying the electrical design somewhat. 

A picture / more detailed description of what you want to achieve would be helpful, especially if construction has already begun.

Hope this helps :)