If a small movement (10x10mm) with big force and high speed is required the current design is the ideal solution. By using an eccentric and bearing the rotation of the servo is converted into linear movement.
Similar movement of the milling head is used in Shaper tool - https://shapertools.com/
A video showing how the design is working could be found on:
Converting the rotation angle of the servos into Cartesian coordinate system requires some trigonometry calculations.