Hi there,
This is Bogdan from Romania. I'm a mechanical engineer specialized in the Industrial Robotic field.
In the first place is necessary to have more info about your system (is neither a robot or stabilizing platform)
What you show as reference is a 2 axis compliance mechanical joint.
Your device could be developed in two ways:
1 - almost the same as the device shown in the video
2 - using gyroscopic stabilization - which is way much better
before to chose the best solution is necessary to know the working environment (required payload, frequency and amplitude of the movements, desired extension lengths, tilting angles, etc, etc)
Based on the first discussion with you and your input (ideas/drawing/sketches, etc) I'll make the primal model - the 3D model with all the necessary parts and components.
On this model, with your assistance, I'll make the required adjustments/modifications/changes, until the desired outcome is reached.
Once the final model is approved, we'll discuss about the further development, and the effective building of the prototype.