[![enter image description here][1]][1]

I tend to use SolidWorks assembly to trial and error these types of assemblies, then worry about how to make it work.  Constrain (Angle and Distance) the crap put of components to start and remove them as components are interconnected.

Following the process I outlines in the comments:  

Platform: 100mm.

Upper Arm: 105mm by 50mm, Angle 90°.  The dimensions you will have to tweak at some point.  Your algorithms.  I started with 100mm by 50mm and it's not perfect.

Lower Arm: 50mm by 50mm, Angle 45°.  

Mate all the lower faces.  Place an angle mate of 120° on each of the lower arms.  The angle mates will allow you to construct the mechanism.  Mate faces to top plane.

Mate one of the lower arm origins to the assembly origin.  Origin must be on mated faces.

Mate Upper Arms to Lower Arms.  Place an angle mate of 90° on upper arm to lower arm.  Make sure all  upper arms are in the same direction.

Mate platform to upper arms.  You can see I cheated and mated them to a point.  

It's a *rather complicated and cumbersome device*, but this will get you going in the right direction. Remove all angle mates and you should have the basic mechanics.

Once you get that working, upper arm is identical, but lower connection cannot be in the same place, so bottom 1 will be at 50mm by 50mm at 45°.  Next 60mm by 40mm at 45°.  Last at 70mm by 30mm at 45°.  The actual numbers will depend on your actual dimensions.

This has been solved on the web with 3D printed files [3D Print Timelapse Build: Spherical Parallel Manipulator][2].

[Thingiverse project.][3]


  [1]: https://i.sstatic.net/XTp0v.png
  [2]: https://www.youtube.com/watch?v=hj0oFC73A5c
  [3]: https://www.thingiverse.com/thing:3951917