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I found an article that can help in my investigations.

Forward kinematics of spherical parallel manipulators with revolute joints

The problem is solved for the following system of vectors, which is shown in the figure.

enter image description here

The methodology described in the article allows, using simple numerical calculations on a computer, to determine the angles $\psi$, $\psi$'and $\phi$, $\phi$'.

All other angles ($\alpha_2$,$\alpha_3$,$\alpha_4$,$\alpha_5$,$\mu$) are known.

The coordinates of points $B_1$,$B_2$,$B_3$ and $C_1$,$C_2$,$C_3$ are also known.

I do not know how, knowing these parameters, it is possible to determine the coordinates of points $A_1$,$A_2$,$A_3$.

In the same article on page 525 there is such a sentence:

enter image description here

But I do not understand how to do this. I ask for help.

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Vectors $v_i$ are restored very simply.

For example, vector $v_1$.

First we need to rotate the $C_2$ vector by the angle $\phi_1$ around the $C_1$ vector. This way we get the intermediate vector $v*_1$.

And then rotate this vector by the angle $(\alpha_2 - C_1 \cdot C_2)$ around axis of rotation $(v*_1 \times C_1)$.

Vectors calculated in this way are similar to the calculated data given in the publication.

For the remaining vectors, the principle is similar.

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