One of my friends (Vincent Groenhuis) has obtained a PhD for research on medical robots that can position a biopsy needle.
In one of the models, called "Stormram 1", the position of the needle (\(\vec E \)) depends on the positions of 5 pistons, stemming from points \(\vec B_1\) till \(\vec B_5\).
I took up the challenge to work this calculation out, and test its convergence. That could save the purchase of a Matlab licence!

(click on the picture below to rotate it)

(This picture was taken from Robotic systems for breast biopsy
using MRI and ultrasound imaging ISBN: 978-90-365-4892-2 by Vincent Groenhuis. I thank the author for his permission)
The calculation took some programming.
If we know the distance from \(B_5\) to \(A\), we can calculate the position of \(\vec A\). To know this distance, we must know the position of \(\vec D\). But \(\vec D\) can rotate around the axis \(\vec A-\vec C\), so it depends on the position of \(\vec A\).

This sounds like a circle without end!
The solution is explained here.