I'm trying to modify the project outlined at http://blog.dzl.dk/2018/08/21/3d-digitizer/ with 4 rotary encoders instead of 3. the first encoder $e_0$ is mounted in the base and allows the rest of the arm to rotate along the z axis. the second encoder $e_1$ is a known distance from the base $H_0$ and allows the rest of the arm to swing. The rest of the arm consisting of $e_2$ and $e_3$ do the same as $e_1$. The variable r refers to the distance between the origin (at the bottom of the base) and the point at the end of $a_3$. the supplied equations on the site are
$$r = a_1 \cdot \cos(e_1) + a_2 \cdot \cos(e_1+e_2)$$ $z = h_0 + a_1 \cdot \sin(e_1) + a_2 \cdot \sin(e_1 + e_2)$, $x = r \cdot \cos(e_0)$ , $y = r \cdot \sin(e_0)$. My guess is this can be modified with a third term and say $$r = a_1 \cdot \cos(e_1) + a_2 \cdot \cos(e_1 +e_2) + a_3 \cdot \cos(e_1+e_2+e_3)$$ am I anywhere near correct? note on the image that ARM1 corresponds to $a_1$, ARM2 is $a_2$, etc.