CoreXYZ: Theory

The kinematic model is an extension of the original coreXY, with an additional Z axis. Here is a diagram showing all moving parts and the 3 degrees of freedom:

For a linear motion $(\Delta a, \Delta b, \Delta c)$ on the left, right and back motors respectively, the resulting $(\Delta x, \Delta y, \Delta z)$ motion of the end effector is given by:

$$ \begin{cases} \Delta x = \frac{\Delta a + \Delta b}{2} - \Delta c\\ \Delta y = \frac{\Delta a - \Delta b}{2}\\ \Delta z = \Delta c \end{cases} $$

Solving for $(\Delta x, \Delta y, \Delta z)$ yields:

$$ \begin{cases} \Delta a = \Delta x + \Delta y + \Delta z\\ \Delta b = \Delta x - \Delta y + \Delta z\\ \Delta c = \Delta z \end{cases} $$

Those two set of equations provide the motor actuation necessary for a given end effector displacement, and vice versa.