3 motors from the base decide the xyz position of the end effector and the other 3 on the tip decide the rotation of it. This is a really simple model as Steve mentioned. I don't know how much speed do you need but it will easily reach 10kHz and over to numerically solve the inverse kinematics with a modern computer and modern lib such as ceres.
Closed-form inverse kinematic equations are used to determine joint angles based on the end effector pose. These equations are well established in the literature. For example, the classic robotics books by Craig or by Paul show the equations for typical 6dof industrial arms.