Steve I'm amazed by your passion and how you manage to come up with all these new articles. Thank you for always doing the hard work and calculations for us. I've learnt and grown so much just by following and reading all the articles in here. Cheers!
Most of us probably have never used the hyperbolic tangent function tanh(x) or the sin(arctan(x)) reference function since our university days. But these functions enable a kinematic controller. Kinematics determines the position and orientation movement of an end-effector on a robotic arm as a function of the joint coordinates.