Consider a 7-DOF robotic manipulator. I want the end effector to follow a predefined trajectory and suppose I care only about the position and not about the orientation. So, the rotation part of the forward kinematics will always be the identity matrix. Now, here is my question. Is it possible for each of the end effector's position coordinates $ (x,y,z) $ to impose a sinusoidal trajectory of the same frequency ? I mean it like this:
$$ x(t) = A_1\sin(\omega \pi t) $$ $$ y(t) = A_2\sin(\omega \pi t) $$ $$ z(t) = A_3\sin(\omega \pi t) $$ So, at each time instant the desired cartesian position of the end effector is obtained by the above sinusoids. If this is possible, I assume that I could also find the desired trajectory of each one of the $7$ joints by solving the inverse kinematics, right ?