Sinusoidal trajectory of each cartesian coordinate for the end effector

Question

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 ?

Answer 1

The short answer is yes, it is possible.

The long answer is:

• The all xyz coordinates have to be inside workspace boundaries of the robot
• Solving IK for 7 DOF is a bit harder then any other non redundant structure, you need an additional equation.
• If you are using an industrial robot arm, with its own programming language, you have to find out how to get a "time signal" or create your own by incrementing a variable in a loop.
• The frequency of the sinusoids has to be very low, I would start with 0.05Hz and increase gradually to see what happens. Maybe 0.1 if the amplitudes are small.
• You should avoid exiting the eigenfrequency of the robot.