Let's say I have a Jacobian which gives the relationship between the velocity of a robot's joints, and the Cartesian velocity of its end-effector, relative to the end-effector's local frame. And let's say I also have a target pose to move the end-effector to, expressed relative to the end-effector's local frame, in Euler angles (a, b, c).
I want to move the end-effector to this target pose, and I want this rotation to happen in 1 second. To achieve this, I need to calculate the Cartesian velocity of the end-effector, so that I can use inverse kinematics to calculate the required joint velocities.
My first idea is to say that the Cartesian velocity is equal to the Euler angles (a, b, c), divided by the time (which is 1). So the angular velocity around the x-axis is a, the angular velocity around the y-axis is b, and the angular velocity around the z-axis is c.
After doing some reading, I'm not sure whether this is correct, and even if it is correct, if it is optimal.
Below, I have written three different things that might happen if I were to implement my controller like this. Which of these three is correct?
1) The end-effector rotates to the target pose in a smooth line (the most direct path).
2) The end-effector rotates to the target pose, but the motion is not smooth (it does not follow the most direct path).
3) The end-effector rotates but it does not end up at the target pose.