I mean if I minimize the second norm for the acceleration for each joint that means I'm minimizing the the second norm for the torque ? and if this is true that means I'm minimizing the energy consumption in the robots?
This will not be true always. For example, if the robot is used to push against a rigid wall, or if the robot is simply supporting a load, the acceleration in the joints will be zero, but the motor torques will not be. In the case of pushing against the wall, you could be spending a lot of energy without observing any acceleration in the joints.
what is the answer if the robot is a material handling robot like SCARA robot and it uses to execute the pick and place tasks?
One could try and prove your proposition for the movement case. I do not know for sure. However, my hunch is that the one does not imply the other, and you are better off minimizing torques rather than joint accelerations if that is what you ultimately want. There are two reason. The first is that in a serial chain, the acceleration measured at one joint is not due to the torque produced at that joint alone. There are interaction effects (powering one joint in a serial chain produces acceleration at ALL joints in that chain). The second is that the mass distribution of your arm will perhaps play a critical role. Imagine a point-to-point movement task which can be accomplished by either (1) a small movement of a proximal joint, or (2) large movements of distal joints. In (1) you will have small joint accelerations but large moments of inertia, and in (2) you may have larger joint accelerations but smaller moments of inertia. The mass distribution could be such that in (1) you have larger torque requirement despite the smaller acceleration as compared to (2) which could have lower torque requirement despite the higher acceleration.
One could simulate such a problem and, if lucky, disprove your hunch and thus avoid the trouble of proving stuff.
The path taken decides the usage of energy in an activity like pick and place whereas the acceleration is how fast the task is done. Both are linked because the higher the acceleration, the higher the energy involved.
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