Solution:

Pick point:

The pick point is the first point in the pick-and-place trajectory.The coordinate frame is denoted by $ T_{\text {base }}^{\text {pick }} $

This represents the initial position and orientation of the part being manipulated.

The pick position $p^{\text {pick }}$ is considered to be the center of mass or centroid.

The object is picked up using the approach vector $r^3$ which is perpendicular to the work surface $x^0 y^0$ plane on which the part is resting and the sliding vector $r^2$ moving inwards along the closing axis of the gripper or tool.

$d^{pick}$represents the vertical distance from the horizontal work surface to the pick point of the object.

Lift-off points:

The second point of the pick-and-place trajectory is the lift-off point. The coordinate frame is denoted by $T_{\text {base }}^{\text {lift }}$.

The lift-off point is a point near the pick point that the robot moves to before the robot reaches down to pick up the part.

The lift-off point is an intermediate point that is inferred from the pick point. The tool orientation at the lift-off point is identical to the tool orientation at the pick point. The tool orientation remains fixed as the tools move from the lift-off point to the pick point.

The tool position at the lift-off point is obtained by starting at the pick position and moving backward a safe distance $v$ along the approach vector, away from the pick point.