This is in context of a robotic arm whose tip is attached to a vacuum gripper. The theoretical holding force of a vacuum gripper is given by,

$$ F_H = m(g+a)S$$

which is, $$F_H = (F_g + F_a)S $$

where, $m$ is mass of the payload, $g$ is the acceleration due to gravity, $a$ is the acceleration experienced by the payload when the arm is moving and $S$, is the safety factor. My understanding of $F_H$ is that it is a resultant of gravity and acceleration of the payload being held by vacuum gripper. If the vacuum suction force doesn’t equal this then the object will slip when the robotic arm is in motion. Is this understanding correct?

The above formula varies slightly when the vacuum gripper is moving vertically or horizontally.

For example, when the suction cup is in horizontal position, and the movement is in vertical direction

$$F_H = m (g + frac {a} {Î¼}) S $$ which can also be written as:

$$F_H = (F_g + frac {F_a} {Î¼}) S $$

$Î¼$ is the frictional coefficient of the payload. Now, how do I account for the gripper orientation when it is at an arbitrary angle holding the payload. I want to understand how this angle influences the $F_H$.

If I get the end effector acceleration $a_x, a_y, a_z$ from Jacobian:

$$ ddot{x} = J(q).q + frac {d} {dt}left(J(q)right)dot{q} $$

Would the following calculation be right to calculate the $F_H$ accounting for orientation?

$$F_H = left(mbegin{bmatrix}0 \ 0 \ -9.8 end{bmatrix} + frac {1}{Î¼}begin{bmatrix}a_x \ a_y \ a_z end{bmatrix}right) S $$

I would be glad if you could point me to some right links.

Reference: https://www.festo.com/wiki/en/Holding_and_break-away_forces_on_suction_cups