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Robot payload is really a wrist moment limit. The rated payload applies at a stated centre of gravity distance, so moving the load further out reduces the allowable mass in inverse proportion. The gripper's own mass comes off that budget before any part can be carried.
Payload at offset
allowable mass = rated payload x rated CoG / actual CoG, then subtract the tool mass
The wrist axes are torque limited, and torque is mass times distance. Doubling the offset halves the mass that can be carried.
No. Acceleration adds inertial load, so a robot at full payload usually has to be slowed, and the manufacturer's inertia limits must also be met.