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Does the weight of the object to be picked by an arm manipulator, influence the trajectory/path planning by OMPL?

No. It does not. Or at least: not in the default situation/configuration/MoveIt integration, and not with the way OMPL is currently used.

I say currently, as more recent OMPL releases start getting more extensive support for cost functions, constraints and optimisation goals. Those could allow more advanced types of planning which do take these things into account, but for now that is not done/used in the MoveIt integration.

I would like to understand what is the cause of failure to pick an heavier object ( >3 kg in this case). How does MoveIt! check this?

MoveIt doesn't check this and I doubt MoveIt (or OMPL) is the cause of what you're describing in #q294326.