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If I understand what you're suggesting, wouldn't this result in a lot more swinging since the gripper would be suspended from a single point directly above it?
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You seem to. That is a good point I hadn't considered - the tension from the sides holds it in place. Maybe we need to use a telescoping arm (which doesn't seem practical, I'm mostly joking). It might still work, but I imagine if you don't grab it on the first try, you may have to go back to the ceiling. And hope there are no strong drafts in the room.

EDIT: maybe lower it on three lines instead of one?

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That too is a solvable problem if you have the sensors for it. You can critically damp or choose a motion profile to have essentially no swing.
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How do you prevent having to grab things perfectly in their center of mass? (Because that requirement seems limiting)

Rigid objects would result in the gripper twisting, but the current design could offset some of that by pulling harder in one direction?

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With the video, it is casually dropping things like clothes and toys into a bucket, so I'm not sure it would be a big deal. But you could also lower it via three lines instead of one, so there is something to counteract that.
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