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It's hard for me to see anything from your description in your screenshot.

the robot dances around... which is fun to watch, but unacceptable for the application.

This sounds like a plan produced by the default planner configuration in indigo - we changed it in kinetic. Try to choose RRTConnect as your planner, if you didn't do that already..

One of which is shown in the following Screenshot. It shows the start State (not touching the frame), the state which is somewhere in the Trajectory and touches the frame, and the goal state. As you can see, there must clearly be a valid path!

MoveIt uses the FCL library for collision testing and until now only tests discrete states for collisions. If you have thin collision structures, it might miss collisions if the arm can move through them between collision checks.

Try to increase the amount of checks. You can do that by reducing longest_valid_segment_fraction in the group section of your ompl_planning.yaml. We currently advise a value of 0.005 which should be good enough for your setup.