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I haven't used Moveit! much, and this might not be what is causing your specific problem, but KDL, which is what Moveit! uses for inverse kinematics by default, is really only meant for arms with at least 6 degrees of freedom.

For arms with less than 6 dof, I would recommend either using an IKFast solution or solving the inverse kinematics by hand. Your arm sounds like an articulated manipulator, so you could google for "inverse kinematics articulated manipulator" and you will likely find a solution that is ready to use (but solving it by hand isn't too hard). Personally, if it were me, I'd just use IKFast because it is straight forward and requires less thinking lol.

As for solving using Cylinderical and Cartesian Coordinates, just solve using Cartesian Coordinates, then have a wrapper functions that converts cylinderical into cartesian before being input to the inverse kinematics solvers...and then convert the output of the IK solver from cartesian back to cylinderical before returning.