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If a 6 DOF robotic manipulator does not have a spherical wrist (the axis of rotation of joints 4, 5, and 6 coincide at the same point), the inverse kinematics are incredibly difficult.

Is ROS (Robot Operating System) capable of creating a closed loop control system for that arm? I know that it can do inverse kinematics, but I don't think it can do general geometry.

Thanks so much for your help!

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Yes, ROS (or really, MoveIt) has general purpose IK plugins that can solve inverse kinematics for arbitrary geometry (both joint configuration and degrees of freedom). There are a number of plugin options for IK solvers, but I'll mention that the general purpose solvers are not analytic, closed form solutions - they are numerical approaches (e.g. the default is KDL).

There are other plugin options as well (just google "ROS Inverse Kinematics" and you should find a nice list), including some solvers that can provide analytical solutions for some geometries (but perhaps more than you might think). IKFast isn't terribly user friendly and is a bit out of date, but if you can get it working, it is undeniably fast.

Outside of the ROS realm, I also stumbled across and had some success with the IKBT project which can provide more human-friendly analytic IK solutions for surprising number of geometries when provided with DH Parameters.

Again, there are some robot geometries that are as of yet, "unsolved" in terms of a closed form IK solution. That's why general purpose numerical solvers are so prevalent and useful. So it really depends on your use case and your needs

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