Redundancy parameterization and inverse kinematics of 7-DOF revolute manipulators

Seven-degree-of-freedom (DOF) robot arms have one redundant DOF for obstacle and singularity avoidance which must be parameterized to fully specify the joint angles for a given end effector pose. Commonly used 7-DOF revolute (7R) industrial manipulators from ABB, Motoman, and KUKA and space manipulators like SSRMS or FREND are conventionally parameterized by the shoulder-elbow-wrist (SEW) angle for path planning and teleoperation. We introduce the general SEW angle which generalizes the conventional SEW angle with an arbitrary reference direction function. Redundancy parameterizations such as the conventional SEW angle encounter an algorithmic singularity along a line in the workspace. We introduce a reference direction function choice called the stereographic SEW angle which has a singularity only along a half-line which can be out of reach, enlarging the usable workspace. We prove all parameterizations have an algorithmic singularity. Finally, using the general SEW angle and subproblem decomposition, we provide efficient singularity-robust inverse kinematics solutions which are often closed-form but may involve a 1D or 2D search. Search-based solutions may be converted to finding polynomial roots. Examples are available in a publicly accessible repository.