The ABB YuMi is a 7-DOF collaborative robot arm with a complex, redundant kinematic structure. Path planning for the YuMi is challenging, especially with joint limits considered. The redundant degree of freedom is parameterized by the Shoulder-Elbow-Wrist (SEW) angle, called the arm angle by ABB, but the exact definition must be known for path planning outside the RobotStudio simulator. We provide the first complete and validated definition of the SEW angle used for the YuMi. It follows the conventional SEW angle formulation with the shoulder-elbow direction chosen to be the direction of the fourth joint axis. Our definition also specifies the shoulder location, making it compatible with any choice of reference vector. A previous attempt to define the SEW angle exists in the literature, but it is incomplete and deviates from the behavior observed in RobotStudio. Because our formulation fits within the general SEW angle framework, we also obtain the expression for the SEW angle Jacobian and complete numerical conditions for all algorithmic singularities. Finally, we demonstrate using IK-Geo, our inverse kinematics (IK) solver based on subproblem decomposition, to find all IK solutions using 2D search. Code examples are available in a publicly accessible repository.

Seven degree-of-freedom (DOF) robot arms have one redundant DOF which does not change the translational or rotational motion of the end effector. The redundant DOF offers greater manipulability of the arm configuration to avoid obstacles and steer away from singularities, but it must be parameterized to fully specify the joint angles for a given end effector pose. For 7-DOF revolute (7R) manipulators, we introduce a new concept of generalized shoulder-elbow-wrist (SEW) angle, a generalization of the conventional SEW angle but with an arbitrary choice of the reference direction function. The SEW angle is easy for human operators to visualize as a rotation of the elbow about the line from the shoulder to the wrist and has been used in the teleoperation of space robot arms. Since the conventional SEW angle formulation is prone to singularities, we introduce a special choice of the reference direction function called the stereographic SEW angle which has a singularity in only one direction in the workspace. We prove that such a singularity is unavoidable for any parameterization. We also include expressions for the SEW angle Jacobian along with singularity analysis. Finally, we provide inverse kinematics solutions for most known 7R manipulators using the general SEW angle and the subproblem decomposition method. These solutions are often closed-form but may sometimes involve a 1D or 2D search. Inverse kinematics solutions, examples, and evaluations are available in a publicly accessible repository.