Redundancy Parameterization of the ABB YuMi Robot Arm

TL;DR

This work resolves a key gap in YuMi autonomy by delivering a complete, validated SEW-angle definition that reconciles the ABB controller with a general SEW-angle framework. It enables closed-form forward kinematics, a SEW-angle Jacobian, and reliable IK via 2D search, facilitating outside-RobotStudio path planning for a nontrivial 7-DOF redundant arm. The authors thoroughly analyze singularities, distinguishing kinematic, augmentation, and SEW-angle-related algorithmic cases, and propose the stereographic SEW variant to mitigate coordinate singularities. Empirical comparisons against RobotStudio validate the correct SEW definition and reveal limitations in previous formulations, guiding improved IK strategies and future analytical methods for high-DOF redundant manipulators.

Abstract

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.

Figures (5)

• Figure 1: Kinematics of YuMi including locations for shoulder $\mathcal{O}_S$ and wrist $\mathcal{O}_W$. The SEW angle is calculated using the direction of joint axis 4 $\vec{h}_4$, so no elbow point is defined. In this case, the reference vector $\vec{e}_r$ is chosen to be equal to $\vec{e}_{z,0} = \vec{h}_1$. The joint angle vector is $q = [0\ {-\pi}/3\ 0\ \pi/3\ 0\ 0\ 0]$. The unit vectors $\vec{e}_i$ are shown with length 100 mm.
• Figure 2: SEW angle calculation. The 4th joint axis direction $\vec{h}_4$ and the reference direction $\vec{e}_r$ are projected onto the plane normal to the shoulder-wrist direction $\vec{p}_{SW}$. The conventional SEW angle is the angle between these two projected vectors. The SEW angle used by ABB is offset by a quarter turn.
• Figure 3: Singularity regions using the conventional and stereographic SEW angles. With the conventional SEW angle, the arm encounters a coordinate singularity when the wrist is on the line passing through the shoulder in the $\vec{e}_r$ direction (blue and red arrows). With the stereographic SEW angle, the arm only encounters a coordinate singularity on a half-line. The half-line can be placed in the $-\vec{e}_r$ direction (red arrow) which is inside the robot base and out of reach.
• Figure 4: Finding all IK solutions using 2D search. The error function is multi-valued, and color indicates the minimum error among the four branches. The 10 IK solutions for this end effector pose and SEW angle occur where the error is zero and are marked with circled crosses. Black crosses lie within joint limits; red crosses lie outside.
• Figure 5: Finding all IK solutions using nested 1D search. For each $q_1$, all IK solutions achieving the desired end-effector pose are found. The 10 IK solutions lie on the horizontal line representing the desired SEW angle are marked with circled crosses. Black crosses lie within joint limits; red crosses lie outside.