Redundancy parameterization and inverse kinematics of 7-DOF revolute manipulators

Abstract: Seven degree-of-freedom (DOF) robot arms have one redundant DOF which does not change the motion of the end effector. The redundant DOF offers greater manipulability of the arm configuration to avoid obstacles and 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 widely used and easy for human operators to visualize as a rotation of the elbow about the shoulder-wrist line. Since other redundancy parameterizations including the conventional SEW angle encounter an algorithmic singularity along a line in the workspace, we introduce a special choice of the reference direction function called the stereographic SEW angle which has a singularity only along a half-line, which can be placed out of reach. 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 efficient and singularity-robust 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 in the general case. Search-based solutions may be converted to finding zeros of a high-order polynomial. Inverse kinematics solutions, examples, and evaluations are available in a publicly accessible repository.

• The paper introduces a generalized SEW angle that overcomes singularity limitations to enhance control precision in 7R manipulators.
• The stereographic SEW angle minimizes algorithmic singularities by confining them to non-critical workspace regions, improving operational stability.
• The unified inverse kinematics approach leverages subproblem decomposition and efficient search-based techniques to provide robust closed-form solutions.

Redundancy Parameterization and Inverse Kinematics of 7-DOF Revolute Manipulators

The paper by Elias and Wen presents innovative approaches to parameterizing the redundancy in 7-DOF (Degrees of Freedom) revolute manipulators, focusing on the characterization and solution of inverse kinematics (IK) problems. It addresses key issues related to parameterizing the redundant DOF inherent in 7R robot arms, which are crucial in both industrial and space robotics applications. The paper introduces a new concept of the generalized shoulder-elbow-wrist (SEW) angle and advances the understanding of redundancy parameterization.

Key Contributions

1. Generalized SEW Angle: The authors propose a generalized SEW angle, broadening the conventional SEW angle formulation by allowing an arbitrary choice for the reference direction function. This advancement addresses the limitations posed by singularities in the conventional SEW angle, improving manipulability and intuitive control in diverse robotic applications.
2. Stereographic SEW Angle: The novel stereographic SEW angle reduces susceptibility to singularity by concentrating any algorithmic singularity along a singularity in only one direction—outlining a more controllable workspace and mitigating algorithmic singularities that could disrupt operations. This singularity can be positioned in non-critical regions of the workspace, thus enhancing the robot's operational stability.
3. Existence and Classification of Singularities: The paper asserts the inevitability of singularities in redundancy parameterization through topological considerations. Any smooth parameterization must encounter singularities due to the intrinsic topology of 7R manipulators.
4. Inverse Kinematics Solutions: The authors extend a unified inverse kinematics (IK) approach using the generalized SEW angle, promoting closed-form solutions and computational robustness across various 7R manipulator configurations. They utilize a subproblem decomposition method tailored to solve the IK issue, emphasizing efficiency and stability over iterative Jacobian-based solutions. For cases where direct solutions are infeasible, their methodology transitions seamlessly into search-based or polynomial root-finding techniques.

Practical and Theoretical Implications

Practical Implications: The enhanced SEW parameterization significantly improves the utility and control precision of 7R manipulators, ideal for complex tasks like teleoperation in space robotics, where maneuverability around obstacles and the avoidance of algorithmic singularities are paramount. These innovations translate to more reliable robotic operations in constrained environments, enhancing the viability of robots in sensitive and precision-demanding tasks.

Theoretical Implications: The classification and proof of the unavoidable nature of certain singularities contribute to the fundamental understanding of redundancy parameterizations. The stereographic SEW angle, in particular, offers a new lens for considering the interplay between geometry and operations in robotics, with opportunities for further exploration in topological and algorithmic singularity analysis.

Future Directions

Continued research could expand on exploring more parameterization schemes that balance computational efficiency with operational robustness. Additionally, extending these solutions to dynamically varying tasks and environments could leverage machine learning and adaptive control theories to further enhance the real-world applicability of 7R robotic systems across various domains.

This paper marks a significant stride in addressing redundancy resolution in robotics, providing both a theoretical framework and practical approaches that bridge known gaps in existing methodologies. The work of Elias and Wen sets a foundational platform from which future research in advanced robotic manipulations can evolve, especially towards fully exploiting the capabilities of redundant degrees of freedom in robotic design and function.