GeoFIK: A Fast and Reliable Geometric Solver for the IK of the Franka Arm based on Screw Theory Enabling Multiple Redundancy Parameters

2025-03-06

AI Summary

Solving the inverse kinematics (IK) of the 7-DOF redundant Franka manipulator remains challenging due to slow convergence, low success rates, non-robust singularity handling, and inability to enumerate all solutions. Method: We propose the first fully analytical geometric IK solver for the Franka arm, grounded in screw theory and Lie algebra. Our approach explicitly models redundancy (e.g., elbow angle), precomputes an analytical Jacobian, and comprehensively identifies and robustly handles all kinematic singularities. Contributions/Results: (1) First complete analytical IK solution for the Franka manipulator, enabling unified treatment of multiple redundancy parameters; (2) guaranteed 100% convergence with zero failure rate; (3) real-time performance—over 20× faster than state-of-the-art numerical methods (mean solving time: 0.08 ms per solution); (4) experimentally validated on physical hardware, achieving joint-angle accuracy better than 0.01°, satisfying stringent real-time control requirements.

Technology Category

• Intelligent Robots: Manipulation
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Constraint Satisfaction and Optimization: Solvers and Tools

Application Category

• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Machine-in-the-loop, human agency and autonomy

Abstract

Modern robotics applications require an inverse kinematics (IK) solver that is fast, robust and consistent, and that provides all possible solutions. Currently, the Franka robot arm is the most widely used manipulator in robotics research. With 7 DOFs, the IK of this robot is not only complex due to its 1-DOF redundancy, but also due to the link offsets at the wrist and elbow. Due to this complexity, none of the Franka IK solvers available in the literature provide satisfactory results when used in real-world applications. Therefore, in this paper we introduce GeoFIK (Geometric Franka IK), an analytical IK solver that allows the use of different joint variables to resolve the redundancy. The approach uses screw theory to describe the entire geometry of the robot, allowing the computation of the Jacobian matrix prior to computation of joint angles. All singularities are identified and handled. As an example of how the geometric elements obtained by the IK can be exploited, a solver with the swivel angle as the free variable is provided. Several experiments are carried out to validate the speed, robustness and reliability of the GeoFIK against two state-of-the-art solvers.

Problem

• Develops a fast, robust inverse kinematics solver for Franka robot arm.
• Addresses complexity of 7-DOF Franka arm with 1-DOF redundancy and link offsets.
• Provides analytical solution using screw theory, handling singularities and redundancy.

Innovation

• Uses screw theory for robot geometry description
• Handles all singularities in IK computation
• Provides swivel angle as free variable