Fast and Robust Inverse Kinematics of Serial Robots Using Halley’s Method

Abstract

This paper proposes a novel numerical inverse kinematics algorithm called the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Quick Inverse Kinematics</i> or <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">QuIK</i> method. The QuIK method is a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">third-order</i> algorithm that uses both the first- and second-order derivative information to iteratively converge to a solution. Numerical inverse kinematics methods are readily implemented on any serial robot and do not rely on joint alignment. However, they typically are slower and less robust. The second-order derivative term allows the QuIK algorithm to converge more rapidly and more robustly than existing algorithms. A damped extension to the QuIK method is also proposed to increase reliability near singularities. The QuIK methods are tested in terms of evaluation speed, reliability, and singularity robustness against the Newton–Raphson method and several other modern algorithms. The proposed QuIK methods outperform all other tested algorithms in terms of speed and robustness, and have strong performance near singularities. The QuIK algorithms are proposed as faster and more robust “drop-in” replacements to the Newton–Raphson methods in inverse kinematics. C++ and MATLAB codebases are made available.