A novel second subproblem for two arbitrary axes of robots

The Paden–kahan subproblem is a simple and flexible method to solve the closed-form inverse resolution but limited by the geometrical structure of robots, which is very difficult to be kept because of processing and installation. Therefore, a closed-form solution on arbitrary configurations is an important issue in the field of robotic inverse kinematics. A novel second subproblem is firstly proposed in this study based on the product-of-exponentials model adapting to the two arbitrary axes without geometric constraints (parallel, vertical, disjoint, and so on). Furthermore, the algebraic methods involving the basic properties of the screw theory and Rodrigues’ rotation formula are employed for the solution, which makes the constraint equations of the second subproblem solvable for arbitrary configurations. This methodology can be applied to the inverse solutions of 5-degree-of-freedom robots that satisfies the Pieper criterion and can express the inverse solutions via two common formulas. Finally, the simulation and the real-world experiment demonstrated the accuracy of the method and the validity, respectively.