General frame for arbitrary 3R subproblems based on the POE model

Inverse kinematic (IK) problems based on the product of exponentials (POE) model are often transformed to a series of Paden-Kahan subproblems, which are based on geometric methods with geometric constraints. The focus of subproblem is to solve the 3rd order subproblems, among which the 3R-type subproblem is the most prevalent. In this paper, a general frame for the inverse solution of arbitrary 3R types, without geometric constraints, is presented. It can be applied in different 3R types of practical cases, including those with parallel, intersecting, and skewed relationships. This paper mainly focuses on: (1) developing a real algebraic geometric (RAG) method based on the properties of the screw theory and the Rodrigues' rotation formula; (2) obtaining the closed-form solutions for arbitrary 3R subproblems, and ensuring the accuracies of these solutions; (3) expanding the Paden-Kahan subproblems and meeting the demands of online real-time applications; and finally, (4) verifying the effectiveness of the RAG method, through comparisons with the geometric method, using simulations and real experiments. The proposed frame can be widely applied in series, reconfigurable, and other types of robots.