Polynomial Method for Inverse Kinematics using Subproblem Decomposition

Solving robot IK by finding zeros of a polynomial, from "IK-Geo: Unified Robot Inverse Kinematics Using Subproblem Decomposition" and "Redundancy parameterization and inverse kinematics of 7-DOF revolute manipulators".

Whenever search is used with subproblems to solve inverse kinematics problems, an alternative method can be used that guarantees finding all solutions. Each subproblem can be written as a multivariate polynomial in $x_i = \tan(q_i/2)$. The error function can also be written as a similar polynomial. Then, the resultant of this system of polynomials can be found to eliminate all but one variable. The zeros of this polynomial can be easily obtained, and these zeros correspond to the zeros of the error function. Subproblems can then be used to find the remaining joint angles.

For closed-form and search-based solutions for 6R robots, see ik-geo.

For closed-form and search-based solutions for 7R robots parameterized by the conventional or stereographic SEW angle, see stereo-sew.

Code in this repo depends on the above two repos.

Folder Breakdown

+polynomial_IK: Functions to generate system of polynomial equations, and functions to find remaining joint angles given 1 or 2 joint solutions

examples: Code to generate systems of polynomials for specific robots

examples_papers: Complete examples from the IK-Geo and Stereographic SEW angle papers. See READMEs in each subfolder.

maple: Code to solve systems of polynoimals in Maple (Reference code)

mathematica: Code to solve systems of polynomials in Mathematica (Not as good as Maple code)

polynomial_subproblems: Subproblem solutions and rotation matrix in terms of tangent half angle

robot_definitions: Kinematic parameters for example robots

unit_tests: Code to test certain MATLAB files