Numerically stable algorithm for Inverse Kinematics of 6R problem and its applications to macrocycles

In the protein loop closure problem, two ends of the loop region are constrained and the internal parameters need to be specified to close the loop. This problem is geometrically similar to the Inverse Kinematics problem in robotic arms where several internal parameters need to be determined to put the robotic hand at specific position and orientation. The analogous ring closure problem arises when the two ends coincide, such as in macrocycles. In this poster, we present a numerically stable algorithm R6B6[1] to compute all solutions of the inverse kinematics problem of a 6 revolute manipulator chain. A system of equations is constructed based on the fundamental closure conditions, leading to a closed algebraic system of 20 equations involving 16 quantities, composed of trigonometric functions of five among the six unknown joint angles. Two among these five are stably eliminated using singular value decomposition (SVD) avoiding the need to consider special cases. The resulting system of equations involving three unknowns is solved by conversion to a generalized eigenvalue problem. The remaining three unknown angles are obtained using the previously computed pseudoinverse. In this formulation we exploit the inherently complex form of the system reducing it to 10 complex equations in 9 quantities, which substantially accelerates the SVD computation. The method's robustness is demonstrated through a comparison to current methods and several examples including known problematic cases where some axis or link lengths vanish, or some joint angles are 180 degrees, as well as cases where multiple eigenvalues arise. The algorithm is incorporated in a macromolecular modeling server for macrocycles, combining the BRIKARD and ClusPro LigTBM suites for macrocycle conformational sampling, macrocycle structure determination using NMR distance restraints and peak data, automatic resolution of uncertainties in the NOESY peak assignment, and protein macrocycle docking. The web-server is freely available for noncommercial purposes at https://brikard.cluspro.org.