0.2 Protein inverse kinematics and the loop closure problem  (Page 3/5)

While the above example offers equations that are easy to solve, general inverse kinematics problems require solving systems of nonlinear equations for which there are no generalalgorithms. Some inverse kinematics problems cannot be solved analytically. In robotics, it is sometimes possible to design systems to have solvable inverse kinematics, but in the general case, we must rely on approximation methods in order to keep the problem tractable, or, in some cases, even solvable. For examples on how to address inverse kinematics in particular robotic systems, please read chapter 4 of . An illustration of the solutions of the inverse kinematics problem for a robot which is widely used in industry is shown below.

Inverse kinematics methods

Inverse kinematics methods are categorized into two main groups:

• exact, classic, or algebraic methods
• heuristic, or optimization methods

Classic inverse kinematics methods

It is known that for manipulators with no more than six degrees of freedom, there is a finite number of solutions to the inverse kinematics problem . There is, however, no analytical method that can find these solutions forall types of manipulators. For manipulators with only revolute joints, which is the case for biomolecules with idealized geometry, thenumber of unique solutions is at most 16, when the number of degrees of freedom does not exceed six . An efficient solution was proposed in and later applied to the conformational analysis of small molecular chains , . Methods based on curve approximation were proposedin for the inverse kinematics of hyper-redundant robots, where the number of regularly distributed joints is very large.

Specialized solutions to inverse kinematics in biochemistry appeared as early as 1970 , where fragments of up to 6 degrees of freedom were predicted by solving a set of polynomialequations representing geometric transformations. These equations were applied to building tripeptide loops under the ideal geometry assumption. Later work , , , offered efficient analytical solutions for three consecutive residues through spherical geometry and polynomial equations. Boundinginverse kinematic solutions for chains with no more than six degrees of freedom within small intervals has also been shown relevant in the context of drugdesign . A new formulation that extends thedomain of solutions to any three residues, not necessarily consecutive, and with arbitrary geometry, was recently proposed . Current work that pushes the dimensionality limit from six to nine degrees of freedom makes use of an efficientsubdivision of the solution space .