Let's take a 6 DOF robotic structure. It's consisting of the 3 DOF global structure for the position - and the 3 DOF local structure for the orientation of the endeffector.

If the last 3 axis (of the local structure) are coincident in one point, the inverse kinematics can be solved analytically by decomposing it into a position- and orientation-problem.

But is it possible to solve the inverse kinematics analytically if the last 3 axis are NOT coincident in one point? I've read several papers that claim that due to high non-linearity of the trigonometric functions and motion complexity in 3D-space, a 6 DOF serial chain cannot be solved analytically.

Does anybody know if this is right?

  • $\begingroup$ Can you indicate which papers? $\endgroup$
    – Mhz4.77
    Apr 17, 2014 at 18:10

2 Answers 2


This paper seems to agree with you on the fact that 6 DOF arms exist that are not analytically solvable using inverse kinematics, but it also implies there are arm structures that can be analytically solved for, so I would recommenced sticking to those. Most 6 DOF robot arms don't have their last 3 axes coincident in one point, but they are still incredibly precise. Analytical solutions must exist for standard 6 DOF robotic arms.


The problem of inverse kinematics to a general 6-degrees-of-freedom serial robot was considered hard for a long time. Nevertheless it was solved and the solution in Raghavan and Roth (1993) is a widely acknowledged method, and improvements have also been made since (see e.g., Husty, Pfurner and Schröcker (2007)).

Although they provides a strategy to solve the inverse kinematics analytically, they do not give the solutions in closed form. All methods stop at a point where a single equation in one unknown variable, but a polynomial of degree 16 is obtained. The solutions to the remaining five variables are expressed in terms of this unknown, which can be found once the polynomial is solved numerically. Further, this polynomial is of degree 16 only in the worst case scenario, where all the joints are rotary. Any further simplification in the architecture only reduce the degree of this polynomial.

These methods use advanced mathematical techniques to solve the problem, which are beyond the scope of this space but a simplified outline of the steps followed in Raghavan and Roth (1993) can be seen in slides 82-91 of this article.


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