Open problem: solving the Forward Kinematics Model in real time

Firstly, this problem was proven to have 40 complex solutions by Ronga and Vust for the general 6-6 Gough platform [Ronga and Vust 1992] .

Since the time when Raghavan solved some FKP cases of a 6-6 Gough Platform using homothopy continuation [Raghavan 1993], a large number of publications tried to address this problem. At Montreal´s McGill University, Husty applied resultants for variable elimination and solved several cases, [Husty 1996], however, I have proven how this method can lead to extraneous roots in my recently published book chapter [Rolland 2008]. One of the usual misconceptions is to consider that we can calculate an equivalent univariate equation derived from a system of six to nine equations non-linear multi-variate equations. In the same paper, I have also shown how homothopy can easily miss some complex solutions.

The issue became to find a proven method to solve the FKP in all instances. More recently, a method based on Groebner bases was developped in a joint venture between the INRIA (Nancy) and Laboratoire Informatique de Paris of the University Paris 6, this method allows to produce an equivalent triangular or even band equation system. One equation system happens to be a univariate equation of degree 40, [Rolland 2005]. This method is algebraic and this means that it can only handle parameters which are rational. However, the results are guaranteed and exact.

Since, real numbers cannot be handled by any computer and relying on floating numbers can be problematic, the Coprin team led by Merlet studied the idea of applying intervals which can contain any real number, [Merlet 2004]. Adapting the methods developped by Hansen, he implemented interval arithmetics with Newton methods which is renowned very fast. The method was certified through the application of Kantorovich theorem as proposed in the thesis of Olivier Chetelat but they used a strong version of it as I have reviewed them in my own PhD thesis. One open problem is to find a version of Kantorovich which can increase the size of the calculated conversion ball which is always smaller than the real one and hence this leads that some problems cannot be solved by this method. It is therefore certified but it does not guarantee to find an exact solution as it is defined by Lazard to all 6-6 problems. The response times of this method are still larger than the Groebner basis one.

To document, in my own tests on a singularity free SSM manipulator (designed by Merlet), David Daney of Coprin asked me to analyse and verify (using Groebner bases and the RUR) his first calibration method using optimization techniques based on Least Squares which also used Newton's method to compute the FKP. It is a pity that I was not put in the author of his publication but anyway I could make observations on the usual problems of numerical instabilities and also Jacobian inversion which resulted in the failure of 5% of the FKP cases. This means that Newton is NOT suitable for proper real time operation.

There exists a test which allows to verify the validity in order to truly certify a method. You just have to try the selected FKP solving method on the SSM and see if it can find the 36 complex solutions. I did introduce this result in my seminar that I gave in 1999 in Oxford and minutes after, I saw Merlet arguing with me that my result was false to the point that Rouillier had to intervene to explain that my result was proven since the Groebner method used is proven and the results obtained with it are then guaranteed. To this date, Merlet never recognised this, it is still not written in his book and leaving an error in it. Hence, for this reason, I am not sure if Merlet's Newton method with Intervals can be considered a certified method as he advocates (it will not be the first area where I disagree with him and I still have to pay the price).

Now, these two methods (Algebraic and Interval) can be used to verify other methods but I am still not sure if the Interval Newton one can find the 36 complex solutions to the SSM, therefore maybe I have reserves with the Newton method with .

We are now looking into solving the FKP in real time for control applications.

That is the reason why I have worked professor Hernandez of Bilbao Superior School of Engineering to eventually apply his Geometric Iterative Method to solve any Kinematics problems since it as fast as Newton or caan become even faster as complexitty of the manipulator increases.

I am also now studying neural networks with Rohitash Chandra as a research assistant to be tuned to real time solving of FKPs.
 
Methods available to solve the FKP:
- Neural networks
- The Geometric Iterative Method which can be faster than the Newton one, [Petuya et al. 2005].

References:

[Husty 1996] M. Husty. An algorithm for solving the direct kinematic of stewart-gough-type platforms. J. of Mechanism and Machine Theory, 31(4):365{379, 1996.

[Merlet 2004] J.P. Merlet. Solving the forward kinematics of a gough-type parallel manipulator with interval analysis. The International Journal of Robotics Research, 23(3):221{235, March 2004.

[Petuya et al. 2005] V. Petuya, A. Alonso, O. Altazurra and A. Hernandez. Resolution of the direct position problem of the parallel kinematic platforms using the geometric-iterative method. In IEEE Intern. Conf. on Robotics and Automation, Barcelona, pages 3255-3260, 2005.

[Raghavan 1993] M. Raghavan. The stewart platform of general geometry has 40 configurations. J. of Mech. Design, Trans ASME, 115(2):277-282, 1993.

[Ronga and Vust 1992] F. Ronga and T. Vust. Stewart platforms without computer ? In Proc of the Intern. Conf. of real, analytic and algebraic Geometry, pages 197-212, 1992.

[Rolland 2005] L. Rolland. Certified solving of the forward kinemactics problem with an exact method for the general parallel manipulator. Advanced Robotics, 19(9):995{1025, October 2005.

[Rolland 2008] Synthesis on modeling and certified solving of the kinematics problems of Gough-Type parallel Manipulator with an exact algebraic method, book chapter in Parallel Manipulators, Towards New Applications, Huapeng Wu Editor, I-Tech Education and Publishing, Vienna, 2008.