I have successfully formulated the direct kinematic model of eight degrees of freedom serial manipulator (PPRPRRRR) where no three consecutive joint axes are either parallel or coinciding, but closed form solution techniques are not helping me out to carry out inverse kinematic modelling. I ended up with total 6 independent nonlinear transcendental equations in eight unknowns. Which numerical techniques will be helpful?
I am adding the figure of the manipulator as an attachment.
It is better to use d-h parameters table for analysing and formylating 6 dof cartaseian robot manipulator. There is book which has been written by craig. The title of the book is introduction to robotics
Numerical techniques based on Jacobian inverse or Jacobian transpose could save the day.
Some related papers:
http://synthetica.eng.uci.edu/~mccarthy/BernieRothCD/PapersOld/Siciliano.pdf
http://math.ucsd.edu/~sbuss/ResearchWeb/ikmethods/iksurvey.pdf
http://www.dcs.shef.ac.uk/intranet/research/public/resmes/CS0406.pdf
Often in such scenarios, it is always advisable to break down the system to represent them as Denavit-Hartenberg parameters. if you are new to the concept, you may follow the Wiki Link
http://en.wikipedia.org/wiki/Denavit%E2%80%93Hartenberg_parameters
Or you may utilise this online resource to understand a step-by-step process of the DH Parameters
http://www4.cs.umanitoba.ca/~jacky/Robotics/Papers/spong_kinematics.pdf
As previously suggested, Denavit-Hartenberg parameteres will be the best and easiest option to solve this.
I always thought that DH parameters helped in the forward kinematic solutions and the inverse kinematic solution may use the resulting Homogeneous Transformation matrix but not exclusively. The difficulty with this robot is that it is redundant and so finding a unique solution is difficult. Like Peter said, I would calculate the Jacobian of the system and then solve numerically. Try to look into using damped leased squares methods or other techniques similar to this like or the SC (Singularity Consistent) approach. Note that this would not require representation using DH but its always good practice to use DH for standards compliance and maybe for writing papers(laugh).
Since this is a redundant robot with 8 dof, you have 8 unknowns and 6 equations describing the pose of the end effector. For such situations, you need a numerical approach such as pseudo-inverse method.
I think there are some problems of using Pseudo_inverse method where the solution near the singularity region will be unstable or oscillating about this region, the use of Singularity-Robust by Nakamura is very useful for this kind of redundant robot
I *believe* that the singularity-robust approach (in Nakamura's book) is based on the the pseudo-inverse of the Jacobian, but with the introduction of a matrix C which regulates the pseudo-inverse to deal with cases where det(J * transpose(J)) = 0. Typically, the problem is formulated as a constrained optimization problem. BTW, Nakamura's book is an excellent source to learn more about how to deal with singularity.
I think DH is the best option for this. And as suggested by Jugesh Sundram, we split the system from end effector connection and then it is easy to solve the inverse. for end effector we can apply Euler's approach also.
The robot is a kind of Festo applications, It should be better to get some details from Festo Germany. It seems all joints are linear. You should ask them about forward and inverse kinematics such robots.
Also, The solution will be written in matrix form using D-H parameters and table.
If you have already have the Jacobian matrix of kinematic velocities, you either can obtain its pseudoinverse by More-Penrose (J^T J)^{-1} J^T, or by singular value decomposition (SVD), due to its kinematic redundancy.
When You have a manipulator with redundancy You must use a numerical iterative method. In my practice I use the coordinate descent method.
For more see the next papers:
http://www.researchgate.net/publication/234014763_Manipulator_Motion_Planning
http://www.researchgate.net/publication/234014789_Computer_Simulation_of_Industrial_Robots
Conference Paper Manipulator Motion Planning.
Conference Paper Computer Simulation of Industrial Robots.
I think the method as Genetics and other evolutionary search method are could not to be online method (Time Consuming) therefore there are many method after try to solve this problem(s) using the geometric and/or algebraic methods such as Nakamura's method ...etc
I solely agree with @Ra'aed Sadoon. I presently solve IK of a similiar robot configuration believe me GA and most iterative methods will no work online. Geometric methods seem better over pure algebraic methods also.
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