Caveat: I am shooting from the hip here and as I *don't* have experience with PKMs, and ignoring your specific request for a closed-form solution. Oh, well.
My experience in solving nonlinear multivariable problems is primarily with ordinary Levenbeg-Marquardt style approaches. In these methods the iteration is to (1) linearize the problem at the current approximate solution, (2) solve the linear problem (can be a direct Gaussian reduction of the matrix in many cases) producing a new approximate solution, (3) rinse and repeat.
My suggestion is to investigate whether the linearization needs to be done very often in your application. If you are working in a moderately-linear region of parameter space (e.g. *not* near gimbal lock of a rotary system) then a single linear model might serve with adequate accuracy for hundreds or thousands of cycles. A separate task could be updating the linear model parameters on a millisecond time scale; depending on how fast your PKM is moving, that might be adequate.
Can you share any more information about the application, e.g. how many degrees of freedom?
On Sun, Mar 3, 2013 at 11:52 PM, Pete Miles <robots@...>
Anyone here have any experience in solving Parallel
Kinematic Mechanisms? Stewart Platforms are one example of a PKM
I am working on a different type of a mechanism where I
need to drive the computation time to solving the system to less than 1us.
Dr Dan Zang, author of Parallel Kinematic Machine Tools tells me that the
inverse models of PKMs are not itereated but generally have closed form
solutions. The mechanism I am working on, I have yet to find a closed form
solution. I am wondering if there are methods do determin based on the
number of unknowns verses the number of drive motors and linkages if there is a
possible closed form solution verses an iteration requirement.
The papers I have been reading on solving techniques have
been based on mapping techniques and clever math techiques. But they
generally ignore the topic of pure computational time to solve the
Solving the models, is not the issues here. Its the
computational time needed to solve them. Iteration techniques work, and
work well, but they take too long when compared to direct closed form