Newton Rhapson Failure Analysis

[hariharan82]

Hi,

My name is Hari and i am new to this forum. I am currently working on a research problem using Newton rhapson method to find solution for a system of non-linear functions. The precise application is Inverse Kinematics of manipulators. However, i have three non linear equations function of three variables. the equations represent location of a point in a robot. i need to find the three variables that will satisfy the given location of the point.

I have a certain trajectory for the point in the robot. i need to solve for the variables at every point in the trajectory using the NR method. the NR works for 90% of the points but has some spots where it fails. These points are generated very close to each other. So the initial condition for the NR is the solution from the previous point. i am not able to understand why the NR method fails at certain spots and not everywhere else. I would like some help with regards to this as i have spent significant amount of time with no solution. Please let me know if you need any other information to make this as clear as possible. I am sure the problem is vague now.

thanks
Hari

 

[I like Serena]

Hi,

My name is Hari and i am new to this forum. I am currently working on a research problem using Newton rhapson method to find solution for a system of non-linear functions. The precise application is Inverse Kinematics of manipulators. However, i have three non linear equations function of three variables. the equations represent location of a point in a robot. i need to find the three variables that will satisfy the given location of the point.

I have a certain trajectory for the point in the robot. i need to solve for the variables at every point in the trajectory using the NR method. the NR works for 90% of the points but has some spots where it fails. These points are generated very close to each other. So the initial condition for the NR is the solution from the previous point. i am not able to understand why the NR method fails at certain spots and not everywhere else. I would like some help with regards to this as i have spent significant amount of time with no solution. Please let me know if you need any other information to make this as clear as possible. I am sure the problem is vague now.

thanks
Hari

Welcome to MHB, Hari! :)

Most numerical algorithms fail for badly conditioned problems.
I would suggest to use the Levenberg-Marquardt algorithm instead.
It finds the optimal solution, and if there is no unique solution, it finds the solution closest to the initial guess.

 

[hariharan82]

Hi,

thanks for you reply. there is one reason why i am not using LM method, it is the time of execution. i am pressing on real-time aspect of the execution. i know a solution exist but for some reason the NR method does not converge. i also checked the condition number of the NR jacobian. it looks fine, similar to points it converges.

Hari

 

[I like Serena]

Hi,

thanks for you reply. there is one reason why i am not using LM method, it is the time of execution. i am pressing on real-time aspect of the execution. i know a solution exist but for some reason the NR method does not converge. i also checked the condition number of the NR jacobian. it looks fine, similar to points it converges.

Hari

Here are the common reasons why NR would fail or be slow to converge.

Btw, what's your reason to think that NR is faster than LM?
For multidimensional problems, NR is usually not the best choice, certainly not for speed of convergence.

 

[hariharan82]

Admin,
thanks once again for the prompt reply. i will give LM a shot and compare the execution times.

Hari

 

[chisigma]

Hi,

My name is Hari and i am new to this forum. I am currently working on a research problem using Newton rhapson method to find solution for a system of non-linear functions. The precise application is Inverse Kinematics of manipulators. However, i have three non linear equations function of three variables. the equations represent location of a point in a robot. i need to find the three variables that will satisfy the given location of the point.

I have a certain trajectory for the point in the robot. i need to solve for the variables at every point in the trajectory using the NR method. the NR works for 90% of the points but has some spots where it fails. These points are generated very close to each other. So the initial condition for the NR is the solution from the previous point. i am not able to understand why the NR method fails at certain spots and not everywhere else. I would like some help with regards to this as i have spent significant amount of time with no solution. Please let me know if you need any other information to make this as clear as possible. I am sure the problem is vague now.

thanks
Hari

A precise condition of convergence of the NRM is reported here...

http://mathhelpboards.com/discrete-mathematics-set-theory-logic-15/difference-equation-tutorial-draft-part-i-426.html#post2492

In most cases of non convergence the following alternative recursive equation...

$\displaystyle x_{n+1} = x_{n} - a\ \frac{f(x_{n})}{f^{\ '}(x_{n})}\ (1)$

... where 0< a< 1 is an appropriate constant can overcome the problem...

Kind regards

$\chi$ $\sigma$