Newton Rhapson Failure Analysis

In summary, Hari is trying to solve a system of three equations for the location of a point in a robot. He has a certain trajectory for the point, but the NR algorithm fails at some points. He asks for help, and provides some additional information.
  • #1
hariharan82
3
0
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
 
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  • #2
hariharan82 said:
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.
 
  • #3
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
 
  • #4
hariharan82 said:
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.
 
  • #5
Admin,
thanks once again for the prompt reply. i will give LM a shot and compare the execution times.

Hari
 
  • #6
hariharan82 said:
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

$\chi$ $\sigma$
 

FAQ: Newton Rhapson Failure Analysis

What is Newton Rhapson Failure Analysis?

Newton Rhapson Failure Analysis is a mathematical method used to find the roots of a function. It is often used in engineering and scientific fields to solve complex equations and identify potential failures in systems.

How does Newton Rhapson Failure Analysis work?

This method uses an iterative process to estimate the root of a function. It starts with an initial guess and then uses the derivative of the function to refine the guess until a satisfactory approximation is reached.

What are the benefits of using Newton Rhapson Failure Analysis?

One of the main benefits of this method is its efficiency in finding roots of a function. It can quickly converge on a solution, making it a valuable tool for solving complex problems. It is also a versatile method that can be applied to a wide range of functions.

What types of problems can Newton Rhapson Failure Analysis solve?

This method is commonly used in engineering and scientific fields to solve problems involving non-linear equations, such as finding the maximum or minimum of a function, determining the stability of a system, and identifying potential failure points in a system.

Are there any limitations to Newton Rhapson Failure Analysis?

While this method is efficient and versatile, it does have some limitations. It may not converge on a solution if the initial guess is too far from the actual root, and it may also fail if the function has multiple roots or is discontinuous.

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