Solving Linear Systems for Multibody Systems

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Good afternoon,
I'm currently coding a Multibody Foward Dynamcis Simulator in MATLAB, for my master thesis, and I am having issues with solving linear systems due to low RCOND values, ill-condition of the matrices.
This problem is expected in this type of problems since each column entry in a line corresponds to the elements of a joint equation for two bodies resulting in a highly sparse matrix and consequently ill-conditioned matrices.
What my algorithm does is it calculates the initial accelerations of bodies in order for them to be integrated through an ode solver for position and velocity. My problems arise in solving the linear system for the initial accelerations that will be integrated. The system I have to solve is the following:
Where: M is a mass matrix, Phiq is a Jacobian, alpha/omega/mu are scalars, Phi, gamma and upsilon are vectors.
For my algorithm I tried using pinv(A)*B or lsqr(A,B), but I am not able to get the correct results (only get constant values from initial time until the last integration). On the other hand I get good results by using mldivide or \, but i get the following message:
Warning: Matrix is close to singular or badly scaled. Results may be inaccurate. RCOND = 1.394430e-19.
I wanted to know if there is anyway to avoid this issue, since I know it can cause significant noise in my results, I usually use pinv for this but it seems to not be working this time.
Thank you for your time and attention.
Tiago
(Attached files are the left and right hand side of the linear problem)
  4 Comments
Tiago Carvalho
Tiago Carvalho on 21 Jul 2022
Edited: Tiago Carvalho on 21 Jul 2022
Hello Torsten,
I don't think I can do that in this case in particular, since my algorithm follows the flowchart in the Picture. The process of this function it self is iterative inside the ode, in order to ensure that the constraints of my specific problem are solved, I don't see how I could integrate that outside the ode to allow him to do the inversion.
(EDIT) What the ode integrates is the product of the equation that I previously shared and the velocity that is an input of the odefun.

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Accepted Answer

Bjorn Gustavsson
Bjorn Gustavsson on 22 Jul 2022
As best I can interpret your flow-chart it seems that it ought to be "reasonably straightforward" to follow Torsten's advice by converting the equation for into one for both and . If I get it right it should be something like:
function Mout = modified_massmatrix(M)
Mout = [eye(size(M,1)),zeros(size(M));
zeros(size(M)),M];
end
or if the mass-matrix is a function of t, q and :
function Mout = modified_massmatrix(t,qqdot,M)
Mnow = M(t,qqdot);
Mout = [eye(size(Mnow,1)),zeros(size(Mnow));
zeros(size(Mnow)),Mnow];
end
Then you should be good to go with the ODE-integrating functions.
  8 Comments
Tiago Carvalho
Tiago Carvalho on 22 Jul 2022
Hello @Sam Chak, no I use the standard Newton-Euler's equation and Lagrange Method, not Kane's.
My main source of information was the P.E Nikravesh book titles Computer Aided Analysis of Mechanical Systems, which suggested the use of the Newton-Euler's, to be 100% honest I am not familiriazed with Kane's method but I will research it to see if it can simplify my issues!

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