MATLAB: Curve-fitting by Levenberg–Marquardt method?

planar rigid body model

I am new to Matlab, but I want to write a matlab code to estimate the values of spring and damper coefficients to fit a function to data curve. I have a simultaneous ode equation for acceleration of a planar rigid body(mass=m) connected to a pair of spring and damper in parallel in x-axis and another pair in y-axis directions as follows.

M*D2X(t)=-Cx*DX(t)-Kx*X(t)+w*DY(t)

M*D2Y(t)=-Cy*DY(t)-Ky*Y(t)+w*DX(t)

where: M=mass C=damper coefficient K=spring coefficient w =yaw rate (constant value) In addition to the functions above, I have x-axis and y-axis direction acceleration data from experimental measurement. So, how can I estimate four parameters Kx ,Cx , Ky and Cy?? I appreciate any hint or help!!

Best Answer

So X,Y are positions or accelerations? If they are positions and you have acceleration data, D2X and D2Y you can integration that data numerically to obtain X,Y, DX,DY. Then, your equations become linear and you can use total lienar least squares to solve them.

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You need to use element-wise division:

Z = (cx*x.^2+cy*y.^2)./(1+sqrt(1-(1+Kx)*cx^2*x.^2-(1+Ky)*cy^2*y.^2)); % biconic
                     ↑ ← INSERT ‘.’ HERE

This will work.

MATLAB: How to solve this coupled ODE problem using ode solvers

If you want to use the odeNN functions you need to rewrite your 2 2nd-order ODEs into 4 1st-order ODEs, something like this:

function dxdydvxdvydt = your_eqs_of_motion(t,xyvxvy,other,parameters)
x = xyvxvy(1);  % If my ode-equations are complicated 
y = xyvxvy(2);  % I prefer to explicitly extract the
vx = xyvxvy(3); % variables into something that's
vy = xyvxvy(4); % easily human-readable
m = other;
Fx = force_functionx(t,x,y,vx,vy,other,parameters); % Adjust inputs as necessary
Fy = force_functiony(t,x,y,vx,vy,other,parameters);
% or possibly preferable:
% [Fx,Fy] = force_function_xny(t,x,y,vx,vy,other,parameters);
dxdydvxdvydt = [vx;vy;Fx/m;Fy/m];
end

Then you can integrate the equations of motion something like this:

m = 12; % you know what to set for this one
x0y0vx0vy0 = [1e-10,5.08e-5,0,0]; % Your initial conditions
t_span = [0,2*pi];                % a time-span of interest
[t_out,xyvxvy] = ode45(@(t,xyvxvy) your_eqs_of_motion(t,xyvxvy,m,[]),t_span,x0y0vx0vy0);

When integrating equations of motion you typically also have to carefully check that the constants of motion we know should be conserved (total energy, angular momentum, etc.) are reasonably conserved - the general ODE-integrating functions are not guaranteed to do this.

HTH

Best Answer

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