Solve the differential eqation and then implement nonlinear least square?

I have 3 dimentional positional data of a pitched baseball.

I would like to estimate the acceleration from the data but I want to avoid differentiating the positional data.

Therefore, I am wondering if I can get the instant acceleration without differentiation of the positions.

The, I firstly tried to fit the data with differential equations model and to determine the coefficients of each term.

The differential equations model were as follows:

diff(rx(t),t,2) == K*sqrt(diff(rx(t),t)^2+diff(ry(t),t)^2+diff(rz(t),t)^2)*(CL/(w)*(diff(rz(t),t)*wy-wz*diff(ry(t),t))-CD*diff(rx(t),t));% X-component

diff(ry(t),t,2) == K*sqrt(diff(rx(t),t)^2+diff(ry(t),t)^2+diff(rz(t),t)^2)*(CL/(w)*(diff(rx(t),t)*wz-wx*diff(rz(t),t))-CD*diff(ry(t),t));%Y-component

diff(rz(t),t,2) == K*sqrt(diff(rx(t),t)^2+diff(ry(t),t)^2+diff(rz(t),t)^2)*(CL/(w)*(diff(ry(t),t)*wx-wy*diff(rx(t),t))-CD*diff(rz(t),t));%z-component

%where rx,ry,rz are each component of the ball position.

%CL and CD are drag and lift coefficients.

%K is the constant value.

%w is the magnitude of the angular velocity and wx, wy, and wz are each of components of the angular velocity.

I tried to solve the differential equations yet there were no positive answers so I guess the equations should be solved numerically.

When I get the equations, which is like "rx(t)=", I guess I could fit the positional data what I have.

I am not good at math so the way I try might be entirely wrong so if there is any possible way to answer my question, it would be great, so please help me get a any hints.

Kind regards,

[Nicolas Schmit]

Are you trying to estimate the acceleration from real experimental data? If this is the case, calculating the acceleration using the differential equations will be only as good as your model (equations) is. Is there a specific reason why you want to avoid differentiating the data?

[Kate]

Thank you for your comment, Mr. Nicolas. I would like to avoid differentiating the data as the data includes errors due to digitizing. I got the positional data from video analysis, then when I differentiated the positional data, the acceleration was not really true. Therefore, I would like to get the instantaneous accecleration at any instant somehow. Then, I thought if I can get the equation as "rx(t)=", I can estimate the value of the acceleration from the coefficient of the terms of the equation. The way that I though might be totally wrong so if there is any other way, please tell me.

Kind regards,

[Torsten]

Difficult problem because of the noise in the data:

https://dsp.stackexchange.com/questions/26248/derive-velocity-and-acceleration-from-position

Best wishes

Torsten.

[Kate]

Dear Torsten,

Thank you for the comment. I could try to use these filters for differentiation, but I hope there is another way to determine the value of the acceleration rather than the approach you suggested.

Sincerely,

[Torsten]

In my opinion, working directly on the data is the way to go.

If you want to fit parameters within your model to best approximate your data, you will have to couple an ODE solver and a curve-fitting tool.

I suggest you study Star Strider's reference example:

https://de.mathworks.com/matlabcentral/answers/43439-monod-kinetics-and-curve-fitting

Best wishes

Torsten.

[Kate]

I will have a look at what you pasted and see how I can solve the problem. Thank you for the reply and the comment kindly.

Sincerely,