Liearization of Lagrange EOMs of an Inverted Pendulum

[road_king]

Linearization of Lagrange EOMs of an Inverted Pendulum

Hi Folks,

I am modelling a state space model of an Inverted Pendulum mounted on a cart over a balancing seesaw.

I developed the equations of motion using the Lagrangian approach an obtained 3 PDEs. I solved them using Mathematica 8 for the second order derivatives of the generalized coordinates, and using one of the new Mathematica functions obtained a linearized representation of the system. The model seems to me to be pretty convincing, but not for my Thesis director, who ask me to obtain a linearized model by myself

I try to use Mathematica to calculate the second order Taylor series expansion of the equations of movement but it only takes as variables the first two coordinates instead of the four that appeared in the equation.

Iif anyone may give me any idea it will be very appreciated.

First I defined the first equation of movement as a function of the 3 generalized coordinates [x[t], θ[t],α[t]] and vm[t], which is the voltage applied to the servo-motor. This 3 coordinates represent the position of the cart along the rack over the seesaw, the rotation of the seesaw around his fulcrum, and the rotation of the pendulum around the cart.


I believe I am using the correct syntax for that as I read in the program help...

Series[ expr, var1, var2, ... ]

Any suggestions of how to do the work?

Thank in advance!

Edit: I attach a pdf file with the mathematica code, the text is in Spanish but I think it is easyly readable. I will ask in the Mathematica subforum later, the expecific question with Series function. Excuse my bad English, please.The expressions I am asking for are on page 5.

I am sorry I have not any drawing of the system, but it it like this image

but without the c1 cart (my system only has one cart with the pendulum, and the names of the parameters and variables are different also)

 

[jackmell]

Any suggestions of how to do the work?

Thank in advance!

I have some suggestions:

(1) You mean linearization right?

(2) Picture. That would be nice but not necessary

(3) That code is way too messy to do anything with. Right off the bat, don't use capital letters to start user-variable names. Try too to make the code pretty, use spaces, comments, better-meaning names for the variables.

(4) Would be nice to just write down the equations nicely but that requires learning latex and I'm not so mean to expect you to do that just starting. Still though, if maybe you took a little bit of time to learn how to write it in latex and then show us then we would have some idea what you're doing.

(5) We have a very good Mathematica forum down below in the Computer and Technology/Math and Science Software sub-forum.

This is my best advice then: if you want to improve your chances of getting help here, spend some time learning latex. See "Math and Science Learning Material" forum above, then the "Latex" thread in there to help you. Then just post the equations, neatly, and then ask how might they be linearized, and if you have specific questions about Matheamtica, post them in the science software sub-forum.

 

[road_king]

Ok! thanks, I going to edit my post.

 

[road_king]

I have some suggestions:

(1) You mean linearization right?

(2) Picture. That would be nice but not necessary

(3) That code is way too messy to do anything with. Right off the bat, don't use capital letters to start user-variable names. Try too to make the code pretty, use spaces, comments, better-meaning names for the variables.

(4) Would be nice to just write down the equations nicely but that requires learning latex and I'm not so mean to expect you to do that just starting. Still though, if maybe you took a little bit of time to learn how to write it in latex and then show us then we would have some idea what you're doing.

(5) We have a very good Mathematica forum down below in the Computer and Technology/Math and Science Software sub-forum.

This is my best advice then: if you want to improve your chances of getting help here, spend some time learning latex. See "Math and Science Learning Material" forum above, then the "Latex" thread in there to help you. Then just post the equations, neatly, and then ask how might they be linearized, and if you have specific questions about Matheamtica, post them in the science software sub-forum.

I tried to improve the notebook file, I hope it will be more friendly!

 

[jackmell]

Ok road-king, that's nice. Really. However I should have mentioned that I might not be the one to help you with this. That's still confussing to me. I can't even pick out clearly what the PDEs are in that code, not without more work anyway. I guess what I was suggesting, if no one else helps you, was to simply write down the PDEs in latex, the same way they might appear in a textbook but with no extra trappings, no hard to read notation, just in their canonical forms like for example reaction-diffusion:

$$\frac{\partial a}{\partial t}=s(k(16-ab)+d_a \nabla^2 a)$$
$$\frac{\partial b}{\partial t}=s(k(ab-b-12-\beta)+d_b \nabla^2 b)$$

Also, your problem looks very interesting. However, I'm not what I would consider very good with PDEs so may not be able to help you with it.