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June 9th, 2010, 09:04 AM   #1
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Polar coordinates trajectory parametrization.

Hi everybody, my name is Daniel, i'm a undergraduate of materials science and engineering at UFSC - Brazil. Here I work at the Precision Mechanics Lab.
Sorry if I post in the wrong section. I think this is a multidisciplinary problem, evolving calculus, cartesian and polar coordinates. And it's engineering application.

Here, I'm working on the development of a software that controls a machine,
This machine is a multi-axial scratch tester, for surfaces scratching. (http://www.youtube.com/watch?v=MHMfqiBKpDI)

As you can see in the video, the machine has two stages, one that executes a horizontal movement, and the other a rotatory movement.
With the machine we scratch surfaces in polar trajectorys (r(x)), were the horizontal stage is the 'r", or radius, and the rotatory stage is the "x", or angle, or theta...

Well, we are having success to describe the entered trajectory.

But our next challenge it's being pretty hard to solve.
We gotta control the tangential velocity of the scratch, and for that, we gotta control the velocity of each stage.
The main objective is to make a scratch with a constant tangential velocity.

For example:

With a trajectory like r(x)=1*cos(x), a cardioid like in the video, how can I get velocity equations for each stage for a constant tangential velocity?

Do i have do parametrize in function of the time?

I appreciate your help very much.
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June 9th, 2010, 03:05 PM   #2
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Re: Polar coordinates trajectory parametrization.

Fala Daniel,

aqui quem fala é outro Daniel aqui de Niterói (RJ) e sou mestrando em Matemática da UFF. Não entendi muito bem o que você quer fazer, tem como mandar mais exemplos.


Um abraço

Daniel
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June 9th, 2010, 04:16 PM   #3
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Re: Polar coordinates trajectory parametrization.

Ok, i`ll answer in Portuguese and English,

Em Português:
Cara que massa, algum brasileiro pra me ajudar, é o seguinte:

As duas mesas possuem controle de velocidade e de posição, juntando esses dois, eu descrevo uma trajetória.
Por exemplo, pra r(x)=1+cos(x) com 0=<x=<2PI, a mesa horizontal vai descrever os pontos em "r", e a rotatória vai descrever um giro de 0 a 2PI. As duas precisam estar em sincronia,para acabarem o movimento de maneira coordenada, e riscar a cardiode com perfeição. Estou tendo sucesso em gerar a sincronia das velocidades, e descrever a tragetória de maneira correta, mas o que ocorre é a minha velocidade tangencial de riscamento está variando.

Eu quero descrever a trajectória com velocidade constante, para tanto eu preciso controlar as velocidades de cada mesa, de maneira distinta.

Por exemplo, queria chegar em um dr(x(t))/dt = velocidade linear, e um dx(t)/dt = velocidade da mesa rotatória, e que a resultante entre essas duas fosse uma constante para todos os instantes de t.

Só que r não é uma função do tempo, nem x, precisaria temporizar.

Daniel, tu consegue ler em inglês né? prefiro escrever em inglês para todos entenderem. podemos trocar emails também se tu preferir.
valeu pela força, abraços.
--

In English:
Both stages have a velocity and position control, assembling these two coordinates, i can describe a trajectory.
For example, for "r(x)=1+cos(x)" with 0=<x=<2PI, the horizontal stage will describe the trajectory appointed by "r", and the rotatory will spin 360 degrees (or 2*PI). Both gotta be synchronized, for end the movement together, and scratch the cardioid correctly. I'm having success in generating the velocity synchronism, and describe the trajectory correctly , but the problem is that the tangential velocity is not a constant.

I want to describe the trajectory with a constant velocity, for so i have to control each stage velocity separately .

For example, I would like to achieve a "dr(x(t))/dt"= linear velocity, and a "dx(t)/dt"= rotatory velocity, and the resultant between these two would be a constant for all instants of "t".
But neither "x"or "r"are time dependent functions.

[]`s
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June 9th, 2010, 05:08 PM   #4
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Re: Polar coordinates trajectory parametrization.

Se você parametrizar com a parametrização

;


Não te ajudaria?
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June 10th, 2010, 06:48 AM   #5
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Re: Polar coordinates trajectory parametrization.

Ok, alright, could be.

But how can i get to these results in a generically way, it should fit for every polar coordinates equation.

Valeu
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June 10th, 2010, 10:14 PM   #6
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Re: Polar coordinates trajectory parametrization.

You must study each case to find a parametrization even in a simple case such as the hyperbole you need to use two parametrizations, one for the part that contains (-1,0) and one for the part that contains (1,0).
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