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Math Help - Finding time of detection of 2 moving objects.

  1. #1
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    Finding time of detection of 2 moving objects.

    Hi all,

    I'm currently doing this java application to find the time of intersection of 2 moving objects in a 2D vector space.

    Description: Object A moves in a straight line with a constant velocity with a circular sensor of known radius. This circular sensor is centered at object A and a detection occurs when object B crossed the sensor radius. Object B on the other hand, rotates continuously around a known point.

    The time of detection is calculated using the formula below:
    X^2 + Y^2 = R^2
    where X and Y are the x and y coordinates of the location at which a detection occurs. And R is the radius of the sensor of object A.

    All the positional and angular information of the 2 objects are known except for the time of detection.

    After working out the long and tedious workings, it was found that terms like tcos (wt) and tsin (wt) emerged in the equation and Laplace transformation was applied to change everything to the S-domain.

    Question:
    1.) Can Laplace transformation be applied in this case?
    2.) How do i find infomation on how to find the roots of a polynomial in S-domain that has a degree of 6?
    3.) How to convert the value of s into the correct value in the time-domain?
    4.) Is it possible that someone can shed some light about how i should go about tackling this issue?? Like advices, previous posts or some links where i can read up?

    Sorry for the abruptness in the description, I hope to hear some advices from the experts in this forum. Thank you in advance.

    santouryu
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  2. #2
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    Quote Originally Posted by santouryu View Post
    Hi all,

    I'm currently doing this java application to find the time of intersection of 2 moving objects in a 2D vector space.

    Description: Object A moves in a straight line with a constant velocity with a circular sensor of known radius. This circular sensor is centered at object A and a detection occurs when object B crossed the sensor radius. Object B on the other hand, rotates continuously around a known point.

    The time of detection is calculated using the formula below:
    X^2 + Y^2 = R^2
    where X and Y are the x and y coordinates of the location at which a detection occurs. And R is the radius of the sensor of object A.

    All the positional and angular information of the 2 objects are known except for the time of detection.

    After working out the long and tedious workings, it was found that terms like tcos (wt) and tsin (wt) emerged in the equation and Laplace transformation was applied to change everything to the S-domain.

    Question:
    1.) Can Laplace transformation be applied in this case?
    2.) How do i find infomation on how to find the roots of a polynomial in S-domain that has a degree of 6?
    3.) How to convert the value of s into the correct value in the time-domain?
    4.) Is it possible that someone can shed some light about how i should go about tackling this issue?? Like advices, previous posts or some links where i can read up?

    Sorry for the abruptness in the description, I hope to hear some advices from the experts in this forum. Thank you in advance.

    santouryu
    You have an equation that you want to solve for the time of first detection. I don't see that taking the Laplace transform will help in finding the solution. You should possibly be looking for a numerical solution method.

    .
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  3. #3
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    Hi Constatine11,

    Thank you for your reply, I would very much agree with you because the laplace equation that I have arrived in results in a polynomial of degree 6, which makes it difficult to solve.

    Does your numerical approach refer to plotting the function out and then use either newton's method or other meothds to estimate the root when the function crosses 0?

    regards,
    Santouryu
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  4. #4
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    Quote Originally Posted by santouryu View Post
    Hi Constatine11,

    Thank you for your reply, I would very much agree with you because the laplace equation that I have arrived in results in a polynomial of degree 6, which makes it difficult to solve.

    Does your numerical approach refer to plotting the function out and then use either newton's method or other meothds to estimate the root when the function crosses 0?

    regards,
    Santouryu
    More or less yes.

    You are looking for a root of f(t)=d^2(t)-R^2=0, where d is the distance between the sensor and the target, and R is the detection range (in fact the smallest positive root, there may be more than one or zero if the target never crosses the detection boundary, and assuming that at t=0,\ d^2>R^2).

    (plotting f(x) is always a good idea, that way at least you have an idea where the root lies, and often a graphical solution is good enough)

    .
    Last edited by Constatine11; December 30th 2008 at 10:20 PM.
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