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Math Help - Help me with my special problem

  1. #1
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    Question Help me with my special problem

    I know it has something to do with differential equations and dynamical systems.

    This is how it goes:

    Taikonauts in training are required to practice a docking maneuver under manual control. As a part of this maneuver, it is required to bring an orbiting spacecraft to rest relative to another orbiting craft.

    The hand controls provide for variable acceleration and deceleration, and there is a device on board that measures the rate of closing between the two spacecrafts.

    The following strategy has been proposed for bringing the craft to rest:

    First, look at the closing velocity, if it's zero we are done. Otherwise, remember the closing velocity and look at the acceleration control.

    Move the acceleration control so that it is opposite to the closing velocity (i.e. if closing velocity is positive, slow down, and speed up if it is negative) and proportional in magnitude (i.e. brake twice as hard if closing velocity becomes twice as fast).

    After a time, look at the closing velocity again and repeat the procedure. Under what circumstances will this strategy be effective?


    I tried equating acceleration to be equal to velocity but I found out that the acceleration should depend on the closing velocity not the velocity of the docking spacecraft. Closing velocity is defined as the difference between the two spacecraft.

    Can you guys give me some hints and tips?

    Thanks!
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by jigogwapo16 View Post
    I know it has something to do with differential equations and dynamical systems.

    This is how it goes:

    Taikonauts in training are required to practice a docking maneuver under manual control. As a part of this maneuver, it is required to bring an orbiting spacecraft to rest relative to another orbiting craft.

    The hand controls provide for variable acceleration and deceleration, and there is a device on board that measures the rate of closing between the two spacecrafts.

    The following strategy has been proposed for bringing the craft to rest:

    First, look at the closing velocity, if it's zero we are done. Otherwise, remember the closing velocity and look at the acceleration control.

    Move the acceleration control so that it is opposite to the closing velocity (i.e. if closing velocity is positive, slow down, and speed up if it is negative) and proportional in magnitude (i.e. brake twice as hard if closing velocity becomes twice as fast).

    After a time, look at the closing velocity again and repeat the procedure. Under what circumstances will this strategy be effective?


    I tried equating acceleration to be equal to velocity but I found out that the acceleration should depend on the closing velocity not the velocity of the docking spacecraft. Closing velocity is defined as the difference between the two spacecraft.

    Can you guys give me some hints and tips?

    Thanks!
    You are going to observe the closing speed at epocs i=1, 2, .. and we will assume these are separated in time by \tau seconds.

    Let the closing speed at epoc i be v_i and the corresponding acceleration a_i=-kv_i for some k>0.

    Then:

    v_{i+1}=v_i+a_i \tau=v_i(1-k\tau)

    The task is achieved if \lim_{i \to \infty}v_i=0, which requires that:

    |1-k\tau|<1

    Notes: We have assumed this is a 1-D problem. I have ignored the requirement to avoid impact.

    CB
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