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== Application in Satellite Control ==
The first task a spacecraft attitude control system must perform after separation from the launcher is to detumble the spacecraft, i.e., to bring it to a state with a sufficiently small angular momentum. Magnetic control has been used for decades to fulfill this task (in all its variants developed and flown through the years). <br \>
Not only for detumbling, the B-Dot law is used to damp the motion of the satellite in the case of very high angular rates.<ref>http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=7171005 </ref>
== Derivation ==
m = magnetic moment <br \>
b = magnetic field <br \>
From this equation, it is clear that the torque is maximized for m ⊥ b and that no torque is produced when m ∥ b. Another useful consideration for the development of a stabilizing control law is that the control torque has to be proportional to -ω, in order to decrease the kinetic energy of the object. <br \>The b-dot control law takes advantage of the fact that the derivative of the magnetic field vector is both perpendicular to b and proportional to ω, hence the commanded magnetic dipole can be expressed as: <ref>https://www.researchgate.net/publication/263008407_Comparison_of_Control_Laws_for_Magnetic_Detumbling</ref> <br \>
[[File:Equation56.gif|frame|center]]
There are multiple variations of the B-dot control law, like the B-dot bang-bang controller<ref>http://www.diva-portal.org/smash/get/diva2:1018210/FULLTEXT02.pdf </ref>, which instead of calculating the commanded magnetic moment with a proportional gain factor, uses the maximum torquer strength.
[[File:Equation57.gif|frame|center]]
Details on the other types of B-Dot control laws which can be used to control the satellite during detumbling can be found [http://www.diva-portal.org/smash/get/diva2:1018210/FULLTEXT02.pd here].
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== References ==