B Dot Law
Introduction
The average magnetic field is commonly used in control schemes to simplify the control gain calculation because the Earth’s magnetic field is a periodic function. As the name suggests, B-dot law uses the variation of the magnetic field over a small time period to determine the control torque to be employed on the body.
It relates the change in magnetic field observed in the body frame to the magnetic moment required in space to counter the angular rates. The magnetic moment produced by actuator in the satellite will lead to control torque due to earth’s magnetic field, thus damping the motion.
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).
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. [1]
Derivation
ζ = m x b
ζ =Torque
m = magnetic moment
b = magnetic field
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.
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: [2]
There are multiple variations of the B-dot control law, like the B-dot bang-bang controller [3], which instead of calculating the commanded magnetic moment with a proportional gain factor, uses the maximum torquer strength.
Details on the other types of B-Dot control laws which can be used to control the satellite during detumbling can be found here.
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