Kinetic equations

Question

The kinetic equation of rotational motion of a satellite using Newton’s law can be expressed as follows: $$ \frac{\partial H_b}{\partial t}+\omega_b\times H_b=T_d \qquad(1) $$ In the above relation $T_d$ is the disturbance torque applied to the satellite, $H_b$ is the angular momentum vector expressed in the body coordinate system and $ω_b$ is the angular velocity of the Body Coordinate System relative to the Inertial Coordinate System that is expressed in the Body Coordinate System. For a satellite with three reaction wheels, $H_b$ can be obtained from the angular momentum of the satellite and reaction wheels as follows: $$ H_b=J\omega_b+J_{w1}\omega_{w1}+J_{w2}\omega_{w2}+J_{w3}\omega_{w3} \qquad(2) $$ J is Moment of Inertia Matrix. $\omega_{wi}$ is the angular velocity of the $i$th wheel relative to the Body Coordinate System.

My question is how is equation (2) obtained.

Answer 1

Equation 2 is just applying the definition. For any object, angular momentum is moment of inertia tensor times angular velocity vector. Equation 2 says there are four spinning things, and this is how their contributions to the total angular momentum are named in this paper.