In space there are disturbances (like aerodynamic torque(in LEO), gravity gradient torque, solar radiation pressure torque) that provide external torques (Ne) to the satellite which tend to destabilize the satellite. Therefore it is necessary to understand the dynamics of the satellite and model the same to actively control the attitude of satellite.<br \>The total angular momentum of the system (satellite (h_{tot}body + reaction wheels) about center of mass of the system can be described written as <ref>Engineering Mechanics Dynamics book by the equation belowJ. L. Meriam and L. G. Kraige</ref>[[File:Equation51RW1.pnggif|frame|center|Equation 1]]While dealing with satellite dynamics we need a [https://en.wikipedia.org/wiki/Rotating_reference_frame rotating reference frame].The rate of change of a vector in rotating and stationary two frames can be related by the equation:<ref>Classical Mechanics Textbook by Herbert Goldstein</ref>[[File:Equation52RW2.pnggif|frame|center|Equation 2]]Here subscript s Where I and B are two frames and [[File:RW3.gif]] is for observation in stationary the angular velocity of B frame and r is for observation in rotating with respect of N frame. <br \>Substituting h_{tot} Now we have another relation relating torque applied to the system and the change in angular momentum of the above equation, system: <ref name = "ref2">Analytical Mechanics of Space Systems Book by Hanspeter Schaub and John Junkins</ref>[[File:Equation53RW4.pnggif|frame|center]]Where subscript p is any arbitrary point in space and c is for center of mass. [[File:RW5.gif]] is the total external moment applied on the system. <br \>Now since we wrote our total angular momentum of the system about the center of mass, the second term in the above equation can treat rate be dropped. <br \>Hence we are left with [[File:RW6.gif]]. <br \>By now, you must have observed that [[File:RW7.gif]] is nothing but the inertial derivative of [[File:RW8.gif]]. Again, we have the relation <ref name = "ref2" /> [[File:RW9.gif]], where:G is the center of mass of the body, <br \>[I] is the inertia matrix in frame fixed to G <br \>[[File:RW10.gif]] is the angular rate of change body. <br \>Note that the challenge is to find [[File:RW11.gif]] and [[File:RW12.gif]] about the center of angular momentum mass of wheels as the control torque system, rather than their own centers of mass, which can be easily done. Please refer to [https://pdfs.semanticscholar.org/d2b5/e126c3d4bd54e39ee134b1cc28227b99a2b8.pdf this]. <br \>Now we have everything, but before moving ahead let’s state some assumptions made to during the final derivation. It’s assumed that the center of mass of satellite in body and center of mass of the system are very close. Also the wheel of reaction wheel is assumed to be uniform. <br \>[[File:RW13.gif|frame|center|Schematic for one wheel mounted on the satellite body coordinate system]]Let's substitute equation 1 in equation 2.<br \>After several manipulations, we get,[[File:Equation54RW20.pnggif|frame|center]]Note: [[File:RW15.gif]] and [[File:RW16.gif]] in the above expression are the derivatives in the B frame and the W frame respectively. [[File:RW19.gif]] is the unit vector along the axis of rotation of the wheel and [[File:RW21.gif]] is the inertia about the axis of rotation of the wheel. <br \>In the above equation it is visible that we are able to relate the angular speed of the wheel and the disturbance torque acting on the satellite. And what we have control is on the speed of the reaction wheel. Therefore it can be accelerated and decelerated suitably to actuate the satellite.== Limitations of Reaction Wheels ==* Has moving parts* Can’t produce external torque.* Size and mass constraints* Limited by max rotation rate* Accumulation of momentum (momentum saturation).* It can’t be independently used (it needs momentum unloading, which is typically achieved by magentorquers)* Shouldn’t be operated near 0 rpm (due to dominating friction).