3-axis reaction wheels, star trackers and momentum dumping in CubeSat ADCS

Rigid-Body Momentum Exchange#

A CubeSat equipped with an internal reaction wheel triad of angular momentum vector ${\mathbf{h}}_w=[h_x,h_y,h_z{]}^T$ obeys the coupled Euler equation[1][2]:

$$\mathbf{I}\dot{\mathbf{\omega }}+{\dot{\mathbf{h}}}_w+\mathbf{\omega }\times (\mathbf{I}\mathbf{\omega }+{\mathbf{h}}_w)={\mathbf{\tau }}_{\text{ext}}$$

To slew a 3U CubeSat ($I_{xx}\approx 0.035\text{kg}\cdot {\text{m}}^2$) at ${\omega }_{\text{slew}}=3^{\circ }/\text{s}$ ($0.052\text{rad/s}$), each wheel must store at least $h_w=I_{xx}{\omega }_{\text{slew}}\approx 1.83\text{mN}\cdot \text{m}\cdot \text{s}$.

Comparison of CubeSat ADCS Tiers#

Continuous Magnetorquer Momentum Desaturation#

Because secular aerodynamic and solar radiation torques accumulate net momentum $\Delta \mathbf{H}$, the wheels eventually approach their maximum speed ${\omega }_{\text{max}}$ ($6,000\text{–}10,000\text{RPM}$). Using the local geomagnetic field $\mathbf{B}$, the magnetorquers command dipole moment

$${\mathbf{m}}_{\text{desat}}=-\frac{k_d}{\Vert \mathbf{B}{\Vert }^2}\left(\mathbf{B}\times ({\mathbf{h}}_w-{\mathbf{h}}_{\text{ref}})\right)$$

which produces an external torque ${\mathbf{\tau }}_{\text{mag}}={\mathbf{m}}_{\text{desat}}\times \mathbf{B}$ strictly in the plane perpendicular to $\mathbf{B}$, bleeding wheel speed back to zero over an orbital revolution.