Reaction Wheels

Characteristics

pg. 148

• rotating flywheel supported by ball bearings, internal brushless DC electric motor, associated electronics
• back EMF developed at high speeds needed to provide high angular momentum → difficult to provide high torque in same wheel
• motor drives of some reaction wheels accept torque command → may also hold at commanded value
• usually has digital/analog tachometer → get rotational speed
• wheel friction (drag) → modeled as viscous and Coulomb components (coefficients $\tau_v$ and $\tau_c$ experimental) → model does not give good results at low speed

$$ L^w_\text{drag} =\tau_v\omega^w-\tau_c\text{sign}(\omega^w) $$

• other effects on the wheel occur due to imbalances in alignment of the wheel on it’s spin axis

Redundant Wheel Configurations

pg. 152

• create transformation from wheel frame to body frame (given by 3 by n distribution matrix) → columns are unit vectors of reaction wheel spin axes in the body frame

$$ \mathcal{W}_n=\begin{bmatrix}\vec{w_1} & \vec{w_2} & \dots & \vec{w_n}\end{bmatrix} $$

• total wheel torque and angular momentum in body frame → W denotes n-dimensional wheel frame

$$ \vec{L}_B^w=\mathcal{W_n}\begin{bmatrix}L_1^w & L_2^w & \dots L_n^w\end{bmatrix}^T=\mathcal{W_n}\vec{L}_W^w $$

$$ \vec{H}_B^w=\mathcal{W_n}\begin{bmatrix}H_1^w & H_2^w & \dots H_n^w\end{bmatrix}^T=\mathcal{W_n}\vec{H}_W^w $$

• note that torque and angular momentum are parallel (hence following equations are analogous)
• $\vec{H}_W^w$ and $\vec{H}_B^w$ do not have same magnitude as $\mathcal{W_n}$ is not an orthogonal matrix
• if $n=3$ and wanting to convert from body frame angular momentum to wheel angular momentum → $\vec{H}_W^w = \mathcal{W_3}^{-1}\vec{H}_B^w$ → requires distribution matrix to be full rank