Dynamics: Reaction Wheel

Description

Reaction wheels are used on satellites as a means of controlling their orientation and maintaining stability. The purpose of a reaction wheel is to store and transfer angular momentum, allowing a satellite to adjust its orientation without using external forces. The principle behind reaction wheels is based on Newton's third law of motion, which states that every action has an equal and opposite reaction. The reaction wheel consists of a spinning flywheel mounted inside a satellite. As the flywheel spins, its angular momentum is conserved, which means that any change in the direction or speed of the wheel will cause an equal and opposite change in the satellite's orientation. This module implements the balanced, simple jitter and fully coupled jitter models described in [2]. Reaction wheel jitter caused by mass imbalances is particularly important to model when considering complex dynamic behaviour caused by flexible structures and fuel slosh dynamics. It’s also important to consider when investigating pointing accuracy requirements for precision instruments. The model also implements an optional friction model.

Example Use Cases

• Add RWs as Actuators: Add reaction wheels as actuators to an ADCS system.
• Analyze the impact of jitter on pointing: Use jitter and friction models to consider the impact of jitter on pointing accuracy and sensor performance.

Module Implementation

Back Substitution Method

To integrate the contribution of spacecraft components into the total system dynamics in a modular way, the simulation system uses a back substitution method described below, where satellite equations of motion (EOMs) are expressed using the following matrix representation:

\[ \begin{bmatrix} [A] & [B]\\ [C] & [D] \end{bmatrix} \begin{bmatrix} \ddot{\textbf{r}}_{B/N}\\ \dot{\omega}_{\cal B/N} \end{bmatrix} = \begin{bmatrix} \textbf{v}_{\text{trans}}\\ \textbf{v}_{\text{rot}} \end{bmatrix} \]

The spacecraft body, as well as each additional component that affects the spacecraft dynamics, make contributions to matrices \([A]\), \([B]\), \([C]\), \([D]\) and vectors \(\textbf{v}_{\mathrm{trans}}\) and \(\textbf{v}_{\mathrm{rot}}\).

Substituting the contributions of the spacecraft body, the spacecraft equations of motion can be written as follows:

\[ \Big(m_{\text{sc}} [I_{3\times3}] +\sum_{i=1}^{N}[A_{\mathrm{contr},i}]\Big)\ddot{\textbf{ r}}_{B/N}+\Big(-m_{\text{sc}} [\tilde{\textbf{r}}_{Cm/B}] +\sum_{i=1}^{N}[B_{\mathrm{contr},i}]\Big) \dot{\omega}_{\cal B/N} \\ = \textbf{F}_{\mathrm{ext}} - 2 m_{\text{sc}} [\tilde{\omega}_{\cal B/N}] \dot{r}_{Cm/B} -m_{\text{sc}} [\tilde{\omega}_{\cal B/N}][\tilde{\omega}_{\cal B/N}]\textbf{r}_{Cm/B} +\sum_{i=1}^{N}\textbf{v}_{\mathrm{trans,contr},i} \]

\[ \Big[m_{\text{sc}}[\tilde{\textbf{r}}_{Cm/B}] +\sum\limits_{i=1}^{N}[C_{\mathrm{contr},i}]\Big]\ddot{\textbf{r}}_{B/N}+\Big[[I_{\text{sc},B}]+\sum\limits_{i=1}^{N}[D_{\mathrm{contr},i}]\Big]\dot{\omega}_{\cal B/N} \\ = -[{\tilde{\omega}}_{\cal B/N}] [I_{\text{sc},B}] \omega_{\cal B/N} - \dot{[I_{\text{sc},B}]} \omega_{\cal B/N} + \textbf{L}_{ext} +\sum\limits_{i=1}^{N}\textbf{v}_{\mathrm{rot,contr},i} \]

Here:

• \([A_{\mathrm{contr},i}]\), \([B_{\mathrm{contr},i}]\), \([C_{\mathrm{contr},i}]\), \([D_{\mathrm{contr},i}]\), \(\textbf{v}_{\mathrm{trans,contr},i}\) and \(\textbf{v}_{\mathrm{rot,contr},i}\) are the contributions of component \(i\) to the terms in the back-substitution equations of motion
• \(m_{sc}\) is the total mass of the spacecraft.
• \([I_{3\times3}]\) is a 3x3 identity matrix.
• \([I_{sc,B}]\) is the spacecraft’s total moment of inertia in the body frame \(\mathcal{B}\).
• \(\dot{[I_{sc,B}]}\) is the rate of change spacecraft’s total moment of inertia in the body frame \(\mathcal{B}\).
• \(\ddot{r}_{B/N}\) is the acceleration of the spacecraft body frame \(\mathcal{B}\) with with respect to the inertial frame \(\mathcal{N}\).
• \({r}_{Cm/B}\) is the vector that describes the offset between the body frame \(\mathcal{B}\) and the spacecraft's total centre of mass.
• \(\dot{r}_{Cm/B}\) is the rate of change of the total spacecraft centre of mass in the body frame \(\mathcal{B}\).
• \(\omega_{B/N}\) is the angular velocity of the spacecraft body frame \(\mathcal{B}\) with respect to the inertial frame \(\mathcal{N}\).
• \(\dot{\omega}_{B/N}\) is the angular acceleration of the spacecraft body frame \(\mathcal{B}\) with respect to the inertial frame \(\mathcal{N}\).
• \(\mathbf{F}_{ext}\) are external forces being applied to the body in the body frame \(\mathcal{B}\).
• \(\mathbf{L}_{ext}\) are external torques being applied to the body in the body frame \(\mathcal{B}\).
• \([\tilde{x}]\) is the skew-symmetric matrix representation of a cross-product operation.

The goal is to structure a component EOM contributions into this form to determine its contributions to \([A]\), \([B]\), \([C]\), \([D]\), \(\textbf{v}_\mathrm{trans}\) and \(\textbf{v}_\mathrm{rot}\), which can be super-imposed to propagate the spacecraft state forwards.

Balanced Model

Given a reaction wheel with its principle moments of inertia aligned with the wheel’s spin axis \(\mathbf{\hat{g}}_s\) and with its centre of mass co-incident with \(\mathbf{\hat{g}}_s\), the rotation rate of the wheel (\(\Omega\)) will contribute to the angular momentum of the spacecraft through its contributions to the system moment of inertia. Similarly, the rate of change of wheel speed (\(\dot{\Omega}\)) will generate a torque on the spacecraft. These can be added to the spacecraft’s rotational EOM as shown below [1]:

\[ m_{\text{sc}}[\tilde{\textbf{r}}_{CmB}] \ddot{\textbf{r}}_{B/N} +[I_{\text{sc},B}] \dot{\omega}_{\cal B/N} +\sum_{i=1}^{N}J_{\text{s}_i} \hat{\textbf{g}}_{\text{s}_i}\dot{\Omega}_i = -[{\tilde{\omega}}_{\cal B/N}] [I_{\text{sc},B}] \omega_{\cal B/N} - \sum_{i=1}^{N}(\omega_{\cal B/N}\times J_{\text{s}_i}\Omega_i\hat{\textbf{ g}}_{\text{s}_i}) + \textbf{L}_{ext} \]

where \(J_{s_i}\) is the moment of inertia of the \(i^{th}\) wheel about its spin axis \(^\mathcal{B}\hat{\mathbf{g}}_{s_i}\).

We can also relate \(\dot{\Omega}_i\) to the commanded motor torque \(\mu_{s_i}\) as,

\[ \dot\Omega_i = \frac{\mu_{s_i}}{J_{s_i}} - \hat{\mathbf{g}}_{s_i}^T \dot\omega_{B/N} \]

Note: here the rotation rate of the wheel is in the inertial frame, so we must project the body’s rotation rate onto \(\dot\Omega_i\).

Substituting this equation into the above EOM, we can define the contributions of the \(i^{th}\) RW to the back substitution matrices as:

\[ [A_{contr}] = [0_{3\times3}] \]

\[ [B_{contr}] = [0_{3\times3}] \]

\[ [C_{contr}] = [0_{3\times3}] \]

\[ [D_{contr}] = - \sum_{i=1}^N J_{s_i} \hat{\mathbf{g}}_{s_i} \hat{\mathbf{g}}_{s_i}^T \]

\[ v_{trans,contr} = \mathbf{0} \]

\[ v_{rot,contr} = - \sum_{i=1}^N ( \mu_{s_i} \hat{\mathbf{g}}_{s_i} + \omega_{B/N} \times J_{s_i} \Omega_i \hat{\mathbf{g}}_{s_i} ) \]

Note that the implication here is that the balanced reaction wheel model does not influence the translational state of the spacecraft. This is generally true. See [1] for more details.

Simple Jitter Model

Reaction wheels typically have some offset between the spin-axis and the reaction wheel centre of mass. These offsets are typically measured and provided by the manufacturer as a static imbalance \(U_s\) and dynamic imbalance \(U_d\) coefficient [2].

\(U_s\) is typically given in units of \(g\cdot cm\) or \(kg \cdot m\) and defines a proportionality between the square of the wheel speed to the magnitude of the disturbance caused by an offset between the geometric centre of the reaction wheel and its centre of mass i.e.

\[ \mathbf{F}_{s} = U_s \Omega^2 \hat{\mathbf{u}} \]

where \(\hat{\mathbf{u}}\) is an arbitrary unit vector normal to the reaction wheel spin axis \(\hat{\mathbf{g}}_s\).

If the centre of mass of the reaction wheel is not coincident with the centre of mass of the parent spacecraft, then the torque on the spacecraft resulting from this mass imbalance is given by the simplified model,

\[ \mathbf{L}_{s} = \mathbf{r}_{W/B} \times \mathbf{F}_s = U_{s_i}\Omega_i^2 [\tilde{\mathbf{r}}_{W/B}]\hat{\mathbf{u}} \]

This uses the approximation where the \(\mathbf{r}_{Wc/B} \approx \mathbf{r}_{W/B}\) because the centre of mass offset of the reaction wheel in the wheel frame is usually very small.

\(U_d\) is typically given in \(g\cdot cm^2\) or \(kg\cdot m^2\) and defines the proportionality between the square of the wheel speed to the magnitude of disturbance torque caused by off-diagonal terms in the RW moment of inertia i.e.

\[ \mathbf{L}_d = U_d \Omega^2 \hat{\mathbf{v}} \]

where \(\hat{\mathbf{v}}\) is an arbitrary unit vector normal to the spin axis of the mass imbalance, which may not correspond to the reaction wheel spin axis.

Note

\(\hat{\mathbf{u}}\) and \(\hat{\mathbf{v}}\) are only required to be normal to their corresponding spin axes. This is because they are lumped parameters and contain no information on the orientation/location of the mass imbalances. They are also independent of the initial wheel angle.*

The simple jitter model then adds the contribution of these forces and torques to the spacecraft EOMs as external perturbations, giving [2]:

\[ m_{\text{sc}} [I_{3\times3}] \ddot{\textbf{r}}_{B/N}-m_{\text{sc}} [\tilde{\textbf{r}}_{Cm/B}] \dot{\omega}_{\cal B/N} = \textbf{F}_{\mathrm{ext}} - 2 m_{\text{sc}} [\tilde{\omega}_{\cal B/N}] \dot{r}_{Cm/B} -m_{\text{sc}} [\tilde{\omega}_{\cal B/N}][\tilde{\omega}_{\cal B/N}]\textbf{r}_{Cm/B} + U_s \Omega^2 \hat{\mathbf{u}} \]

\[ m_{\text{sc}}[\tilde{\mathbf{r}}_{CmB}] \ddot{\mathbf{r}}_{B/N} +[I_{\text{sc},B}] \dot{\mathbf\omega}_{\cal B/N} +\sum_{i=1}^{N}J_{\text{s}_i} \hat{\mathbf{g}}_{\text{s}_i}\dot{\Omega}_i = -[\mathbf{\tilde{\omega}}_{\cal B/N}] [I_{\text{sc},B}] \mathbf\omega_{\cal B/N} - \sum_{i=1}^{N}(\mathbf\omega_{\cal B/N}\times J_{\text{s}_i}\Omega_i\hat{\mathbf{g}}_{\text{s}_i}) + U_{s_i}\Omega_i^2 [\tilde{\mathbf{r}}_{Wi/B}]\hat{\mathbf{u}_i} + U_{d_i}\Omega_i^2 \hat{\mathbf{v}_i} + \mathbf{L}_{ext} \]

Making the same substitution for motor torques as in the balanced model, the contributions to the back substitution matrices are:

\[ [A_{contr}] = [0_{3\times3}] \]

\[ [B_{contr}] = [0_{3\times3}] \]

\[ [C_{contr}] = [0_{3\times3}] \]

\[ [D_{contr}] = - \sum_{i=1}^N J_{s_i} \hat{\mathbf{g}}_{s_i} \hat{\mathbf{g}}_{s_i}^T \]

\[ v_{trans,contr} = - \sum_{i=1}^N U_{s_i} \Omega^2 \hat{\mathbf{u}}_{s_i} \]

\[ v_{rot,contr} = - \sum_{i=1}^N ( \mu_{s_i} \hat{\mathbf{g}}_{s_i} + \omega_{B/N} \times J_{s_i} \Omega_i \hat{\mathbf{g}}_{s_i} + U_{s_i}\Omega_i^2 [\tilde{\mathbf{r}}_{Wi/B}]\hat{\mathbf{u}_i} + U_{d_i}\Omega_i^2 \hat{\mathbf{v}_i}) \]

Note that unlike in the balanced case, the jitter impacts both the rotational and translational dynamics of the spacecraft. Also note that, as we’re applying external forces and torques on the body due to a change in the spacecraft state, this method does not conserve momentum and energy. To do this, the forces and torques due to mass imbalance must be considered as an internal disturbance (i.e. perturbations caused by change in system mass properties). This is done in the fully coupled jitter model below.

This model is primarily useful for considering the impact of jitter on system performance based on manufacturer specifications.

Fully Coupled Jitter Model

To consider the impact of the reaction wheel jitter on spacecraft dynamics in a momentum and energy-conserving way requires the actual state of the reaction wheels to be tracked. That is, given known mass imbalances on wheels and their initial state \(\theta_i\), compute the internal forces and torques generated on the spacecraft.

To use this model accurately requires a timestep to sufficiently resolve the motion of the reaction wheels and their corresponding contributions to system dynamics. This is most applicable for applications involving flexible structures, fuel slosh, etc.

The full derivation of this model can be found in [2]. For brevity, we shall only report the resulting contributions to the back substitution matrix here:

\[ [A_\text{contr}] = \sum\limits_{i=1}^{N}m_{\text{rw}_i}d_i\hat{\mathbf{w}}_{3_i}\mathbf{a}_{\Omega_i}^T \]

\[ [B_\text{contr}] = \sum\limits_{i=1}^{N}m_{\text{rw}_i}d_i\hat{\mathbf{w}}_{3_i}\mathbf{b}_{\Omega_i}^T \]

\[ [C_\text{contr}] = \sum\limits_{i=1}^{N}\Big([I_{\text{rw}_i,W_{c_i}}]\hat{\mathbf{g}}_{s_i} + m_{\text{rw}_i}d_i[\tilde{\mathbf{r}}_{W_{c_i}/B}]\hat{\mathbf{w}}_{3_i}\Big)\mathbf{a}_{\Omega_i}^T \]

\[ [D_\text{contr}] = \sum\limits_{i=1}^{N}\Big([I_{\text{rw}_i,W_{c_i}}]\hat{\mathbf{g}}_{s_i} + m_{\text{rw}_i}d_i[\tilde{\mathbf{r}}_{W_{c_i}/B}]\hat{\mathbf{w}}_{3_i}\Big)\mathbf{b}_{\Omega_i}^T \]

\[ \mathbf{v}_\text{trans,contr} = \frac{1}{m_{\text{sc}}}\sum\limits_{i=1}^{N}m_{\text{rw}_i}d_i\left(\Omega_i^2\hat{\mathbf{w}}_{2_i}-c_{\Omega_i}\hat{\mathbf{w}}_{3_i}\right) \]

\[ \mathbf{v}_\text{rot,contr} = \sum\limits_{i=1}^{N} \Big[ m_{\text{rw}_i}[\tilde{\mathbf{r}}_{W_{c_i}/B}]d_i \Omega_i^2\hat{\mathbf{w}}_{2_i}-[I_{\text{rw}_i,W_{c_i}}]'\Omega_i \hat{\mathbf{g}}_{s_i} -[\tilde{\mathbf\omega}_{\cal B/N}]\Big([I_{\text{rw}_i,W_{c_i}}]\Omega_i \hat{\mathbf{g}}_{s_i}+ m_{\text{rw}_i}[\tilde{\mathbf{r}}_{W_{c_i}/B}]\mathbf{r}'_{W_{c_i}/B}\Big) \\-\Big([I_{\text{rw}_i,W_{c_i}}]\hat{\mathbf{g}}_{s_i} + m_{\text{rw}_i}d_i[\tilde{\mathbf{r}}_{W_{c_i}/B}]\hat{\mathbf{w}}_{3_i}\Big)c_{\Omega_i} \Big] \]

where \([I_{rw_i}]\) is the moment of inertia of the \(i^{th}\) wheel, \(d_i\) is the offset from the wheel centre of mass, and

\[ \mathbf{a}_{\Omega_i} = -\frac{m_{\text{rw}_i} d_i \hat{\mathbf{w}}_{3_i} }{J_{11_i} + m_{\text{rw}_i} d_i^2} \]

\[ \mathbf{b}_{\Omega_i} = -\frac{(J_{11_i} + m_{\text{rw}_i} d_i^2)\hat{\mathbf{ g}}_{\text{s}_i} + J_{13_i}\hat{\mathbf{w}}_{3_i} +m_{\text{rw}_i} d_i [\tilde{\mathbf{r}}_{W_i/B}]\hat{\mathbf{w}}_{3_i}}{J_{11_i} + m_{\text{rw}_i} d_i^2} \]

\[ c_{\Omega_i} = \frac{1}{J_{11_i} + m_{\text{rw}_i} d_i^2} \left[\omega_{w_{2_i}} \omega_{w_{3_i}} (J_{22_i} - J_{33_i} - m_{\text{rw}_i} d_i^2 ) - J_{13_i} \omega_{w_{2_i}}\omega_{s_i} - \\m_{\text{rw}_i} d_i \hat{\mathbf{w}}_{3_i}^T [\tilde{\mathbf\omega}_{\cal B/N}] [\tilde{\mathbf\omega}_{\cal B/N}] \mathbf{ r}_{W_i/B} + u_{s_i} \right] \]

Friction Model

All reaction wheel models can choose to use a built-in friction model, which includes static, coulomb and viscous components. This implementation follows the Stribeck model described in [3] where the computation of the net friction torque (\(\tau_f\)) is treated differently depending on whether the wheel has broken free of the static friction term (\(\tau_{st}\)).

Case 1: \(\tau_f \leq \tau_{st}\)

The Stribeck model from [2] is used to compute the net friction torque (\(\tau_f\)) on a reaction wheel using the following equation:

\[ \tau_f = \sqrt{2e}\left(\tau_{st} - \tau_c \right)^{\left[ -\left( \frac{\Omega}{\beta_{st}} \right)^2 \right]} + \tau_c \tanh{\left[10\frac{\Omega}{\beta_{st}} \right]} + c_v \Omega \]

where \(\tau_{st}\) is the static friction torque, \(\tau_c\) is the coulomb friction torque, \(\beta_{st}\) is the Stribeck coefficient, which modifies the peak of the Stribeck curve, and \(c_v\) is the viscous damping coefficient.

Case 2: \(\tau_f > \tau_s\)

The Stribeck function is only valid for cases where \(\tau_f \leq \tau_{st}\). For all other cases, the following equation is used:

\[ \tau_f = - \tau_c \text{sgn}({\Omega}) - c_v\Omega \]

References

[1] Hanspeter Schaub and John L. Junkins. Analytical Mechanics of Space Systems. AIAA Education Series, Reston, VA, 3rd edition, 2014.

[2] John Alcorn, Cody Allard, and Hanspeter Schaub. Fully coupled dynamical modelling of a rigid spacecraft with imbalanced reaction wheels. In AIAA/AAS Astrodynamics Specialist Conference, Long Beach, CA, Sept. 12–15 2016

[3] H. Olsson, K.J. A strom, C. Canudas de Wit, M. Gafvert, and P. Lischinsky. Friction models and friction compensation. European Journal of Control, 4(3):176 { 195, 1998.