Hinged Rigid Body

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Description

A hinged rigid body is a rigid planar panel, which attaches to the Spacecraft body through a one-degree-of-freedom hinge. The hinge allows for dynamic motion of the panel, modelled using a linear spring term and a linear damping term. An applied torque can also be added to simulate an active actuator such as a motor or latch.

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Example Use Cases

• Deployable solar panel. Use to simulate the dynamics of a spacecraft during solar panel deployment. The panel may be initially positioned against the spacecraft for launch, and then actuated out away from the spacecraft body
• Rotating solar panel. Use to rotate a deployed solar panel axially to point at the Sun.

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 in [1], where a satellites equations of motion (EOMs) are expressed using the following matrix representation:

$$\left[\begin{array}{cc}[A]& [B]\\ [C]& [D]\end{array}\right]\left[\begin{array}{c}{\ddot{\text{r}}}_{B/N}\\ {\dot{\omega }}_{\mathcal{B}/\mathcal{N}}\end{array}\right]=\left[\begin{array}{c}{\text{ v}}_{\text{trans}}\\ {\text{v}}_{\text{rot}}\end{array}\right]$$

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

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

$$\begin{gathered} (m_{\text{sc}}[I_{3\times 3}]+\sum _{i=1}^N[A_{\mathrm{contr},i}]){\ddot{\text{ r}}}_{B/N}+(-m_{\text{sc}}[{\tilde{\text{r}}}_{Cm/B}]+\sum _{i=1}^N[B_{\mathrm{contr},i}]){\dot{\omega }}_{\mathcal{B}/\mathcal{N}} \\ ={\text{F}}_{\mathrm{ext}}-2m_{\text{sc}}[{\tilde{\omega }}_{\mathcal{B}/\mathcal{N}}]{\dot{r}}_{Cm/B}-m_{\text{sc}}[{\tilde{\omega }}_{\mathcal{B}/\mathcal{N}}][{\tilde{\omega }}_{\mathcal{B}/\mathcal{N}}]{\text{r}}_{Cm/B}+\sum _{i=1}^N{\text{v}}_{\mathrm{trans},\mathrm{contr},i} \end{gathered}$$ $$\begin{gathered} [m_{\text{sc}}[{\tilde{\text{r}}}_{Cm/B}]+\sum _{i=1}^N[C_{\mathrm{contr},i}]]{\ddot{\text{r}}}_{B/N}+[[I_{\text{sc},B}]+\sum _{i=1}^N[D_{\mathrm{contr},i}]]{\dot{\omega }}_{\mathcal{B}/\mathcal{N}} \\ =-[{\tilde{\omega }}_{\mathcal{B}/\mathcal{N}}][I_{\text{sc},B}]{\omega }_{\mathcal{B}/\mathcal{N}}-\dot{[I_{\text{sc},B}]}{\omega }_{\mathcal{B}/\mathcal{N}}+{\text{L}}_{ext}+\sum _{i=1}^N{\text{v}}_{\mathrm{rot},\mathrm{contr},i} \end{gathered}$$

Here:

• $[A_{\mathrm{contr},i}]$, $[B_{\mathrm{contr},i}]$, $[C_{\mathrm{contr},i}]$, $[D_{\mathrm{contr},i}]$, ${\text{v}}_{\mathrm{trans},\mathrm{contr},i}$ and ${\text{v}}_{\mathrm{rot},\mathrm{contr},i}$ 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.
• $[I_{3\times 3}]$ 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 total center of mass.
• ${\dot{r}}_{Cm/B}$ is the rate of change of the total spacecraft center 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 $\mathcal{B}$.

Note: $[\tilde{x}]$ is the skew-symmetric matrix representation of a cross product operation.

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

Contributions for the Hinged Rigid Body

The EOMs are as developed in [2], resulting in the following contributions:

$$\begin{array}{rl}\left[A_{\text{contr }}\right]& =m_{{\text{sp }}_i}{d}_i{\hat{s}}_{i,3}{\mathbf{a}}_{{\theta }_i}^T\\ \left[B_{\text{contr }}\right]& =m_{{\text{sp }}_i}{d}_i{\hat{s}}_{i,3}{\mathbf{b}}_{{\theta }_i}^T\\ \left[C_{\text{contr }}\right]& =\left(I_{s_{i,2}}{\hat{s}}_{i,2}+m_{{\text{sp }}_i}{d}_i\left[{\tilde{\mathbf{r}}}_{S_{c,i}/B}\right]{\hat{s}}_{i,3}\right){\mathbf{a}}_{{\theta }_i}^T\\ \left[D_{\text{contr }}\right]& =\left(I_{s_{i,2}}{\hat{s}}_{i,2}+m_{{\text{sp }}_i}{d}_i\left[{\tilde{\mathbf{r}}}_{S_{c,i}/B}\right]{\hat{s}}_{i,3}\right){\mathbf{b}}_{{\theta }_i}^T\\ {\mathbf{v}}_{\text{trans },\text{ contr }}& =-\left(m_{{\text{sp }}_i}{d}_i{\dot{\theta }}_i^2{\hat{s}}_{i,1}+m_{{\text{sp }}_i}{d}_i{c}_{{\theta }_i}{\hat{s}}_{i,3}\right)\\ {\mathbf{v}}_{\text{rot, }\text{ contr }}& =-\left\{\left({\dot{\theta }}_i\left[{\tilde{\mathbf{\omega }}}_{\mathcal{B}/\mathcal{N}}\right]+c_{{\theta }_i}\left[I_{3\times 3}\right]\right)\left(I_{s_{i,2}}{\hat{s}}_{i,2}+m_{{\text{sp }}_i}{d}_i\left[{\tilde{\mathbf{r}}}_{S_{c,i}/B}\right]{\hat{s}}_{i,3}\right)+m_{{\text{sp }}_i}{d}_i{\dot{\theta }}_i^2\left[{\tilde{\mathbf{r}}}_{S_{c,i}/B}\right]{\hat{s}}_{i,1}\right\}\end{array}$$

with the following definitions:

$$\begin{array}{rl}{\mathbf{a}}_{{\theta }_i}& =-\frac{m_{{\mathbf{sp}}_i}{d}_i}{\left(I_{s_{i,2}}+m_{{\mathbf{sp}}_i}{d}_i^2\right)}{\hat{\mathbf{s}}}_{i,3}\\ {\mathbf{b}}_{{\theta }_i}& =-\frac{1}{\left(I_{s_{i,2}}+m_{{\mathbf{sp}}_i}{d}_i^2\right)}\left[\left(I_{s_{i,2}}+m_{{\mathbf{sp}}_i}{d}_i^2\right){\hat{s}}_{i,2}+m_{{\mathbf{sp}}_i}{d}_i\left[{\tilde{\mathbf{r}}}_{H_i/B}\right]{\hat{s}}_{i,3}\right]\\ {\mathbf{c}}_{{\theta }_i}& =\frac{1}{\left(I_{s_{i,2}}+m_{{\mathbf{sp}}_i}{d}_i^2\right)}\left(u_i-k_i{\theta }_i-c_i{\dot{\theta }}_i+{\hat{s}}_{i,2}\cdot {\tau }_{\mathrm{ext},H_i}+\left(I_{s_{i,3}}-I_{s_{i,1}}+m_{{\mathbf{sp}}_i}{d}_i^2\right){\omega }_{s_{i,3}}{\omega }_{s_{i,1}}-m_{{\mathrm{sp}}_i}{d}_i{\hat{\mathbf{s}}}_{i,3}^T\left[{\tilde{\mathbf{\omega }}}_{\mathcal{B}/\mathcal{N}}\right]\left[{\tilde{\mathbf{\omega }}}_{\mathcal{B}/\mathcal{N}}\right]{\mathbf{r}}_{H_i/B}\right)\end{array}$$

Here:

• $d_i$ is the length of the moment arm from the hinge to the panel center of mass, measured along ${\hat{s}}_{i,1}$

Since the model introduces another dynamic variable ${\theta }_i$, we obtain an additional equation of motion:

$${\ddot{\theta }}_i={\mathbf{a}}_{{\theta }_i}^T{\ddot{\mathbf{r}}}_{B/N}+{\mathbf{b}}_{{\theta }_i}^T{\dot{\omega }}_{\mathcal{B}/\mathcal{N}}+{\mathbf{c}}_{{\theta }_i}$$

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Assumptions/Limitations

• The hinged rigid body must have a diagonal inertia tensor with respect to the $S_i$ frame as seen in Figure 1
• Hinge dynamics is modelled as a linear spring and linear damping term. There is no maximum or minimum deflection; there is no maximum or minimum torque. If the spring is not stiff enough the hinged rigid body will unrealistically travel through bounds such as running into the spacecraft body.
• An arbitrary number of panels can be added to the spacecraft, but the model cannot support attaching hinged rigid bodies to other hinged rigid bodies.

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References

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

[2] C. Allard, Hanspeter Schaub, and Scott Piggott. General hinged solar panel dynamics approximating first-order spacecraft flexing. In AAS Guidance and Control Conference, Breckenridge, CO, Feb. 5–10 2016. Paper No. AAS-16-156.