# Modeling and Disturbance Analysis of Spinning Satellites with Inflatable Protective Structures

^{*}

## Abstract

**:**

## 1. Introduction

- (1)
- A novel IPS is implemented to guarantee the smooth operation of spinning satellites;
- (2)
- The dynamic model of the spinning satellite with IPSs in the inflatable stage is decoupled into two separate models: the spring hinge unfolding model and the spring expansion model;
- (3)
- The multi-body dynamics method based on the Newton–Euler equations is utilized to develop both the spring hinge unfolding model and the spring expansion model;
- (4)
- Various operating conditions are taken into consideration to thoroughly analyze the effects on the spinning satellite during the unfolding or expansion of IPSs.

## 2. Description of the Spinning Satellite with IPSs

**Remark 1**. In the realm of practical engineering, simultaneous separation of all fixed covers is unfeasible. Consequently, it is postulated that the fixed covers are sequentially separated at predetermined intervals. A similar principle applies to the inflation of the IPSs. In addition, when the spinning satellite with IPSs is in a compressed state, the flexible IPSs are vacuumed (with some residual gas remaining), and then the structure is folded in the pattern shown in Figure 3.

## 3. Dynamic Modeling of the Spinning Satellite with IPSs

#### 3.1. Spring Hinge Unfolding Model

_{u}

_{0}and O

_{uk}.

**F**

_{udk}, ${\mathit{M}}_{udk,0}$ and ${\mathit{M}}_{udk,k}$ in (1), we can obtain

_{uk}into O

_{u}

_{0}. According to the geometric constraint relationship between the kth IPS and the spinning satellite, we have

_{u}

_{0}into O

_{u}, ${\mathit{A}}_{uk}$ is the coordinate conversion matrix that transforms O

_{uk}into O

_{u}, ${\alpha}_{uk}$ is the angular displacement at the kth spring hinge, ${\rho}_{uk,0}$ and ${\rho}_{uk,k}$ are the installation positions on the kth line segment in O

_{u}

_{0}and O

_{uk}, ${l}_{uk,0}$, and ${l}_{uk,k}$ are the unit directions on the kth line segment in O

_{u}

_{0}and O

_{uk}. $\delta {q}_{uk}$ is the quaternion that describes O

_{uk}with respect to O

_{u}

_{0}, which is expressed as $\delta {q}_{uk}={\left[\begin{array}{cc}\mathrm{cos}\frac{{\alpha}_{uk}}{2}& {l}_{uk,0}\mathrm{sin}\frac{{\alpha}_{uk}}{2}\end{array}\right]}^{\mathrm{T}}$. Taking the derivative of (3), we obtain

_{uk}with respect to O

_{u}

_{0}, which is expressed as $\delta {\omega}_{uk,k}=2\delta {q}_{uk}{}^{-1}\delta {\dot{q}}_{uk}={l}_{uk,0}{\dot{\alpha}}_{uk}$. Based on (2) and (4), we have

#### 3.2. Spring Expansion Model

**F**

_{edk}is the force at the kth spring, ${\mathit{M}}_{edk,0}$ and ${\mathit{M}}_{edk,k}$ are the torques at the kth spring hinge in O

_{e}

_{0}and O

_{ek}.

**F**

_{edk,k}is the force at the kth spring in O

_{ek}. ${\xi}_{ek,0}$ is the installation position of the kth spring in O

_{e}

_{0}. ${\mathit{A}}_{ek}$ is the coordinate conversion matrix that transforms O

_{ek}into O

_{e}, and $\delta {A}_{ek}$ is the coordinate conversion matrix that transforms O

_{ek}into O

_{e}

_{0}. $\delta {q}_{ek}$ is the quaternion that describes O

_{ek}with respect to O

_{e}

_{0}. $\delta {\theta}_{ek}$, $\delta {q}_{e}$, ${\mathit{F}}_{edk}$ and ${\mathit{M}}_{edk,0}$ can be expressed as

## 4. Simulation Results and Analysis

#### 4.1. IPS Unfolding Analysis

- The deviation motion curve of the spinning satellite between IPSs in the orbital coordinate system and free flight in the same orbital coordinate system is

_{u}

_{0}during free flight, ${\mathit{e}}_{{r}_{u0}}$ is the displacement deviation of the spinning satellite between IPSs and free flight, ${\mathit{e}}_{{v}_{u0}}$ is the velocity deviation of the spinning satellite between IPSs and free flight, ${\mathit{e}}_{{\mathit{a}}_{u0}}$ is the Euler angle deviation of the spinning satellite between IPSs and free flight, and ${\mathit{e}}_{{\omega}_{u0,0}}$ is the angular velocity deviation of the spinning satellite between IPSs and free flight.

- The disturbance force and torque curve of the spinning satellite in the spinning satellite coordinate system are

_{u}

_{0}during IPSs, and ${\mathit{M}}_{ud,0}$ is the disturbance torque during IPSs.

#### 4.1.1. Unfolding Interval of 0.01 s

^{−2}m in displacement, 10

^{−2}m per second in velocity, 10

^{0}degrees in angular displacement, and 10

^{−1}revolutions per second in angular velocity.

^{1}Newtons in magnitude, while the disturbance torques reach an approximate magnitude of 10

^{1}Newton meters. The resulting impact of these disturbance forces and torques on the satellite’s velocities and angular velocities induces changes in its momenta and angular momenta, respectively, which manifest as impulses and angular impulses. Through rigorous calculations, the impulses caused by the disturbance are approximately 10

^{−1}Newtons per second in magnitude, and the angular impulses amount to an approximate magnitude of 10

^{−1}Newton meters per second.

#### 4.1.2. Unfolding Interval of 0.4 s

^{−2}m, the velocity deviations are on the order of 10

^{−2}m per second, the angular displacement deviations are on the order of 10

^{0}degrees, and the angular velocity deviations are on the order of 10

^{−1}revolutions per second.

^{1}Newtons, while the disturbance torques reach a magnitude of 10

^{1}Newton meters.

#### 4.2. IPS Expansion Analysis

- The deviation motion curve of the spinning satellite between IPS expansion in the orbital coordinate system and free flight in the same orbital coordinate system is

- The disturbance force and torque curve of the spinning satellite in the spinning satellite coordinate system are

_{u}

_{0}during IPS expansion, and ${\mathit{M}}_{ed,0}$ is the disturbance torque during IPS expansion.

#### 4.2.1. Expansion Interval of 0.01 s

^{−1}m per second. Once the IPSs reach their maximum capacity and undergo sudden tightening, the spinning satellite’s velocities experience significant deviations on the order of 10

^{0}m per second. Throughout the procedure, the spinning satellite exhibits displacement deviations on the order of 10

^{−2}m, while its angular velocities experience deviations of approximately 10

^{−3}rotations per second. Additionally, the angular displacements encounter deviations on the order of 10

^{−2}degrees.

^{2}newtons. Once the IPSs are fully expanded, the disturbance forces are expected to escalate to the order of 10

^{5}newtons, leading to a sudden tightening of the flexible body. Moreover, the expansion procedure induces the disturbance torques on the order of 10

^{1}Newton meters.

#### 4.2.2. Expansion Interval of 0.08 s

^{−1}m per second. Upon reaching its limit and experiencing instantaneous tension, the velocity deviations increase to the order of 10

^{0}m per second. Throughout the entire expansion process, the displacements result in deviations on the order of 10

^{−2}m. Additionally, the spinning satellite’s angular velocities experience deviations on the order of 10

^{−3}rotations per second, while the angular displacements encounter deviations on the order of 10

^{−2}degrees.

^{2}Newtons. When the IPSs reach their maximum expansion, the disturbance forces reach an order of approximately 10

^{5}Newtons, causing the IPSs to tighten instantaneously. Additionally, the expansion process generates disturbance torques on the order of approximately 10

^{1}Newton meters.

#### 4.3. Dynamic Model Verification

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

u | spring hinge unfolding | e | spring expansion |

O_{u} | the composite center-of-mass orbital coordinate system (the reference frame) | O_{e} | the composite center-of-mass orbital coordinate system (the reference frame) |

O_{u}_{0} | the spinning satellite coordinate system | O_{e}_{0} | the spinning satellite coordinate system |

m_{u}_{0} | the mass of the spinning satellite | m_{e}_{0} | the mass of the spinning satellite |

J_{u}_{0} | the moment of inertia of the satellite | J_{e}_{0} | the moment of inertia of the satellite |

r_{u}_{0} | the displacement of the spinning satellite | r_{e}_{0} | the displacement of the spinning satellite |

q_{u}_{0} | the quaternion of the spinning satellite | q_{e}_{0} | the quaternion of the spinning satellite |

v_{u}_{0} | the velocity of the spinning satellite | v_{e}_{0} | the velocity of the spinning satellite |

w_{u}_{0}_{,}_{0} | the angular velocity of the satellite in O_{u0} | w_{e}_{0}_{,}_{0} | the angular velocity of the satellite in O_{e0} |

k | number of IPSs | k | number of IPSs |

O_{uk} | the kth IPS coordinate system | O_{ek} | the kth IPS coordinate system |

m_{uk} | the mass of the kth IPS | m_{ek} | the mass of the kth IPS |

J_{uk} | the moment of inertia of the kth IPS | J_{ek} | the moment of inertia of the kth IPS |

r_{uk} | the displacement of the kth IPS | r_{ek} | the displacement of the kth IPS |

q_{uk} | the quaternion of the kth IPS | q_{ek} | the quaternion of the kth IPS |

v_{uk} | the velocity of the kth IPS | v_{ek} | the velocity of the kth IPS |

w_{uk}_{,k} | the angular velocity of the kth IPS in O_{uk} | w_{ek}_{,k} | the angular velocity of the kth IPS in O_{ek} |

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**Figure 6.**The deviation curves of (

**a**) ${\mathit{e}}_{{r}_{u0}}$, (

**b**) ${\mathit{e}}_{{v}_{u0}}$, (

**c**) ${\mathit{e}}_{{\mathit{a}}_{u0}}$, and (

**d**) ${\mathit{e}}_{{\omega}_{u0,0}}$ in the orbital coordinate system (t

_{e}= 0.01 s).

**Figure 7.**The disturbance curves of (

**a**) ${\mathit{F}}_{ud}$, (

**b**) ${\mathit{M}}_{ud,0}$ in the spinning satellite coordinate system (t

_{e}= 0.01 s).

**Figure 8.**The deviation curves of (

**a**) ${\mathit{e}}_{{r}_{u0}}$, (

**b**) ${\mathit{e}}_{{v}_{u0}}$, (

**c**) ${\mathit{e}}_{{\mathit{a}}_{u0}}$, and (

**d**) ${\mathit{e}}_{{\omega}_{u0,0}}$ in the orbital coordinate system (t

_{e}= 1.2 s).

**Figure 9.**The disturbance curves of (

**a**) ${\mathit{F}}_{ud}$, (

**b**) ${\mathit{M}}_{ud,0}$ in the spinning satellite coordinate system (t

_{e}= 0.4 s).

**Figure 10.**The deviation curves of (

**a**) ${\mathit{e}}_{{r}_{u0}}$, (

**b**) ${\mathit{e}}_{{v}_{u0}}$, (

**c**) ${\mathit{e}}_{{\mathit{a}}_{u0}}$, and (

**d**) ${\mathit{e}}_{{\omega}_{u0,0}}$ in the orbital coordinate system (t

_{e}= 0.01 s).

**Figure 11.**The disturbance curves of (

**a**) ${\mathit{F}}_{ud}$, (

**b**) ${\mathit{M}}_{ud,0}$ in the spinning satellite coordinate system (t

_{e}= 0.01 s).

**Figure 12.**The deviation curves of (

**a**) ${\mathit{e}}_{{r}_{u0}}$, (

**b**) ${\mathit{e}}_{{v}_{u0}}$, (

**c**) ${\mathit{e}}_{{\mathit{a}}_{u0}}$, and (

**d**) ${\mathit{e}}_{{\omega}_{u0,0}}$ in the orbital coordinate system (t

_{e}= 0.08 s).

**Figure 13.**The disturbance curves of (

**a**) ${\mathit{F}}_{ud}$, (

**b**) ${\mathit{M}}_{ud,0}$ in the spinning satellite coordinate system (t

_{e}= 0.08 s).

Parameter | Description | Value |
---|---|---|

m_{0} | the mass of the spinning satellite | 150 kg |

m_{k} | the mass of the kth IPS | 2 kg |

r_{0} | the displacement of the spinning satellite | [0 m 0 m 0 m]T |

v_{0} | the velocity of the spinning satellite | [0 m/s 0 m/s 0 m/s]T |

a_{0} | the Euler angle of the spinning satellite | [0° 0° 0°]T |

w_{0}_{,}_{0} | the angular velocity of the spinning satellite | [10 rad/s 0.03 rad/s 0.03 rad/s]T |

h | the height of the spinning satellite | 1.7 m |

r | the radius of the spinning satellite | 0.6 m |

${\alpha}_{k}$ | the Euler angle of the kth IPS | 0° |

w_{k} | the angular velocity of the kth IPS | 0 rad/s |

**Table 2.**RMS comparison of calculation results of the finite element model and the proposed dynamic model.

Parameter | The Finite Element Model | The Proposed Dynamic Model |
---|---|---|

F_{x} | 74.5123 N | 72.2888 N (2.9841%) |

F_{y} | 15,395 N | 14,895 N (3.2478%) |

F_{z} | 18,063 N | 17,463 N (3.3217%) |

M_{x} | 2.6426 N*m | 2.5355 N*m (4.0528%) |

M_{y} | 6.2578 N*m | 6.0007 N*m (4.1084%) |

M_{z} | 8.2648 N*m | 8.1714 N*m (1.1301%) |

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## Share and Cite

**MDPI and ACS Style**

Shang, Y.; Deng, Y.; Cai, Y.; Chen, Y.; He, S.; Liao, X.; Jiang, H.
Modeling and Disturbance Analysis of Spinning Satellites with Inflatable Protective Structures. *Aerospace* **2023**, *10*, 971.
https://doi.org/10.3390/aerospace10110971

**AMA Style**

Shang Y, Deng Y, Cai Y, Chen Y, He S, Liao X, Jiang H.
Modeling and Disturbance Analysis of Spinning Satellites with Inflatable Protective Structures. *Aerospace*. 2023; 10(11):971.
https://doi.org/10.3390/aerospace10110971

**Chicago/Turabian Style**

Shang, Yuting, Yifan Deng, Yuanli Cai, Yu Chen, Sirui He, Xuanchong Liao, and Haonan Jiang.
2023. "Modeling and Disturbance Analysis of Spinning Satellites with Inflatable Protective Structures" *Aerospace* 10, no. 11: 971.
https://doi.org/10.3390/aerospace10110971