# Microsatellite Uncertainty Control Using Deterministic Artificial Intelligence

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Rigid Body Motion

#### 2.2. PDI Control

#### 2.3. Luenberger Observers

_{d}is the discretized state matrix, ${B}_{d}$ is the discretized input matrix, and ${L}_{d}$ is the observer gain matrix.

#### 2.4. Deterministic Artificial Intelligence (DAI)

## 3. Results

#### 3.1. Thirty-Degree Yaw

#### 3.2. 30 Degree Yaw with Perturbations

#### 3.3. One-Hundred-Degree Yaw Maneuver with Perturbations and Simulated Damage

## 4. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Simulink Model

**Figure A2.**Trajectory subsystem depicted in Figure A1.

**Figure A3.**controllers’ and observers’ subsystem depicted in Figure A1.

#### Appendix A.2. Learn [J] Subsystem Function File Depicted in Figure A3

#### Appendix A.3. Feedforward Control Subsystem Function File Depicted in Figure A3

**Figure A4.**Observer subsystem depicted in Figure A3.

**Figure A5.**(

**a**) Feedback controller subsystem depicted in Figure A3; (

**b**) proportional, derivative, integral controller depicted in subfigure (

**a**).

**Figure A6.**Actuators’ subsystem depicted in Figure A1.

**Figure A8.**Dynamics Calc subsystem depicted in Figure A1.

**Figure A9.**Disturbances’ subsystem depicted in Figure A1.

**Figure A10.**Quaternion Calc subsystem depicted in Figure A1.

#### Appendix A.4. [ω]_([4 × 4])Subsystem Depicted in Figure A10 Constructing Direction Cosine Matrix from Angular Velocity

#### Appendix A.5. [q]_([4 × 4]) Subsystem Depicted in Figure A10 Constructing Direction Cosine Matrix from Quaternion Vector

**Figure A11.**Orbital Frame Calc subsystem depicted in Figure A1.

**Figure A12.**DCM subsystem depicted in Figure A1.

**Figure A13.**Euler Angle Calc subsystem depicted in Figure A1.

## Appendix B. Wrapper MATLAB Line Code

## Appendix C. Simulation Input Parameters

## References

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**Figure 1.**NASA’s Cyclone Global Navigation Satellite System (CYGNSS) mission, a constellation of eight microsatellites, will improve hurricane forecasting by making measurements of ocean surface winds in and near the eye wall of tropical cyclones, typhoons and hurricanes throughout their life cycle. Figure taken from [1] in compliance with NASA’s image use policy [2].

**Figure 2.**(

**a**) Thirty degree yaw feedforward plus feedback (FFD + FB) control, (

**b**) thirty degree yaw hybrid deterministic artificial intelligence (DAI) control. Both figures include display of tracking errors in the zoomed-inset graphic. Notice the ordinate scale, respectively of insets in subfigure (

**a**) and (

**b**) to reveal the relative comparison.

**Figure 3.**Control with perturbations. (

**a**) Thirty degree yaw using feedforward plus feedback (FFD + FB) control; (

**b**) thirty degree yaw hybrid deterministic artificial intelligence control. Both figures include display of tracking errors in the zoomed-inset graphic. Notice the ordinate scale, respectively of insets in subfigure (

**a**,

**b**) to reveal the relative comparison.

**Figure 4.**(

**a**) One hundred degree yaw feedforward plus feedback (FFD + FB) control; (

**b**) one hundred degree yaw with hybrid deterministic artificial intelligence (DAI) control. Both figures include display of tracking errors in the zoomed-inset graphic. Notice the ordinate scale, respectively of insets in subfigure (

**a**,

**b**) to reveal the relative comparison.

**Table 1.**Definitions of proximal variables for Section 2.1.

Variable | Definition | Variable | Definition |
---|---|---|---|

$T$ | Resultant applied torque | $\omega $ | Angular velocity (radians/second) |

$\dot{H}$ | Timed rate of change of ${H}_{S}$ | ${H}_{S}$ | Spacecraft angular momentum |

**Table 2.**Definitions of proximal variables for Section 2.2.

Variable | Definition | Variable | Definition |
---|---|---|---|

${u}_{fb}$ | Feedback control signal | ${\theta}_{d}$ | Desired attitude angle |

${k}_{p}$ | Proportional gain | ${\omega}_{d}$ | Desired angular rate |

${k}_{d}$ | Derivative gain | $\theta $ | Actual attitude angle |

${k}_{i}$ | Integral gain | $\omega $ | Actual angular rate |

${e}_{\theta}$ | Angular position error | J | Mass moment of inertia |

${\dot{e}}_{\theta}$ | Angular velocity error | dt | Differential element of time |

**Table 3.**Definitions of proximal variables for Section 2.3.

Variable | Definition | Variable | Definition |

$x\left(k+1\right)$ | State at following timestep | $u\left(k\right)$ | Control at present timestep |

${A}_{d}$ | Discretized state matrix | $y\left(k\right)$ | Output at present timestep |

$\widehat{x}\left(k\right)$ | Present state estimate | $\widehat{y}\left(k\right)$ | Present timestep output estimate |

${B}_{d}$ | Discretized input matrix | k | Present timestep |

${L}_{d}$ | Observer gain matrix |

**Table 4.**Definitions of proximal variables for Section 2.4.

Variable | Definition | Variable | Definition |
---|---|---|---|

$u$ | Total control | ${\dot{\mathsf{\omega}}}_{x}$ | Acceleration about body x-axis |

$\widehat{J}$ | Estimated mass moment of inertia | ${\dot{\mathsf{\omega}}}_{y}$ | Acceleration about body y-axis |

${\dot{\omega}}_{d}$ | Desired angular acceleration vector | ${\dot{\mathsf{\omega}}}_{z}$ | Acceleration about body z-axis |

${\omega}_{d}$ | Desired angular rate vector | ${\mathsf{\omega}}_{x}$ | Angular rate about the body x-axis |

${\mathsf{\Phi}}_{d}$ | Regression matrix of “knowns” | ${\mathsf{\omega}}_{y}$ | Angular rate about the body y-axis |

$\widehat{\mathsf{\Theta}}$ | Regression vector of “unknowns” | ${\mathsf{\omega}}_{z}$ | Angular rate about the body z-axis |

FFD | Feedforward control | DAI | Deterministic artificial intelligence |

FB | Feedback control |

${\mathit{K}}_{\mathit{p}}$ | ${\mathit{K}}_{\mathit{d}}$ | ${\mathit{K}}_{\mathit{i}}$ | |
---|---|---|---|

PDI control | 1000 | 10 | 0.1 |

Luenberger Observer | 10,000 | 500 | 0.1 |

^{1}These gains will remain constant for all data sets.

Method | $\mathbf{Mean}\mathbf{Tracking}\mathbf{Error}\left(\mathit{\mu}\right)$ | $\mathbf{Tracking}\mathbf{Error}\mathbf{Standard}\mathbf{Deviation}\left(\mathit{\sigma}\right)$ | Control Effort |
---|---|---|---|

Feedforward + feedback (PDI) | 2.1424 × 10^{−4} | 2.3 × 10^{−3} | 2.13 × 10^{1} |

Hybrid deterministic artificial intelligence | 1.5147 × 10^{−5} | 2.0181 × 10^{−4} | 1.82 × 10^{1} |

^{1}Illustration of performance improvement.

Method | $\mathbf{Mean}\mathbf{Tracking}\mathbf{Error}\left(\mathit{\mu}\right)$ | $\mathbf{Tracking}\mathbf{Error}\mathbf{Standard}\mathbf{Deviation}\left(\mathit{\sigma}\right)$ | Control Effort |
---|---|---|---|

Feedforward + feedback (PDI) | 2.1537 $\times {10}^{-4}$ | 2.3 $\times {10}^{-3}$ | 2.46 $\times {10}^{1}$ |

Hybrid deterministic artificial intelligence | 6.0185 $\times {10}^{-6}$ | 4.3078 $\times {10}^{-5}$ | 2.13 $\times {10}^{1}$ |

^{1}Illustration of performance improvement.

Method | $\mathbf{Mean}\mathbf{Tracking}\mathbf{Error}\left(\mathit{\mu}\right)$ | $\mathbf{Tracking}\mathbf{Error}\mathbf{Standard}\mathbf{Deviation}\left(\mathit{\sigma}\right)$ | Control Effort |
---|---|---|---|

Feedforward + feedback (PDI) | $6.2142\times {10}^{-5}$ | $1.40\times {10}^{-3}$ | $2.29\times {10}^{1}$ |

Hybrid deterministic artificial intelligence | $5.4427\times {10}^{-6}$ | $4.0650\times {10}^{-5}$ | $1.99\times {10}^{1}$ |

^{1}Illustration of performance improvement.

**Table 9.**Percent performance improvements in DAI vs Feedforward+Feedback for thirty degree yaw with perturbations and simulated damage

^{1}.

Method | Mean Tracking Error | Tracking Error Standard Deviation | Control Effort |
---|---|---|---|

Hybrid deterministic artificial intelligence | 91.24% | 97.10% | 13.10% |

^{1}Illustration of performance improvement.

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Wilt, E.; Sands, T.
Microsatellite Uncertainty Control Using Deterministic Artificial Intelligence. *Sensors* **2022**, *22*, 8723.
https://doi.org/10.3390/s22228723

**AMA Style**

Wilt E, Sands T.
Microsatellite Uncertainty Control Using Deterministic Artificial Intelligence. *Sensors*. 2022; 22(22):8723.
https://doi.org/10.3390/s22228723

**Chicago/Turabian Style**

Wilt, Evan, and Timothy Sands.
2022. "Microsatellite Uncertainty Control Using Deterministic Artificial Intelligence" *Sensors* 22, no. 22: 8723.
https://doi.org/10.3390/s22228723