# Angular Rate Sensing with GyroWheel Using Genetic Algorithm Optimized Neural Networks

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Overview of GyroWheel System

#### 2.1. GyroWheel Mechanical Configuration

#### 2.2. Dynamic Models of GyroWheel System

## 3. GyroWheel Rate Sensing for Small Tilt Conditions and Error Analysis

#### 3.1. Rate Sensing Equation for Small Tilt Conditions

#### 3.2. Error Analysis of GyroWheel Rate Sensing

#### 3.2.1. Linearization Error

- The rate sensing errors caused by linearization at zero tilt are significantly correlated to the tilt angles of the GyroWheel rotor and the spin rate. The rate sensing errors increase with the increasing of the tilt angles and the increasing of the spin rate.
- The rate sensing Equation (14) can be applied to measure spacecraft angular rates under small tilt conditions where the rotor tilt angles are less than 0.5°. However, when the GyroWheel is operated at a tilt angle of 4°, the rate sensing errors are up to 10
^{−2}rad/s. Obviously, the rate sensing accuracy is far from satisfactory under large tilt conditions. - In an effort to ensure the rate sensing accuracy, the linearization errors should be compensated. The compensation terms are functions of tilt angles and spin rate, and can be denoted as ${\delta}_{nlx}\left({\phi}_{x},{\phi}_{y},{\omega}_{s}\right),{\delta}_{nly}\left({\phi}_{x},{\phi}_{y},{\omega}_{s}\right)$.

#### 3.2.2. Parameter Error

- Measurable values, including the tilt angles ${\phi}_{x},\text{}{\phi}_{y}$, the spin rate ${\omega}_{s}$, and the coil currents ${i}_{x},\text{}{i}_{y}$.
- System parameters, including the moments of inertia ${I}_{gt},\text{}{I}_{gs},\text{}{I}_{rs}$, the stiffness coefficients ${K}_{x},\text{}{K}_{y}$, the damping coefficient ${C}_{g}$, and the torque factors ${k}_{tx},\text{}{k}_{ty}$.

## 4. GyroWheel Rate Sensing Using Genetic Algorithm Optimized Neural Networks

#### 4.1. Rate Sensing Principle Based on Torque Balance Theory

#### 4.2. Identification of Torque Factors and Equivalent Rates

#### 4.3. Rate Sensing Using Genetic Algorithm Optimized Neural Networks

#### 4.3.1. Methodology: Genetic Algorithm Optimized Neural Network

#### 4.3.2. GAANN-Based Rate Sensing for GyroWheel

#### 4.3.3. Simulation Results and Analysis

^{−6}rad/s using the proposed method. Therefore, the rate sensing method given in Equation (28) is an effective way to estimate the spacecraft angular rates under various operating conditions.

## 5. Conclusions

- The GAANN-based method provides a high rate sensing accuracy even under large tilt conditions. Therefore, it can be applied to measure angular rates in the whole operating range of the GyroWheel.
- The GAANN-based method does not depend on the GyroWheel parameters that are difficult to identify. Instead, explicit ANN models are established using experimental data. Once the weights and biases of the ANN models are determined, the spacecraft angular rates can be estimated with the measurable tilt angles, spin rate and coil currents of the GyroWheel.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

DTG | Dynamically Tuned Gyroscope |

ANN | Artificial Neural Network |

GA | Genetic Algorithm |

CMG | Control Moment Gyroscope |

DC | Direct Current |

MLP | Multi-layer Perception |

BP | Back-propagation |

GAANN | Genetic Algorithm Optimized Neural Network |

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**Figure 3.**Relationship between rate sensing errors and tilt angles: (

**a**) X-axis rate sensing error versus x-axis tilt; (

**b**) Y-axis rate sensing error versus y-axis tilt; (

**c**) X-axis rate sensing error versus y-axis tilt; (

**d**) Y-axis rate sensing error versus x-axis tilt.

**Figure 6.**GA optimized ANN algorithm: (

**a**) Flowchart; (

**b**) An example of storing weights and biases of an ANN model in the genes of a chromosome.

**Figure 9.**GAANN correlation performance: (

**a**) ANN models for predicting equivalent rates; (

**b**) ANN models for predicting torque factors.

**Figure 10.**Relationship between rate sensing errors and tilt angles: (

**a**) X-axis rate sensing error versus x-axis tilt; (

**b**) Y-axis rate sensing error versus y-axis tilt; (

**c**) X-axis rate sensing error versus y-axis tilt; (

**d**) Y-axis rate sensing error versus x-axis tilt.

Parameters | Values |
---|---|

Rotor transverse inertia I_{rt} | 3.458 × 10^{−3} kg·m^{2} |

Rotor spin inertia I_{rs} | 6.402 × 10^{−3} kg·m^{2} |

Gimbal transverse inertia I_{gt} | 1.276 × 10^{−5} kg·m^{2} |

Gimbal spin inertia I_{gs} | 1.805 × 10^{−5} kg·m^{2} |

Stiffness coefficients K_{x}, K_{y} | 0.092 Nm/rad |

Damping coefficient C_{g} | 3.100 × 10^{−8} Nm/(rad/s) |

Tile range $\phi =\sqrt{{\phi}_{x}^{2}+{\phi}_{y}^{2}}$ | $0\xb0\le \phi \le 4\xb0$ |

Range of spin rate ${\omega}_{s}$ | 133.52 rad/s ≤ ${\omega}_{s}$ ≤ 180.64 rad/s |

Parameters | Small Tilt ($\mathit{\phi}=0.5\xb0$) | Large Tilt ($\mathit{\phi}=4\xb0$) | ||
---|---|---|---|---|

$\left|\Delta {\mathit{\omega}}_{\mathit{c}\mathit{x}\mathit{m}}\right|\left(\mathbf{rad}/\mathbf{s}\right)$ | $\left|\Delta {\mathit{\omega}}_{\mathit{c}\mathit{y}\mathit{m}}\right|\left(\mathbf{rad}/\mathbf{s}\right)$ | $\left|\Delta {\mathit{\omega}}_{\mathit{c}\mathit{x}\mathit{m}}\right|\left(\mathbf{rad}/\mathbf{s}\right)$ | $\left|\Delta {\mathit{\omega}}_{\mathit{c}\mathit{y}\mathit{m}}\right|\left(\mathbf{rad}/\mathbf{s}\right)$ | |

I_{gt} | 3.137 × 10^{−4} | 3.137 × 10^{−4} | 2.510 × 10^{−3} | 2.510 × 10^{−3} |

I_{gs} | 2.220 × 10^{−4} | 2.220 × 10^{−4} | 1.776 × 10^{−3} | 1.776 × 10^{−3} |

I_{rs} | 1.019 × 10^{−4} | 1.019 × 10^{−4} | 8.150 × 10^{−4} | 8.150 × 10^{−4} |

K_{x} | 4.690 × 10^{−5} | 4.690 × 10^{−5} | 3.752 × 10^{−4} | 3.752 × 10^{−4} |

K_{y} | 4.690 × 10^{−5} | 4.690 × 10^{−5} | 3.752 × 10^{−4} | 3.752 × 10^{−4} |

C_{g} | 4.220 × 10^{−9} | 4.220 × 10^{−9} | 3.376 × 10^{−8} | 3.376 × 10^{−8} |

Parameters | Values |
---|---|

Coding type | Real coding |

Population size | 100 |

Iterations | 50 |

Selection operator | Roulette-wheel selection |

Crossover probability | 60% |

Mutation probability | 0.5% |

Parameters | Values |
---|---|

Number of hidden neurons | 10 |

Epochs | 2000 |

Training algorithm | Bayesian regulation back-propagation |

Activation function of hidden layer | tan-sigmoid |

Activation function of output layer | purelin (linear transfer function) |

ANN | MSE Values | ||
---|---|---|---|

Training | Validation | Testing | |

1 | 1.1142 × 10^{−8} | 7.3956 × 10^{−9} | 1.5940 × 10^{−8} |

2 | 7.7689 × 10^{−9} | 1.6244 × 10^{−8} | 1.0707 × 10^{−8} |

3 | 1.7201 × 10^{−9} | 9.4487 × 10^{−10} | 7.4277 × 10^{−10} |

4 | 5.8538 × 10^{−10} | 8.4108 × 10^{−10} | 1.2611 × 10^{−9} |

ANN | Weights between Input and Hidden Layer | Biases of Hidden Layer | Weights between Hidden and Output Layer | Biases of Output Layer |
---|---|---|---|---|

1 | 0.0100, 0.5545, 0.0777; 0.1455, 0.4150, −0.1270; 0.0704, −0.3346, −0.0711; −0.0391, −0.7435, −0.0994; 0.1792, −0.4262, 0.0458; 0.2559, −0.0055, −0.0170; 0.2264, −0.2593, 0.0561; −0.2273, −0.3106, 0.0165; 0.2551, 0.2819, 0.0432; −0.2071, 0.2051, 0.0241. | −0.5267; −1.4775; −0.3092; 1.7672; −1.4598; −0.7754; 0.4807; −0.6000; −0.2493; −0.1981. | 5.0975, −7.1387, −8.8096, −9.8384, −7.3076, 10.6156, −5.8899, 6.0972, −5.9602, −16.2299. | 3.7141 |

2 | 0.5141, −0.3526, 0.0901; 0.2502, 0.0662, −0.0692; −0.7629, 0.4477, 0.0198; 0.3041, 0.1787, −0.0073; −0.6078, −0.2979, −0.1073; 0.3246, 0.1595, −0.0853; −0.4315, 0.3101, −0.0164; 0.1087, −0.0002, 0.1244; −0.4581, 0.2426, 0.0308; 0.2989, −0.0997, −0.2243. | −1.9411; 0.8813; −1.1333; 0.3888; 1.4684; −0.1584; −0.8051; −0.1484; −0.2198; −1.6234. | −5.3511, −10.8303, −1.5636, 8.1233, 4.6847, −4.7499, 4.5676, −4.1065, −3.9801, 2.6404. | −2.1126 |

3 | 0.1551, 0.0128, 0.2639; 0.0151, 0.0003, −0.3749; −0.3603, −0.0390, −0.4786; 0.0009, 0.0318, −0.3257; 0.0303, 0.0207, −0.9521; 0.0565, 0.0221, 0.2269; −0.1718, −0.0120, −0.2306; −0.0226, −0.0192, 0.6862; 0.0200, −0.1314, −0.2398; −0.0373, −0.0052, 0.2146. | 0.1712; 0.2737; 0.1333; 0.3065; −1.7030; −0.2572; −0.5986; 0.4407; −1.0285; −0.1963. | 0.5813, 0.6213, 0.0341, 0.5309, 0.9480, −0.4722, 0.6712, −0.4977, 0.2949, −0.4503. | 0.8952 |

4 | 0.0064, −0.0065, 0.3470; 0.0150, −0.0038, 0.6597; 0.0137, −0.0044, 0.8656; −0.0123, −0.0312, 0.1601; −0.0019, 0.0316, −0.0801; 0.0036, −0.0433, −0.3119; −0.0196, 0.1975, 0.0048; −0.1069, −0.0316, 0.1904; −0.0218, 0.1599, 0.1726; 0.0258, −0.0250, 0.2503. | −0.1904; 0.4451; 1.6821; −0.3732; 0.1029; 0.1148; 0.4758; 0.9927; 0.0577; −0.2785. | −0.5990, −0.4535, −1.1659, −0.6272, 0.2505, 0.5259, −0.3702, −0.4228, 0.2553, −0.5580. | 1.0207 |

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## Share and Cite

**MDPI and ACS Style**

Zhao, Y.; Zhao, H.; Huo, X.; Yao, Y.
Angular Rate Sensing with GyroWheel Using Genetic Algorithm Optimized Neural Networks. *Sensors* **2017**, *17*, 1692.
https://doi.org/10.3390/s17071692

**AMA Style**

Zhao Y, Zhao H, Huo X, Yao Y.
Angular Rate Sensing with GyroWheel Using Genetic Algorithm Optimized Neural Networks. *Sensors*. 2017; 17(7):1692.
https://doi.org/10.3390/s17071692

**Chicago/Turabian Style**

Zhao, Yuyu, Hui Zhao, Xin Huo, and Yu Yao.
2017. "Angular Rate Sensing with GyroWheel Using Genetic Algorithm Optimized Neural Networks" *Sensors* 17, no. 7: 1692.
https://doi.org/10.3390/s17071692