# A New Bias Error Prediction Model for High-Precision Transfer Alignment

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. TA Approach

#### 2.1.1. Introduction to the Coordinate Frame

_{i}) at the center of the earth and is fixed with inertial space. The x,y-axes are defined in the mean equatorial plane of the earth and the x-axis points to the first point of Aries. The z-axis is directed along the mean rotation axis of the earth. The ship body coordinate frame (b-frame) is rigidly attached to the ship and its origin (O

_{b}) is at the ship’s center of gravity. The x-axis points starboard, the y-axis is in the direction of ship bow, and the z-axis points upward. The peripheral sensor body frame (s-frame) has origin (O

_{s}) at the orthogonal sensor center and the coordinates are in accordance with the sensor measurement coordinate frame.

#### 2.1.2. Acceleration Matching Function

_{m}(x

_{m}, y

_{m}, z

_{m}) is aligned with respect to the b-frame, and the SINS coordinate frame O

_{s}(x

_{s}, y

_{s}, z

_{s}) is in accordance with s-frame.

_{m}(x

_{m}, y

_{m}, z

_{m}) coordinates, which can be written as ${\stackrel{\rightharpoonup}{f}}_{im}^{m}$, whereas the SINS measures the ship acceleration projected onto the O

_{s}(x

_{s}, y

_{s}, z

_{s}) coordinates, which can be written as. The acceleration relationship can be expressed as:

_{m}(x

_{m}, y

_{m}, z

_{m}) coordinates, and ${\stackrel{\rightharpoonup}{r}}_{ms}^{m}$ is the total displacement or lever-arm of SINS relative to MINS, which includes a static component ${\stackrel{\rightharpoonup}{r}}_{0}$ and a dynamic component ${\stackrel{\rightharpoonup}{r}}_{d}^{m}$. The dot operator $\dot{(\u2022)}$ represents the derivation with respect to time t. If the misalignment angle $\overrightarrow{\psi}$ is small, ${C}_{s}^{m}$ can be written as:

#### 2.1.3. Kalman Filtering Formulation

**z**is a measurement vector,

**H**is a measurement matrix,

**x**is a state vector, and

**v**is a measurement error vector. The total misalignment Euler angle includes a static component $\stackrel{\rightharpoonup}{\phi}$ and a dynamic component $\stackrel{\rightharpoonup}{\theta}$. The bias of the accelerometers of SINS is considered. The dynamic lever-arm velocity ${\ddot{\stackrel{\rightharpoonup}{r}}}_{d}^{m}$ is considered an unobservable disturbance. Then, the state vector is specified by:

**H**takes the form:

**F**is given by:

**w**has the covariance matrix:

#### 2.2. Correlation Between Linear Motion and Dynamic Lever-Arm

#### 2.2.1. Bernoulli-Euler Beam Model

_{n}is a constant of the nth spatial part $q\left(y\right)$. Substituting Equations (23)–(25) into Equation (22) yields:

_{0}tend to negative infinite, Equation (28) can be rewritten as:

#### 2.2.2. Beam Linear Motion and Dynamic Lever-arm Model

_{1}and y

_{2}can be expressed as:

#### 2.3. Bias Error Analysis and Modeling

#### 2.3.1. Bias Error Analysis

**X**and

**Y**are given by:

#### 2.3.2. Bias Error Prediction Model

## 3. Experiment Validation

#### 3.1. Simulation Conditions and Model Establishment

^{T}. The dynamic misalignment angles were treated as three independent second-order Markov processes whose parameters are illustrated in Equation (16). The parameters $\sigma ,\beta $, and $\alpha $ used to depict the dynamic misalignment angles are listed in Table 1. The parameters were identified from the real measurement data.

^{T}. However, this additional acceleration caused by ${\stackrel{\rightharpoonup}{r}}_{0}$ was completely compensated in our simulation.

#### 3.2. Result and Analysis

^{2}, the bias error in yawing angle increased from −0.6 to −2.4 mrad. This result fits the second inference about Equation (50) well. Figure 5b shows that the bias error in pitching angle changed slowly as the amplitude of ${\ddot{r}}_{dy}^{m}$ increased. Similarly, the reason for this finding is that the gravity in ${f}_{isz}^{s}$ makes the autocorrelation of ${f}_{isz}^{s}$ much higher than that of ${\ddot{r}}_{dy}^{m}$, which was discussed through Figure 4b.

^{2}, the bias error in yawing angle decreased from −0.6 to −0.2 mrad. This result matches the third inference for Equation (50) well.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Kain, J.; Cloutier, J. Rapid Transfer Alignment for Tactical Weapon Applications. In Proceedings of the Guidance, Navigation and Control Conference, Boston, MA, USA, 14–16 August 1989. [Google Scholar]
- Schneider, A.M. Kalman Filter Formulations for Transfer Alignment of Strapdown Inertial Units. J. Navig.
**1983**, 30, 72–89. [Google Scholar] [CrossRef] - Berke, L. Master reference system for rapid at sea alignment of aircraft inertial navigation systems. In Proceedings of the Guidance and Control Conference, Seattle, WA, USA, 15–17 August 1966. [Google Scholar]
- Browne, B.H.; Lackowski, D.H. Estimation of Dynamic Alignment Errors in Shipboard Firecontrol Systems. In Proceedings of the Conference on Decision and Control including the 15th Symposium on Adaptive Processes, Clearwater, FL, USA, 1–3 December 1976. [Google Scholar]
- Spalding, K. An Efficient Rapid Transfer Alignment Filter. In Proceedings of the Astrodynamics Conference on Guidence, Navigation and Control Conference, Hilton Head Island, SC, USA, 10–12 August 1992. [Google Scholar]
- Wu, W.; Chen, S.; Qin, S.Q. Online estimation of ship dynamic flexure model parameters for transfer alignment. IEEE Trans. Control Syst. Technol.
**2013**, 21, 1666–1678. [Google Scholar] [CrossRef] - Hagan, C.E.; Lofts, C.S. Accelerometer based alignment transfer. In Proceedings of the IEEE PLANS 92 Position Location and Navigation Symposium Record, Monterey, CA, USA, 23–27 March 1992. [Google Scholar]
- Sinpyo, H.; Man Hyung, L.; Ho-Hwan, C.; Sun-Hong, K.; Speyer, J.L. Experimental study on the estimation of lever arm in GPS/INS. IEEE Trans. Veh. Technol.
**2006**, 55, 431–448. [Google Scholar] - Chattaraj, S.; Mukherjee, A.; Chaudhuri, S.K. Transfer alignment problem: Algorithms and design issues. Gyroscopy Navig.
**2013**, 4, 130–146. [Google Scholar] [CrossRef] - Xiong, Z.; Peng, H.; Liu, J.Y.; Wang, J.; Sun, Y. Online calibration research on the lever arm effect for the hypersonic vehicle. In Proceedings of the IEEE/ION Position, Location and Navigation Symposium, Monterey, CA, USA, 5–8 May 2014. [Google Scholar]
- Gao, W.; Zhang, Y.; Sun, Q.; Ben, Y. Error analysis and compension for lever-arm effect in transfer anlgnment. Chin. J. Sci. Instrum.
**2013**, 34, 560–564. [Google Scholar] - Yang, D.; Wang, S.; Li, H.; Liu, Z.; Zhang, J. Performance Enhancement of Large-Ship Transfer Alignment: A Moving Horizon Approach. J. Navig.
**2012**, 66, 17–33. [Google Scholar] [CrossRef] [Green Version] - Gao, W.; Ben, Y.; Sun, F.; Xu, B. Performance comparison of two filtering approaches for INS rapid transfer alignment. In Proceedings of the 2007 International Conference on Mechatronics and Automation, Harbin, China, 5–8 August 2007. [Google Scholar]
- Qingwei, G.; Guorong, Z.; Xibin, W.; Fang, W. Incorporate modeling and simulation of transfer alignment with flexure of carrier and lever-arm effect. Chin. J. Aeronaut.
**2009**, 11, 2172–2177. [Google Scholar] - Chen, W.; Zeng, Q.; Liu, J.; Wang, H. Research on shipborne transfer alignment under the influence of the uncertain disturbance based on the extended state observer. Opt. Int. J. Light Electron Opt.
**2017**, 130, 777–785. [Google Scholar] [CrossRef] - Groves, P.D.; Wilson, G.G.; Mather, C.J. Robust rapid transfer alignment with an INS/GPS reference. In Proceedings of the 2002 National Technical Meeting of The Institute of Navigation, San Diego, CA, USA, 28–30 January 2002. [Google Scholar]
- Mochalov, A.V. Optical Gyros and Their Application. In A System for Measuring Deformations of Large-Sized Objects; LouKianov, D., Rodloff, R., Sorg, H., Stieler, B., Eds.; Canada Communication Group Inc.: Hull, QC, Canada, 1999; pp. 1–9. [Google Scholar]
- Groves, P.D. Optimising the Transfer Alignment of Weapon INS. J. Navig.
**2003**, 56, 323–335. [Google Scholar] [CrossRef] - Petovello, M.G.; Keefe, K.O.; Lachapelle, G.; Cannon, M.E. Measuring aircraft carrier flexure in support of autonomous aircraft landings. IEEE Trans. Aerosp. Electron. Syst.
**2009**, 45, 523–535. [Google Scholar] [CrossRef] - Pehlivanoğlu, A.G.; Ercan, Y. Investigation of Flexure Effect on Transfer Alignment Performance. J. Navig.
**2012**, 66, 1–15. [Google Scholar] [CrossRef] [Green Version] - Wu, W.; Qin, S.; Chen, S. Coupling influence of ship dynamic flexure on high accuracy transfer alignment. Int. J. Model. Ident. Control
**2013**, 19, 224–234. [Google Scholar] [CrossRef] - Heffes, H. The effect of erroneous models on the Kalman filter response. IEEE Trans. Autom. Control
**1966**, 11, 541–543. [Google Scholar] [CrossRef] - Abei, X.; Qingming, G.; Songhui, H. Analysis of model error effect on kalman filtering. J. Geod. Geodyn.
**2008**, 28, 101–104. [Google Scholar] - Kailath, T. An innovations approach to least-squares estimation—Part I: Linear filtering in additive white noise. IEEE Trans. Autom. Control
**1968**, 13, 646–655. [Google Scholar] [CrossRef] - Kailath, T.; Frost, P. An innovations approach to least-squares estimation—Part II: Linear smoothing in additive white noise. IEEE Trans. Autom. Control
**1968**, 13, 655–660. [Google Scholar] [CrossRef] - Sorenson, H.W. Least-squares estimation: From Gauss to Kalman. IEEE Spectr.
**1970**, 7, 63–68. [Google Scholar] [CrossRef] - Ishihara, J.Y.; Terra, M.H.; Campos, J.C.T. Robust Kalman filter for descriptor systems. IEEE Trans. Autom. Control
**2006**, 51, 1354. [Google Scholar] [CrossRef] - Luo, X.; Wang, H. Robust Adaptive Kalman Filtering—A method based on quasi-accurate detection and plant noise variance–covariance matrix tuning. J. Navig.
**2016**, 70, 137–148. [Google Scholar] [CrossRef] - Nayfeh, A.H.; Mook, D.T.; Marshall, L.R. Nonlinear Coupling of Pitch and Roll Modes in Ship Motions. J. Hydronaut.
**1973**, 7, 145–152. [Google Scholar] [CrossRef] - Juncher Jensen, J.; Dogliani, M. Wave-induced ship full vibrations in stochastic seaways. Mar. Struct.
**1996**, 9, 353–387. [Google Scholar] [CrossRef] - Lee, J.; Whaley, P.W. Prediction of the Angular Vibration of Aircraft Structures. J. Sound Vib.
**1976**, 49, 541–549. [Google Scholar] [CrossRef] - Huddle, J.R.; Chueh, V.K. Transfer Alignment of Navigation Systems. U.S. Patent 7206694B2, 16 July 2004. [Google Scholar]

**Figure 3.**Alignment error using Kalman filtering (KF) for different correlation between the SINS acceleration data in X (or Z)-direction and the dynamic lever-arm acceleration in Y-direction: (

**a**) Uncorrelated (

**b**) Partly correlated.

**Figure 4.**Bias error as a function of the correlation coefficient. (

**a**) Error for the estimation of yawing angle. (

**b**) Error for the estimation of pitching angle. The vertical lines indicate the corresponding standard deviations or error bars.

**Figure 5.**Bias error as a function of the amplitude of dynamic lever-arm acceleration ${\ddot{r}}_{dy}^{m}$. Error for the estimation of (

**a**) yawing angle and (

**b**) pitching angle. The vertical lines indicate the corresponding standard deviations or error bars.

**Figure 6.**Bias error as a function of the amplitude of acceleration measured by SINS. Error for the estimation of (

**a**) yawing angle and (

**b**) pitching angle. The vertical lines indicate the corresponding standard deviations or error bars.

$\mathit{\sigma}$ (mrad) | $\mathit{\beta}$ (Hz) | $\mathit{\alpha}$ (1/s) | |
---|---|---|---|

X-axis | 0.0776 | 0.16 | 0.1 |

Y-axis | 0.1551 | 0.19 | 0.1 |

Z-axis | 0.1551 | 0.18 | 0.1 |

$\mathit{\sigma}$ (mrad/s) | $\mathit{\beta}$ (Hz) | $\mathit{\alpha}$ (1/s) | |
---|---|---|---|

X-axis | 4.7 | 0.17 | 0.1 |

Y-axis | 7.2 | 0.16 | 0.1 |

Z-axis | 2.2 | 0.18 | 0.1 |

$\mathit{\sigma}$ (mm/s^{2}) | $\mathit{\beta}$ (Hz) | $\mathit{\alpha}$ (1/s) | |
---|---|---|---|

X-axis | 3.1 | 0.17 | 0.1 |

Y-axis | 0.77 | 0.16 | 0.1 |

Z-axis | 1.5 | 0.17 | 0.1 |

$\mathit{\sigma}$ (mm/s^{2}) | $\mathit{\beta}$ (Hz) | $\mathit{\alpha}$ (1/s) | |
---|---|---|---|

X-axis | 0.6 | 0.16 | 0.1 |

Y-axis | 0.4 | 0.17 | 0.1 |

Z-axis | 0.3 | 0.17 | 0.1 |

**Table 5.**Driven noise of different models in each direction. (WNx, WNy, and WNz denote independent Gaussian white noises in three directions, and the weight factor ξ takes a value from 0 to 1).

X-Direction | Y-Direction | Z-Direction | |
---|---|---|---|

Ship inertial angular velocity (${\stackrel{\rightharpoonup}{\omega}}_{im}^{m}$) | WNx | WNy | WNz |

Dynamic misalignment angle ($\stackrel{\rightharpoonup}{\theta}$) | WNx | WNy | WNz |

Dynamic lever-arm acceleration (${\ddot{\stackrel{\rightharpoonup}{r}}}_{d}^{m}$) | WNx | WNy | WNz |

SINS acceleration data (${\stackrel{\rightharpoonup}{f}}_{is}^{s}$) | (1 − ξ)*WNx + ξ*WNy | WNy | (1 − ξ)*WNz + ξ*WNy |

MINS | SINS | |||
---|---|---|---|---|

Bias | White Noise | Bias (mGal) | white Noise (SD) (mGal) | |

X | 0 | 0 | 30 | 10 |

Y | 0 | 0 | 30 | 10 |

Z | 0 | 0 | 30 | 10 |

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**MDPI and ACS Style**

Zhang, Y.; Yang, S.; Qin, S.; Hu, F.; Wu, W.
A New Bias Error Prediction Model for High-Precision Transfer Alignment. *Sensors* **2018**, *18*, 3277.
https://doi.org/10.3390/s18103277

**AMA Style**

Zhang Y, Yang S, Qin S, Hu F, Wu W.
A New Bias Error Prediction Model for High-Precision Transfer Alignment. *Sensors*. 2018; 18(10):3277.
https://doi.org/10.3390/s18103277

**Chicago/Turabian Style**

Zhang, Yutong, Shuai Yang, Shiqiao Qin, Feng Hu, and Wei Wu.
2018. "A New Bias Error Prediction Model for High-Precision Transfer Alignment" *Sensors* 18, no. 10: 3277.
https://doi.org/10.3390/s18103277