# Application of Updated Sage–Husa Adaptive Kalman Filter in the Navigation of a Translational Sprinkler Irrigation Machine

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## Abstract

**:**

## 1. Introduction

## 2. Kinematic Model of the Sprinkler Irrigation Machine

#### 2.1. The Self-Developed SIM

#### 2.2. Establishment of Kinematic Model

## 3. Updated Kalman Filter Algorithm

#### 3.1. Conventional Kalman Filter

_{0}and P

_{0}—are given, the state estimation and its covariance matrix at time k—i.e., X(k) and P(k), namely ${X}_{k}$ and ${P}_{k}$, (k = 1,2, …)—can be obtained recursively according to the observation Z(k), namely ${Z}_{k}$, at time k. The implementation steps of the Kalman filter are given below:

#### 3.2. Sage–Husa Adaptive Kalman Filter

#### 3.3. Updated Sage–Husa Adaptive Kalman Filter

## 4. Design of Kalman Filter for Navigation of SIM

#### 4.1. Calculation of Navigation Position Based on Dead Reckoning

_{0}, y

_{0}), the position of the next moment of the SIM—i.e., (x

_{1}, y

_{1})—can be calculated by using the sampling time, the speed and heading angle of the SIM, where the speed and heading angle of the SIM are obtained by the speed sensor and electronic compass, respectively. Then, the known position (x

_{1}, y

_{1}) can be used to calculate the position of the next moment of the SIM; i.e., (x

_{2}, y

_{2}). The real time position during the movement of SIM can be deduced by analogy. The recurrence formula of navigation position can be written as

#### 4.2. Design of Kalman Filter

#### 4.2.1. Establishment of State-Transition Matrix

#### 4.2.2. Establishment of Observation Matrix

_{k}is written as

## 5. Application in the Navigation of the Self-Developed SIM

## 6. Conclusions

- On the platform of the self-developed translational sprinkler irrigation machine, the kinematic model for the SIM is established.
- The updated Sage–Husa adaptive Kalman filter is applied to the navigation of the SIM. Experiment verifications were carried out, and the results show that the self-developed SIM has good navigation performance. Besides this, the influence of abnormal observations on the positioning accuracy of the system can be restrained by using the updated Sage–Husa adaptive Kalman filter.
- The maximum deviation between the sprinkler irrigation machine and the predetermined path is 0.18 m and the average deviation is 0.08 m after using the updated filtering algorithm; the deviations are within a reasonable range. This indicates that the updated Sage–Husa adaptive Kalman filter is suitable for sprinkler irrigation machine signal processing.

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Barawid, O.C., Jr.; Mizushima, A.; Ishii, K.; Noguchi, N. Development of an autonomous navigation system using a two-dimensional laser scanner in an orchard application. Biosyst. Eng.
**2007**, 96, 139–149. [Google Scholar] [CrossRef] - Surmann, H.; Nüchter, A.; Hertzberg, J. An autonomous mobile robot with a 3D laser range finder for 3D exploration and digitalization of indoor environments. Robot. Auton. Syst.
**2003**, 45, 181–198. [Google Scholar] [CrossRef] - Nagasaka, Y.; Umeda, N.; Kanetai, Y.; Taniwaki, K.; Sasaki, Y. Autonomous guidance for rice transplanting using global positioning and gyroscopes. Comput. Electr. Agric.
**2004**, 43, 223–234. [Google Scholar] [CrossRef] - Nagasaka, Y.; Saito, H.; Tamaki, K.; Seki, M.; Kobayashi, K.; Taniwaki, K. An autonomous rice transplanter guided by global positioning system and inertial measurement unit. J. Field Robot.
**2009**, 26, 537–548. [Google Scholar] [CrossRef] - Welch, G.; Bishop, G. An Introduction to the Kalman Filter; Department of Computer Science University of North Carolina at Chapel Hill: Chapel Hill, NC, USA, 1995; pp. 41–95. [Google Scholar]
- Gao, T.; Huang, K.; Yang, J.; Hu, Q.; Zhao, F. An Altitude Location System for Vehicle Based on Federated Kalman Filter. In Proceedings of the 5th International Conference on Systems and Informatics (ICSAI), Nanjing, China, 10–12 November 2018; IEEE: Piscataway, NJ, USA, 2018; pp. 1235–1239. [Google Scholar]
- Fan, Z.; Sun, Q.; Du, L.; Bai, J.; Liu, J. Application of adaptive Kalman filter in vehicle laser Doppler velocimetry. Opt. Fiber Technol.
**2018**, 41, 163–167. [Google Scholar] [CrossRef] - Ko, N.; Youn, W.; Choi, I.; Song, G.; Kim, T. Features of Invariant Extended Kalman Filter Applied to Unmanned Aerial Vehicle Navigation. Sensors
**2018**, 18, 2855. [Google Scholar] [CrossRef] [PubMed] - Tradacete, M.; Sáez, Á.; Arango, J.F.; Huélamo, C.G.; Revenga, P.A.; Barea, R.; López-Guillén, E.; Bergasa, L.M. Positioning System for an Electric Autonomous Vehicle Based on the Fusion of Multi-Gnss Rtk and Odometry by Using an Extented Kalman Filter. In Proceedings of the Workshop of Physical Agents, Madrid, Spain, 22–23 November 2018; Springer: Cham, Switzerland, 2018; pp. 16–30. [Google Scholar]
- Kim, D.Y.; Jeon, M. Data fusion of radar and image measurements for multi-object tracking via Kalman filtering. Inf. Sci.
**2014**, 278, 641–652. [Google Scholar] [CrossRef] - Chen, Y.; Huang, T.; Rui, Y. Parametric Contour Tracking Using Unscented Kalman Filter. In Proceedings of the International Conference on Image Processing, Rochester, NY, USA, 22–25 September 2002; IEEE: Piscataway, NJ, USA, 2002; Volume 3, pp. 613–616. [Google Scholar]
- Li, S.E.; Li, G.; Yu, J.; Liu, C.; Cheng, B.; Wang, J.; Li, K. Kalman filter-based tracking of moving objects using linear ultrasonic sensor array for road vehicles. Mech. Syst. Signal Proc.
**2018**, 98, 173–189. [Google Scholar] [CrossRef] - Shantaiya, S.; Verma, K.; Mehta, K. Multiple object tracking using Kalman filter and optical flow. Eur. J. Adv. Eng. Technol.
**2015**, 2, 34–39. [Google Scholar] - Narasimhappa, M.; Rangababu, P.; Sabat, S.L.; Nayak, J. A Modified Sage-Husa Adaptive Kalman Filter for Denoising Fiber Optic Gyroscope Signal. In Proceedings of the Annual IEEE India Conference (INDICON), Kochi, India, 7–9 December 2012; IEEE: Piscataway, NJ, USA, 2012; pp. 1266–1271. [Google Scholar]
- Chen, X.; Xu, Y.; Li, Q. Application of adaptive extended Kalman smoothing on INS/WSN integration. Math. Probl. Eng.
**2013**, 2013, 130508. [Google Scholar] - Zhao, L.; Qiu, H.; Feng, Y. Analysis of a robust Kalman filter in loosely coupled GPS/INS navigation system. Measurement
**2016**, 80, 138–147. [Google Scholar] [CrossRef] - Liu, M.; Lai, J.; Li, Z.; Liu, J. An adaptive cubature Kalman filter algorithm for inertial and land-based navigation system. Aerosp. Sci. Technol.
**2016**, 51, 52–60. [Google Scholar] [CrossRef] - Chen, X.; Shen, C.; Zhang, W.; Tomizuka, M.; Xu, Y.; Chiu, K. Novel hybrid of strong tracking Kalman filter and wavelet neural network for GPS/INS during GPS outages. Measurement
**2013**, 46, 3847–3854. [Google Scholar] [CrossRef] - Farrell, J.; Givargis, T.; Barth, M. Differential Carrier Phase GPS-Aided INS for Automotive Applications. In Proceedings of the 1999 American Control Conference (Cat. No. 99CH36251), San Diego, CA, USA, 2–4 June 1999; IEEE: Piscataway, NJ, USA, 1999; Volume 5, pp. 3660–3664. [Google Scholar]
- Chui, C.K.; Chen, G. Kalman Filtering; Springer International Publishing: Berlin/Heidelberg, Germany, 2017. [Google Scholar]
- Zhao, D.; Jia, W.; Zhang, Y.; Zhao, Y.; Ji, W.; Liu, Y. Design of agricultural robot autonomous navigation control based on improved self-adaptive filter. Trans. Chin. Soc. Agric. Mach.
**2015**, 46, 1–6. (In Chinese) [Google Scholar] - Guo, L. Develop of a Low-Cost Navigation System for Autonomous Off-Road Vehicles; University of Illinois at Urbana-Champaign: Urbana-Champaign, IL, USA, 2003. [Google Scholar]
- Lu, P.; Zhao, L.; Chen, Z. Improved Sage-Husa adaptive filtering and its application. J. Syst. Simul.
**2007**, 15, 3503–3505. (In Chinese) [Google Scholar]

**Figure 1.**Structural diagrams of the self-developed translational sprinkler irrigation machine (SIM).

**Figure 7.**Deviations between the predetermined path and the observation points, the path obtained by using the conventional Kalman filter (CKF) and the updated algorithm (UA).

Parameters | Value |
---|---|

Dimension (length × width × height)/mm | 70,000 × 4200 × 5000 |

Weight/kg | 3500 |

Spray range/m | 72–76 |

Nozzle type | Nelson D3000 |

Speed/(m/min) | ≤1.0 |

Rate of flow/(m^{3}/h) | ≤48 |

Nozzle number | 24 |

Nozzle spacing/m | 3 |

Inlet pressure/MPa | 0.1 |

Clearance from the ground/mm | 1800 |

Items | Maximum Deviation/m | Average Deviation/m | Deviation Variance |
---|---|---|---|

Before filtering | 0.28 | 0.1 | 0.004 |

After filtering | 0.18 | 0.08 | 0.003 |

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**MDPI and ACS Style**

Liu, K.; Zhao, W.; Sun, B.; Wu, P.; Zhu, D.; Zhang, P. Application of Updated Sage–Husa Adaptive Kalman Filter in the Navigation of a Translational Sprinkler Irrigation Machine. *Water* **2019**, *11*, 1269.
https://doi.org/10.3390/w11061269

**AMA Style**

Liu K, Zhao W, Sun B, Wu P, Zhu D, Zhang P. Application of Updated Sage–Husa Adaptive Kalman Filter in the Navigation of a Translational Sprinkler Irrigation Machine. *Water*. 2019; 11(6):1269.
https://doi.org/10.3390/w11061269

**Chicago/Turabian Style**

Liu, Kenan, Wuyun Zhao, Bugong Sun, Pute Wu, Delan Zhu, and Peng Zhang. 2019. "Application of Updated Sage–Husa Adaptive Kalman Filter in the Navigation of a Translational Sprinkler Irrigation Machine" *Water* 11, no. 6: 1269.
https://doi.org/10.3390/w11061269