# 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

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**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