# Determining the Initial Orientation for Navigation and Measurement Systems of Mobile Apparatus in Reforestation

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## Abstract

**:**

## 1. Introduction

## 2. Task Definition

- The local geographical CS OLNE is Earth-related, the beginning of which coincides with the UMA’s center of mass, located at latitude φ. The axis OL is directed along the radius R of the sphere of the Earth from its center, the axis ON lies in the plane of the local meridian (towards the north), and the axis OE complements the CS to the right (towards the east). In Figure 1, the OLNE CS axes are marked in red.
- The dash CS OIJK, the beginning of which coincides with the UMA’s center of mass. The axes OI, OJ, and OK, prior to the implementation of the system, deployed an initial orientation relative to the axes of CS OLNE unknown angles. In Figure 1, the OIJK CS axes are marked in green.

^{−6}– 4.8 · 10

^{−6}rad) using precision accelerometers. Measurement errors lie within 1 · 10

^{−4}– 1 · 10

^{−5}m s

^{-2}, and the axis of sensitivity is oriented respectively long the axes OJ and OK. Thus, this stage is implemented by the combination of the axis A with the axis OL. The final solution initial orientation—the definition of the azimuthal angle A of dash CS—given the inevitable AVS interference measurements is implemented next, using the gyrocompassing algorithm described below.

- ω = Ω cosφ cosA—the Earth’s angular velocity projection on the AVS sensitivity axis;
- Ω—the angular velocity of the Earth’s rotation (see Figure 1);
- φ—the latitude;
- A—the azimuthal angle of the AVS sensitivity axis;
- S—the constant random interference, with a priori uncertain statistical characteristics;
- W—the broadband random interference, with a priori uncertain statistical characteristics.

_{i}becomes equal to:

_{i}= ω

_{i}+ S,

_{i}= Ω cosφ cosA

_{i}, A

_{i}—the unknown azimuthal angle of the AVS sensitivity axis in the i-th position.

_{i}obtained in the neighboring angular positions of the AVS sensitivity axis (the rotation angles which are different with respect to the local meridian on Δ value), the difference δ

_{i}is

_{i}= Z

_{i+1}− Z

_{i}= ω

_{i+1}− ω

_{i}= Ω cosφ (cosA

_{i+1}− cosA

_{i})

_{i}, the termination of rotation is determined either by reaching a given value of n, or by changing the sign of the difference δ

_{i}. The difference of the cosines (cosA

_{i+1}− cosA

_{i}) is calculated before the gyrocompassing process for all possible values of angles A

_{n}in a given interval of their change with the required accuracy. The maximum values of the boundaries of the interval here will be equal to [0, π/2]—for example, when calculating the angles A

_{n}with a sampling step d = 10″, the maximum size of the calculated values array of cosines difference will be only 32,400 values, which is not difficult for modern computers. In addition, to begin the gyrocompassing process, the reference array calculates accurate values of ${\mathsf{\delta}}_{i}^{Et}$ for a particular latitude of gyrocompassing by multiplying the values of the array computed cosines difference on the Ω cosφ.

_{i}values, because of the measurements carried out, the selection of a series (n − 1) of consecutive δ

_{i}

^{Et}values from the reference array of exact values that coincide as much as possible with the series (n–1) of δ

_{i}values is carried out by brute force. The comparison is based on the specified matching criteria, for example:

_{1}—set values,$min{{\displaystyle \sum}}_{i=1}^{n-1}\left|{\mathsf{\delta}}_{i}^{Et}-{\mathsf{\delta}}_{i}\right|\forall {A}_{n}$, $min{{\displaystyle \sum}}_{i=1}^{n-1}{\left({\mathsf{\delta}}_{i}^{Et}-{\mathsf{\delta}}_{i}\right)}^{2}\forall {A}_{n}$ etc.

_{n}, corresponds to the value ${\mathsf{\delta}}_{n-1}^{Et}$. The angle A

_{n}, with high precision (due to independence of the values δ

_{i}from the above-mentioned interference), is the azimuth angle of the n-th position of the AVS sensitivity axis.

## 3. Results

^{2}°/hour and a broadband noise of W (Gaussian random sequence with zero expectation and dispersion of (0.15)

^{2}°/hour), generated with a time step of 0.01 s for the number of implementations equal to 90. The azimuth of the initial position of the AVS sensitivity axis, relative to the plane of the local meridian (with latitude 47°35′), was chosen to be 28°–29°, and the sampling step d to calculate all possible values of angles A

_{n}in the range [0, π/2] was chosen to be 10″. The rotation of the AVS sensitivity axis was carried out in 60 steps and a time of each step of 1 s. At each step, the AVS output signal was filtered using a sixth-order Butterworth filter. Following the determination of differences δ

_{i}using Equation (3), the choice of a number of values of δ

_{i}

^{Et}from the reference array of exact values that coincide as much as possible with a number of obtained values of δ

_{i}was carried out by criterion ${{\displaystyle \sum}}_{i=1}^{n-1}\left|{\mathsf{\delta}}_{i}^{Et}-{\mathsf{\delta}}_{i}\right|\le \alpha $, where parameter α was selected to be 5″. The minimum values of errors in determining the azimuth AVS sensitivity axis at different azimuths of its initial position, obtained after processing the δ

_{i}

^{Et}values for all implementations, are presented in Table 1.

## 4. Conclusions

_{n}with a sampling step d = 10″ (corresponding to the modern requirements of the accuracy of the azimuth of the UMA’s navigation–measuring complex) in the interval [0, π/2] limits the maximum size of the array of the calculated values of the difference of the cosines to just 32,400 values (π/2d)—if the bitness of the computer is equal to 64, then it only requires 2592 Kbyte of RAM. For an exhaustive search of the array data within a time-step of AVS rotation axis sensitivity (at least one second in modern requirements of speed systems for the initial orientation), this does not pose difficulties for modern computers.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Coordinate Systems (CS) used in study. The OL, ON, and OE axes form a local geographic coordinate system. The OI, OJ, and OK axes form a dush coordinate system. A is the angle of rotation of the axis OJ relative to the plane of the local meridian; R is the radius of the Earth’s sphere from its center; φ is a latitude.

Number | Initial Azimuth | Error |
---|---|---|

1 | 28°10′ | 7.3″ |

2 | 28°30′ | 9.6″ |

3 | 28°50′ | 6″ |

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**MDPI and ACS Style**

Sokolov, S.; Novikov, A.; Ivetić, V. Determining the Initial Orientation for Navigation and Measurement Systems of Mobile Apparatus in Reforestation. *Inventions* **2019**, *4*, 56.
https://doi.org/10.3390/inventions4040056

**AMA Style**

Sokolov S, Novikov A, Ivetić V. Determining the Initial Orientation for Navigation and Measurement Systems of Mobile Apparatus in Reforestation. *Inventions*. 2019; 4(4):56.
https://doi.org/10.3390/inventions4040056

**Chicago/Turabian Style**

Sokolov, Sergey, Arthur Novikov, and Vladan Ivetić. 2019. "Determining the Initial Orientation for Navigation and Measurement Systems of Mobile Apparatus in Reforestation" *Inventions* 4, no. 4: 56.
https://doi.org/10.3390/inventions4040056