# Introduction of a New Index of Field Operations Efficiency

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Overview

- (1)
- Generation of the fixed entities. This step includes the generation of the headland passes, and the field-work tracks (presented in Section “Fixed Entities”).
- (2)
- Generation of connectors. Four types of connectors that connects the fixed entities are generated (presented in Section “Generation of Connectors”).
- (3)
- Generation of a continuous path for field coverage. Formulation of the field coverage problem as the problem of traversing an undirected weighted graph (presented in Section “Continuous Path Generation”).

- The coordinates of the edges of the polygon representing the field boundary (B);
- The machine’s effective operating width (w);
- The driving angle (θ), which defines the direction of tracks (in relation to the Universal Transverse Mercator (UTM)-Easting axis);
- The number of headland passes (h).
- The minimum turning radius of the vehicle (r).
- The coordinates of the location of the field gate where vehicles can enter and exit the field (E).
- The fieldwork pattern (F), which determines the traversing order of the track sequence.

- (a)
- Path generation for fields with obstacles is not considered.
- (b)
- The methodology can only be applied to non-capacitated field operations such as tillage, plough, and so on.

#### 2.2. Generation of Continuous Path

#### 2.2.1. Fixed Entities

- (a)
- Headland pass (H): A headland pass is a concentric path covering the headland area at the same width as the operating width $\mathrm{w}$, of the implement which is made up of a set of sequentially clockwise ordered points. An inner boundary between the headland area and the work area is created at a distance half of the operating width w/2 from the last headland passes, the area enclosed by the inner boundary is denoted the field body area.
- (b)
- Row (R): The field body area is covered by parallel rows that transect the area. The width of each row equals the operating width w of the machine.
- (c)
- Field-work track (T): A track is represented by two ending points is the central line of a row and is used as the guidance line for the machine to cover each row.

#### 2.2.2. Generation of Connectors

- Gate-to-headland connector (G2H): A connection path between the field gate and the first headland pass. The G2H connectors provide the path for the agricultural vehicles to enter and exit the field area.
- Headland-to-headland connector (H2H): A connection path between two adjacent headland passes. These connectors are used for agricultural vehicles to move between headland passes.
- Track-to-headland connector (T2H): A connection path between a track end and a headland pass for agricultural vehicles to drive from a track to a headland pass or vice versa.
- Track-to-track connector (T2T): A connection path between a track end and another track end.

#### 2.2.3. Continuous Path Generation

_{H}) that connects all headland passes ${\mathrm{H}}_{\mathrm{i}=\left\{1,\dots ,\mathrm{h}\right\}}$ as one path that starts from the gate, the fieldwork tracks traversing path (P

_{T}) that connects the ordered tracks and the field exiting path (P

_{B}) for vehicle exiting field from the last track in $\mathsf{\rho}$ when the operation is done.

- $\mathrm{adj}\left({\mathrm{v}}_{\mathrm{i}},\mathrm{m}\right):$ is a function for headland traversing path ${\mathrm{P}}_{\mathrm{H}}$ generation, which finds all adjacent vertices {$\mathrm{v}$} of ${\mathrm{v}}_{\mathrm{i}}$ that directly link with ${\mathrm{v}}_{\mathrm{i}}$ in the graph based on the given direction $\mathrm{m}$.
- $\mathrm{seq}\left({\mathrm{v}}_{\mathrm{i}},{\text{}\mathrm{v}}_{\mathrm{j}}\text{}\right):$ finds the shortest path with a sequence of vertices from a source vertex ${\mathrm{v}}_{\mathrm{i}}$ to a target vertex ${\mathrm{v}}_{\mathrm{j}}$ on the graph (by implementing the Dijkstra’s algorithm [16].
- $\mathrm{get}\left(\mathrm{e},\mathrm{m}\right):$ returns the two vertex ids which correspond to edge $\mathrm{e}$ in sequence $\left[\mathrm{s},\text{}\mathrm{t}\right]$ when $\mathrm{m}=1$ or in sequence $\left[\mathrm{t},\mathrm{s}\right]$ when $\mathrm{m}=-1$.
- $\mathrm{find}\left({\mathrm{v}}_{\mathrm{i}},{\mathrm{v}}_{\mathrm{j}}\right):$ returns the edge $\mathrm{e}$ that links ${\mathrm{v}}_{\mathrm{i}}\text{}\mathrm{to},{\mathrm{v}}_{\mathrm{j}}$.
- $\mathsf{\Theta}\left(\mathrm{p}\right):$ returns the last vertex of path $\mathrm{p}$.

Algorithm 1. Pseudo codes for continuous path generation. | ||

Initialization: | ||

Path$p$is initialized as set$p$=$\left\{{v}_{0}\right\}$that starts at gate. | ||

${\mathit{P}}_{\mathit{H}}$path generation: | ||

Get all edges with type$H$,$se{t}_{H}=\left\{{e}_{i}\right\}where{e}_{i}\xb7type==H;$ | ||

While$se{t}_{H}\ne \varnothing $ | ||

Apply$adj\left(\Theta \left(p\right),1\right)\to \left\{v\right\}$ to get all adjacent vertices of the last vertex of path $p$; | ||

$If\forall x\in \left\{v\right\},e=find\left(\Theta \left(p\right),x\right)suchthate\in se{t}_{H}ande\xb7type==H$; | ||

Then add e to path$p=p{\cup}^{\text{}}e$; and remove$e$from$se{t}_{H}:se{t}_{H}=se{t}_{H}\setminus e$; | ||

Else if$e\xb7type==H2H$,then add headland connector e to path$p=p{\cup}^{\text{}}e$. | ||

End | ||

${\mathit{P}}_{\mathit{T}}$path generation: | ||

${p}_{temp}=get\left({e}_{1},{d}_{{e}_{1}}\right)$; | ||

$p=p{\cup}^{\text{}}seq\left(\Theta \left(p\right),\Theta \left({p}_{temp}\right)\right)$# Find the path from last vertex of$p$to starting vertex of track edge${e}_{1}$; | ||

For$i=1:(n-1$) do: | ||

$p=p{\cup}^{\text{}}get({e}_{i},{d}_{{e}_{i}})$; # Add vertices of track edge${e}_{i}$to path$p$ | ||

$p=p{\cup}^{\text{}}seq\left(\Theta \left(p\right),{e}_{i+1}\right)$; # Add path that is from end vertex of${e}_{i}$to start vertex of${e}_{i+1}$to path$p$ | ||

$p=p{\cup}^{\text{}}get({e}_{i},{d}_{{e}_{i+1}})$; # Add vertices of track edge${e}_{i+1}$to path$p$ | ||

End | ||

${P}_{B}$path generation: | ||

$p=p{\cup}^{\text{}}seq\left(\mathsf{\Theta}\left(p\right),{v}_{0}\right)$; # Exiting field from last vertex of path$p$ |

#### 2.2.4. Distance-Based Field Efficiency

#### 2.3. Sample Fields Scenarios

- (a)
- (b)
- Machinery system: Three different types of machinery with different implements were considered in terms of sizes and maneuverability (minimum turning radius). Specifically, a large sized machine with a minimum turning radius of 6 m, a medium sized machine with a minimum turning radius of 4.5 m, and a small sized unit with minimum turning radius of 3 m, were selected. The selected implements’ width ranged from 3 m to 12 m. The specific test setups of machinery and implements (Table 1) have been selected as typical configurations implemented in field operation management assessments [17,18].
- (c)
- Fieldwork pattern: Three different common used fieldwork patterns (Figure 9a, AB; Figure 9b, SF; and Figure 9c, BL) were selected for the assessment. Each fieldwork pattern is represented mathematically with the traversal function, which produces the traversal sequence of the field tracks. The specific traversal functions of these three patterns are provided in Bochtis et al. (2013) [17]. The fieldwork pattern defines the traversal sequence of field-work tracks, thereby determining the total non-working turning distance in the headland area, and subsequently determining how efficient the machinery performs, in terms of distance covered [21].
- (d)
- Driving direction: The driving direction is an important factor in determining the number of tracks and their length, and subsequently affecting the field efficiency. Four driving directions ($\mathsf{\theta}$ = $0\xb0$, $30\xb0$, $60\xb0$, and $90\xb0$) were selected.

## 3. Results

#### 3.1. Effect of Field Shape on FTE

#### 3.2. Effect of Fieldwork Pattern on FTE

#### 3.3. Effect of Driving Direction on FTE

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 4.**Illustration of procedures of generating gate to headland connector: (

**a**) densify headland pass based on given length ($\mathrm{l}$); (

**b**) generate a circle centered at gate $\mathrm{E}$ with radius $\mathrm{t}$; (

**c**) generated $\mathrm{G}2\mathrm{H}$ connector.

**Figure 6.**An illustrative example of track-to-track connectors (red curved paths) from one track end to all other tracks’ end.

**Figure 7.**The fixed entities and all connectors (

**a**), and their conversion to an undirected graph (

**b**).

**Figure 9.**Three conventional fieldwork patterns, (

**a**) AB—pattern, (

**b**) SF—pattern and (

**c**) BL—pattern used in the tests.

**Figure 10.**The minimum (min), average (ave), and maximum (max) field traversing efficiency (FTE) for each template field.

**Figure 11.**Average FTE for the three selected fieldwork patterns (AB, SF, and BL) in all template field shapes.

**Figure 13.**Demonstration of the effect of the field-work pattern on FTE. Other parameters are the same for all cases (operating width: 9 m; turning radius: 6 m; number of headlands: 2).

**Figure 14.**Demonstration of generated continuous paths with FTE on two selected fields (F: pattern; w: width; r: radius; h: headlands).

$\mathbf{Minimum}\text{}\mathbf{Turning}\text{}\mathbf{Radius}\mathit{r}\text{}\left(\mathbf{m}\right)$ | $\mathbf{Operating}\text{}\mathbf{Width}\text{}\mathit{w}\text{}\left(\mathbf{m}\right)$ | |||||
---|---|---|---|---|---|---|

Large size machine | 6 | 4.5 | 7.5 | 9 | 10.5 | 12 |

Medium size machine | 4.5 | 4.5 | 6 | 7.5 | 9 | - |

Small size machine | 3 | 3 | 4.5 | 6 | - | - |

Field Shape | Scenario Returning Minimum FTE | Scenario Returning Maximum FTE | ||||||
---|---|---|---|---|---|---|---|---|

Pattern | Width (m) | Radius (m) | Direction (Degrees) | Pattern | Width (m) | Radius (m) | Direction (Degrees) | |

SQR | SF | 12 | 6 | 30 | BL | 4.5 | 3 | 0 |

STD | SF | 12 | 6 | 30 | BL | 3 | 3 | 0 |

R41 | AB | 4.5 | 6 | 30 | AB | 6 | 3 | 90 |

R21 | SF | 12 | 6 | 30 | AB | 4.5 | 3 | 90 |

REN | SF | 12 | 6 | 30 | BL | 3 | 3 | 0 |

BPL | SF | 9 | 6 | 60 | AB | 6 | 3 | 0 |

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**MDPI and ACS Style**

Zhou, K.; Bochtis, D.; Jensen, A.L.; Kateris, D.; Sørensen, C.G. Introduction of a New Index of Field Operations Efficiency. *Appl. Sci.* **2020**, *10*, 329.
https://doi.org/10.3390/app10010329

**AMA Style**

Zhou K, Bochtis D, Jensen AL, Kateris D, Sørensen CG. Introduction of a New Index of Field Operations Efficiency. *Applied Sciences*. 2020; 10(1):329.
https://doi.org/10.3390/app10010329

**Chicago/Turabian Style**

Zhou, Kun, Dionysis Bochtis, Allan Leck Jensen, Dimitrios Kateris, and Claus Grøn Sørensen. 2020. "Introduction of a New Index of Field Operations Efficiency" *Applied Sciences* 10, no. 1: 329.
https://doi.org/10.3390/app10010329