# Metric Map Generation for Autonomous Field Operations

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Geometric Primitives

**Polyline**. A polyline, $l$, is an ordered path composed of a collection of contiguous line segments $l=\langle {l}_{{x}_{1}{x}_{2}},{l}_{{x}_{2}{x}_{3}},\dots ,{l}_{{x}_{n-1}{x}_{n}}\rangle $. A line segment is a special case of a polyline with $n=2$.

**Polygon**. A polygon, $p$, is defined by a collection of rings, $p=\langle {r}_{1},{r}_{2}\dots {r}_{n}\rangle ,n\ge 1$, where a ring, $r$, is defined by a collection of contiguous ordered line segments, $r=\langle {l}_{{x}_{1}{x}_{2}},{l}_{{x}_{2}{x}_{3}}\dots {l}_{{x}_{n}{x}_{1}}\rangle $, such that the start and end points are the same point, as depicted in Figure 2a.

**Envelope.**An envelope, $e$, is a bounding rectangle that encloses a geometry $g\in \left\{l,r,p\right\}$. A given geometry $g$ can have multiple bounding rectangles with different orientation angle. The minimum bounding rectangle ${e}_{min}$ (as shown in Figure 3) has the smallest area among all the bounding rectangles.

#### 2.2. Geometric Operators

**Intersection ($\mathbf{\cap}\mathbf{\left(}A\mathbf{,}B\mathbf{\right)}$).**The intersection operator $\cap $ is used to find the overlapped area $g$ between two geometries $\mathsf{{\rm A}}$ and $\mathsf{{\rm B}}$. The geometries can be line segments, polylines, rings or polygons. If $g$ is the intersection between $\mathsf{{\rm A}}$ and $\mathsf{{\rm B}}$ it satisfies that $g\subseteq \mathrm{A}\wedge g\subseteq \mathrm{B}$. Intersection examples are illustrated in Table 1 where the green marked geometry corresponds to the intersection between two examples of geometries $\mathsf{{\rm A}}$ and $\mathrm{B}$.

**Subtraction ($\mathsf{\Delta}\mathbf{\left(}A\mathbf{,}B\mathbf{\right)}$).**The subtraction operator $\mathsf{\Delta}$ determines a geometry $g$, which is the difference between the two input geometries $\mathrm{A}$ and $\mathrm{B}$. The subtracted geometry $g$ satisfies the condition that $g\subseteq \mathrm{A}\wedge g\u2288B$ and. Examples are given in Table 1. Note, that subtraction does not obey the commutative law ($\mathsf{\Delta}\text{}\left(\mathrm{A},\text{}\mathrm{B}\right)\ne \mathsf{\Delta}\text{}\left(\mathrm{B},\mathrm{A}\right)$).

**Offset (${\mathbf{\parallel}}_{\mathbf{\pm}\mathit{d}}^{\mathit{g}}$).**The offset operator ${\parallel}_{\pm d}^{g}$ creates a geometry that is parallel to the input geometry $g\in \left\{l,r,p\right\}$ at a specified distance $d$. The distance can be either positive $(+)$ or negative $(-)$. Examples of offset for a polygon, polyline and line segment are presented in Table 2.

**Buffer (${\mathsf{\Pi}}_{\mathit{d}}^{\mathit{l}}$).**The buffer operator $\mathsf{\Pi}$ creates a polygon around the input line $l$, which may be a line segment or a polyline. First, two parallel lines to both sides of $l$ at a specified distance $d$ are generated with the offset operator ${\parallel}_{+d}^{l}$ and ${\parallel}_{-d}^{l}$, and then, these two lines are joined in the ends to compose the buffer polygon ${p}_{buf}$ around $l$. In the example in Figure 4, the black line is the input line$l$ and the green polygon is a buffer polygon.

**Projection ($\mathbf{\perp}\mathbf{\left(}\mathit{m}\mathbf{,}\mathit{l}\mathbf{\right)}$).**The projection operator, $\perp $, projects a point $m$ to a line or polyline $l$ and determines the projected point ${m}^{\prime}$ on $l$ such that the line ${l}_{m{m}^{\prime}}$ has the shortest distance to $l$. Figure 5a. illustrates how the points ${m}_{1},{m}_{2},{m}_{3}$ and ${m}_{4}$ are projected on the polyline $l=\langle {l}_{{x}_{1}{x}_{2}},{l}_{{x}_{2}{x}_{3}}\rangle $. Note, that ${m}_{2}$ projects to two points on $l$, ${m}_{2}^{\prime}$ and ${m}_{2}^{\u2033}$, since ${m}_{2}$ is equidistant to the line segments ${l}_{{x}_{1}{x}_{2}}$ and ${l}_{{x}_{2}{x}_{3}}$ in Figure 5a. The projection point ${m}^{\prime}$ can be either interior or exterior of $l$, as illustrated in Figure 5a, where the projection points ${m}_{1}^{\prime},{m}_{2}^{\prime},{m}_{2}^{\u2033}$ and ${m}_{3}^{\prime}$ are interior to $l$, while ${m}_{4}^{\prime}$ is exterior.

**Point union ($\mathsf{\u2a03}\mathbf{\left(}\mathbf{\left\{}\mathit{m}\mathbf{\right\}}\mathbf{\right)}$).**The point union operator $\mathsf{\u2a03}$ takes an ordered set of points $\left\{m\right\}$ and generates a polyline consisting of line segments with end points in $\left\{m\right\}$. In the context of this paper, the point union operator is always used along with the projection operator $\perp $ to create a line segment or a polyline that links all the projection points and points of the original line that lie in between these projection points. This is illustrated in Figure 5b, where the point union operator is used to create a new polyline ${l}^{\prime}=\mathsf{\u2a03}\left(\left\{m\right\}\right)$, where $m=\{{m}_{1}^{\prime},{m}_{2}^{\prime},{x}_{2},{m}_{2}^{\u2033},{m}_{3}^{\prime},{x}_{3},{m}_{4}^{\prime}\}$.

**Boolean intersect ($\mathbf{\&}\mathbf{\left(}A\mathbf{,}B\mathbf{\right)}$).**The boolean intersect operator & is used to check if two geometries $A$, $B$ intersect or not, i.e., if they share at least one point.

#### 2.3. Metric Map Generaiton

#### 2.3.1. Inputs

- The boundary of the field area and the boundaries of the in-field obstacles, which are represented by a polygon, ${\mathit{p}}_{\mathit{field}}$.
- The number of headland passes ($\mathit{\eta}$) for the main field and each obstacle.
- The line (${\mathit{l}}_{\mathit{A}\mathit{B}}$), which is used as $\mathit{A}\mathit{B}$ reference line for generation of the parallel field-work tracks. ${\mathit{l}}_{\mathit{A}\mathit{B}}$ can be either a straight-line segment or a polyline.
- The operating width ($\mathit{w}$). This is the effective operating width of the implement.

#### 2.3.2. Metric Map Entities

#### 2.3.3. Generation of Headland Passes and Headland Areas

#### 2.3.4. Tracks and Rows Generation

**Straight field-work tracks**

- Firstly, each initial track ${\mathit{t}}_{\mathit{i}}$ is used to create a buffer polygon ${\mathit{p}}_{\mathit{buf}}$ using the buffer operator ${\mathsf{\Pi}}_{\mathit{w}/\mathbf{2}}^{{\mathit{t}}_{\mathit{i}}}$ (Figure 8a). Then, the intersection operator is used to find the intersection area between the buffer polygon ${\mathit{p}}_{\mathit{buf}}$ and the inner field boundary ${\mathit{b}}_{\mathit{f}}$ disregarding obstacles: ${\mathit{p}}_{\mathit{int}}=\cap \left({\mathit{p}}_{\mathit{buf}},{\mathit{b}}_{\mathit{f}}\right)$ (Figure 8b).
- Next, the boolean intersect operator $\&$ is applied to check whether the ${\mathit{p}}_{\mathit{int}}$ intersects with the obstacles. Following this, there are three potential cases, as illustrated in Figure 8c: Case 1: if ${\mathit{p}}_{\mathit{int}}$ is not intersected with any obstacle, then ${\mathit{p}}_{\mathit{int}}$ remains unchanged; Case 2: if ${\mathit{p}}_{\mathit{int}}$ only partially intersects with obstacles, then the subtraction operator $\mathsf{\Delta}$ is employed to obtain the subtracted area; and Case 3: if ${\mathit{p}}_{\mathit{int}}$ crosses through an obstacle, then the subtraction operator $\mathsf{\Delta}$ divides the polygon ${\mathit{p}}_{\mathit{int}}$ into sub-polygons. Other cases such as a ${\mathit{p}}_{\mathit{int}}$ partially intersects one obstacle and crosses through another obstacle are considered as the case of the combination of basic cases 2 and 3.
- After the subtraction operation has been performed, each ${\mathit{p}}_{\mathit{fin}}$ area is enclosed by a ring with multiple points denoted as a set $\left\{\mathit{m}\right\}$ (red points in Figure 8c). To obtain the final track, ${\mathit{t}}_{\mathit{fin}}$, first, points in the set $\left\{\mathit{m}\right\}$ are projected to the initial track $\mathit{t}$ by the projection operator $\perp $ to acquire the projection points set $\left\{{\mathit{m}}^{\prime}\right\}$. Afterwards, the final track ${\mathit{t}}_{\mathit{fin}}$ (black dotted lines in Figure 9a) is created using the union operator $\mathsf{\u2a03}\left(\left\{{\mathit{m}}^{\prime}\right\}\right)$. In addition, the row ${\mathit{R}}_{\mathit{fin}}$ of the final track ${\mathit{t}}_{\mathit{fin}}$ (green area in Figure 9b) is generated by employing the buffer operator ${\mathsf{\Pi}}_{\mathit{w}/\mathbf{2}}^{{\mathit{t}}_{\mathit{fin}}}$.

Algorithm 1. Pseudo-code for the generation of straight tracks and corresponding rows. |

For each initial track t $\u03f5\text{}\left\{{t}_{1},{t}_{2}\dots {t}_{\tau}\right\}$ do |

${p}_{buf}$ $\leftarrow {\Pi}_{w/2}^{t}$ # Create a buffer around a initial track t |

${p}_{int}$ $\leftarrow \text{}\cap ({p}_{buf},{b}_{f})$ # Obtain the intersection ${p}_{int}$ between ${p}_{buf}$ and field inner boundary area ${b}_{f}$ |

If $\&({p}_{int},{b}_{i=\left\{1,2,\dots ,\theta \right\}}$) == false |

${p}_{fin}\leftarrow {p}_{int}$; |

Else |

${p}_{fin}\leftarrow \Delta \text{}({p}_{int},\text{}{b}_{i=\left\{1,2,\dots ,\theta \right\}})$; # Find the subtracted area between ${p}_{int}$ and inner boundary ${b}_{i=\left\{1,2,\dots ,\theta \right\}}$ of obstacle ${r}_{i}$. |

End |

For each ${p}_{fin}$ do |

Get a set of projection points to $t$, {m′} ← $\perp \left(m,t\right)$, $m$ is the set of points that constitute ${p}_{fin}$; |

Final track ${t}_{fin}$ ← $\mathsf{\u2a03}\left(\left\{{m}^{\prime}\right\}\right)$; |

Row of final track ${R}_{fin}\leftarrow {\Pi}_{w/2}^{{t}_{fin}}$; |

End |

End |

**Curved field-work tracks**

Algorithm 2. Pseudo-code for generation of curved tracks with corresponding rows. |

While until no track intersects with ${e}_{min}$ |

If $i==1$ |

${l}_{A{B}^{\prime}}\leftarrow $ Extend ${l}_{AB}$ to intersect with ${e}_{min}$; |

${t}_{1}$ = ${\parallel}_{w/2}^{{\mathit{l}}_{\mathit{A}{\mathit{B}}^{\mathbf{\prime}}}};$ |

${p}_{buf}\leftarrow {\Pi}_{w/2}^{\text{}{t}_{1}}$ # Create a buffer around ${t}_{1}$; |

Else |

${t}_{i}$ = Extend ${t}_{i-1}^{\prime}$ to intersect with ${e}_{min}$; |

${p}_{buf}\leftarrow {\Pi}_{w}^{{t}_{i}}$ # Create a buffer around ${t}_{i}$; |

End |

${p}_{int}\leftarrow {p}_{buf}\text{}\cap {b}_{f}$ # Obtain the intersection ${p}_{int}$ between ${p}_{buf}$ and field inner boundary area ${b}_{f}$; |

If $\&({p}_{int},{b}_{i=\left\{1,2,\dots ,\theta \right\}}$) == false |

${p}_{fin}\leftarrow {p}_{int}$; |

Else |

${p}_{fin}\leftarrow \Delta \text{}({p}_{int},\text{}{b}_{i=\left\{1,2,\dots ,\theta \right\}})$; # Find the subtracted area between ${p}_{int}$ and inner boundary ${b}_{i=\left\{1,2,\dots ,\theta \right\}}$ of obstacle ${r}_{i}$. |

End |

Get set of projection points {m′} ← $\perp \left(m,{t}_{i}\right)$, $m$ is the set of points that constitute ${p}_{fin}$; |

Final track ${t}_{1}^{\prime}\leftarrow \mathsf{\u2a03}\left(\left\{m\prime \right\}\right)$; |

Row of final track ${R}_{fin}\leftarrow {\Pi}_{w/2}^{{t}_{1}^{\prime}}$; |

$i=i+1$; |

End |

#### 2.3.5. Overlapped Area Quantification

## 3. Implementation and Results

## 4. Discussion

- Not all information about field shapes is often available, so the operator has to drive around the field and obstacles boundaries to acquire the path coordinates, and so the map generation process has to take place on-site.
- The field metric map should be generated based on the effective working width instead of the specified working width which in many cases the operator must adjust to field conditions.
- Factors also such as soil condition in terms of moisture content also affects field shapes and thus affecting the field representation scheme. That might be the case, when, for example, a water area inside the field resulting from rain constitutes an operational obstacle for the proposed method and would require a new field representation to be generated in real-time, on site, for the remainder of the field operation.

- Regarding infield obstacles, the proposed method does not handle cases such as two obstacles in close proximity. These types of obstacles, from the operational point of view, should be considered as one obstacle. Furthermore, cases of small obstacles (e.g., trees, electricity pylons, etc.) that cause a local deviation from the designed track are not taken into account. The categorization of the obstacles depending on the driving direction, working width, the shape of field, size and location of obstacle [18] is an important step in the map generation since it affects the accuracy and feasibility of the generated field representation scheme.
- In the case of curved tracks, each track consists of sequentially connected line segments. There are cases where it is not feasible for an agricultural machine to perform a sharp turn between two line segments due to operational limitations on the maximum turning curvature. Therefore, there is the need for the implementation of curved polyline smoothing algorithms to generate an easier steerable path for agricultural machines [12].
- As part of the tool, the machine operator has to set the value of the number of headland passes manually for generation of the required headland space. It might be difficult for the user to determine an appropriate value due to the fact that the headland space depends on a number of factors such as field shapes, field condition, machine’s kinematic features and so on. Therefore, an algorithm or model should be developed for this tool to improve the reliability of the field representation.
- This tool needs further elaboration so that the user can interactively divide the field into subfields, where each subfield can have its own driving direction.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Tracks are represented by red lines and green areas are the covered areas by following the designated tracks. In (

**a**) there are uncovered areas (yellow); in order to cover these areas, the tracks have to be extended (

**b**).

**Figure 4.**Buffer polygons (green) with offset distance $\mathit{d}$, where $\mathit{l}$ is either a straight line segment (

**a**) or a polyline (

**b**).

**Figure 5.**Illustrated example of operator projection $\perp $ (

**a**) and point union $\mathsf{\u2a03}$ (

**b**).

**Figure 6.**Headland area consisting of rings inside the field and outside the obstacles. Black rings are field and obstacle boundaries, blue rings are headland passes and red rings are the resultant headland area boundaries.

**Figure 7.**The bounding envelope $\mathit{e}$ of the field is covered by a set of straight lines that are parallel to the reference line l.

**Figure 8.**Generation of polygons from initial tracks for three case steps. (

**a**) generate buffer polygon ${\mathit{p}}_{\mathit{buf}}$ using operator ${\mathsf{\Pi}}_{\mathit{w}/\mathbf{2}}^{\text{}{\mathit{t}}_{\mathit{i}}}$ for an initial track ${\mathit{t}}_{\mathit{i}}$; (

**b**) find ${\mathit{p}}_{\mathit{int}}$ as the intersection of ${\mathit{p}}_{\mathit{buf}}$ and inner field border ${\mathit{b}}_{\mathit{f}}$; (

**c**) obtain the subtracted area ${\mathit{p}}_{\mathit{int}}$ using operator $\mathsf{\Delta}$.

**Figure 9.**(

**a**) The (blue) projection points of corresponding (red) initial points; (

**b**) The final tracks and corresponding rows; (

**c**) The generated field representation.

**Figure 12.**The selected fields 2 (

**a**) and 10 (

**b**) (presented also in Table 3) and the selected direction reference lines (red marked lines) used in the scenario tests.

**Table 1.**Illustration of intersection $\cap \left(A,B\right)$ and subtraction $\mathsf{\Delta}\text{}\left(A,\text{}B\right)$. Results of the operations are marked green.

Input Geometries | Intersection $\cap \left(A,B\right)$ | Subtraction $\mathsf{\Delta}\left(A,B\right)$ |
---|---|---|

**Table 2.**Illustration of the offset operator for polygon, polyline and line segment with offset width $\mathit{d}$ . The direction of both lines is from bottom left to top right. The green lines and rings are the generated geometries by the offset operator ${\parallel}_{\pm \mathit{d}}^{\mathit{g}}$.

Polygon Offset | Polyline Offset | Line Offset |
---|---|---|

Field ID | Area (ha) | Shape and Reference Line | Computation Time (s) | Track Type | ||
---|---|---|---|---|---|---|

$\mathit{w}$ = 2 m | $\mathit{w}$ = 4 m | $\mathit{w}$ = 6 m | ||||

1 | 10.27 | 0.98 | 0.73 | 0.60 | Straight | |

2 | 22.78 | 9.63 | 5.42 | 2.59 | Curved | |

3 | 53.06 | 19.62 | 6.41 | 2.72 | Curved | |

4 | 37.27 | 14.22 | 4.83 | 2.86 | Curved | |

5 | 15.56 | 4.82 | 2.12 | 0.63 | Curved | |

6 | 63.27 | 2.01 | 1.28 | 1.12 | Straight | |

7 | 30.27 | 24.51 | 6.81 | 3.13 | Curved | |

8 | 37.41 | 1.78 | 1.22 | 1.01 | Straight | |

9 | 27.58 | 18.2 | 6.22 | 3.62 | Curved | |

10 | 34.51 | 1.58 | 0.97 | 0.73 | Straight | |

11 | 74.98 | 1.57 | 0.94 | 0.27 | Straight | |

12 | 36.3 | 1.95 | 1.23 | 1.01 | Straight |

$\mathit{w}$ = 4 m | $\mathit{w}$ = 6 m | |||||||
---|---|---|---|---|---|---|---|---|

d1 | d2 | d3 | d4 | d1 | d2 | d3 | d4 | |

Total length of tracks (m) | 80,441 | 80,649 | 80,787 | 80,541 | 51,652 | 51,799 | 51,967 | 51,655 |

Number of tracks | 260 | 272 | 262 | 216 | 176 | 184 | 180 | 149 |

Overlapped area (ha) | 0.28 | 0.37 | 0.42 | 0.32 | 0.48 | 0.57 | 0.67 | 0.49 |

w = 4 m | w = 6 m | |||||||
---|---|---|---|---|---|---|---|---|

d1 Curved | d2 Curved | d3 Straight | d4 Straight | d1 Curved | d2 Curved | d3 straight | d4 Straight | |

Total length of tracks (m) | 52,027 | 51,797 | 52,080 | 52,068 | 33,350 | 33,131 | 33,358 | 33,355 |

Number of tracks | 137 | 178 | 189 | 99 | 91 | 119 | 125 | 99 |

Overlapped area (ha) | 0.20 | 0.11 | 0.23 | 0.33 | 0.22 | 0.18 | 0.34 | 0.33 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Zhou, K.; Jensen, A.L.; Bochtis, D.; Nørremark, M.; Kateris, D.; Sørensen, C.G.
Metric Map Generation for Autonomous Field Operations. *Agronomy* **2020**, *10*, 83.
https://doi.org/10.3390/agronomy10010083

**AMA Style**

Zhou K, Jensen AL, Bochtis D, Nørremark M, Kateris D, Sørensen CG.
Metric Map Generation for Autonomous Field Operations. *Agronomy*. 2020; 10(1):83.
https://doi.org/10.3390/agronomy10010083

**Chicago/Turabian Style**

Zhou, Kun, Allan Leck Jensen, Dionysis Bochtis, Michael Nørremark, Dimitrios Kateris, and Claus Grøn Sørensen.
2020. "Metric Map Generation for Autonomous Field Operations" *Agronomy* 10, no. 1: 83.
https://doi.org/10.3390/agronomy10010083