# Introduction of a New Index of Field Operations Efficiency
## 2 Materials and Methods
### Generation of Continuous Path
#### 2.2.2 Generation of Connectors  
Connectors (as shown in Figure 3) are curved paths that inter-connect the gate points, the tracks, and the headland passes. The connectors are generated both in clockwise and anti-clockwise directions. The different types of connectors are explained as follows:  
1. 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.
2. 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.
3. 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.
4. Track-to-track connector (T2T): A connection path between a track end and another track end.  
In the followings section, the procedure of generation of the aforementioned four types of connectors by using the Dubins curves path method [15] is described. A Dubins curve path is the  
Figure 3: An illustrative example of four types of connectors.  
Figure 2: Illustrative example of fixed entities.  
shortest path between an initial point \((x_{i},y_{i})\) with an initial heading \(i\) and a goal point \((x_{g},y_{g})\), with goal heading \(\mathrm{o_{g}}\) under the constraints of the bounded curvature not exceeding 1/r.  
As the first step of generation of a G2H connector, the first headland pass is divided into segments by inserting points (red points in Figure 4a) between existing vertices based on a specified length (\(l\)), so that no segments are longer than this given threshold value. Additionally, a circle C centred at gate E with radius t (t is set to 2*d, where d is the perpendicular distance of gate to the first headland pass) (Figure 4b) is created and only those headland pass points (black points in Figure 4c) within C are considered as an ending point of the G2H connector. Then, the shortest connection path (black curved path in Figure 4c) among all connection paths is generated (by implementing the Dubins method) between the gate, and the set of selected points within the circle C is selected as the G2H connector.  
To generate H2H and T2H connectors, the same procedure for generation of a connector G2H is implemented. The circle C is centred at the ending point of a G2H connector, and at the end of a track, for generation of a H2H and a T2H connector, respectively. Figure 5a,b illustrates the generated H2H and T2H connectors, respectively.  
Figure 4: Illustration of procedures of generating gate to headland connector: (**a**) densify headland pass based on given length (\(l\)); (**b**) generate a circle centered at gate E with radius t; (**c**) generated G2H connector.  
Figure 5: (**a**) The H2H connector (blue path), (**b**) the T2H connector (black path).  
Regarding the generation of T2T connectors, only the connectors that are not crossing the field body area are generated. Figure 6 illustrates all the possible T2T connectors (red paths) from a track end to all other tracks' ends inside the headland area.