## 2 HDD Trajectory Designs
### Geometric Designs
If the central segment of the trajectory is considered to be horizontal and straight in design, the geometric parameters curve angle \(\gamma\) and deviation  
Figure 3: The geometric profile of alternating straight and curvilinear segments. Figure 2: Graphic representation of catenary trajectory by Equation (3). angle \(\theta\) will be united (\(\theta_{1}=\gamma_{1}\), \(\theta_{2}=\gamma_{2}\)), so that the geometrical design parameters of the five-segment trajectory can be simplified and containing the following:  
* Length of entry straight segment, \(L_{1}\)
* Inclined angle of entry inclined straight segment, \(\theta_{1}\)
* Radius of entry curved segment, \(R_{2}\)
* Central straight segment length, \(L_{3}\)
* Inclined angle of exit inclined straight segment, \(R_{4}\)
* Radius of exit curved segment, \(\theta_{2}\)  
In practice, the straight segments of drill path, \(L_{1}\), \(L_{3}\), and \(L_{5}\), are generally determined by the gyratory drilling processes of the drill head, while the curved segments are determined by the direct jacking [16]. The entry angle \(\theta_{1}\) (for the entry straight segment \(L_{1}\)) and exit angle \(\theta_{2}\) (and the exit straight segment \(L_{5}\)) of the trajectory are commonly restricted in valid ranges. [PERSON] et al. [21] suggested the ranges could be defined between 5 and 18 degrees for trajectory design, while [PERSON] et al. [19] recommended it could be set between 6 and 16 degrees. Another study gave higher acceptable values of deviation angles between 8 and 30 degrees [26]. Its determination is usually constrained by constructability limitations which impart conditions of geometric design. The central straight segment \(L_{3}\) is commonly the core part of the trajectory, which must satisfy the requirements of specific burial depths and crossing distance for the engineering. To avoid surface disturbance and damage to installed products or pipelines, the minimum depth of cover along the trajectory should not be less than the minimum allowable H in the standards. The bending radius (\(R_{2}\) and \(R_{4}\)) of curved segments mostly depends on the pipeline strength which is limited by the material properties or diameter. To prevent excessive bending stress in the pipelines, the radius of the curved segments should be limited to the minimum bending radius. More details of the minimum bending radius can be found in the published literatures [16; 19; 21]. In this study, the bending radius adopted was between 2000 and 3000 m for pipelines with diameter over 1000 mm. After preliminary investigation, the assigned domains of the locations of entry and exit points are decided upon for the predesigns. Project risks are evaluated as well, which might result in more stringent limitations in the predesigns. Based on these limitations, possible trajectories are designed (fitted) to ensure the maximum similarity with the catenary trajectory in unifying the plane coordinate system by the least squares method. The sum of squares \(SOS\) is determined as the similarity measurement function to find the best fit (Equation (7)). \(z_{i}\) and \(zc_{i}\) are the depths of the five-segment trajectory and catenary trajectory determined by Equations (3)-(6) above. The lower the difference between the depths, the higher is the similarity of the two trajectories. \[Fitness\_S=SOS=\Sigma(\Delta z_{i})^{2}=\Sigma(z_{i}-zc_{i})^{2} \tag{7}\]  
According to the six parameters obtained above, the length of the preliminary design trajectory can be given in Equation (8) with specific geometric constraints. \[Fitness\_L=\Sigma L_{i}=L_{1}+R_{2}\times\theta_{1}+L_{3}+R_{4}\times\theta_{2 }+(A-L_{1}\times sin\;\theta_{2}-L_{3}-R_{4}\times cos\;\theta_{2})/sin\;\theta _{2} \tag{8}\]