# An Improved NSGA-II-Based Method for Cutting Trajectory Planning of Boom-Type Roadheader
## 2 Methodology
### Design of Cutting Space Under Kinematic Constraints  
#### 2.2.1 Development of Roadheader Kinematic Model  
The boom-type roadheader comprises a crawler-mounted mobile body and a longitudinal-axis cutting arm. Upon reaching the preset position, the roadheader engages the cutting arm to initiate excavation. To facilitate the planning and control of the section forming trajectory, a global coordinate system within the excavation tunnel for the roadheader is established, along with a corresponding kinematic model.  
In order to accurately describe the relative pose of the roadheader and its working components within the tunnel, we have defined several coordinate systems: the tunnel coordinate system \(O_{h}X_{h}Y_{h}Z_{h}\), the section coordinate system \(O_{d}X_{d}Y_{d}Z_{d}\), the roadheader body coordinate system \(O_{0}X_{0}Y_{0}Z_{0}\), and the cutting arm coordinate system \(O_{i}X_{i}Y_{i}Z_{i}(i=1,2,3,4)\). The relationship between these coordinate systems is shown in Figure 2.  
Where \(C_{h}\), \(H_{h}\) and \(R_{h}\), respectively, represent the depth of the target section, the height of the roadway and the length of the roadway that has been cut. The \(a_{1}\sim a_{4}\) and \(b_{1}\sim b_{3}\) represent the DH parameter of the boom-type roadheader. \(\theta_{1}\), \(\theta_{2}\) and \(d\) represent the  
Figure 1: Framework of cutting trajectory planning based on improved NSGA-II for boom-type roadheader.  
Figure 2: Kinematic model of roadheader.  
lifting angle of the cutting arm, the turning angle and the elongation of the cutting head, respectively.  
We derived the transformation matrices among the various coordinate systems of the roadheader to describe the movement and positioning more precisely. The transformation matrix between the cutting head coordinate system \(O_{i}X_{i}Y_{i}Z_{i}(i=1,2,3,4)\) and the roadheader body coordinate system \(O_{\text{o}}X_{o}Y_{o}Z_{\text{o}}\) is derived as follows:  
\[\begin{split}\begin{split}\begin{matrix} 0_{T}=\begin{bmatrix}\cos\theta_{1}&-\sin\theta_{1}&0&a_{1}\\ \sin\theta_{1}&\cos\theta_{1}&0&0\\ 0&0&1&b_{2}\\ 0&0&0&1\end{bmatrix}\end{split}\begin{bmatrix}\begin{matrix} 0_{T}=\begin{bmatrix}\cos\theta_{2}&-\sin\theta_{2}&0&a_{2}\\ 0&0&1&0\\ -\sin\theta_{2}&-\cos\theta_{2}&0&0\\ 0&0&0&1\end{bmatrix}\end{bmatrix}\end{split}\begin{bmatrix}\begin{matrix} 0_{T}=\begin{bmatrix}1&0&0&b_{3}\\ 0&0&1&a_{3}+d\\ 0&-1&0&0\\ 0&0&0&1\end{bmatrix}\end{bmatrix}\end{split}\begin{bmatrix}\begin{matrix} 0_{T}=\begin{bmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&1&a_{4}\\ 0&0&0&1\end{bmatrix}\end{split}\end{split} \tag{1}\]  
We derived the homogeneous transformation matrix for the cutterhead relative to the roadheader body coordinate system as:  
\[\begin{split}\begin{matrix}\begin{matrix} 0_{T}=\begin{bmatrix} 0_{T}=\begin{bmatrix} \begin{matrix} 0_{T}=\begin{bmatrix} \begin{matrix} 0_{T}=\begin{matrix} 0 of the cutting section to a geometric surface formed by rotating a curve, which is composed of a parabola and line segments, around a rotational axis.  
A piecewise curve \(\Gamma\), composed of a parabola and line segments, can be expressed as:  
\[\Gamma:\begin{cases}l_{3}y^{2}=r^{2}z&-r<y\leq 0,-l_{3}<z\leq 0\\ l_{2}y=(R-r)(l_{2}\ +l_{3})+(R-r)z&-R<y\leq-r,-(l_{2}+l_{3})<z\leq-l_{3}\\ y=-R&-(l_{1}+l_{2}+l_{3})\leq z\leq-(l_{2}+l_{3})\end{cases} \tag{5}\]  
where \(R\) denotes the large radius of the cutting section, \(r\) the small radius, and \(l_{1}\ +l_{2}\ +l_{3}\) the total length of the cutting head. By rotating this piecewise curve around the rotational axis \(O_{4}L_{1}\) (designated as the \(Z_{4}\)-axis) of the cutting head coordinate system for one full circle, an outer envelope surface of the cutting head is formed, denoted as surface \(\Pi\).  
\[\Pi:\left\{\begin{array}{l}l_{3}(x^{2}+y^{2})=r^{2}z,-r<y\leq r\\ l_{2}^{2}\ (x^{2}+y^{2})=((R-r)(l_{2}\ +l_{3})+(R-r)z)^{2},-R<y\leq-r\ \text{and}\ r<y\leq R\\ x^{2}+y^{2}=R^{2},-R<y\leq R\end{array}\right. \tag{6}\]