# Research on Trajectory Planning and Tracking Methods for Coal Mine Mobile Robots
## 2 Trajectory Planning Based on MINCO Trajectory Classes
### Trajectory Planning Based on MINCO Trajectories and Safe Corridors
#### 2.2.3 Unconstrained Processing
Then, it can be expressed as:  
\[\mathcal{P}^{B}=\left\{\mathbf{x}\in R^{m}\|\left\|\mathbf{x}-\mathbf{o}\right\|_{2}\leq r\right\} \tag{34}\]  
The transformation can be used to map the whole space \(R^{m}\) into \(\mathcal{P}^{B}\) to obtain an unconstrained optimization variable in space \(R^{m}\). The whole process is illustrated in two-dimensional and three-dimensional forms and is shown in Figure 10.  
Map the entire space \(R^{m}\) to a unit sphere in space \(R^{m+1}\) without the North Pole as follows:  
\[\mathcal{S}^{m}_{\odot}=\left\{\mathbf{x}\in R^{m+1}\middle|\left\|\mathbf{x}\right\|_ {2}=1,x_{m+1}<1\right\} \tag{35}\]  
Denote this mapping as \(f_{s}\):  
\[f_{s}(\mathbf{x})=\frac{\left(2\mathbf{x}^{T},\mathbf{x}^{T}\mathbf{x}-1\right)^{T}}{\mathbf{x}^{T }\mathbf{x}+1}\in\mathcal{S}^{m}_{\odot},\forall\mathbf{x}\in R^{m} \tag{36}\]  
The unit sphere \(\mathcal{S}^{m}_{\odot}\) defined in space \(R^{m+1}\) is then projected into space \(R^{m}\) to obtain the unit sphere \(\mathcal{B}^{m}\) defined in space \(R^{m}\):  
\[\mathcal{B}^{m}=\left\{\mathbf{x}\in R^{m}\middle|\left\|\mathbf{x}\right\|_{2}\leq 1\right\} \tag{37}\]  
Denote this mapping as \(f_{o}\):  
\[f_{o}(x)=\left(x_{1},\ldots,x_{m}\right)\in\mathcal{B}^{m},\forall\mathbf{x}\in \mathcal{S}^{m}_{\odot} \tag{38}\]  
Figure 10: Spherical constraint variable mapping. Figure 9: Trajectory processing outside the safe area. Then, all positions except the midpoint in \(B^{m}\) can correspond to two positions on the sphere \(\mathcal{S}_{\odot}^{m}\). Finally, a linear mapping is added to change the radius so that \(\mathcal{B}^{m}\) can map to \(\mathcal{P}^{B}\). By combining the mapping \(f_{s}\), \(f_{o}\), and the linear mapping, a composite mapping \(f_{B}\) from the whole space \(R^{m}\) to the sphere \(\mathcal{P}^{B}\) in space \(R^{m}\) can be obtained:  
\[f_{B}=o+\frac{2\pi x}{x^{T}x+1}\in\mathcal{P}^{B},\forall x\in R^{m} \tag{39}\]  
Represent the variables in Equation (39) as a new variable \(\xi\). For arbitrary \(\xi\in R^{m}\), \(q\) mapped by \(f_{B}\) always results in a spherical constraint. Thus, the cost function can be optimized in space \(R^{m}\) for unconstrained variables \(\xi\) instead of optimizing \(q\) in the constraints \(\mathcal{P}^{B}\). However, the optimization on \(\xi\) requires a gradient \(\partial/\partial\xi\). If we denote the \(i\)th element of \(\partial/\partial q_{i}\) by \(g_{i}\), then the gradient of \(l\) to \(\xi_{i}\) (the \(i\)th element in \(\xi\)) can be obtained by computing the differential of the mapping \(f_{B}\):  
\[\frac{\partial l}{\partial\xi_{i}}=\frac{2r_{i}g_{i}}{\xi_{i}T^{\xi}_{i}+1}- \frac{4r_{i}(\xi_{i}^{T}g_{i})\xi_{i}}{\left(\xi_{i}^{T}\xi_{i}+1\right)^{2}} \tag{40}\]  
Compute \(\partial/\partial\xi_{i}\) (\(1\leq i\leq M\)) to obtain the \(\partial/\partial\xi\). Then, the constrained optimization problem becomes an unconstrained optimization problem over the variable \(\xi\). * Convex polyhedral constraints The set of convex polyhedra is denoted as \(\mathcal{P}^{\mathcal{H}}\), which can be expressed as: Figure 10.0 (41)  
For a general convex polyhedron, all points \(q\) in the interior can be represented by their vertex coordinates. When the dimension is low and the vertex coordinates are easy to obtain, then all points \(q\) in the convex polyhedron constraint can be parameterized by their vertex coordinates. If the convex polyhedron in \(\mathcal{P}^{\mathcal{H}}\) has \(n+1\) vertex, denote its vertex by \((\mathbf{v}_{0},\ldots,\mathbf{v}_{n})\), where \(\mathbf{v}_{i}\in R^{m}\). To visualize the convex polyhedron vertices in terms of \(q\), let \(\overline{\mathbf{v}}_{i}=\mathbf{v}_{i}-\mathbf{v}_{0}\) and \(\overline{\mathbf{V}}=(\overline{\mathbf{v}}_{1},\ldots,\overline{\mathbf{v}}_{n})\); then, the relationship between \(q\) with the vertices can be expressed as:  
Figure 10.1  
where \(\mathbf{\omega}=(\omega_{1},\ldots,\omega_{n})^{T}\in R^{n}\).