# 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  
The most important thing for the robot during the execution of tasks in the underground coal mine environment is safety. In practical applications, most CMRRs usually have a speed of less than 1 m/s. The goal of trajectory planning is that the robot avoids obstacles to reach the target safely, and the MINCO trajectory itself can guarantee the smoothness of the trajectory. Hence, the dynamic constraints \(\mathcal{G}\) are not given, and the main considerations are the temporal regularity constraint \(\rho\) and the safety constraints \(\widetilde{\mathcal{F}}\). For problem (11), if we want to deform the MINCO trajectory in time and space to obtain the optimal trajectory, we need to put the cost function, the constraints \(\rho\) and \(\widetilde{\mathcal{F}}\), into the general optimizer to obtain the MINCO trajectory corresponding to the minimum control cost under the constraints. For the optimizer, the computational cost of optimization with constraints is usually much larger than that of optimization without constraints. Therefore, in order to obtain the optimal trajectory faster by using the spatio-temporal deformation of MINCO, some processing of the constraints \(\rho\) and \(\widetilde{\mathcal{F}}\) are needed to transform the problem into an unconstrained optimization problem for faster solving. Time regular term constraints  
Figure 8: Safe corridor. Figure 6: Setting the boundary box. Figure 7: Polyhedron collection after setting the bounding box. For multiband polynomial curves, the cost function in problem (11) is a function of the polynomial coefficient matrix \(c\) and the time vector \(T\), denoted as \(J_{c}(c,T)\). The expressions for \(J_{c}\), \(\partial J_{c}/\partial c\), and \(\partial J_{c}/\partial T\) are given in [30], and the MINCO trajectory itself belongs to a special subset of multiband polynomials. So, the cost function for this problem can also be written in the following form:  
\[J(\mathbf{q},\mathbf{T})=J_{q}(\mathbf{q},\mathbf{T})+\rho(\left\lVert\mathbf{T}\right\rVert_{1}) \tag{30}\]  
where \(J_{q}\) is defined as \(J_{q}(\mathbf{q},\mathbf{T})=J_{c}(c(\mathbf{q},\mathbf{T}),\mathbf{T})\), i.e., the form \(\mathcal{W}(\mathbf{q},\mathbf{T})\) of the cost function on the MINCO trajectory in the previous section. So, \(J_{q}\partial J_{q}/\partial q\) and \(\partial J_{q}/\partial T\) are all solvable. First, for the case in Equation (12), it is defined as follows:  
\[\mathcal{T}_{f}=\left\{\mathbf{T}\in R^{M}\middle|\left\lVert\mathbf{T}\right\rVert_{ 1}=T_{\Sigma},\mathbf{T}\succ 0\right\} \tag{31}\]  
That is, for any \(1\leq i\leq M\), the domain of definition of \(T_{i}\) is \(0<T_{i}<T_{\Sigma}\). The following mapping transformations are performed on \(\mathbf{T}\) using variables \(\mathbf{\tau}=(\tau_{1},\ldots,\tau_{M-1})\), which can represent all the elements in the space \(R^{M-1}\):  
\[ \begin{split} T_{i}=\frac{e^{\tau_{1}}}{1+\sum\limits_{j=1}^{M-1 }e^{\tau_{j}}}T_{\Sigma},\\ T_{M}=T_{\Sigma}-\sum\limits_{j=1}^{M-1}T_{j},1\leq i<M\end{split}  \tag{32}\]  
where the domain of definition of \(\mathbf{\tau}\) is the whole space of Based on the above approach, it is necessary to constrain the path points in the overlapping convex polyhedral region of the safety corridor or in the artificial spherical region. The cost function \(J\) is also a function of the path point vector \(\mathbf{q}\). So, the problem becomes optimizing the cost function \(J\) when the path point vectors \(\mathbf{q}\) are under the safety constraints. The problem is converted to an unconstrained optimization problem before solving it, and the constraints to be dealt with are spherical constraints and polyhedral constraints. * Spherical constraints  
A spherical constraint has \(\mathbf{q}\in\mathcal{P}^{B}\subset R^{m}\) for any intermediate path point, where \(\mathcal{P}^{B}\) is a closed sphere. Its radius is \(r\) and the center point is \(o\).