# Research on Trajectory Planning and Tracking Methods for Coal Mine Mobile Robots
## 2 Trajectory Planning Based on MINCO Trajectory Classes
### Differential Flatness-Based Control Optimization Trajectories
#### 2.1.5 MINCO Trajectory Class  
After the analysis of the above content, it can be seen that in the case of certain initial and termination boundary conditions, if a set of intermediate path point vectors \(\mathbf{q}=(\mathbf{q}_{1},\dots,\mathbf{q}_{M-1})\) and the corresponding time vectors \(\mathbf{T}=(T_{1},\dots,T_{M})^{T}\) of the intermediate paths of the mobile robot trajectories are given (where \(\mathbf{q}_{i}=R^{m}\) is the 0 th-order derivative at the intermediate time point \(t_{i}\), which is the position, and \(T_{i}\) is the time interval between \(t_{i-1}\) and \(t_{i}\)), then according to the sufficiency and necessary conditions for optimality the only corresponding multisegment polynomial trajectory with the optimal control cost can be obtained as shown in Figure 1.  
Each set of \(\mathbf{q}\) and \(\mathbf{T}\) corresponds to a control-optimized multiband polynomial trajectory, and such trajectories are defined as the class of MINCO trajectories, which can be denoted as:  
\[\Sigma_{MINCO}=\left\{p(t):[0,T]\Big{|}\mathbf{c}=\mathbf{c}(\mathbf{q},\mathbf{T}),\forall\mathbf{q }\in R^{m\times(M-1)},\mathbf{T}\in R_{>0}^{M}\right\} \tag{26}\]  
where \(\mathbf{c}=\mathbf{c}(\mathbf{q},\mathbf{T})\) is determined by the optimality condition. And the MINCO trajectory can be optimized by spatio-temporal deformation of the cost function in problem (11); the process is roughly as follows: because the cost function of a polynomial trajectory is a function of its coefficient matrix and the time vector, one can denote the artificially-defined cost function or the constraint function as \(\mathcal{K}(\mathbf{c},\mathbf{T})\), which can be denoted by \(\mathcal{W}(\mathbf{q},\mathbf{T})=\mathcal{K}(\mathbf{c}(\mathbf{q},\mathbf{T}),\mathbf{T})\) by MINCO trajectory. Subsequently, as long as \(\mathcal{W}\) is known and its gradient is \(\partial\mathcal{W}/\partial\mathbf{q}\), then \(\partial\mathcal{W}/\partial\mathbf{T}\) can optimize the cost function by spatio-temporal deformation of the MINCO trajectory. The computation of \(\mathcal{W}\) is the positive generating process of MINCO trajectory. The final form of \(\partial\mathcal{W}/\partial\mathbf{q}\) and \(\partial\mathcal{W}/\partial\mathbf{T}\) shows as follows [29]:  
\[\begin{array}{c}\frac{\partial\mathcal{W}}{\partial\mathbf{q}}=(G_{1}^{T}\mathbf{c} _{1},\ldots,G_{M-1}^{T}\mathbf{c}_{1})\\ \\ \frac{\partial\mathcal{W}}{\partial\mathbf{T}_{i}}=\frac{\partial\mathcal{K}}{ \partial\mathbf{T}_{i}}-\textit{Tr}\Big{\{}\mathbf{G}_{i}^{T}\frac{\partial\mathbf{c}_{i} }{\partial\mathbf{T}_{i}}\mathbf{c}_{i}\Big{\}}\end{array} \tag{27}\]  
\(\partial\mathcal{W}/\partial\mathbf{T}\) can be obtained by solving for all \(\partial\mathcal{W}/\partial\mathbf{T}_{i}\), where \(1\leq i\leq M\), and \(\mathbf{G}=(\mathbf{G}_{0}^{T},\mathbf{G}_{1}^{T},\ldots,\mathbf{G}_{M-1}^{T},\mathbf{G}_{M}^{T}) ^{T}\) is determined by the following equation:  
\[\mathbf{M}^{T}\mathbf{G}=\frac{\partial\mathcal{K}}{\partial\mathbf{c}} \tag{28}\]  
\(\mathbf{M}\) has been shown in (25). \(\textit{Tr}\{\cdot\}\) represents the trace of the matrix. \(\mathbf{E}_{i}\) has been shown in (22). The above computational procedure constitutes a gradient computation of \(O(M)\) linear complexity. Thus, the MINCO trajectory can be spatial-temporally deformed to obtain the optimal control trajectory with the given constraints by a generalized solver.