## 5 D MMAEO for Engineering Application Problems
### UGV Multi-Target Path Planning (EA7)  
The fourth engineering application problem used in this section is the UGV multi-target path planning problem, which is common in tasks such as warehousing logistics, data collection, and security patrols. A map for the multi-target path planning problem is shown in Figure 11, where the black area represents the infeasible region formed by different types of obstacles, the white area represents the feasible region, and the pentagrams represent the multiple predefined targets. For this problem, the UGV needs to start from one of the multiple predefined targets, navigate sequentially through a series of predefined targets while avoiding obstacles, stop at each predefined target to perform the corresponding task operation, and finally return to the starting location. Therefore, this problem aims to obtain the shortest feasible path for the UGV through multiple predefined targets in an obstacle environment so as to achieve the efficient processing of related tasks. The detailed description of the mathematical model for this UGV multi-target path planning problem are given through Equation (50). \[\min f_{\text{EA7}}(\mathbf{Q})=\sum_{k=1}^{K}\|\mathbf{Q}_{k}\|,\] (50) s.t. \[\mathbf{Q}\
ot\in\text{O}_{\text{b}}, \tag{51}\]  
where \(\mathbf{Q}\) represents the total path connecting all predefined targets in turn, which consists of a series of path points \((\mathbf{Q}_{x},\mathbf{Q}_{y})\); \(K\) is the overall amount for sub-paths, which is also the same as the overall amount for predefined targets; \(\|\mathbf{Q}_{k}\|\) denotes the path length for the \(k\)-th sub-path in the total path; and \(\text{O}_{\text{b}}\) represents the infeasible region in the obstacle environment. In accordance with the features from the UGV multi-target path planning problem, the problem can be transformed into two types of sub-problems: a path planning problem between any two predefined targets and a traveling salesman problem. In this way, the optimal solution of path planning problem between any two predefined targets yields the shortest path between any two predefined targets. Then, based on the results of the shortest path between any two predefined targets, the optimal solution of traveling salesman problem yields the shortest total path through all predefined targets. In addition, a smooth path is helpful for the subsequent tracking control of UGV to avoid unnecessary overdriving and slipping. For this purpose, the cubic quasi-uniform B-spline curve approach is applied to generate the smooth path for the UGV multi-target path planning problem, which can guarantee driveability and smoothness for the generated path [50]. The B-spline curve is constructed through  
\[\mathbf{Q}(t)=\sum_{l=0}^{L}F_{l,m}(t)\mathbf{q}_{l},l=0,1,\ldots,L, \tag{52}\]  
where \(\mathbf{Q}(t)\) indicates the B-spline curve; \(m\) is the order of B-spline curve, and the corresponding degree of B-spline curve is \(m-1\); the overall control points amount is \(L+1\), which needs to be greater than or equal to the order \(m\); \(\mathbf{q}_{l}\) represents the \((l+1)\)-th control point of B-spline curve; \(F_{l,m}(t)\) indicates the B-spline basic function with \(m\)-order for \((l+1)\)-th control point, which is calculated through Cox-de Boor recursion formulas as \[F_{l,m}(t) = \begin{cases}1,&t_{l}\leq t<t_{l+1}\\ 0,&\text{otherwise}\end{cases} ,m=1, \tag{53}\] \[F_{l,m}(t) =\frac{t-t_{l}}{t_{l+m-1}-t_{l}}F_{l,m-1}(t)+\frac{t_{l+m}-t}{t_{l+m }-t_{l+1}}F_{l+1,m-1}(t),m\geq 2,\]  
where \({}^{\prime}0/0=0^{\prime}\) is defined in Equation (53); and the B-spline basic function is formed through a non-decreasing sequence with continuously varying values \(\{t_{0},t_{1},\ldots,t_{n+k}\}\) called the knot vector, whose values at the beginning and end are commonly fixed at 0 and 1, respectively. For a cubic quasi-uniform B-spline curve, the degree needs to be set at 3, the knots at both ends in the knot vector have a repetition of 4, and the middle knots are uniformly distributed.