## 3 System Models
### Sensing Coverage Model
To avoid the above situation, researchers put particles into quantum space and proposed the QPSO algorithm. At this time, the particle motion is described by the attraction generated by the quantum potential energy. Particles under quantum attraction can appear at any point in space with a certain probability. The optimal global solution can be obtained with a certain probability. Therefore, the QPSO algorithm has successfully solved various optimization problems [29; 30; 31]. In the QPSO algorithm, \(\alpha\) is a parameter named 'contract expansion' coefficient [31] in updating the average optimal position of particles through quantum potential energy, which is concerned with the convergence of the QPSO algorithm. Therefore, for the MSSQ problem, we propose an IQPSO algorithm. In IQPSO, as the number of iteration steps increases, we adaptively adjust \(\alpha\) according to the iterations to improve the performance of the QPSO algorithm. ### Proposed IQPSO Algorithm

In the \(N\)-dimensional target search space, the IQPSO algorithm \(\tilde{X}=\{X_{1},X_{2},\ldots,X_{o}\}\) consists of \(o\) particle swarms, which represents the possible suboptimal mobile charging sequence of MSSQ problem. At the \(k\) step, the position of the \(i\)-th particle is \(X_{i}(k)=\{x_{i,1}(k),x_{i,2}(k),\ldots,x_{i,N}(k)\}\). In IQPSO algorithm, the particle does not have velocity vector, and the best position of the personal particle \(i\) is expressed as \(p_{i}=\{X_{1},X_{2},\ldots,X_{o}\}\). At the \(k\) step, the \(p_{i}(k)\) of the \(i\)-the particle is

\[p_{i}(k)= \begin{cases}X_{i}(k)&\Delta TSCNS\geq 0\\ p_{i}(k-1)&\Delta TSCNS<0\end{cases} , \tag{8}\]

where \(\Delta TSCNS= TSCNS(X_{i}(k))- TSCNS(p_{i}(k-1))\). The position with the largest \(TSCNS\) searched in particle represents personal experience represented by \(p_{best}\), and set \(p_{best}=p_{i}\) changing with particle optimization process. At the \(k\) step, the global best position of the population \(G_{b}\) is

\[m_{1}= \begin{array}{l}\arg\max\limits_{1\leq i\leq o}\left\{TSCNS(p_{i}(k)) \right\}\\ G_{b}(k)=p_{m_{1}}(k)\end{array} , \tag{9}\]

where \(m_{1}\in\{1,2,\ldots,o\}\) is subscript of the global optimal particle position in the population. The global best position should be calculated before each update of the particle's position. Thus, only the \(TSCNS\) of the current personal position of each particle needs to be compared with the global best position, if the \(p_{i}\) is better, update the \(G_{b}\). To get the evolution equation of particles, the following settings are needed

\[p_{i,j}(k)=\varphi_{j}(k)p_{i,j}(k)+(1-\varphi_{j}(k))G_{bj}(k),(\varphi_{j}(t )\in U(0,1)). \tag{10}\]

The IQPSO algorithm assumes that there is a one-dimensional potential well at the local attraction of each dimension and that every particle in the group has quantum behavior. [PERSON] et al. [32] demonstrated that the interaction of individual and public search of each particle is pulled towards its local attractor \(q_{i}=\{q_{i,1},q_{i,2},\ldots,q_{i,N}\}\) to ensure convergence. The probability density function of each particle flight position can be derived from the Schrodinger equation. The next step of the IQPSO iteration is described

\[X_{i,j}(k+1)=q_{i,j}(k)\pm(h_{j}(k)/2)\ln(1/\beta),(\beta=rand(0,1)) \tag{11}\]

where \(h_{j}(k)\) is delta potential well characteristic length, the average position of all the particles is expressed as \(r_{j}(k)\), defined as \(h_{j}(k)=2\alpha|r_{j}(k)-X_{i,j}(k)|\), \(r_{j}(k)=\frac{1}{o}\sum\limits_{i=1}^{o}p_{i,j}(k)\)where \(\alpha\) is a parameter named the 'contraction-expansion' coefficient [31]. It is the only parameter need to be set artificially, which is concerned with the convergence performance of the algorithm. The improved methods for \(\alpha\) include the fixed value and linear variation methods. In this paper, we adaptively adjust \(\alpha\) according to the iterative steps within the upper and lower bounds to ensure that the algorithm obtains better global and local search ability in the whole iteration.