# An Improved PSO Algorithm and Its Application in GNSS Ambiguity Resolution
## 3 Experiments and Result Analysis
### Search Procedure of IPSO-AR  
We select one BDS six-dimensional DD ambiguity float resolution and its corresponding VC matrices as the input information to illustrate the search procedure of IPSO-AR as introduced in Section 2.2.3. Here, we use the ambiguity dilution of precision (ADOP) to assess the precision of ambiguity float resolution. The specific definition and calculation method of ADOP are provided by [38]. In general, a small ADOP value corresponds to a high precision float resolution. In this test,  
Figure 1: Flow diagram of IPSO–AR (improved particle swarm optimization–ambiguity resolution). VC: variance–covariance matrix.  
the ADOP of the ambiguity float resolution is 0.28. The IPSO-AR parameter settings are (90, 30, 0.5, 0.5, 0.4, 0.8, 0.001, 10). Specifically, the total population size is 90. The sizes of the optimal, suboptimal, and poor populations are all 30. The cognitive learning rate and social learning rate are both 0.5. The inertia weight is adjusted over the range of 0.4 to 0.8. The iterative convergence threshold is 0.001. The ambiguity search range is restricted to a finite interval from \(-\)10 to 10.  
Figure 2 shows the change trend of the total population maturity and adaptive inertia weight with respect to the number of generations. Notably, the initial value of population maturity is relatively small and is approximately 0.63 because the initial particles are randomly generated with a scattered distribution. Meanwhile, particle distribution becomes increasingly concentrated as the position of individual and global optimum particles is continuously updated. The corresponding population similarity gradually increases. Figure 2 shows that when the number of generations is approximately 10, the value of the total population maturity tends to stabilize (approximately 0.95). The adaptive inertia weight calculated using population maturity also tends to stabilize (approximately 0.5).  
Figure 3 shows the evolution of the global optimum. The global optimal fitness completely stabilizes when the number of generations reaches 12. Theoretically, the PSO algorithm has already found the estimated optimal ambiguity integer. However, we utilize several strategies, which are introduced in Section 2.2.2, to avoid falling into a local optimum. The IPSO-AR algorithm satisfies the convergence condition with the strict evolution convergence judgment until the number of generations reaches 47. Thus, the ambiguity search is completed.  
Figure 3: Trend of global optimal fitness.  
Figure 2: Trend of Population maturity and adaptive weight  
For a straightforward illustration of the search procedure of IPSO-AR, we select the former second dimension DD ambiguities (i.e., C02-C01 and C07-C01) as an example to represent the evolution of \(S_{1}\) particles. As shown in Figure 4, the planar distribution of the initial particles, which are shown as solid blue dots (total of 30), is scattered and irregular. Meanwhile, as the number of generations increases, particle evolution became increasingly homogeneous and finally converges to the point (0, \(-1\)).