3.2. Performance Analysis of the Proposed AR Method with Pseudolite System
Several experiments including both static and kinematic tests were conducted to verify the validity of the proposed AR method.
The base station receiver antenna is accurately placed at a fixed point in the laboratory, and the rover station was located approximately 0.6 m away from the base station. Because it lacks a strict centering device, the rover station coordinate can only be approximately pre-given, namely, 0.6, 0.6, and 0.01 m, within a positioning error of 0.03 m. Static observation was performed for about half a minute. The data sampling interval is 0.1 s. Five pseudolite signals can be captured without interruption during the experiment.
AFM is performed with a conventional grid search method by using the data of the first epoch carrier phase observations. The initial search approximate coordinate is given as 0.6, 0.6, and 0.01 m, the search range is 0.3 m × 0.3 m, and the search step is 0.005 m, consuming about 0.43 s. Figure 2
shows the 2-D contour map of pseudolite AFV for this epoch. Multiple AFV peaks can be observed in the corresponding search regions. The highest peak corresponds to an AFV value of 0.994, which is close to theoretical value of one. The corresponding optimal coordinates are 0.60 m and 0.58 m, which are within the centering error of the rover station.
The IPSO search method was conducted using the same data as above, and the particle convergence threshold is set to 0.001. Figure 3
shows the evolution of the AFV of the global optimal particle (gBest
). In the initial state, the particles are generated randomly, and the corresponding AFV of gBest
is only 0.76, with the continuous positioning and speed update of pBest
(personal optimal particle) and gBest
. The homoplasy of particle evolution becomes increasingly apparent until the evolution times reach 38, indicating that the IPSO algorithm satisfies the convergence condition. The total time consumed is 0.034 s.
shows the evolution of the 3-D coordinates of the global optimal particles (gBest
), where the 3-D component value is the difference with the initial pre-given coordinates. Compared with Figure 3
, we can see that the fitness of gBest
is smaller, and the difference between the gBest
position and the actual position is greater. In the gradual convergence process of the 3-D coordinate components of gBest
, in the early stage of particle evolution, the convergence speed is very fast and the latter is relatively stable. When the AFV difference between the two continuous iterations is less than 1 mm, the IPSO search is complete, indicating that when using the IPSO algorithm to make a coordinate domain search, the search resolution can theoretically reach a submillimeter level.
The multi-peak characteristic of the ambiguity function, particularly in the case of less observation, often makes it difficult to absolutely ensure that the highest peak in the search area corresponds to the true optimal coordinates, especially when the search range is too large. In fact, the AFM search range is determined by the accuracy of the initial approximate coordinate. The higher the accuracy, the smaller the search range needed and the higher the reliability of the searched optimal coordinates. The relationship between the AFM reliability and accuracy of the initial approximate coordinate in an indoor environment was verified by randomly generating 1000 points in the 0.3 m × 0.3 m region around the point (0.6, 0.6 and 0.01). Every point is taken as the initial approximate coordinate to conduct an IPSO search using the same data. The search range is set to 0.3 m × 0.3 m. Figure 5
shows the AFM search results for the 1000 points corresponding to plane 2-D scatter distribution, and the 2-D coordinate of every point (both red and blue point) can be obtained from the x-axis and y-axis value. The red points denote that the search optimal solution and the corresponding DD ambiguity is correct, while the blue point is the opposite. From Figure 5
, we can see that the blue point appears in the 2-D region far from the initial coordinates, within a 0.2 m × 0.2 m region around the true position, and all points are taken as the approximate coordinate that can search the correct DD ambiguity, indicating that when the accuracy of the approximate coordinate is better than 0.2 m, the AFM exhibits a high reliability and performance for an indoor pseudolite system.
To verify the conclusion that the AFM method is not in need of a very high accurate pre-given initial coordinate compared with the LAMBDA method, we carry out the comparison experiment to demonstrate the influence of initial coordinate bias (ICB) on the proposed AR method and LAMBDA method. The comparison experiment is conducted by using the above data of the first epoch carrier phase observations. Table 1
shows the results of the influence of ICB on the two different AR methods. When the ICB reaches the dm-level, the LAMBDA fails to obtain the correct ambiguity resolution (cannot pass the Ratio validation), while the proposed AR method is still competent, although at the expense of a lower computational efficiency.
In fact, one apparent difference when using LAMBDA and AFM is that, usually when adopting the LAMBDA method, only one pre-given initial coordinate will be used; however, when adopting AFM, many possible initial coordinates within a pre-defined search region will be used to decide the best coordinate by a certain criterion. Therefore, we can say that the search procedure is one strategy, not an AR method, and we can also apply this search strategy when using the LAMBDA method. For example, using many possible initial coordinates within a pre-defined search region, we can decide on one best candidate coordinate which has a maximum corresponding ratio value, and thus, even if the accuracy of a pre-given initial coordinate is not sufficiently high, it may still search for the true position of the rover station and obtain the correct ambiguity resolution.
Considering the high computational efficiency and reliability of LAMBDA, we can further improve our data processing strategy when conducting the Known Point Initialization (KPI) for the pseudolite system. The LAMBDA method can initially be used preferentially, if passing the ratio validation, and the correct ambiguity will be obtained, meaning that the pre-given initial coordinate has a relatively high accuracy and the search strategy does not need to be adopted; after all, when conducting the search strategy, it inevitably wastes unnecessary computing time. If this step fails to obtain the correct ambiguity, meaning the accuracy of the pre-given initial coordinate may be relatively poor, then the search strategy will be used. In this step, both the proposed method and the ‘LAMBDA + search strategy’ model are theoretically feasible, although considering the computational efficiency, the proposed method may have a certain advantage, and considering the reliability, ‘LAMBDA + search strategy’ model may have a certain advantage. This is what we expected to conclude, and in further work, we will pursue an in-depth study on this topic. In this paper, we select an alternative method, namely our proposed method, to deal with the situation when the initial coordinate accuracy is not very high.
3.2.1. Static Test
The above static observation data were processed epoch-by-epoch using our proposed AR method. The initial approximate coordinates of the first-time epoch were pre-given as 0.6, 0.6, and 0.01 m, and the subsequent epochs adopt the first method introduced in Section 2.3.2
. The corresponding search space is set to 0.2 m × 0.2 m × 0.2 m. The traditional grid search is carried out at the same time, to highlight the superiority of the IPSO search method. The corresponding search step is set to 0.005 m.
shows the AFM efficiency for two different search methods over the entire observation period. The IPSO method is more efficient than the traditional grid search by an order of magnitude. Compared with the traditional mechanical grid search, IPSO is a new intelligent search method with a high computational efficiency. In fact, the grid search time is closely related to the search space and search step. Table 2
shows the average efficiency of the two search methods for different search spaces and steps. As the search space increases, the efficiency of the grid search decreases rapidly, while the efficiency of the IPSO method was not greatly affected. When the search step is 0.2 m and the search step is set to 0.001 m, the traditional grid search completely loses the search function because of computer memory overflow, while the IPSO is still able to maintain a high computational efficiency.
shows the single-epoch positioning results for the IPSO and grid search methods. The two search results are in good agreement and all of the 3-D coordinates for each epoch fluctuated slightly within 3 cm near the true position, indicating that the IPSO search method has a good global search ability and does not appear to fall into the local optimal situation. In addition, the minimum resolution of the grid search result is 5 mm, whereas the IPSO can perform a more detailed search than the grid search, which is another significant advantage of IPSO.
As we obtained the optimal coordinate solution of the rover station, the corresponding DD ambiguity can be gained from the integer round (IR) method. Considering the DD ambiguity, we can obtain another coordinate solution using LS estimation. Theoretically, the AFM and LS methods are equivalent. The former is nonlinear, while the latter is a linear estimation method. For GNSS positioning, we can ignore the first-order linearization error, and thus, the positioning results using AFM and LS should be generally consistent. However, the indoor space is relatively small, and it can be expected that a coordinate linear error will exist when using the LS estimation method. Figure 8
shows the difference between the final positioning results obtained through the AFM and LS methods. These two methods have a 2 mm difference on the plane and an elevation difference of approximately 1.2 cm, which means that the first-order linear error can cause a cm-level positioning deviation for indoor pseudolite positioning.
The validity of the proposed AR method is verified by selecting four additional points for a further static test. The initial approximate coordinates, observation epoch number, pseudolite number, and position dilution of precision (PDOP) values of the four fixed points are shown in Table 3
. The search space is set to 0.2 m × 0.2 m × 0.2 m, in which the initial approximate coordinates given are within the accuracy of 3 cm.
shows the 2-D positioning results that correspond to the four fixed points. For the four different spatial distribution points, the IPSO method can search for the optimal solution with a good performance, further verifying the robustness of the IPSO search method. The pseudolites and rover station are in a static state under static observation, and thus, theoretically speaking, the search optimal coordinates of each epoch for every point should be consistent. Figure 9
shows that the search results of the four points are in a discrete distribution, with a standard deviation of about 0.005 m, which can be attributed to the noise error of the carrier phase observation and the effects of multipath errors.
3.2.2. Kinematic Test
A kinematic test is carried out to verify the validity of the proposed single-epoch AR method. The mobile car was placed in a fixed rail, as shown in Figure 10
. The length of the rail is 2.2 m. During the entire experimental procedure, the mobile car’s moving speed is from slow to fast. The data sampling interval is 0.1 s. A total of five pseudolite signals can be captured without interruption. The initial coordinate on the first epoch is pre-given with an accuracy of 3 cm, and the subsequent epochs adopt the first method introduced in Section 2.3.2
, because the data sampling rate is high and the speed of the mobile car is relatively slow. The corresponding AFM search space is set to 0.2 m × 0.2 m × 0.2 m.
shows the final pseudolite positioning results for this kinematic test. The overall positioning trajectory is relatively smooth. The relative deviation from the straight line of most points is within 5 cm. The calculated baseline length using the positioning results of the starting and stopping points is 2.224 m, with an actual rail length difference of 0.024 m, which means that the kinematic positioning precision of the indoor pseudolite can reach a cm-level using the proposed single-epoch AR method.
The wavelength of the GPS L1-band is 19 cm; that is, if the ambiguity is solved incorrectly with one cycle bias, it can cause cm- or even dm-level positioning errors. Generally, we can use the variances of the baseline parameters estimated using the LS method to validate the search optimal solution of the rover station during each epoch. Figure 12
shows the 3-D positioning error sequence of each epoch for this kinematic test. The 2-D positioning error is within 1 cm and the elevation direction is relatively large, that is, less than 2 cm, indicating that the solved DD ambiguity for every epoch is correct.