# The First Result of Relative Positioning and Velocity Estimation Based on CAPS
## 3 Experiment Result and Analysis
### Post-Processing Experiment  
In this post-processing experiment, the relative positioning and velocity estimation was carried out based on a very short baseline in static mode from 9:00 to 14:00 local time, on 1 August 2017, which is the same time period as for the zero-baseline described in Section 3.1; there are more than 70,000 epochs in the data collection. Two antennas were installed on the roof of AOE, one of which is the same antenna mentioned in Section 3.1, while the other antenna was about 6 m away from the first one. These two right-hand circularly polarized microstrip antennas are specially designed for the evaluation and processing of the CAPS signal. They can receive the L-band and C-band signals simultaneously and have passed the microwave darkroom test. The standard antenna test results showed that the offset of the phase centers of L-band and C-band can be almost considered coincide in the horizontal component and it is about several millimeters in the vertical component. In our experiment, this mm-level offset of phase center between the signals of C- and L-band has been ignored in comparison with the cm-level positioning validation. The position of the antenna used in our experiment was precisely estimated over a long enough period of averaging of the estimated position series using the RTK technology based on GPS with a nearby reference station in the reference frame of ITRF2000, which is in line with CAPS03; it could therefore be considered the reference standard. The real velocity could be considered to be zero. The ground track of the CAPS satellites at the time of our experiment is shown in Figure 2. The code and carrier-phase measurements were individually used in the relative positioning, and the doppler measurement was used for the velocity estimation. The carrier-phase ambiguities in the relative positioning were fixed by the approach described in Section 2. Using the results in Section 3.1, we can analyze the ambiguity estimation error in Equation (9) and determine the value of the smooth period L. Since the trend term error caused by the multipath cannot be eliminated by smoothing, we first determine the value of \(\sigma_{\Delta N^{i}}\) in the absence of the multipath error. Assuming that the residual of the float ambiguity obeys the zero-mean Gaussian distribution and that the observations in two neighboring epochs are independent from each other, if \(3\sigma_{\Delta N^{i}}<0.5\), the correct ambiguity can be obtained with a probability of 99.73% through the rounding algorithm. For the sake of conservation, we take the standard deviations of the code and carrier phase measurement noise as 0.25 m and 0.17 cm, respectively. According to Equation (10), a period averaging approximately 3 min is needed while the sampling rate is 4 Hz. If one takes the multipath error into account, the mean of the residual may be non-zero. Taking the multipath error of 1 m as an example, the mean of the ambiguity residual distribution will deviate by a 0.5 cycle, which will cause a 1 cycle error on \(\Delta N^{i}\) with a 50% probability. The integer error on \(\Delta N^{i}\) will affect the positioning result, and its impact on the positioning error will depend on the geometric factor corresponding to the satellite \(i\). In our experiment, six minutes of data were used; the data series from the first three minutes were used to fix an initial ambiguity, while the ambiguity fixed with the last three minutes was used for verification. If the value of these two ambiguities was the same, then the integer value was considered to be the correct ambiguity. The positioning error can be calculated from the differences of the rover coordinates between the value estimated from the CAPS relative positioning and the'real' value. The distribution of the series of positioning errors in the east, north and up components based on the code measurements are illustrated in Figure 7, while the corresponding results for the carrier-phase measurements are provided in Figure 8, respectively.