## 3 Algorithm and Strategies of Data Processing
### Absolute PCO and PCV Separated from Relative Observation
We actually used a Personal Computer (PC) to record the robot's moving time and sets of spatial coordinates and attitudes w.r.t the robot local orthonormal coordinate system and further implemented a straight-forward approach of time synchronization to correct the used PC clock (synchronization accuracy better than 0.1 s) in an hourly fashion using the GPS GGA time tag output from one of GPS receivers in our calibration facility. After coordinate transformation and time synchronization, the GPS observables and robot motion data can be easily merged together, _i.e._, any vector \(\mathbf{a}\) in robot local orthonormal coordinate system can be transformed to \(\mathbf{b}\) in the GPS WGS-84 coordinate system using well-known seven parameter Helmet transformation model, read as:  
\[\mathbf{b}=x_{0}+(1+m)R(\alpha,\beta,\gamma)\mathbf{a} \tag{2}\]  
Figure 4: Observations coverage on antenna hemisphere when the antenna was fixed (24 h static, DoY2014207 in Wuhan). where, \(x_{0}\) is coordinate shift components, \(m\) is a scale factor (\(m=0\) in this paper), and \(\alpha,\beta,\gamma\) are the Eulerian angles. \(R\) is rotation matrix from robot local coordinate system to GPS WGS-84 coordinate system. For a very short baseline, the linearized phase double difference (DD) observation equations in meters from the location of ARP between reference antenna and the antenna under test can be simply read as:  
\[\
abla\Delta\phi_{1,2}^{i,j}=-\Delta a\cdot\mathbf{r_{1}}+\Delta b\cdot\mathbf{r_{2}}- \Delta PCV_{1}^{i,j}(az,el)+\Delta PCV_{2}^{i,j}(az,el)+\
abla\Delta MP_{1,2}^ {i,j}+\
abla\Delta N_{1,2}^{i,j}+\
abla\Delta\varepsilon_{1,2}^{i,j} \tag{3}\]  
The subscripts 1, 2 stand for reference and test receiver antennas and \(i,j\) for satellites. Notations of \(\
abla\Delta N\), \(\
abla\Delta PCV\) and \(\
abla\Delta\varepsilon\) stand for DD ambiguities, DD phase center variations and some expected random phase noise for real observations, respectively. \(az,\ el\) are azimuth and elevation w.r.t antenna ARP. \(\Delta a\) and \(\Delta b\) are the SD linearized coefficients of line of sight between receiver and satellite. Vectors \(\mathbf{r_{1}}\) and \(\mathbf{r_{2}}\) stand for the PCO from the location of ARP for reference antenna and test antenna. The coordinate vector between locations of ARP of reference antenna and test antenna at any epoch are well-determined with high precise robot motion data and coordinate transformation parameters (see Equation (2)) and this geometry quantity is removed when Equation (3) is linearized. Additionally, the atmospheric propagation delay error (ionosphere and troposphere) and both of receiver and satellite clock biases are considered equivalent for each antenna and these common errors are eliminated since the distance of baseline in our procedure is as short as about 3 m. However, the multipath term \(\
abla\Delta MP\) errors may be on the order of a few millimeters in terms of phase path, and therefore cannot be neglected in the DD linearization. Intuitively, the expected test antenna absolute PCO and PCV cannot be extracted if the reference antenna PCO and PCV are not available in Equation (3). Nevertheless, an absolute antenna calibration can be accessible if the reference antenna phase center is properly eliminated or greatly reduced and it does not move during the calibration procedure. This can be achieved by rotating and tilting the test antenna mounted on the end of a robot (see Figure 2). The time difference (also Triple difference, TD) of successive two set of antenna orientations in a short time, generally several seconds, can separate the test antenna PCO and PCV from DD observables (see Equation (3)). We expand Equation (3) for times \(t_{A}\) and \(t_{B}\), and ignore superscript satellite symbol \(i\) and \(j\), reads:  
\[