#### 4.2. Generation of Measurements

The first simulation step was to propagate the GEO satellite’s orbit. We propagated the GEO using Cowell’s method, which propagates the position and velocity of the satellite by integrating the accelerations caused by perturbations at each time step [

5]. We included the geopotential, solar pressure, and third-body gravity (the Sun and Moon). We chose the EGM-96 model as the geopotential, and the degree and order were both set to 20. The Runge-Kutta 68 algorithm was chosen as the numerical integrator, and the integral step was set to 10 s. The initial orbital elements for propagation are listed in

Table 2.

The GPS satellites’ position and velocity were generated every 10 s for 24 h using the Almanac data. After generating the positions and velocities of the GEO and GPS satellites, we checked whether the GPS satellites were observable at each epoch. A GPS satellite was only visible when it was not blocked by the Earth and its signal power was sufficiently strong to be processed by the GPS receiver.

After determining the visibility of the GPS satellites, the C/A code pseudorange and pseudorange rate were calculated. The C/A code pseudorange is given by [

12]

where

$\text{\rho}$
is the C/A code pseudorange in L1,

$RX$
is the signal reception time,

$TX$
is the signal emission time,

${\text{\sigma}}_{r}$
is the receiver clock bias,

${\text{\sigma}}_{t}$
is the GPS satellite clock bias,

$n$
is noise,

$\text{\tau}$
is the time delay,

${t}_{0}$
is the reference time,

${a}_{0}$
and

${a}_{1}$
are the polynomial coefficients of the GPS satellite clock bias, and

${b}_{0}$
and

${b}_{1}$
are the polynomial coefficients of the receiver clock bias.

The geometric distance from the GPS satellite to the receiver was calculated using

${\overrightarrow{r}}_{t}(TX)$
and

$\overrightarrow{r}(RX)$; we did not use

$\overrightarrow{r}(TX)$ [

13]. The GPS signal travels through space at the speed of light, and thus, the receiver does not instantly receive the signal emitted from the GPS satellite. Therefore, the signal reception time is later than the signal emission time. The equation of the elapsed time from the emission to reception is given by

$\text{\tau}$
appears on both sides of Equation (30). Thus, an iteration technique was used to calculate the proper

$\text{\tau}$. First,

${\overrightarrow{r}}_{t}(RX)$
was used instead of

${\overrightarrow{r}}_{t}(RX-\text{\tau})$
to calculate the temporary

$\text{\tau}$
on the left side of the equation, Then, the temporary

$\text{\tau}$
was used on the right side of Equation (30) to update the temporary

$\text{\tau}$, and the iterations continued until

$\text{\tau}$
converged [

13,

14].

The equation for the pseudorange rate is similar to that for the pseudorange and is given by

where

${\dot{\text{\delta}}}_{r}$
and

${\dot{\text{\delta}}}_{t}$
are the clock bias rates of the receiver and GPS satellite, respectively. These variables are calculated from the derivatives of

${\text{\sigma}}_{r}$
and

${\text{\sigma}}_{t}$.

#### 4.3. Simulation Results

The algorithm was tested using data from the four points (A, B, C and D) selected from the 24 h simulated GEO orbit. We selected four points at 6 h intervals to rigorously validate the algorithm. The simulation times of the selected points A, B, C and D were UTC 00:00:00, 06:00:00, 12:00:00 and 18:00:00, respectively, on 1 January 2006. The position and velocity vector of each point is given in

Table 3.

**Table 3.**
Position and velocity of the COMS at the four selected points.

**Table 3.**
Position and velocity of the COMS at the four selected points.
Point | A | B | C | D |
---|

UTC time | 00:00:00 | 06:00:00 | 12:00:00 | 18:00:00 |
---|

x (km) | −27,828.9136 | 31,792.4080 | 27,551.1020 | −32,042.5366 |

y (km) | −31,685.0220 | −2769.1326 | 31,911.9004 | 27,494.4159 |

z (km) | 3.5110 | −27.4958 | −5.3747 | 28.3683 |

v_{x} (km/s) | 2.3098 | 2.0194 | −2.3276 | −1.9989 |

v_{y} (km/s) | −2.0286 | 2.3185 | 2.0093 | −2.3364 |

v_{z} (km/s) | −0.0019 | −0.0003 | 0.0019 | 0.0004 |

Semi-major axis (km) | 42,165.029 | 42,166.897 | 42,165.112 | 42,167.097 |

Eccentricity | 0.000140 | 0.000108 | 0.000142 | 0.000192 |

Inclination (deg) | 0.0360 | 0.0378 | 0.0378 | 0.0393 |

Ascending node (deg) | 56.3009 | 58.8100 | 60.3141 | 61.9844 |

Argument of perigee (deg) | 348.0949 | 325.9801 | 12.1476 | 17.0192 |

True anomaly (deg) | 172.4067 | 260.1279 | 348.8802 | 77.4768 |

The observable GPS satellites over 24 h are depicted in

Figure 4; however, the simulations were performed under controlled conditions in which only two or three GPS satellites were visible. The initial error in the reference position was set to 1000 km at each of the four points, and the reference receiver clock bias was set to 100 km. The noise terms in Equations (28) and (31) were selected from Gaussian distributions with standard deviations of 0.01 km and 0.0001 km/s, respectively.

The positions of the COMS and its visible GPS satellites at each point are shown in

Figure 5,

Figure 6,

Figure 7,

Figure 8,

Figure 9,

Figure 10,

Figure 11 and

Figure 12. The red spot represents the COMS, and the yellow spots represent GPS satellites. At each point, we produced scenarios such that two or three GPS satellites were visible at the COMS. Thus, we intentionally chose GPS satellites among those visible to control the number of visible satellites if more than three GPS satellites were visible. For consistency, we simply removed one satellite from the scenario where three GPS satellites were visible such that only two GPS satellites were visible. As shown in the figures, the visible GPS satellites are located behind the Earth and are located closely together.

**Figure 5.**
Point A with two visible GPS satellites.

**Figure 5.**
Point A with two visible GPS satellites.

**Figure 6.**
Point A with three visible GPS satellites.

**Figure 6.**
Point A with three visible GPS satellites.

**Figure 7.**
Point B with two visible GPS satellites.

**Figure 7.**
Point B with two visible GPS satellites.

**Figure 8.**
Point B with three visible GPS satellites.

**Figure 8.**
Point B with three visible GPS satellites.

**Figure 9.**
Point C with two visible GPS satellites.

**Figure 9.**
Point C with two visible GPS satellites.

**Figure 10.**
Point C with three visible GPS satellites.

**Figure 10.**
Point C with three visible GPS satellites.

**Figure 11.**
Point D with two visible GPS satellites.

**Figure 11.**
Point D with two visible GPS satellites.

**Figure 12.**
Point D with three visible GPS satellites.

**Figure 12.**
Point D with three visible GPS satellites.

We tested the developed algorithm using a single-epoch measurement at each point. The calculated state vectors of the COMS were compared to the true values, and the differences of each are summarized in

Table 4 and

Table 5. We also tested the influence of the position of the third GPS satellite at point D; more than three GPS satellites are observable at point D. We ran simulations with two fixed GPS satellites and a third satellite placed at several different positions. The results are summarized in

Table 6. The residual refers to the difference between the calculated and true values. The results indicate that the residual of the calculated position is less than 40 km in range, and three of the four points have residuals of less than several kilometers in range when using three visible GPS satellites.

**Table 4.**
Estimated error when using two visible GPS satellites.

**Table 4.**
Estimated error when using two visible GPS satellites.
Residuals | A | B | C | D |
---|

x (km) | 4.453 | 13.415 | −2.984 | 16.474 |

y (km) | 5.113 | 16.693 | 8.600 | 20.910 |

z (km) | −3.511 | 27.495 | 5.374 | −28.268 |

V_{x} (km/s) | 0.000325 | −0.000971 | 0.000001 | −0.000963 |

V_{y} (km/s) | 0.000331 | 0.000796 | −0.000543 | 0.001067 |

V_{z} (km/s) | 0.001920 | 0.000348 | −0.001994 | −0.000458 |

clock bias (km) | 6.019 | 2.108 | −5.550 | −9.404 |

clock bias rate (km/s) | 0.000001 | −0.000519 | −0.000276 | 0.000424 |

The residuals increase as the z-component of orbital state increases. The maximum error occurs when the z-component of the orbital state is greatest, and the errors are small when the z-components of the orbital state are small. This relationship occurs because the developed algorithm does not include the z-component in the state vector, and thus, the z-component in the real orbit influences the x- and y-components in the state vector. The z-component value increases with the inclination and the relationship between the inclination and error at point D are presented in

Figure 13. Based on

Figure 13, we can conclude that the accuracy level of the proposed algorithm is high when the GEO satellite’s inclination is small.

**Table 5.**
Estimated error when using three visible GPS satellites.

**Table 5.**
Estimated error when using three visible GPS satellites.
Residuals | A | B | C | D1 |
---|

x (km) | 1.908 | 4.642 | 4.951 | 17.425 |

y (km) | 7.349 | 6.615 | 1.750 | 22.021 |

z (km) | −3.511 | 27.495 | 5.374 | −28.268 |

V_{x} (km/s) | 0.000162 | −0.000236 | 0.000514 | −0.001044 |

V_{y} (km/s) | −0.000517 | 0.000156 | 0.000003 | 0.001136 |

V_{z} (km/s) | 0.001920 | 0.000348 | −0.001994 | −0.000458 |

clock bias (km) | 7.156 | −1.598 | −4.319 | −10.088 |

clock bias rate (km/s) | −0.000194 | −0.000261 | −0.000472 | 0.000194 |

**Table 6.**
Estimated errors when using two fixed GPS satellites and a third satellite at various positions at point D.

**Table 6.**
Estimated errors when using two fixed GPS satellites and a third satellite at various positions at point D.
Residuals | D1 | D2 | D3 |
---|

x (km) | 17.425 | 5.2519 | −7.992 |

y (km) | 22.021 | 7.799 | −7.787 |

z (km) | −28.268 | −28.268 | −28.268 |

V_{x} (km/s) | −0.001044 | −0.000007 | 0.001122 |

V_{y} (km/s) | 0.001136 | 0.000249 | −0.000716 |

V_{z} (km/s) | −0.000458 | −0.000458 | −0.000458 |

clock bias (km) | −10.088 | −5.146 | −0.468115 |

clock bias rate (km/s) | 0.000194 | −0.000050 | −0.000165 |

No significant difference occurs when using measurements from two or three visible GPS satellites, except for point B, where the residual decreases when using the measurements from three GPS satellites. Furthermore, the geometric relationship among the GPS satellites and the GEO also affected the accuracy of the algorithm, as demonstrated by the simulation results for point D.

**Figure 13.**
Relationship between inclination and the estimated error at point D.

**Figure 13.**
Relationship between inclination and the estimated error at point D.

The EKF was tested using the initial state vector calculated by the developed algorithm at each point. The time update was processed using the Runge-Kutta method. The simulation results of the EKF are shown in

Figure 14,

Figure 15,

Figure 16,

Figure 17 and

Figure 18. The time for filter convergence varied across the simulation points and the error of the initial state; however, the filter converged within 120 min with an accuracy of 100 m in all simulations. The errors expressed in the RIC frame after the filter is stabilized are shown in

Figure 19 and

Figure 20, and the error norm is bounded at 30 m when using three GPS satellites. These results were quite acceptable because the EKF filter for a GEO converges very slowly due to the orbit’s characteristics. For example, the convergence time of the EKF filter for a GEO is approximately one or two hours under the sparse visibility of GPS satellites [

15]. The convergence rate of the filter depends on the geometric location of the GPS satellites and the changing visibility of the GPS satellites, and thus, the convergence rate varies even though the accuracies of the initial conditions are not significantly different.

**Figure 14.**
Test result of the EKF simulation started at point A.

**Figure 14.**
Test result of the EKF simulation started at point A.

**Figure 15.**
Test result of EKF simulation started at point B.

**Figure 15.**
Test result of EKF simulation started at point B.

**Figure 16.**
Test result of the EKF simulation started at point C.

**Figure 16.**
Test result of the EKF simulation started at point C.

**Figure 17.**
Test result of the EKF simulation started at point D for GPS satellite position D1.

**Figure 17.**
Test result of the EKF simulation started at point D for GPS satellite position D1.

**Figure 18.**
Test result of the EKF simulations started at point D for various positions of the third GPS satellite. The Y axis is zoomed for convenience.

**Figure 18.**
Test result of the EKF simulations started at point D for various positions of the third GPS satellite. The Y axis is zoomed for convenience.

**Figure 19.**
The stabilized EKF errors in a RIC frame when using two GPS satellites. (R is radial, I is along-track and C is cross-track direction).

**Figure 19.**
The stabilized EKF errors in a RIC frame when using two GPS satellites. (R is radial, I is along-track and C is cross-track direction).

**Figure 20.**
The stabilized EKF errors in a RIC frame when using three GPS satellites.

**Figure 20.**
The stabilized EKF errors in a RIC frame when using three GPS satellites.