On the basis of the first generation Compass test system, COMPASS is currently composed of mixed types of satellites, among which the GEO and IGSO satellites achieve mainly regional coverage, while MEO satellites will be gradually launched to provide the ultimate global coverage. As COMPASS evolves from a regional system to a global one, the increasing number of MEO satellites in the constellation will enhance the coverage for the users, gradually improve the viewing geometry, and thus upgrade the overall positioning precision [5].

Additionally, in terms of the orbit precision of the navigation satellites, different types of navigation satellites have various orbital characteristics, which lead to unequal orbit determination errors, extrapolation errors and broadcast ephemeris precisions. Speaking more generally, there are rather large differences in the users’ equivalent ranging errors (UERE) for different types of satellites [6]. For instance, the GEO satellites are relatively static to the tracking stations. This essentially makes the orbit determination geometry remain the same, thus the clock errors are difficult to separate. Moreover, GEO orbits are more sensitive to the radiation pressure so that extrapolation cannot be too long ahead. Therefore, under the same conditions the ranging error introduced by the GEO satellites ephemeris is greater than those of MEO and IGSO satellites. As of this writing, the orbit determination for GEO satellites is still a prime difficulty for applications of the system. It may even become the bottleneck that determines whether or not GEO satellites can participate in the navigation and positioning operation along with other types of satellites within the whole constellation. Many research efforts have been made to improve the precision of orbit determination for GEO satellites [7-9].

Therefore, with the ongoing stepwise deployment of the COMPASS constellation, the positioning performance of the system can be improved in at least two aspects: new satellite launches that will enhance the system structure by gradually increasing the number of MEO satellites, and advances in the receiver technology and information processing techniques that can reduce the UERE of GEO satellites. The motivation of this paper is to study the evolution of the positioning performance of the system throughout the COMPASS constellation deployment in terms of these two aspects.

In view of that different navigation satellites in the mixed constellation carry different error characteristics in their own ranging information, we choose the weighted dilution of precision (WDOP) as an evaluation criterion to study the relationship between the internal factors and the positioning precision in the system evolution process. WDOP can assign corresponding weights to the observations according to the ranging errors of each navigation satellite in the case of unequal-precision measurements. It is often used in the evaluation of the system positioning performance and satellite selection because it is more objective than the general DOP (dilution of precision) [10-12].

Suppose that at a certain instant there are m visible navigation satellites, among which there are respectively, , and satellites of different types in sight so that. Then linearizing the nonlinear pseudo-range observation equations leads to the following design matrix

where is the unit directional vector between satellite-i and the approximate location of the user receiver and satisfies

where “” represents “always equal”. Then the WDOP can be defined as

where Tr(·) is the trace of the matrix, and is the weight matrix, which can be chosen as follows. In the case of independent observations, assume that the squared UERE value of each satellite forms the following diagonal matrix

which contains the pseudo-range covariances, reflecting the uncertainty in the pseudo-range measurements. By taking as the basic pseudo-range error, we convert the weight matrix into

where “” means “denotes”. According to the property of diagonal matrices, we have

and the size of the matrix depends on the number of the visible satellites and it varies with time. The selection of is only used for the normalization of and can refer to the pseudo-range error of any type of the satellites in the mixed constellation. Take, and as the UERE values of the satellite types, respecttively. In this paper, we assume. Then influences the weight matrix and eventually the WDOP. By organizing the diagonal elements in the order of, and, we have

In the rest of our analysis, we assume that: 1) The total numbers of satellites in the constellation are, (fixed), and with the stepwise deployment of the constellation, the number of MEO satellites is increasing. The constellation consists of in-orbit satellites, where; 2) The UEREs for same type satellites have equal values, and, (fixed), while the value of decreases gradually with the improvements of hardware and software techniques.

Therefore, we can regard the WDOP at a certain instant as a two-variable function, with as the function forms, that is

In the following we discuss respectively the influences of and on the WDOP to illustrate the changes in positioning precision during the constellation stepwise deployment. For convenience, we denote the joint influence as.

In this section, we fix and study the impact of on the WDOP. By taking as a set of MEO satellites in the constellation, we have the following proposition.

Proposition I At any fixed instant, is monotonically non-increasing with. That is, for any, and, the following holds true:

Proof If we can prove that the WDOP value decreases or remains the same while an MEO satellite is added to the set, i.e.,

then we can get recursively that

To prove Equation (10), let us consider two cases.

1) When the added MEO satellite is not visible to the user, i.e., , then the increase of does not change the total number m of visible satellites, the corresponding design matrix, and the weight matrix. Thus we have

2) When the added MEO satellite is visible to the user, then the total number of visible satellites increases from to. From the fact that the weight matrix is diagonal and satisfies Equation (5), we have, where  

.

Therefore the following relation is true:

where,.

The following part of the proof is the same as given in [10]. To make the proof complete, it is summarized as follows.

Performing the singular value decomposition on produces

where and are orthogonal matrices, and a diagonal matrix.

By substituting Equation (13) into Equation (12), and making the same orthogonal transformation to both sides of the resultant equation, we get the following set of equations based on the fact that an orthogonal transformation does not change the trace of a matrix:

where

,

,

, and

thus. Therefore, we have

By Integrating Equation (11) and Equation (14), we get Equation (10). This concludes the proof of the proposition.

Remark: The contribution of a new satellite to the WDOP value can also be calculated. That is, from Equation (12) and the Sherman-Morrison formula the following holds true:

where is a scalar. So we have

Now, it can be readily seen that

These formulas jointly produce the recursive algorithm from to; and the magnitude of shows the decrement in WDOP as produced by adding a new MEO satellite visible to the user. The recursion lowers the storage loads due to savings in computation incurred in the reduced matrix inversion and repetitious operations.

In this section, with fixed we look at the impact of on WDOP.

Proposition II At any chosen instant, is monotonically non-increasing as decreases. That is, for any, the following holds:

Proof Generally, let. Then is a 4^th order square matrix for any value.

Because, if, i.e., there is no GEO satellite among the satellites visible to the user, then the change of does not make any difference in the matrix. Hence we have

.

In the following, we focus on the proof for the circumstances that. For any, there is such that one of the following three cases occurs.

1) Assume that and. Then. Now is a 4^th order inversible square matrix. Hence we have

.

Let. Then from

it follows that

.

Therefore, from, it follows immediately that.

2) Suppose that a GEO satellite is added on the basis of case 1), so that and. Then. Now because is not square, the method above cannot be applied. Let us consider the block matrix:

.

So we have

.

By the Sherman-Morrison formula we have

therefore

Because is a non-negative definite matrix with eigenvalues 0, 0, 0, and 2, and is a positive definite matrix, , and. Hencefor, holds true. From case 1) we have

, i.e., , so

.

3) Suppose that either an IGSO or a MEO satellite is added on the basis of case 1), so that and

. Then. Let.

Then we can prove in a similar way as in case 2) that according to Equation (17), we have

where is irrelevant to. Therefore,

which leads to.

We can repeat the steps above recursively and eventually lead to the conclusion that for any

Equation (16) holds.

As the MEO satellites are launched in steps, there exists a variety of possible constellation deployment scenarios, each of which will affect the positioning performance of the system. We consider two different schemes as suggested in [4], and see how the characteristics of the positioning precision evolve with varying parameters. In the following, S₁₂ denotes the second satellite on the first orbital plane, and the same notation rule applies for the other satellites.

Scheme I:

S₁₁ → S₁₂ → S₁₃ → S₁₄ → S₁₅ → S₁₆ → S₁₇ → S₁₈ → S₂₁ → S₂₂ → S₂₃ → S₂₄ → S₂₅ → S₂₆ → S₂₇ → S₂₈ → S₃₁ → S₃₂ → S₃₃ → S₃₄ → S₃₅ → S₃₆ → S₃₇ → S₃₈;

Scheme II:

S₂₆ → S₁₃ → S₂₂ → S₁₇ → S₂₈ → S₂₄ → S₃₇ → S₃₃ → S₂₃ → S₂₇ → S₃₂ → S₃₆ → S₂₅ → S₃₁ → S₃₅ → S₃₈ → S₃₄ → S₂₁ → S₁₈ → S₁₆ → S₁₂ → S₁₅ → S₁₁ → S₁₄.

Let us take the Beijing COMPASS station as an example of the users. The relationship between the number of MEO satellites, the UERE of GEO satellites, and the WDOP values under these two schemes are shown in Figure 1, in which the contour maps of WDOP are presented as a function of these two factors.

Several phenomena can be seen out of the simulation results. 1) WDOP decreases with the increase of MEO satellites numbers or with the decrease of GEO satellites UERE. This is consistent with the results of theoretical derivation; 2) the number of satellites poses a greater impact on the decrease of WDOP, while the extent of the decrement declines with the increase of the number. More specifically, the launch of each additional satellite improves the positioning precision at the initial deployment of the constellation, while little improvement can be seen in the WDOP after the number of MEO satellites

reaches a certain level, such as 12 or so. In such circumstances, enhancement in the positioning precision becomes limited; and 3) even if the numbers of satellites in orbit are the same, different constellation configurations affect the evolutionary pattern of the WDOP values during the process of deployment. Therefore, it is essential to consider relevant factors jointly in all aspects in order to design the optimal constellation deployment.

Since future potential users will not be limited to the ground-based ones, let us also analyze the positioning performance for a low Earth orbit (LEO) satellite based on the COMPASS system. Assume that the orbital elements of the LEO satellite are: semi-major axis is 6678 km, eccentricity 0, inclination 28.5^˚, perigee 0^˚, ascending node right ascension 0^˚, mean anomaly 0^˚. Computer simulation shows that under the constellation Scheme I, at least 22 MEO satellites are needed to make the LEO satellite positioning possible at any time; and under Scheme II, at least 20 MEO satellites are required. The reason is that the LEO satellite moves around the Earth in space, thus the regionally augmented function of GEO and IGSO satellites is weakened and the receiver onboard of the LEO satellite relies most of the time on the visibility of the MEO satellites.

Figure 2(a) shows the evolutionary pattern of the WDOP for the LEO satellite by using Scheme II of the stepwise deployment. In particular, due to the satisfactory viewing geometry between the satellites in the constellation and the LEO satellite, the WDOP values are relatively small, and the number of MEO satellites plays a leading role in increasing the magnitude of the WDOP. Figure 2(b) illustrates the variation curves of the WDOP as a function of the GEO satellites UERE with different numbers of MEO satellites. It can be seen that in order to improve the positioning precision, the focus should be on the completion of launching the MEO satellites in the constellation, because the LEO satellite moves at a relatively high speed to the constellation. Consequently, the key to improve the positioning geometry is to enhance the constellation coverage.