In this paper, both the integrity monitoring and fault detection and exclusion (FDE) mechanisms are incorporated into the vector tracking loop (VTL) architecture of the Global Positioning System (GPS) receiver for reliability enhancement. For the VTL, the tasks of signal tracking and navigation state estimation no longer process separately and a single extended Kalman filter (EKF) is employed to simultaneously track the received signals and estimate the receiver’s position, velocity, etc. In contrast to the scalar tracking loop (STL) which utilizes the independent parallel tracking loop approach, the VTL technique is beneficial from the correlation of each satellite signal and user dynamics. The VTL approach provides several important advantages. One of the merits is that the tracking loop can be assisted for overcoming the problem of signal blockage. Although the VTL architectures provide several important advantages, they suffer some fundamental drawbacks. For example, the errors in the navigation solutions may degrade the tracking accuracy. The most significant drawback is that failure of tracking in one channel may affect the entire tracking loop and possibly lead to loss of lock. For reliability enhancement, the EKF based integrity monitoring and FDE algorithms are developed to prevent the error from spreading into the entire tracking loop. The integrity monitoring is utilized to check the possible fault in the pseudorange and the pseudorange rate, followed by the FDE mechanism employed to exclude the abnormal satellite signals. Performance assessment and evaluation for the proposed approach will be presented.

The Global Positioning System (GPS) is a satellite-based navigation system [

As the most vulnerable parts of a receiver, the carrier and code tracking loops play a key role in a GPS receiver. The scalar tracking loop (STL) processes signals from each satellite separately. Specifically, a delay lock loop (DLL) is adopted to track the code phase of the incoming pseudorandom code and a carrier tracking loop, such as a frequency lock loop (FLL) or a phase lock loop (PLL), is adopted to track the carrier frequency or phase. The tracking results from different channels are then combined to perform the navigation state estimate. The drawback of a STL is that it neglects the inherent relationship between the navigation solutions and the tracking loop status. A STL is more like an open loop system and suffered from performance degradation when scintillation, interference, or signal outages occur. The vector tracking loop (VTL) [

Navigation system integrity refers to the ability of the system to provide timely warming to users when the system should not be used for navigation. It is regarded as a risk factor can provide timely warning to users when the position error exceeds a specified limit. The receiver autonomous integrity monitoring (RAIM) [

In addition to the least squares method, the sequential approach that uses the extended Kalman filter (EKF) [

The remainder of this paper is organized as follows. In Section 2, preliminary background on system model for the GPS vector tracking loop is reviewed. The snapshot approach for GPS navigation solution with RAIM is introduced in Section 3. The EKF based integrity monitoring and FDE algorithms are discussed in Section 4. In Section 5, simulation experiments are carried out to evaluate the performance for various scenarios. Conclusions are given in Section 6.

The traditional GPS receiver involves some parallel DLLs, each of which tracks a satellite to estimate the corresponding pseudorange. The parallel pseudorange measurements are sent to the navigation filter to solve for the navigation state vector. The VTL differs from the traditional STL in that the task of navigation solutions, code tracking and carrier tracking loops for all satellites are combined into one loop. The central part of a VTL is the EKF which provides an optimal estimation of signal parameters for all satellites in view and user PVT solutions based on both current and previous measurements from all satellites.

In the VDLL, each channel does not form a loop independently. The vector comprised of outputs of all the code phase discriminators is the measurement of navigation filter. The navigation state vector is estimated by navigation filter, and the error signals arise from the estimated user positions and the satellite positions calculated by the ephemeris. The code loop numerically-controlled oscillator (NCO) as the signal generator in the SDLL is replaced by the estimated user positions, to control the update of the local code. When one channel experiences interference or signal outages in the VTL, the information from other satellites can be used estimate the status of this channel. The system architectures for the STL and VTL are shown as in

The code phase observation of the GPS C/A code can be represented by:

where

In

where _{0} is the carrier to noise ratio of the received signal,

where

Consider the vectors relating the Earth’s center, satellites and user positions. The vector

The distance

where _{b}

where

The states and the measurements are related nonlinearly; the nonlinear ranges are linearized around an operating point using Taylor’s series.

where

The vector

which can be represented as

The matrix

Navigation system integrity refers to the ability of the system to provide timely warning to users when the system should not be used for navigation. The conventional RAIM is usually the “snapshot” type of approaches. While four satellites are sufficient for navigation, at least five satellites in view are needed for integrity monitoring. Otherwise, the geometry is unavailable for GPS RAIM. The linearized GPS pseudorange equation is an over-determined system of linear equations when the number of visible satellites is more than four. Three RAIM methods have received special attention in recent literatures on GPS integrity, including the range comparison method, least-squares residual method, and parity method. All three methods are snapshot schemes in that they assume that noisy redundant range-type measurements are available at a given sample point in time.

In the least-squares residuals method, the residuals are formed in much the same manner as was done in the range comparison method. Since the least-squares estimate of the solution is given by

This is the liner transformation that takes the range measurement error into resulting residual vector. The sum of the squares of the elements of

The test statistic employed in the RAIM algorithm in terms of SSE is given by

where SSE is the unnormalized sum of the squared measurement residuals in all-in-view least squares solution and

In addition to the sequential approach, the other method is referred to as the sequential algorithm, where the Kalman filter is commonly employed. The approach is sometimes referred to as the Autonomous Integrity Monitored Extrapolation (AIME). The Kalman filter algorithms used in the linear system can be extended to the nonlinear system via the EKF approach, which is a nonlinear version of the Kalman filter and is widely used for the position estimation in GPS receivers. The process model and measurement model for the EKF can be written as

where the state vector

The discrete-time extended Kalman filter algorithm is summarized as follow:

Correction steps/measurement update equations:

Prediction steps/time update equations:

Implementation of the EKF algorithm starts with an initial condition value,

Further detailed discussion can be referred to Gelb [

The statistic ^{2} (sum of squared residuals, or simply SSR for short) is used to detect failure, in the way that the parity vector squared magnitude ^{2} is used in RAIM. If there are ^{2} is Chi-square distributed with n degrees of freedom, and ^{2} is Chi-square distributed with ^{2} depends on the entire past history of measurements.

When redundant observations have been made, Kalman filter residuals of the pseudoranges:

has zero mean,

Satellite failures are detected by using the magnitude of the normalized residual vector s as the test statistic:

In the process of failure detection, the threshold _{D}

where _{FA}

The parameter ^{2}| as the test statistic. Therefore the normalized threshold for |

After detecting the fault, it is helpful to find out the unhealthy satellites to be eliminated. The pseudorange residuals

where _{FA}

with the threshold of

The relative parameter

Simulation experiments are carried out for confirmation of the effectiveness and performance evaluation of the proposed design. The computer codes were developed using the Matlab

When selecting extended Kalman filter as the navigation state estimator in the GPS receiver, using

where _{1}, _{3}, _{5} represent the east, north, and vertical position; _{2}, _{4}, _{6} represent the east, north, and vertical velocity; and _{7} and _{8} represent the receiver clock offset and drift errors, respectively. The process noise covariance matrix is as follows:

where

If only the pseudorange observables are available, the linearized measurement equation based on

where ^{2}. Let each of the white-noise spectral amplitudes that drive the random walk position states be

The scenarios involved in the numerical experiments cover two aspects. The first one deals with performance comparison for VTL and STL architectures for various numbers of visible satellites. The second one deals with reliability enhancement when the RAIM and FDE mechanisms are incorporated into the VTL.

In the first part of experiment, performance comparison for VTL- and STL-based solutions is presented. Three examples, with good or bad geometry involved, are given to illustrate the effectiveness of the VTL architecture.

In the first example, it is assumed that all the GPS signals are in good condition. There are totally 9 GPS signals available in the open sky.

The second and third examples present the performance comparison in the case of signal blockage. Initially there are only 5 satellites visible, where some of the GPS signals are intentionally blocked out at some time intervals. In this example, we consider one signal is blocked out at certain time interval.

Channel | SV ID | Time interval (s) | |||
---|---|---|---|---|---|

[10–25] | [100–110] | [330–340] | [440–450] | ||

1 | 3 | ||||

2 | 6 | ||||

3 | 7 | ✓ | |||

4 | 9 | ✓ | ✓ | ||

5 | 21 | ✓ |

In this example, it is assumed that the number of visible satellites has been reduced from 5 to 3 at some time intervals. Same as in Example 2, there are initially 5 satellites visible. However, 2 GPS signals are blocked out simultaneously at some time intervals.

Channel | SV ID | Time interval (s) | |||
---|---|---|---|---|---|

[10–25] | [100–110] | [330–340] | [440–450] | ||

1 | 3 | ✓ | ✓ | ||

2 | 6 | ✓ | |||

3 | 7 | ✓ | |||

4 | 9 | ✓ | ✓ | ✓ | |

5 | 21 | ✓ |

In the second part of experiment, reliability enhancement for VTL using the FDE mechanism is presented. Two examples are given for illustration. It is assumed that there are 9 GPS signals available, but some fault signals occur at certain time interval.

The abnormal signals corrupted by bias errors are assumed to occur at the following time intervals: 10–20 s, 220–230 s, 300–310 s, 370–380 s, and 440–450 s, as summarized in

Channel | SV ID | Time interval (s) | ||||
---|---|---|---|---|---|---|

[10–20] | [220–230] | [300–310] | [370–380] | [440–450] | ||

1 | 3 | ✓ | ||||

3 | 6 | ✓ | ||||

5 | 9 | ✓ | ||||

7 | 18 | ✓ | ||||

9 | 21 | ✓ |

The second example investigates the case when 3 abnormal signals occur simultaneously.

Channel | SV ID | Time interval (s) | ||||
---|---|---|---|---|---|---|

[10–20] | [220–230] | [300–310] | [370–380] | [440–450] | ||

1 | 3 | ✓ | ✓ | ✓ | ||

3 | 6 | ✓ | ✓ | ✓ | ||

5 | 9 | ✓ | ✓ | ✓ | ||

7 | 18 | ✓ | ✓ | ✓ | ||

9 | 21 | ✓ | ✓ | ✓ |

The integrity monitoring algorithms in this work is implemented dealing with the reliability enhancement of the tracking loops. Navigation system integrity refers to the ability of the system to provide timely warning to users when the system should not be used for navigation. The most significant drawback in the VTL is that the failure of tracking in one channel may affect the entire system and lead to loss of lock on all satellites. The scenarios involved in the numerical experiments cover two aspects. The first aspect deals with performance comparison for VTL- and STL-based architectures for various numbers of visible satellites. The second one deals with reliability enhancement when the RAIM and FDE mechanisms are incorporated into the VTL.

The RAIM and the FDE mechanisms have been incorporated into the vector tracking loop architecture where the RAIM mechanism is used to check the possible fault in the pseudorange and the pseudorange rate, and the FDE mechanism is employed for excluding the wrong satellite signal. When the FDE algorithm is incorporated, the vector tracking loop can prevent the failure of one channel from spreading into the entire tracking loop. The feasibility of the proposed approach has been demonstrated for various scenarios. Performance evaluation for the VTL with FDE has been presented. The reliability enhancement for the vector tracking loop has been demonstrated.