Non-Linear Modeling of Pseudorange Errors for ...

Non-Linear Modeling of Pseudorange Errors for Centralized Non-Linear Multi-Sensor System Integration Umar Iqbal1,3, Jacques Georgy2, Walid F. Abdelfatah2, Mohammad Tamazin1, Michael Korenberg1, and Aboelmagd Noureldin3,1

1 Electrical and Computer Engineering Department, Queen’s University, Canada, 2 Trusted Positioning Inc., Canada 3 Electrical and Computer Engineering Department, Royal Military College of Canada GNSS signal processing, software receiver design and integrated navigation systems.

BIOGRAPHY Dr. Umar Iqbal is Adjunct Assistant Professor at the Department of Electrical and Computer Engineering (ECE) of Queen’s University. He is also working as Postdoctoral Fellow at Navigation and Instrumentation research group at Royal Military College (RMC) of Canada. His research focuses on the development and improvement of integrated positioning and navigational technologies, and other multisensor systems. He holds Ph.D. degree in ECE (2012) from Queen’s University and Masters of ECE (2008) from RMC. He also holds M.Sc. of Electronics Engineering (2004) from GIK Institute and B.Sc. in ECE (1993) from University of Engineering and Technology, Lahore, Pakistan. He is serving as Secretary of IEEE Kingston Section and Member of Senate Committee on Academic Development at Queen's University.

Dr. Michael Korenberg is a Professor in the Department of Electrical and Computer Engineering at Queen’s University, Kingston, Ontario, Canada. He holds an M.Sc. (Mathematics) and a Ph.D. (Electrical Engineering) from McGill University, Montreal, Quebec, Canada, and has published extensively in the areas of nonlinear system identification and time-series analysis. Dr. Aboelmagd Noureldin is Cross-Appointment Professor at the Departments of Electrical and Computer Engineering of both Queen’s University and the Royal Military College (RMC) of Canada. He is also the founder and the leader of the Navigation and Instrumentation research group at RMC. His research is related to artificial intelligence, digital signal processing, spectral estimation and de-noising, wavelet multi-resolution analysis and adaptive filtering with emphasis on their applications in mobile multisensor system integration for navigation and positioning technologies. Dr. Noureldin holds B.Sc. degree in Electrical Engineering (1993) and M.Sc. degree in Engineering Physics (1997) from Cairo University, Giza, Egypt. In addition, he holds Ph.D. degree in Electrical and Computer Engineering (2002) from The University of Calgary, Alberta, Canada.

Dr. Jacques Georgy is the VP of Research and Development at Trusted Positioning Inc., Calgary, AB, Canada. He received his Ph.D. degree in Electrical and Computer Engineering from Queen’s University, Canada in 2010. He received the B.Sc. and M.Sc. degrees in Computer and Systems Engineering from Ain Shams University, Egypt, in 2001 and 2007, respectively. He is working in positioning and navigation systems for portable, vehicular, and machinery applications. His research interests include linear and nonlinear state estimation, positioning and navigation systems, autonomous mobile robot navigation, and underwater target tracking.

ABSTRACT GPS signal interruption is a primary cause which affects the continuity and reliability of navigation systems that rely on standalone GPS solution. In order to obtain continuous positioning services in all environments, GPS can be integrated with inertial sensors. Micro-Electro-Mechanical-System (MEMS) based inertial sensors are preferred for vehicular navigation due to their low cost, small size, and low power consumption. However, these sensors suffer from severe stochastic error characteristics that can cause large positional error growth. To enhance the performance of vehicular navigation along with reducing the reliance on inertial sensors, three-dimensional reduced inertial sensors system (RISS) relies on measurements from one gyroscope, two accelerometers and a speed sensor (i.e. speed measurements in a vehicle are obtained from the vehicle’s odometer) which are integrated with measurements from a GPS receiver. To address the non-

Walid F. Abdelfatah is the Lead Systems Designer in Trusted Positioning Inc. He received his M.A.Sc. degree in Electrical and Computer Engineering from Queen’s University in 2010 and his B.Sc. degree in 2007 from the Department of Computer and Systems Engineering at Ain Shams University, Egypt. His work in Trusted Positioning Inc. is focused on embedded navigation systems design and navigation algorithms realization. Mohamed Tamazin is a PhD candidate in the Navigation and Instrumentation (NavINST) group of the Department of Electrical and Computer Engineering in Queen’s University, Ontario, Canada. He holds M.Sc. degree in Geomatics Engineering from University of Calgary, Alberta, Canada. In addition, He holds B.Sc. and M.Sc. in Electrical Engineering from Arab Academy for Science and Technology, Alexandria, Egypt. His research interests in the fields of

Footer: The footer should be left align at the left bottom page. In Times New Romans, 9 point size (footer below) 25th International Technical Meeting of the Satellite Division of The Institute of Navigation, Nashville TN, September 17-21, 2012

494

satellites) with good geometry [1-3]. The most recognized operational GNSS is the Global Positioning System (GPS) and the number of users of GPS is growing exponentially due to its applications for land transportation, and the fact that its basic navigational services are free-of-charge for civilians world-wide. Advanced technologies have empowered the manufacturing of compact, inexpensive, and low power consumption receivers providing incentives for innovative applications in numerous domains. However, the signals transmitted by the GPS satellites can suffer frequent interference and signal blockage in urban canyons and thick foliage where an uninterrupted clear view of the sky is not feasible for the receiver causing intermittent or erroneous determination of the position and velocity navigation states [4-6].

linear error characteristics in the integrated navigation system, Mixture Particle filter (M-PF), a nonlinear filtering technique is employed to perform tightly-coupled integration of RISS with GPS. Tightly-coupled systems can provide GPS aiding during limited GPS satellite availability and thus can improve the operation and performance of the navigation system. Tightly-coupled integration utilizes pseudoranges and pseudorange rates measured by the GPS receiver. The accuracy of the positional estimates is highly dependent on the accuracy of the range measurements. This paper introduces a nonlinear system identification technique called Parallel Cascade Identification (PCI) to model pseudoranges correlated errors in the measurement model of the PF filter. When less than four satellites are visible, the identified parallel cascades for the remaining visible satellites will be used to predict the pseudorange errors for these respective satellites and correct the pseudorange value to be provided to M-PF. The proposed technique enhances the overall positioning accuracy, especially during partial GPS outages by modelling the pseudoranges correlated errors using PCI and MEMSbased RISS/GPS integration using M-PF to compute a 3D navigation solution.

To provide continuously accurate and reliable full navigational states (i.e. position, velocity and attitude), the GPS has to be combined with other complementary navigational sensors such as inertial sensors and an odometer [5]. By integrating the GPS with these motion sensors, an always available solution can be obtained that is often more accurate than that of GPS alone. Different manufacturers provide solutions, which fulfill the requirements of continuity, reliability, and availability for an exclusive class of users, as they are ready to spend a hefty amount of money. However, for the average consumer, only smaller, cheaper, and lower power solutions can be feasible. Low cost Micro-ElectroMechanical Systems (MEMS) based inertial sensors fulfill the requirements of common users, but MEMS sensors have composite error characteristics that need to be accounted for before the data is made useful. Full inertial measurement units (IMUs), commonly used in integrated navigation solutions, are composed of three accelerometers and three gyroscopes [4]. To decrease the reliance on MEMS sensors and decrease their error contribution in the integrated navigation system, 3D reduced inertial sensor system (RISS) which is based on only one single axis gyroscope, two accelerometers together with the vehicle’s odometer and integrated with GPS, was proposed earlier, to provide a full 3D land vehicle navigation solution [7]. To further enhance the aiding of the RISS based system, tightly-coupled integration was proposed, which could provide GPS aiding for a navigation solution when the number of visible satellites is three or less, eliminating the foremost requirement of loosely-coupled integration, which is the visibility of at least four satellites [8].

The effectiveness of the proposed algorithm is verified by different real-life road tests during GPS signal degradation and blockage using low cost MEMS-based RISS. Results for scenarios with intentionally inserted GPS outages where the number of visible satellites varies between 3, 2, 1, and 0 are presented to demonstrate the usefulness and performance of the proposed technique. The results are examined and compared with PF tightlycoupled RISS/GPS integration without the pseudorange corrections using the PCI modules. The results are also compared with KF tightly-coupled RISS/GPS integration with and without pseudorange corrections using PCI modules. The comparison demonstrates the advantages of using PCI modules for correcting pseudoranges for tightly-coupled integration using PF. The proposed navigation system, including the PCI modules used with PF, provides continuous determination of the navigational states with higher performance along with considerable reduction in the cost, size, and power consumption due to the use of fewer low-cost MEMSbased inertial sensors, vehicle odometer, and a low-cost GPS receiver. These features make the proposed system a viable product that can provide an accurate, always available positioning solution for the common consumer. 1

Particle filter (PF), a nonlinear filtering technique, is utilized in this paper to perform tightly coupled integration of 3D RISS with GPS to avoid the linearization errors that result from Kalman Filter (KF), the conventional technique used for data fusion. An enhanced version of PF, called Mixture PF, is employed for tightly-coupled RISS/GPS integration utilizing

INTRODUCTION

The determination of accurate and reliable navigational states without interruptions is an important task for many applications. Global Navigation Satellite Systems (GNSS) has made the determination of position and velocity feasible for outdoor scenarios by making range measurements to GNSS satellites (at least four

Footer: The footer should be left align at the left bottom page. In Times New Romans, 9 point size (footer below) 25th International Technical Meeting of the Satellite Division of The Institute of Navigation, Nashville TN, September 17-21, 2012

495

pseudoranges and pseudorange rates measured by the GPS receiver. The accuracy of the position estimates is dependent on the accuracy of the range measurements. The tightly-coupled solutions presented in the literature assume that the pseudorange measurements, after correcting for ionospheric and tropospheric delays, satellite clock errors, and ephemeris errors, only have errors due to the receiver clock errors and white noise. Consequently, the latter two are the only errors modeled inside the measurement model in the tightly-coupled solutions presented in the literature.

case the accelerometers are not utilized. When the vehicle is moving, the forward accelerometer measures the forward vehicle acceleration as well as the component due to gravity. In order to calculate the pitch angle, the vehicle acceleration derived from the odometer measurements is removed from the forward accelerometer measurements. Consequently, the pitch angle is computed as:  f y  aod  (1) p  sin 1    g   Similarly, the transversal accelerometer measures the normal component of the vehicle acceleration as well as the component due to gravity. Thus, to calculate the roll angle, the transversal accelerometer measurement must be compensated for the normal component of acceleration. The roll angle is then given by  f  vod z  (2) r   sin 1  x   g cos p  The single gyroscope aligned with the vertical axis of the vehicle body frame is used together with the pitch and roll information to obtain an accurate azimuth angle in the horizontal East-North plane that is compensated for tilt errors. The computed azimuth, pitch and roll angles allow the transformation of the vehicle’s speed along the forward direction vod (obtained from the odometer measurements) to East, North and Up velocities. Consequently, the latitude, longitude and the altitude of the vehicle are determined yielding a 3D position of the vehicle.

This paper proposes a solution that exploits the fact that PF can accommodate any nonlinear models by incorporating Parallel Cascaded Identification (PCI) models for the correction of residual pseudorange correlated errors in the measurement model used inside the filter. PCI is a nonlinear system identification technique that is used to improve the overall navigation solution by modeling residual pseudorange correlated errors and thus correcting these measurements. In full GPS coverage when four or more satellites are available to the GPS receiver; the PF integrated solution provides adequate position benefiting from both GPS and RISS measurements, and the PCI works on building models for the pseudorange errors for each visible satellite. The measured pseudorange of each visible satellite is used as an input to the model; the target output is the error between the corrected pseudorange (i.e. calculated based on corrected receiver position from the integrated solution) and the measured pseudorange. This target output provides the reference output to build the PCI model of the pseudorange residual errors.

3

PF is the estimation technique used for tightly coupled RISS/GPS integration in this paper.

During partial GPS outages (i.e. characterized by the availability of less than four satellite signals), each PCI model, built during full GPS coverage, estimates the correction for the corresponding pseudorange error. The PCI-corrected GPS pseudoranges of the outlasting visible satellites during the partial outages will be used as measurements in the PF. This improvement in pseudorange measurements result in a more accurate aiding for RISS, and thus more accurate estimates of the navigational states. 2

PARTICLE FILTERING

3.1

PROBLEM STATEMENT

The aim is to estimate the state of the vehicle x k at the current time step k, given a set of measurements (observations) Z k  z 0 ,..., z k  acquired at time steps 0,1,...,k. The nonlinear state transition model (motion model) is

xk  f  xk 1 , uk 1 , w k 1 

RISS/ GPS INTEGRATION

The 2D reduced Inertial Sensor System (RISS) was proposed in [9-11] in order to reduce the cost of the overall positioning system for land vehicles without appreciable performance compromise depending on the fact that land vehicles mostly stay in the horizontal plane. The gyroscopes technology is what mostly contributes to both the overall cost of an IMU and the performance of the INS. Mixture PF for loosely coupled 2D RISS/GPS integration was proposed in [11]. The two horizontal accelerometers can be employed for obtaining the pitch and roll angles of the vehicle, and a 3D navigation solution can be achieved [7] instead of a 2D solution in

where u k is the control input which is coming from the inertial sensors, and w k is the process noise which is independent of the past and present states and it accounts for the uncertainty in vehicle motion and INS readings. The state measurement model is

z k  h  xk ,  k 

(4)

where  k is the measurement noise which is independent of the past and current states and the process noise and it accounts for uncertainty in observations.

Footer: The footer should be left align at the left bottom page. In Times New Romans, 9 point size (footer below) 25th International Technical Meeting of the Satellite Division of The Institute of Navigation, Nashville TN, September 17-21, 2012

(3)

496

sk (i )   xk (i ) ,  k (i )  denotes the i-th sample where x k (i ) is

The errors and uncertainty in sensor readings and in vehicle motion motivates the use of a probabilistic algorithm for this estimation problem. Probabilistic algorithms for estimation calculate a probability distribution instead of calculating a single best estimate only. The state of the vehicle x k is a vector of stochastic processes, and it is required to get

the vehicle state and  k (i ) is the weight of the sample. To initialize the filter at time k=0, the sample set 1 (where  0(i )  ) is S0   x0(i ) ,  0(i )  i  1,..., N N sampled from p  x0  which encodes any knowledge



p  xk Z k  the

probability density function PDF of the state at each time step k conditioned on the whole set of sensors measurements from beginning to time k.



about the vehicle’s initial state. For INS/GPS integration, the algorithm is initialized with samples drawn from a Gaussian density with mean around the GPS position and velocity.

This state estimation problem from sensor observations is an instance of the Bayesian filtering problem because the states are unobserved Markov process and the observations can be assumed independent given the states. Bayesian filtering has a recursive solution that uses Bayes theorem, and it yields multidimensional integrals. According to [12], it is impossible to evaluate this recursive solution analytically except in a few special cases which includes linear Gaussian state space models (using Kalman Filter). For nonlinear nonGaussian systems these multi-dimensional integrals are intractable and approximate solutions are needed.

Each iteration of the SIR algorithm consists of three steps: 1. The prediction phase: starting from the set of samples 1 Sk 1   xk 1(i ) ,  k 1(i )  i  1,..., N (where  k 1(i )  ) of N the previous iteration, the motion model is applied to each 1  one sample sample sk 1(i )   x k 1(i ) ,  and N   1  sk (i )   xk (i ) ,  is drawn from p xk xk 1i  , u k 1 which N  is fully specified by the system model (Equation 1) and the process noise PDF p  w k  . A new sample set S k is







One of these solutions is the Extended Kalman Filter (EKF), which is the traditional method to integrate INS and GPS. EKF linearizes the system and measurement models. As mentioned in [13], while Kalman Filter gives optimal solution for linear Gaussian models, it becomes suboptimal when these assumptions are violated like the problem at hand.



obtained that approximates density p  xk Z k 1  .

Another solution that avoids the linearization of the models is to handle the multidimensional integrals numerically using Monte Carlo integration method. The solution to the Bayesian filtering problem leads to the sequential Monte Carlo methods (PF) [12] which propagates the probability density in the form of a set of random samples or particles.

the

predictive

2. The update phase: the measurement z k is taken into account and each of the samples in S k is weighted



using the observation likelihood p z k xk   i



which is

fully specified by the measurement model (Equation 2) and the measurement noise PDF p νk  , then all weights are

When using KF, the model needs to be linearized which leads to error-state approach where the KF estimates the error in the navigation states not the states themselves. The system model used by KF is the dynamic error model which is a linearized model. Also there is a separate INS mechanization used. On the other hand, the approach used in this paper is a total-state approach not an error-state approach as there is no need for linearization. Also, the method does not require a separate mechanization.

normalized.

The

weighted

sample

set

approximates the density p  xk Z k  . 3.

Sk 

The

 x

(i ) k

obtained

resampling

,  k (i )  i  1,..., N by

resampling

step:



the

sample set 1 (where  k (i )  ) is N from the weighted set

 x ,  i  1,..., N such that p  x    x        . The Sk 

(i )

k

(i )

k

i

k

3.2 THE SAMPLE IMPORTANCE RESAMPLING (SIR) PARTICLE FILTER

Sk

j

k

j

k

obtained

Sk

approximates the density p  xk Z k  .

The presented description for SIR is according to [12, 14-16]. At time k, the PDF p  xk Z k  is represented by a

3.3

set of N random samples or (1) (N ) Sk  sk ,..., sk  distributed according

This modified version of particle filtering was first reported in the area of robotics in [16, 17] and had further elaboration in [18]. In Robotics, the SIR Particle filter

particles to it.

MIXTURE PARTICLE FILTER

Footer: The footer should be left align at the left bottom page. In Times New Romans, 9 point size (footer below) 25th International Technical Meeting of the Satellite Division of The Institute of Navigation, Nashville TN, September 17-21, 2012

497

used for mobile robot localization is called Monte Carlo Localization (MCL), and this modified version is called MCL with planned sampling [16] or Mixture MCL [18].

technique models the input/output behavior of a nonlinear system by a sum of parallel cascades of alternating dynamic linear (L) and static nonlinear (N) elements. The parallel array can be built up one cascade at a time [21].

In the prediction phase, the SIR Particle filter samples





from the importance density p xk xk 1  , u k 1 , which i

Palm [22] proved that any discrete-time Volterra series with finite memory and anticipation can be uniformly approximated by a finite sum of parallel LNL cascades, where the static nonlinearities N are exponentials and logarithmic functions. Korenberg [21] showed that any discrete-time doubly finite (finite memory and order) Volterra series can be exactly represented by a finite sum of LN cascades where the N are polynomials. A key advantage of this technique is that it is not dependent on a white or Gaussian input, but the identified individual L and N elements may vary depending on the statistical properties of the input chosen [21]. The cascades can be found one at a time and nonlinearities in the models are localized in static functions. This reduces the computation as higher order nonlinearities are approximated using higher degree polynomials in the cascades rather than higher order kernels in a Volterra series approximation.

does not depend on the last observation. So, the samples are predicted from the motion model, and then the most recent observation is used to adjust the importance weights of this prediction. The idea used in this enhancement to particle filtering is to add to those samples predicted from the motion model some samples predicted from the most recent observation [18]. The importance weights of these new samples are adjusted according to the probability samples of the last iteration and the latest vehicle motion. These new samples are called planned samples [16] or samples generated from the dual of MCL [18]. The planned samples are drawn from the importance density p  z k xk  which is the observation likelihood. These samples are consistent with the most recent observation but ignorant of the previous belief about the vehicle state p  xk 1 Z k 1  and the motion u k 1 . The



The method begins by approximating the nonlinear system by a first such cascade. The residual (i.e., the difference between the system and the cascade outputs) is treated as the output of a new nonlinear system, and a second cascade is found to approximate the latter system, and thus the parallel array can be built up one cascade at a time. Let yk(n) be the residual after fitting the k-th cascade, so yo(n) = y(n). Let zk(n) be the output of the k-th cascade, so



samples are weighted using p xk   xk 1  , u k 1 . The i

i

version of PF that uses this type of sampling alone is known as likelihood PF [19]. In the version of PF used in this research and as described in [18], a number of samples (a suitably chosen ratio from the total number of samples) are drawn from p  z k xk  and added to the samples drawn



yk (n)  yk 1 (n)  zk (n) where k=1,2,….



The dynamic linear elements in the cascades can be determined in a number of ways [21, 23]. The method employed in this paper uses cross correlations of the input with the current residual. Best fitting of current residual were used to find the polynomial coefficients of the static nonlinearities. These resulting cascades are such as to drive the cross correlations of the input with the residual to zero [21, 23]. However, a few basic parameters have to be specified in order to identify a parallel cascade model, as follows:

from p xk xk 1  , u k 1 . Samples in these two groups are i

weighted with their respective weight update equations, and then the resampling is achieved. According to [18], these two importance densities have complimentary advantages and disadvantages, so their combination gives better performance. This version of PF will be called Mixture Particle filter after the name used in [18], because it samples from a mixture of importance densities. Mixture PF also enables the use of a much smaller number of samples as compared to SIR PF. 4

(1) The memory length of the dynamic linear element that begins each cascade: this can be seen from the following equation

PARALLEL CASCADE IDENTIFICATION

System identification is an effort to infer the dynamic characteristics between system input and output from an analysis of time-varying input-output data. Most of the techniques assume linearity due to the simplicity of analysis as nonlinear techniques make analysis much more complicated and difficult than for the linear case. However, for more realistic dynamic characterization, nonlinear techniques are suggested. Parallel cascade identification (PCI) is a nonlinear system identification technique proposed by M.J. Korenberg [20]. This

R

uk ( n)   g k ( j ) x ( n  j )

(6)

j 0

where the linear element’s output uk (n) depends on input values x(j), x(j-1), ............ x(j-R) so its memory length is R+1, and g k ( j ) , is the impulse response of the linear element. (2) The degree D of the polynomial static nonlinearity

Footer: The footer should be left align at the left bottom page. In Times New Romans, 9 point size (footer below) 25th International Technical Meeting of the Satellite Division of The Institute of Navigation, Nashville TN, September 17-21, 2012

(5)

498

that follows the linear element: this polynomial is best fit to minimize the MSE of the approximation of the residual.

where

frame, and xm  [ xm , y m , z m ]T is the position of the mth satellite in ECEF frame

(3) The maximum number of cascades allowable in the model;

In vector form, equation (9) can be written as:

cm  x  xm  br   m

(4) A threshold based on a standard correlation test for determining whether a cascade’s reduction of the meansquare error (MSE) justifies its addition to the model.

4 zk2 (n)  yk21 (n) T  R 1

(7)

the candidate cascade’s output, and yk21 (n) denotes the mean square of the current residual, i.e., the residual remaining from the cascades already present in the model.

The receiver clock error and the residual errors assumed as white Gaussian noise are the only errors modeled inside the measurement model in the tightlycoupled solutions presented in the literature. Experimental investigation of the GPS pseudoranges in trajectories in different areas and scenarios showed that the residual errors are not just white noise as assumed in the literature, but they are correlated errors. As the GPS observables are used to update the PF, a technique must be developed to adequately model these errors to improve the overall performance of PF. This research proposes using PCI to model these correlated errors. The PCI module models these errors, and then provides correction prior to sending the GPS pseudoranges to aid PF during periods of GPS partial outages.

NONLINEAR MODELS FOR TIGHTLY COUPLED INTEGRATION

In tightly coupled RISS/GPS system architecture instead of using the position and velocity solution from the GPS receiver as input for the fusion algorithm, raw GPS observations such as pseudoranges and Doppler shifts are used. The range measurement is usually known as a pseudorange due to the contamination of errors as well as synchronization errors between the satellite and receiver clocks. The pseudorange measurement for the mth satellite can be written as [2,3].

 m  r m  c tr  c ts  cI m  cT m   m

(8)

where  m : is the mth satellite to receiver measured

6

AUGMENTED PF-PCI FOR INTEGRATED RISS/GPS In section 4, the parallel cascade model was briefly presented, together with a simple method for building up the model to approximate the behavior of a dynamic nonlinear system, given only its input and output. In order to apply PCI to 3D RISS /GPS integration, this paper proposes an M-PF-PCI module where the role of PCI is to model the residual errors of GPS pseudoranges.

th

m

pseudorange (meters), r : the true range between m satellite and receiver (meters), c : speed of light (meters/second),  tr : receiver clock bias (second),  ts : satellite clock bias (second), I m : ionospheric delay from mth satellite (second), T m : tropospheric delay from mth satellite (second),  m : error in range mainly due to receiver noise (meters).

In full GPS coverage when 4 or more satellites are available to the GPS receiver, the PF integrated solution provides adequate position benefiting from both GPS and RISS readings, and the PCI builds the model for the pseudoranges errors for each visible satellite. The input of each PCI module is the pseudorange of the visible mth m GPS satellite ( GPS ), and the reference output  Rm is the

After compensating for satellite clock bias, Ionospheric and Tropospheric errors, we can write the corrected pseudorange as [2],

cm  r m  c tr   m

(9)

where  m is the total effect of residual errors.

difference of GPS observed pseudorange (  Rm ) compared with of the estimated mth pseudorange from the m corrected navigation solution (  Nav  Sol ) to derive the residual error of pseudorange as given in Equation (12).

The geometric range from mth satellite to receiver is

r m  ( x  xm )2  ( y  y m )2  ( z  z m )2  x  xm

(10)

m m Rm   Nav  Sol  GPS

Footer: The footer should be left align at the left bottom page. In Times New Romans, 9 point size (footer below) 25th International Technical Meeting of the Satellite Division of The Institute of Navigation, Nashville TN, September 17-21, 2012

(11)

where br  c tr is the error in range (in meters) due to receiver clock bias. This equation is nonlinear, the traditional techniques relying on KF used to linearize these equations about the pseudorange estimate got from the inertial sensors mechanization, and the details of this operation are described in [24]. Since PF is used in this paper and it can accommodate nonlinear models, there is no need for linearizing this equation [25].

In Equations (10), zk2 (n) denotes the mean square of

5

x  [ x, y, z ]T is the receiver position in ECEF

499

(12)

The vehicle’s forward speed readings were obtained through vehicle built-in sensors via the on-board diagnostics version II (OBD-II) interface using a device called CarChip. The specifications of this device are described in [27], and more details about OBD-II speed readings can be found in [28]. The sample rate for the collection of speed readings was 1 Hz. To demonstrate the performance of the proposed solution GPS receiver used was the NovAtel OEMV-1G [29]. The results were evaluated with respect to a reference solution made by NovAtel, where Honeywell HG1700 tactical grade IMU [30] was integrated with the NovAtel OEM4 GPS receiver[31]. Together the NovAtel and Honeywell systems were integrated with an off-the-shelf unit developed by NovAtel, the G2 Pro-Pack SPAN unit [32]. The NovAtel system provided the reference solution to validate the proposed method and to examine the overall performance during simulated GPS outages.

The reference output  Rm does not apply any correction during full GPS coverage. It is only used as a reference output to build the PCI model. Dynamic characteristics between system input and output help to achieve a residual pseudorange error model. During partial GPS coverage, when there are less than 4 satellites available, the PCI modules for all satellites cease training, and the available PCI model for each visible satellite will be used to predict the corresponding residual pseudorange errors. The PF operates as usual, but in this instance, the GPS observed pseudorange is corrected by the output of the corresponding PCI. The pre-built PCI models only for the visible satellites during the partial outage predict the corresponding residual pseudorange errors  Rm to obtain a correction. Thus, the m can then be obtained as corrected pseudorange Corrected given in Equation (12): m m Corrected  GPS  Rm

Table 1: Crossbow IMU specifications [26]. Accelerometer

(13) Range

During a full GPS outage, when no satellites are available, PF operates in prediction mode and the PCI modules neither provide correction nor operate in training modes. 7

±2g

Bias

± 30 mg

Scale Factor