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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

The paper investigates approaches for loosely coupled GPS/INS integration. Error performance is calculated using a reference trajectory. A performance improvement can be obtained by exploiting additional map information (for example, a road boundary). A constrained solution has been developed and its performance compared with an unconstrained one. The case of GPS outages is also investigated showing how a Kalman filter that operates on the last received GPS position and velocity measurements provides a performance benefit. Results are obtained by means of simulation studies and real data.

The error in an inertial system grows very quickly over time even if when an initial calibration procedure has been performed. Position error bias on startup also significantly affects position error over time. In fact, an initial calibration can correct short term errors only and the position error can become unacceptable after a very short period of time. In order to mitigate the error of inertial devices another sensor can be used in cooperation with the Inertial Measurement Unit (IMU). Typically, an absolute precise position estimate from a GPS receiver can be used to reset an Inertial Navigation System's (INS) solution or may be integrated with it by applying a data fusion algorithm (e.g., Kalman filter). The benefits of a GPS/INS integration are that the INS estimates can be corrected by the GPS data and that the INS can provide position and angle updates at a quicker rate than GPS. For highly dynamic vehicles such as missiles and aircraft, INS navigation solutions can interpolate between the GPS updates. Additionally, GPS signal losses may occur and the INS can continue to calculate position, velocity and orientation angles during outages. The two systems are complementary and are often employed together. Several approaches are possible for the integration of GPS and INS to provide a combined navigation solution. Such integration strategies differ on the type of information that is shared between the systems. There are four different categories of integration approaches: un-coupled [

In this work we address a loosely coupled approach. The paper investigates the performance of such an integration using both simulated and real measurements. For real tests we have used a Sirf-JP13 [

The paper is organized as follows: Section 2 describes the main characteristics of the INS mechanization equations and it provides the main features of the Kalman filter that will be applied in a loosely coupled integration. Results obtained through simulations and in a real scenario are presented. Section 3 deals with the improvements obtained by using known information to constrain the solution, such as the boundaries of the road along which the system is travelling, provided by an external aiding source such as Google Map. A description of the Kalman filter designed to include constraints information is also given. Performance comparisons are presented for an unconstrained loosely coupled system using simulated and real data.

Section 4 demonstrates the performance of a loosely coupled system in case of a 50-s-long GPS outage. An approach that uses a Kalman filter to reduce INS position error has been developed by exploiting the last available GPS position and velocity. Eventually, conclusions are drawn in Section 5.

LC integration combines estimates from GPS and INS outputs, and such integration might be basically performed in two different ways. The first one, referred as open loop, estimates INS error by exploiting GPS information, and does not interfere with the operation of INS. The second approach, named closed loop, involves the use of a Kalman filter to mitigate INS errors.

In our work we have employed the second method where a Kalman filter calculates position and velocity error states to correct the INS solution. The block diagram of the closed-loop LC solution is shown in

Independent position and velocity estimates are calculated within a GPS receiver and are optionally filtered. Then, the output of this filter is used periodically as input to an INS filter. The second Kalman filter uses the difference between the GPS-derived positions, velocities and the ones computed by means of an INS device to get the error estimates.

The design of the Kalman filter for loosely coupled integration is described later in this Section. An INS filter generally consists of nine navigation error states, including three positions Δ^{n}^{n}^{n}

The increased number of the error states could include bias error estimations both of the INS gyros and accelerometers (_{g}_{a}_{g}_{a}

_{a}_{g}

_{a}_{g}

_{a}_{g}

In the paper we have modeled the accelerometers and gyroscopes' noises as white noise components (_{a}_{g}^{®} able to generate accelerometers and gyros raw measurements at different rates (e.g., sampling frequency at 500 or 100 Hz) and with different noise components. Furthermore, the simulator can provide information about the orientation angles (yaw, pitch and roll) of a vehicle that is moving along a trajectory. As far as the simulated path is concerned, we have implemented the same surveyed track of the real scenario in which the tests have performed, giving us the possibility to do accurate comparison between synthetic data and the real ones. In the case of real data we have utilized measurements from a 3DM-Gx2 MEMS-IMU. It has been argued in [

The perturbation of the inertial navigation equations to obtain error states is detailed in [

X^{n} horizontal axis in the direction of increasing latitude;

Y^{n} horizontal axis in the direction of increasing longitude;

Z^{n} to make a right-handed orthogonal coordinate system.

In the following we will refer to the NED coordinate system as n-frame.

The position in the n-frame is expressed in geodetic coordinates, namely

The velocities in the n-frame are given by:
_{N}_{E}_{D}

The motion of a vehicle can be described by equations that involve INS kinematics. The derivations of these equations can be broken up into three parts: position, velocity and attitude. A full derivation is reported in [^{−1} is a diagonal matrix defined as follows:
_{M} is the radius of curvature in the meridian and R_{N} is the prime vertical at certain latitude expressed as:
^{b}

A more detailed explanation of the previous equations can be found in [

_{e}^{5} rad/s) is the magnitude of the Earth rotation rate.

^{n}

Eventually, the equations describing the error dynamics are obtained by perturbing the kinematic

The linearized position error is given by:

The velocity error is given by:

The attitude error can be written as:

In this subsection we recall the traditional design of an error state Kalman filter for a loosely coupled GPS/INS application. An n-frame error state model [_{xx}^{n}

The covariance matrix _{K}_{a}_{g}

In a loosely-coupled integration approach, the filter measurement is the difference between the INS and the GPS navigation solutions. The measurement vector is given by:

Following the approach suggested in [_{M}_{N}

The design matrix becomes:

Finally, the measurement noise covariance matrix is calculated as:

The value of each element in the diagonal matrix R depends on the accuracy of the GPS estimates. A detailed description of Kalman filtering combining models with measurements is reported in [

Tests were carried out along a surveyed track located within CSIRO's site in Pullenvale, QLD, Australia. A 3-D plot of the test track and is shown in

We investigated the algorithm's performance using both simulated data and real data. In the first case we have generated synthetic gyros and accelerometers values along a path that perfectly matches the realistic scenario. This simulation will be used as a term of comparison with the results we obtained by working with real GPS and INS measurements on the field.

We have also corrupted the simulated data, representing the inertial accelerometer and gyro, with noise sources resembling the typical characteristics of a low-cost MEMS IMU, as in

Simulated GPS positions and velocities' estimates were also generated, and the variances of those measurements were set equal to the values reported in

The variances of

We have repeated the test, by using this time, data directly read from the GPS and the IMU sensors:

raw gyro and accelerometer outputs from a 3DM-Gx2 INS after an initial calibration,

positions and velocities' estimates from a Sirf JP13- Falcom GPS receiver.

In our LC GPS/INS implementation, the process noise covariance, Q, was estimated through the Allan Variance technique which is detailed in [

The results of such a GPS/INS hybridization technique are plotted in

It is clear from

It is well known that GPS estimation is affected by a certain error that is strongly dependent on the number of satellites available for the PVT estimation. Thus, as a consequence, the LC solution trusts the GPS position and velocity estimates in a way proportional to number of satellites in view. Therefore, the overall performance of the LC solution will leverage on the availability of the GPS updates. In order to improve the performance with respect to the results shown in

As far as the accuracy of GM is concerned, the last improvements in terms of position precision developed by Google on GE and GM can be found in [

This means the GM can not be used for systems that require a high level of accuracy but it could be fine for mass-market applications. Another concern of using a Google Maps service is the process of receiving the map information should be quick enough to be applied in real-time and the internet connection should always be available for all the duration of the test. If this happens we can build up a more complex LC scheme that also integrates street constraints to estimate the user's position and velocity.

In

The offset in the position in decimal degrees is obtained as

Constraint information can be added by increasing the dimensions of the output mapping matrix

As previously explained also in this case constraint information are computed in LLH coordinate system. The variances of the map measurements need to be well-selected so that the resulting position estimates fall within the constraint boundaries.

The position estimates calculated from simulated data are shown in

It can be seen from

The same approach has been used with Google-Maps-sourced altitude information. In this case we have chosen a noise variance of 4 m^{2} for the height constrained filter. The calculated altitude position errors using simulated GPS and INS measurements are shown in

The mean error without additional altitude information is 3.69 m, whereas with constraints the mean error is 1.05 m, thus yielding an improvement of about 2.5 m.

The powerful synergy between the GPS and INS makes the combination of these two navigation technologies a viable position option. GPS, when combined with MEMS inertial devices, can restrict their error growth over time and allows for online estimation of the sensor errors, while the inertial devices can bridge the position estimates when there is no GPS signal reception. Generally speaking, during GPS outages the LC position estimates follow INS-only navigation solution [

As shown in

We have designed a Kalman filter that, by propagating the last GPS information available, corrects the position and velocity estimation as computed by the INS navigation equation only. A block diagram that depicts this strategy is shown in

As we mentioned before, we exploit the last received GPS position information, namely

The covariance matrix _{K}

The measurement vector

The output mapping matrix is given by:

The noise covariance corresponding to (28) is:

The time-varying measurement noise covariance results in a time-varying gain. At the beginning of a GPS outage, the GPS estimates are weighted more than the INS outputs. As long as the outage time increases, such weight is decreased. An example of with a linear trend over time is depicted in

The performance resulting from the use of a time-varying measurement noise covariance yields is shown in

It can be seen from

For the real scenario we have developed a similar Kalman filter that works by using the last available velocity information of GPS. The state-space matrices for the Kalman filter are:

The definition of all the variables in

The G matrix becomes:

For the measurement we can write:

The design matrix

We assumed that the parameter α increases very slowly with time (as the velocity of the user was almost constant) as shown in

We have run again the LC solution after having included the modified noise covariance matrix in the Kalman filter design. In particular,

For example the first method that uses the last user's position information, as provided by the GPS module, gives an error of about 20 m after 45 s of outage with a reduction of almost 30 m with respect a traditional un-constrained LC algorithm that is able to exploit the GPS data only when available and relying on the INS-only navigation during the period of GPS unavailability. The Kalman filter, that exploits the last received update of the GPS velocity and the time-varying weights, provides the best performance with an error after 40 s of only 12 m.

This paper addresses the subject of loosely-coupled GPS/INS integration. In particular, we have designed a nine states Kalman filter that gives a correction to inertial derived position, velocity and Euler angles by exploiting the available GPS measurements. We have demonstrated the performance of this approach using simulated and real measurements.

In order to improve the LC performance, a constrained approach has been described. Use of maps or altitude constraints can provide benefits in the accuracy of the navigation solution. For example, with simulated measurements, the three-dimensional root-mean-square error (latitude, longitude and altitude) has been reduced of 5m. In the case of real measurements, a 2.5 m reduction in altitude error has been observed.

In addition, we have examined the performance of loosely-coupled integration solutions when GPS outages occur. When GPS information is not available we rely on INS estimates. In order to improve the error during outage times we have developed two solutions to improve performance that exploits an adaptive Kalman filter whose measurement's noise covariance varies over time according to the last GPS update. From the results, it is observed that using the last received GPS position and velocity information can lead to decrease the position error between 30 m and 40 m when a 50-s-long GPS outage occurs.

A closed-loop Loosely Coupled GPS/INS integration scheme [

Theoretical Double Integration of different noise sources for a calibrated Gx2IMU.

Scheme of the INS Mechanization equations [

Test track view and 3D plot.

LC performance obtained by Simulation: Position (LLH), Velocity and Euler Angles.

LC performance obtained with real data: Position (LLH), Velocity and Euler Angles.

A closed-loop Loosely Coupled GPS/INS integration scheme with constraint.

Example of constraints on position. The boundaries of the street are shown (red lines) and the user position has been approximated as affected by a white noise (blue).

LC with and without constraints (

LC without constraints (

LC in case of GPS outage- Error in case of outage with LC algorithm.

New approach in case of GPS outages.

K(t) linear profile.

LC in case of GPS outage with weigthening of the last available GPS solution.

α design for Kalman filter.

LC in case of GPS outage with weighting strategies.

Typical characteristics of a low-cost MEMS IMU.

Noise (ARW)
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3 | |

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Noise (VRW)
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1000 |

Simulated measurement variances.

Position [m^{2}] | |

100 | |

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Velocity [m/sec]^{2} | |

10 |

Noise covariance features.

Position [m^{2}] | |

10 | |

| |

Velocity [m/s]^{2} | |

0.2 |