KVH Industries, Inc., has introduced the 1725 Inertial Measurement Unit (IMU) and the 1775 IMU, advanced sensors designed to be integrated into the most demanding stabilization, pointing, and navigation applications. These two new products complement KVH’s successful 1750 IMU and create a complete range of choices for advanced six-degrees-of-freedom (DOF) sensors with enhanced performance. All three products utilize the E•Core ThinFiber technology of KVH’s DSP-1750 fiber-optic gyro (FOG).
“With these three products, system designers and integrators now have a high-performance solution for every application — ranging from manned and unmanned commercial and defense platforms, optical equipment stabilization systems, and pipeline inspection equipment, to autonomous vehicle control and navigation,” said Jay Napoli, KVH’s vice president of FOG/OEM sales. “This line satisfies the performance, size, and price parameters for IMUs in a way that no competitor can match due to KVH’s control over the design and manufacturing process, from creating the fiber to integrating all of the IMU components into the final design. Maintaining complete control of this process, combined with our proprietary technologies, allows KVH to offer a winning combination of innovative solutions, superior quality, and affordable options for nearly every stabilization or guidance application.”
The 1725 IMU features a flexible user interface, with user programmable data output rates from 1 to 1000 Hz. It delivers excellent FOG performance and stability at a price comparable to competitive MEMS-based IMUs. The 1725 IMU is designed for all platforms and navigation or stabilization systems where low cost, high-performance, and high bandwidth are critical for success.
The 1775 IMU is a premium sensor designed to deliver the highest level of performance to meet the demands of platforms requiring superior performance in the most challenging environments. Providing ease of integration for designers of high-level inertial navigation, guidance, or stabilization systems, the 1775 IMU offers a flexible interface with user-programmable data output rates from 1 to 5000 Hz. It includes three axes of magnetometers for automatic gyro bias compensation even in the presence of strong magnetic fields. The 1775 IMU is designed for sophisticated systems and applications where very high bandwidth, low latency, and extreme stability are critical.
Like KVH’s 1750 IMU, introduced in 2012, the 1725 IMU and the 1775 IMU incorporate three axes of KVH’s DSP-1750 FOG, a tiny high-performance FOG integrated with three axes of advanced accelerometer technology. All three IMUs provide excellent shock, vibration, and thermal performance, as well as a compact form factor, KVH said.
KVH controls the entire production process, from creating its own specially designed polarization-maintaining optical fiber to packaging its gyros together in advanced systems for inertial measurement, inertial navigation, and attitude heading reference. As a result, KVH’s open-loop fiber optic gyros offer outstanding accuracy and excellent durability at a lower cost than competing systems, the company said.
The STIM300 is a small, tactical-grade, low-weight, high-performance non-GPS aided IMU. It contains three highly accurate MEMS gyros, three high-stability accelerometers and three inclinometers. The IMU is factory calibrated and compensated over its temperature operating range.
The STIM series is designed for use below and on the ocean, on land, in the air, and in orbit and space. The STIM300 IMU is well suited for stabilization, guidance and navigation applications in the industrial, aerospace and defense markets. It is a crucial building block for inertial navigation systems in UAVs, AUVs, AGVs, UGVs and ROVs, Sensonor said.
The STIM300 is also used for camera turret stabilization and for use in various handheld devices that require a small IMU to secure operations during GPS outage.
NovAtel Inc. has added the IMU-ISA-100C as an inertial measurement unit (IMU) option to its SPAN GNSS+INS line of positioning products. The IMU-ISA-100C is a high-performance, near navigation-grade IMU designed for platform stabilization, general-purpose navigation, photogrammetry, remote sensing, and ground mobile-mapping applications.
Commercially exportable, the IMU-ISA-100C integrates easily with a NovAtel SPAN capable receiver to provide a tightly coupled 3D navigation solution, NovAtel said. Offering customers continuous position, velocity and attitude (roll, pitch and azimuth) measurements, a SPAN system is stable and available even through periods when satellite signals are blocked or unavailable.
With the IMU-ISA-100C, customers will receive near navigation-grade performance, at an affordable price point. The product also features a new enclosure designed to maximize versatility for a range of applications.
“The compelling aspect of this product is its value — it is a high-end SPAN solution, delivering extremely precise positioning, at an attainable price point,” said Siamak Akhlaghi, NovAtel’s product manager, SPAN. “And without the export constraints, it can be shipped internationally in a timely fashion.”
The Geo-iNAV Advanced is a fully integrated GPS-aided inertial navigation system that utilizes KVH’s 1750 IMU to provide a high-performance navigation solution.
KVH Industries, Inc., has entered into a strategic partnership with Geodetics Inc., developer of real-time, high-precision position and navigation solutions. The goal is to provide high-performance positioning and navigation products for commercial applications requiring high levels of precision, from unmanned platforms to terrestrial navigation.
Geodetics is integrating the KVH 1750 inertial measurement unit (IMU) into two solutions: Geo-iNAV Advanced, a GPS-aided inertial navigation system; and Geo-RelNAV, a high-accuracy relative navigation, positioning, and orientation system. The KVH 1750 IMU provides highly accurate 6-degrees-of-freedom angular rate and acceleration data, contributing to the high performance of the Geodetics products while also providing a commercial off-the-shelf (COTS) solution. The COTS designation means the Geo-iNAV Advanced system is available for commercial applications such as manned and unmanned aircraft and control, security platforms on land, air and sea, surface or subsea unmanned vehicles, mobile mapping systems, and photogrammetry and terrestrial navigation.
As reported April 9, NovAtel, Inc., has added the KVH 1750 as an inertial measurement unit (IMU) option in its SPAN GNSS/INS line of positioning products.
“Geodetics evaluated a number of IMU technologies, and based on our desire to address the needs of the commercial marketplace worldwide without sacrificing performance, we chose the KVH 1750 IMU, says Dr. Jeffrey Fayman, vice president, planning and development for Geodetics Inc. “With the integration of the KVH 1750 IMU in Geo-iNAV Advanced, you have the best inertial navigation system Geodetics can provide worldwide.” The navigation, position, and orientation accuracy of the Geo-iNAV Advanced is centimeter level, according to Fayman, thanks in part to the high accuracy of the KVH 1750 IMU.
“KVH is proud to have a strategic relationship with Geodetics,” says Jay Napoli, vice president, FOG/OEM sales at KVH. “The high performance of the 1750 IMU helps enable Geodetics’ systems to deliver ground-breaking accuracy while remaining available to the commercial marketplace.”
For navigation challenges such as collision avoidance and vehicle-to-vehicle navigation and communication (V2V), the Geodetics Geo-RelNAV system offers a highly accurate, real-time relative positioning and orientation solution that utilizes single- or dual-frequency GPS receivers and the high performance KVH 1750 IMU. The Geo-RelNAV provides precise relative position and orientation between moving platforms such as manned or unmanned air, marine, and ground vehicles. This relative position data is used for such applications as autonomous aerial refueling, autonomous landing, and collision avoidance.
KVH is one of the only fiber optic gyro manufacturers to control the entire production process, from creating its own specially designed polarization-maintaining optical fiber to packaging its gyros together in advanced systems for inertial measurement, inertial navigation, and attitude heading and reference systems. As a result, KVH’s inertial sensors and gyros offer outstanding accuracy and excellent durability at a lower cost than competing systems.
By Malek O. Karaim, Tashfeen B. Karamat, Aboelmagd Noureldin, Mohamed Tamazin, and Mohamed M. Atia
A team of university researchers has developed a technique combining GPS receivers with an inexpensive inertial measuring unit to detect and repair cycle slips with the potential to operate in real time.
INNOVATION INSIGHTS by Richard Langley
DRUM ROLL, PLEASE. The “Innovation” column and GPS World are celebrating a birthday. With this issue, we have started the 25th year of publication of the magazine and the column, which appeared in the very first issue and has been a regular feature ever since. Over the years, we have seen many developments in GPS positioning, navigation, and timing with a fair number documented in the pages of this column.
In January 1990, GPS and GLONASS receivers were still in their infancy. Or perhaps their toddler years. But significant advances in receiver design had already been made since the introduction around 1980 of the first commercially available GPS receiver, the STI-5010, built by Stanford Telecommunications, Inc. It was a dual-frequency, C/A- and P-code, slow-sequencing receiver. Cycling through four satellites took about five minutes, and the receiver unit alone required about 30 centimeters of rack space. By 1990, a number of manufacturers were offering single or dual frequency receivers for positioning, navigation, and timing applications. Already, the first handheld receiver was on the market, the Magellan NAV 1000. Its single sequencing channel could track four satellites. Receiver development has advanced significantly over the intervening 25 years with high-grade multiple frequency, multiple signal, multiple constellation GNSS receivers available from a number of manufacturers, which can record or stream measurements at data rates up to 100 Hz. Consumer-grade receivers have proliferated thanks, in part, to miniaturization of receiver chips and modules. With virtually every cell phone now equipped with GPS, there are over a billion GPS users worldwide. And the chips keep getting smaller. Complete receivers on a chip with an area of less than one centimeter squared are common place. Will the “GPS dot” be in our near future?
The algorithms and methods used to obtain GPS-based positions have evolved over the years, too. By 1990, we already had double-difference carrier-phase processing for precise positioning. But the technique was typically applied in post-processing of collected data. It is still often done that way today. But now, we also have the real-time kinematic (or RTK) technique to achieve similar positioning accuracies in real time and the non-differenced precise point positioning technique, which does not need base stations and which is also being developed for real-time operation. But in all this time, we have always had a “fly in the ointment” when using carrier-phase observations: cycle slips. These are discontinuities in the time series of carrier-phase measurements due to the receiver temporarily losing lock on the carrier of a GPS signal caused by signal blockage, for example. Unless cycle slips are repaired or otherwise dealt with, reduction in positioning accuracy ensues. Scientists and engineers have developed several ways of handling cycle slips not all of which are capable of working in real time. But now, a team of university researchers has developed a technique combining GPS receivers with an inexpensive inertial measuring unit to detect and repair cycle slips with the potential to operate in real time. They describe their system in this month’s column.
“Innovation” is a regular feature that discusses advances in GPS technology and its applications as well as the fundamentals of GPS positioning. The column is coordinated by Richard Langley of the Department of Geodesy and Geomatics Engineering, University of New Brunswick. He welcomes comments and topic ideas.
GPS carrier-phase measurements can be used to achieve very precise positioning solutions. Carrier-phase measurements are much more precise than pseudorange measurements, but they are ambiguous by an integer number of cycles. When these ambiguities are resolved, sub-centimeter levels of positioning can be achieved.
However, in real-time kinematic applications, GPS signals could be lost temporarily because of various disturbing factors such as blockage by trees, buildings, and bridges and by vehicle dynamics. Such signal loss causes a discontinuity of the integer number of cycles in the measured carrier phase, known as a cycle slip. Consequently, the integer counter is reinitialized, meaning that the integer ambiguities become unknown again. In this event, ambiguities need to be resolved once more to resume the precise positioning and navigation process. This is a computation-intensive and time-consuming task. Typically, it takes at least a few minutes to resolve the ambiguities.
The ambiguity resolution is even more challenging in real-time navigation due to receiver dynamics and the time-sensitive nature of the required kinematic solution. Therefore, it would save effort and time if we could detect and estimate the size of these cycle slips and correct the measurements accordingly instead of resorting to a new ambiguity resolution. In this article, we will briefly review the cause of cycle slips and present a procedure for detecting and correcting cycle slips using a tightly coupled GPS/inertial system, which could be used in real time. We will also discuss practical tests of the procedure.
Cycle Slips and Their Management
A cycle slip causes a jump in carrier-phase measurements when the receiver phase tracking loops experience a temporary loss of lock due to signal blockage or some other disturbing factor. On the other hand, pseudoranges remain unaffected. This is graphically depicted in FIGURE 1. When a cycle slip happens, the Doppler (cycle) counter in the receiver restarts, causing a jump in the instantaneous accumulated phase by an integer number of cycles. Thus, the integer counter is reinitialized, meaning that ambiguities are unknown again, producing a sudden change in the carrier-phase observations.
FIGURE 1. A cycle slip affecting phase measurements but not the pseudoranges.
Once a cycle slip is detected, it can be handled in two ways. One way is to repair the slip. The other way is to reinitialize the unknown ambiguity parameter in the phase measurements. The former technique requires an exact estimation of the size of the slip but could be done instantaneously. The latter solution is more secure, but it is time-consuming and computationally intensive. In our work, we follow the first approach, providing a real-time cycle-slip detection and correction algorithm based on a GPS/inertial integration scheme.
GPS/INS Integration
An inertial navigation system (INS) can provide a smoother and more continuous navigation solution at higher data rates than a GPS-only system, since it is autonomous and immune to the kinds of interference that can deteriorate GPS positioning quality. However, INS errors grow with time due to the inherent mathematical double integration in the mechanization process. Thus, both GPS and INS systems exhibit mutually complementary characteristics, and their integration provides a more accurate and robust navigation solution than either stand-alone system. GPS/INS integration is often implemented using a filtering technique. A Kalman filter is typically selected for its estimation optimality and time-recursion properties.
The two major approaches of GPS/INS integration are loosely coupled and tightly coupled. The former strategy is simpler and easier to implement because the inertial and GPS navigation solutions are generated independently before being weighted together by the Kalman filter. There are two main drawbacks with this approach: 1) signals from at least four satellites are needed for a navigation solution, which cannot always be guaranteed; and 2) the outputs of the GPS Kalman filter are time correlated, which has a negative impact upon the system performance. The latter strategy performs the INS/GPS integration in a single centralized Kalman filter. This architecture eliminates the problem of correlated measurements, which arises due to the cascaded Kalman filtering in the loosely coupled approach. Moreover, the restriction of visibility of at least four satellites is removed. We specifically use a tightly coupled GPS/reduced inertial sensor system approach.
Reduced Inertial Sensor System. Recently, microelectromechanical system or MEMS-grade inertial sensors have been introduced for low-cost navigation applications. However, these inexpensive sensors have complex error characteristics.
Therefore, current research is directed towards the utilization of fewer numbers of inertial sensors inside the inertial measurement unit (IMU) to obtain the navigation solution.
The advantage of this trend is twofold. The first is avoidance of the effect of inertial sensor errors. The second is reduction of the cost of the IMU in general. One such minimization approach, and the one used in our work, is known as the reduced inertial sensor system (RISS). The RISS configuration uses one gyroscope, two accelerometers, and a vehicle wheel-rotation sensor. The gyroscope is used to observe the changes in the vehicle’s orientation in the horizontal plane. The two accelerometers are used to obtain the pitch and roll angles. The wheel-rotation sensor readings provide the vehicle’s speed in the forward direction. FIGURE 2 shows a general view of the RISS configuration.
FIGURE 2. A general view of the RISS configuration.
A block diagram of the tightly coupled GPS/RISS used in our work is shown in FIGURE 3. At this stage, the system uses GPS pseudoranges together with the RISS observables to compute an integrated navigation solution. In this three-dimensional (3D) version of RISS, the system has a total of nine states. These states are the latitude, longitude, and altitude errors ( ; the east, north, and up velocity errors ; the azimuth error ; the error associated with odometer-driven acceleration ; and the gyroscope error .
The nine-state error vector xk at time tk is expressed as: (1)
FIGURE 3. Tightly coupled integration of GPS/RISS using differential pseudorange measurements.
Cycle Slip Detection and Correction
Cycle slip handling usually happens in two discrete steps: detection and fixing or correction. In the first step, using some testing quantity, the location (or time) of the slip is found. During the second step, the size of the slip is determined, which is needed along with its location to fix the cycle slip. Various techniques have been introduced by researchers to address the problem of cycle-slip detection and correction. Different measurements and their combinations are used including carrier phase minus code (using L1 or L2 measurements), carrier phase on L1 minus carrier phase on L2, Doppler (on L1 or L2), and time-differenced phases (using L1 or L2). In GPS/INS integration systems, the INS is used to predict the required variable to test for a cycle slip, which is usually the true receiver-to-satellite range in double-difference (DD) mode, differencing measurements between a reference receiver and the roving receiver and between satellites. In this article, we introduce a tightly coupled GPS/RISS approach for cycle-slip detection and correction, principally for land vehicle navigation using a relative-positioning technique.
Principle of the Algorithm. The proposed algorithm compares DD L1 carrier-phase measurements with estimated values derived from the output of the GPS/RISS system. In the case of a cycle slip, the measurements are corrected with the calculated difference. A general overview of the system is given in FIGURE 4.
FIGURE 4. The general flow diagram of the proposed algorithm.
The number of slipped cycles is given by (2)
where is the DD carrier-phase measurement (in cycles) is DD estimated carrier phase value (in cycles). is compared to a pre-defined threshold μ . If the threshold is exceeded, it indicates that there is a cycle slip in the DD carrier-phase measurements.
Theoretically, would be an integer but because of the errors in the measured carrier phase as well as errors in the estimations coming from the INS system, will be a real or floating-point number.
The estimated carrier-phase term in Equation (2) is obtained as follows: (3)
where λ is the wavelength of the signal carrier (in meters) are the estimated ranges from the rover to satellites i and j respectively (in meters) are known ranges from the base to satellites i and j respectively (in meters).
What we need to get from the integrated GPS/RISS system is the estimated range vector from the receiver to each available satellite ( ). Knowing our best position estimate, we can calculate ranges from the receiver to all available satellites through: (4)
where is the calculated range from the receiver to the mth satellite xKF is the receiver position obtained from GPS/RISS Kalman filter solution xm is the position of the mth satellite M is the number of available satellites.
Then, the estimated DD carrier-phase term in Equation (3) can be calculated and the following test quantity in Equation (2) can be applied: (5)
If a cycle slip occurred in the ith DD carrier-phase set, the corresponding set is instantly corrected for that slip by: (6)
where s is the DD carrier-phase-set number in which the cycle slip has occurred.
Experimental Work
The performance of the proposed algorithm was examined on the data collected from several real land-vehicle trajectories. A high-end tactical grade IMU was integrated with a survey-grade GPS receiver to provide the reference solution. This IMU uses three ring-laser gyroscopes and three accelerometers mounted orthogonally to measure angular rate and linear acceleration. The GPS receiver and the IMU were integrated in a commercial package. For the GPS/RISS solution, the same GPS receiver and a MEMS-grade IMU were used. This IMU is a six-degree of freedom inertial system, but data from only the vertical gyroscope, the forward accelerometer, and the transversal accelerometer was used. TABLE 1 gives the main characteristics of both IMUs. The odometer data was collected using a commercial data logger through an On-Board Diagnostics version II (OBD-II) interface. Another GPS receiver of the same type was used for the base station measurements. The GPS data was logged at 1 Hz.
Table 1. Characteristics of the MEMS and tactical grade IMUs.
Several road trajectories were driven using the above-described configuration. We have selected one of the trajectories, which covers several real-life scenarios encountered in a typical road journey, to show the performance of the proposed algorithm. The test was carried out in the city of Kingston, Ontario, Canada. The starting and end point of the trajectory was near a well-surveyed point at Fort Henry National Historic Site where the base station receiver was located. The length of the trajectory was about 30 minutes, and the total distance traveled was about 33 kilometers with a maximum baseline length of about 15 kilometers. The trajectory incorporated a portion of Highway 401 with a maximum speed limit of 100 kilometers per hour and suburban areas with a maximum speed limit of 80 kilometers per hour. It also included different scenarios including sharp turns, high speeds, and slopes.
FIGURE 5 shows measured carrier phases at the rover for the different satellites. Some satellites show very poor presence whereas some others are consistently available. Satellites elevation angles can be seen in FIGURE 6.
FIGURE 5. Measured carrier phase at the rover.FIGURE 6. Satellite elevation angles.
Results
We start by showing some results of carrier-phase estimation errors. Processing is done on what is considered to be a cycle-slip-free portion of the data set for some persistent satellites (usually with moderate to high elevation angles). Then we show results for the cycle-slip-detection process by artificially introducing cycle slips in different scenarios. In the ensuing discussion (including tables and figures), we show results indicating satellite numbers without any mention of reference satellites, which should be implicit as we are dealing with DD data.
FIGURE 7. DD-carrier-phase estimation error, reference satellite with PRN 22.FIGURE 8. Measured versus estimated DD carrier phase, reference satellite with PRN 22.
As can be seen in TABLE 2, the root-mean-square (RMS) error varies from 0.93 to 3.58 cycles with standard deviations from 0.85 to 2.47 cycles. Estimated phases are approximately identical to the measured ones. Nevertheless, most of the DD carrier-phase estimates have bias and general drift trends, which need some elaboration. In fact, the bias error can be the result of more than one cause. The low-cost inertial sensors always have bias in their characteristics, which plays a major role in this. The drift is further affecting relatively lower elevation angle satellites which can also be attributed to more than one reason. Indeed, one reason for choosing this specific trajectory, which was conducted in 2011, was to test the algorithm with severe ionospheric conditions as the year 2011 was close to a solar maximum: a period of peak solar activity in the approximately 11-year sunspot cycle.
Table 2. Estimation error for DD carrier phases (in cycles).
Moreover, the time of the test was in the afternoon, which has the maximum ionospheric effects during the day. Thus, most part of the drift trend must be coming from ionospheric effects as the rover is moving away from the base receiver during this portion of the trajectory. Furthermore, satellite geometry could contribute to this error component. Most of the sudden jumps coincide with, or follow, sharp vehicle turns and rapid tilts. Table 2 shows the averaged RMS and standard deviation (std) DD carrier-phase estimation error for the sample satellite-pairs. We introduced cycle slips at different rates or intensities and different sizes to simulate real-life scenarios. Fortunately, cycle slips are usually big as mentioned earlier and this was corroborated by our observations from real trajectory data. Therefore, it is more important to detect and correct for bigger slips in general.
Introducing and Detecting Cycle Slips. To test the robustness of the algorithm, we started with an adequate cycle slip size. Cycle slips of size 10–1000 cycles were introduced with different intensities. These intensities are categorized as few (1 slip per 100 epochs), moderate (10 slips per 100 epochs), and severe (100 slips per 100 epochs). This was applied for all DD carrier-phase measurement sets simultaneously. The threshold was set to 1.9267 (average of RMS error for all satellite-pairs) cycles. Four metrics were used to describe the results. Mean square error (MSE); accuracy, the detected cycle slip size with respect to the introduced size; True detection (TD) ratio; and Mis-detection (MD) ratio. Due to space constraints and the similarity between results for different satellites, we only show results for the reference satellite with PRN 22. FIGURES 9–12 show introduced versus calculated cycle slips along with the corresponding detection error for sample satellites in the different scenarios. TABLES 3–5 summarize these results.
FIGURE 9. Introduced and calculated cycle slips (upper plot) and detection error (lower plot). Few cycle slips case, reference satellite with PRN 22.FIGURE 10. Introduced and calculated cycle slips (upper plot) and detection error (lower plot). Moderate cycle slips case, reference satellite with PRN 22.FIGURE 11. Introduced and calculated cycle slips (upper plot) and detection error (lower plot). Intensive cycle slips case, reference satellite with PRN 22.FIGURE 12. Introduced and calculated cycle slips (upper plot) and detection error (lower plot). Small cycle slips case, reference satellite with PRN 22.Table 3. Few slips (1 slip per 100 epochs).Table 4. Moderate slips (10 slips per 100 epochs).Table 5. Intensive slips (100 slips per 100 epochs).
All introduced cycle slips were successfully detected in all of the few, moderate, and severe cases with very high accuracy. A slight change in the accuracy (increasing with higher intensity) among the different scenarios shows that detection accuracy is not affected by cycle-slip intensity. Higher mis-detection ratios for smaller cycle-slip intensity comes from bigger error margins than the threshold for several satellite pairs. However, this is not affecting the overall accuracy strongly as all mis-detected slips are of comparably very small sizes. MD ratio is zero in the intensive cycle-slip case as all epochs contain slips is an indicator of performance compromise with slip intensity.
It is less likely to have very small cycle slips (such as 1 to 2 cycles) in the data and usually it will be hidden with the higher noise levels in kinematic navigation with low-cost equipment. However, we wanted to show the accuracy of detection in this case. We chose the moderate cycle slip intensity for this test. TABLE 6 summarizes results for all satellites.
Table 6. Small slips (1–2 cycles) at moderate intensity (10 slips per 100 epochs).
We get a moderate detection ratio and modest accuracy as the slips are of sizes close to the threshold. The MSE values are not far away from the case of big cycle slips but with higher mis-detection ratio.
Conclusions
The performance of the proposed algorithm was examined on several real-life land vehicle trajectories, which included various driving scenarios including high and slow speeds, sudden accelerations, sharp turns and steep slopes. The road testing was designed to demonstrate the effectiveness of the proposed algorithm in different scenarios such as intensive and variable-sized cycle slips.
Results of testing the proposed method showed competitive detection rates and accuracies comparable to existing algorithms that use full MEMS IMUs. Thus with a lower cost GPS/RISS integrated system, we were able to obtain a reliable phase-measurement-based navigation solution. Although the testing discussed in this article involved post-processing of the actual collected data at the reference station and the rover, the procedure has been designed to work in real time where the measurements made at the reference station are transmitted to the rover via a radio link. This research has a direct influence on navigation in real-time applications where frequent cycle slips occur and resolving integer ambiguities is not affordable because of time and computational reasons and where system cost is an important factor.
Acknowledgments
This article is based on the paper “Real-time Cycle-slip Detection and Correction for Land Vehicle Navigation using Inertial Aiding” presented at ION GNSS+ 2013, the 26th International Technical Meeting of the Satellite Division of The Institute of Navigation held in Nashville, Tennessee, September 16–20, 2013.
Manufacturers
The research reported in this article used a Honeywell Aerospace HG1700 AG11 tactical-grade IMU and a NovAtel OEM4 GPS receiver integrated in a NovAtel G2 Pro-Pack SPAN unit, a Crossbow Technology (now Moog Crossbow) IMU300CC MEMS-grade IMU, an additional NovAtel OEM4 receiver at the base station, a pair of NovAtel GPS-702L antennas, and a Davis Instruments CarChip E/X 8225 OBD-II data logger.
Malek Karaim is a Ph.D. student in the Department of Electrical and Computer Engineering of Queen’s University, Kingston, Ontario, Canada.
Tashfeen Karamat is a doctoral candidate in the Department of Electrical and Computer Engineering at Queen’s University.
Aboelmagd Noureldin is a cross-appointment professor in the Departments of Electrical and Computer Engineering at both Queen’s University and the Royal Military College (RMC) of Canada, also in Kingston.
Mohamed Tamazin is a Ph.D. student in the Department of Electrical and Computer Engineering at Queen’s University and a member of the Queen’s/RMC NavINST Laboratory.
Mohamed M. Atia is a research associate and deputy director of the Queen’s/RMC NavINST Laboratory.
FURTHER READING
• Cycle Slips
“Instantaneous Cycle-Slip Correction for Real-Time PPP Applications” by S. Banville and R.B. Langley in Navigation, Vol. 57, No. 4, Winter 2010–2011, pp. 325–334.
“GPS Cycle Slip Detection and Correction Based on High Order Difference and Lagrange Interpolation” by H. Hu and L. Fang in Proceedings of PEITS 2009, the 2nd International Conference on Power Electronics and Intelligent Transportation System, Shenzhen, China, December 19–20, 2009, Vol. 1, pp. 384–387, doi: 10.1109/PEITS.2009.5406991.
“Cycle Slip Detection and Fixing by MEMS-IMU/GPS Integration for Mobile Environment RTK-GPS” by T. Takasu and A. Yasuda in Proceedings of ION GNSS 2008, the 21st International Technical Meeting of the Satellite Division of The Institute of Navigation, Savannah, Georgia, September 16–19, 2008, pp. 64–71.
“Cycle-Slip Detection and Repair in Integrated Navigation Systems” by A. Lipp and X. Gu in Proceedings of PLANS 1994, the IEEE Position Location and Navigation Symposium, Las Vegas, Nevada, April 11–15, 1994, pp. 681–688, doi: 10.1109/PLANS.1994.303377.
Short-Arc Orbit Improvement for GPS Satellites by D. Parrot, M.Sc.E. thesis, Department of Geodesy and Geomatics Engineering Technical Report No. 143, University of New Brunswick, Fredericton, New Brunswick, Canada, June 1989.
• Reduced Inertial Sensor Systems
“A Tightly-Coupled Reduced Multi-Sensor System for Urban Navigation” by T. Karamat, J. Georgy, U. Iqbal, and N. Aboelmagd in Proceedings of ION GNSS 2009, the 22nd International Technical Meeting of the Satellite Division of The Institute of Navigation, Savannah, Georgia, September 22–25, 2009, pp. 582–592.
“An Integrated Reduced Inertial Sensor System – RISS / GPS for Land Vehicle” by U. Iqbal, A. Okou, and N. Aboelmagd in Proceedings of PLANS 2008, the IEEE/ION Position Location and Navigation Symposium, Monterey, California, May 5–8, 2008, pp. 1014–1021, doi: 10.1109/PLANS.2008.4570075.
• Integrating GPS and Inertial Systems
Fundamentals of Inertial Navigation, Satellite-based Positioning and their Integration by N. Aboelmagd, T. B. Karmat, and J. Georgy. Published by Springer-Verlag, New York, New York, 2013.
Aided Navigation: GPS with High Rate Sensors by J. A. Farrell. Published by McGraw-Hill, New York, New York, 2008.
Global Positioning Systems, Inertial Navigation, and Integration, 2nd edition, by M.S. Grewal, L.R. Weill, and A.P. Andrews. Published by John Wiley & Sons, Inc., Hoboken, New Jersey, 2007.
Trimble introduced today the Trimble AP15, the latest member of the AP series of OEM GNSS-Inertial board sets. The introduction was made at the AUVSI 2013 Conference and Exhibition, being held in Washington, D.C.
The AP15 uses a custom Micro Electromechanical Machined (MEMS) based Inertial Measurement Unit (IMU). It is the first product to take advantage of Applanix’ proprietary calibration process — Applanix SmartCal, a new software compensation technology that allows Trimble to achieve exceptional performance from IMUs manufactured specifically for mobile mapping applications.
The AP15 combines high-precision GNSS positioning with Applanix IN-Fusion GNSS-Inertial integration technology, all running on a powerful, dedicated Inertial Engine (IE) board. AP products provide the performance and functionality of Applanix’ POS systems in an embedded form-factor that is specifically designed for third-party manufacturers and systems integrators, Trimble said. The AP Series is designed for a variety of commercial mobile positioning and orientation applications including airborne, terrestrial and marine mapping and guidance for unmanned vehicles.
Combined with a wheel-mounted Distance Measurement Instrument (DMI), the AP15 provides a full 6-degrees-of-freedom navigation solution for land vehicles that is capable of providing robust position and orientation information regardless of obstructions to GNSS-only positioning such as multipath or complete signal loss. Applanix IN-Fusion technology produces uninterrupted position, roll, pitch and true heading measurements of moving platforms by combining IMU data with raw GNSS observables and DMI velocity.
GNSS functionality is provided by a Trimble GNSS module, a dual-antenna, 440 channel, multi-frequency survey-grade GNSS receiver that supports a wide range of satellite signals, including GPS L1/L2/L2C/L5 and GLONASS L1/L2 signals. The module also supports Satellite-Based Augmentation Service (SBAS) corrections, including the U.S. Wide Area Augmentation System (WAAS), European Geostationary Overlay Service (EGNOS), Japan’s Multi-functional Satellite Augmentation System (MSAS) and the OmniStar VBS, HP and XP/G2 corrections.
“Trimble is a leading provider of technology for positioning and orientation solutions and the introduction of the AP15 module continue this tradition,” said Kevin Andrews, product manager. “The AP15 has been designed as a more compact, lighter unit which can deliver excellent performance at lower cost.”
The Trimble AP15 is expected to be available in October of 2013 through Applanix’ sales channel.
When a team of firefighters, first responders, or soldiers operates inside a building, in urban canyons, underground, in foliage, or under the forest canopy, the GPS-denied environment presents unique navigation challenges. An enhanced personal inertial navigation system (ePINS), based on a strapdown navigation solution using a mid-grade IMU and wavelet-based motion-classification algorithms, can track positions with errors of less than 2 percent of distance traveled in both indoor and outdoor environments.
By Yunqian Ma, Wayne Soehren, Wes Hawkinson, and Justin Syrstad
Numerous pedestrian navigation applications are currently available or proposed for development. Some of them include localization for coordinating firefighters, first responders, or soldiers. In these applications, the safety and efficiency of the entire team relies directly on the location and orientation of each team member. Operations in high signal interference areas such as cities, rugged terrain, forest, or indoor spaces deliver intermittent or no GPS signal. An alternative to GPS-based location is required.
In this article, we introduce an enhanced personal inertial navigation system (ePINS) solution specifically designed for environments where GPS is unavailable. ePINS combines an array of state-of-the-art sensors and fusion algorithms into a personal navigation system that provides accurate location information for pedestrian applications.
The ePINS concept.
The ePINS solution has the following benefits:
Accurate positioning in GPS-denied environments;
Small, lightweight unit can be easily carried by first responders, rescue workers, or soldiers;
Ruggedized packaging to withstand difficult first responder and military environments.
Features of the ePINS unit include:
State-of-the-art micro-electromechanical systems (MEMS) gyros and accelerometers, barometric altitude sensor, and advanced navigation software;
Advanced motion classification algorithms that accurately identify and measure user activity;
Immunity to magnetic disturbances.
Related Work
In the field of personal navigation, it is common to find systems that rely on sensors that need infrastructure (for example, Wi-Fi positioning) or sensors that actively emit electro-magnetic radiation (such as Doppler radar). These requirements are major drawbacks for communities such as dismounted soldiers in hostile environments.
Other approaches exploit the so-called Zero-velocity update (ZUPT) mechanism, which resets the inertial measurement unit (IMU) velocity errors during the stationary phase of motion. However, implementation of such schemes relies on sensors embedded in footwear, which is not readily accepted in many user communities.
To address these drawbacks, Honeywell has been developing advanced aiding techniques for personal navigation that do not rely on infrastructure and compute a self-contained, relative-navigation solution based only on passive sensors. One technique that Honeywell has developed uses displacement estimation from human-motion models. This technology has been implemented in the ePINS prototype and shows promising performance.
The human-motion model uses IMU measurements as inputs and was developed to infer distance traveled. It generates a displacement estimate that is used as a measurement in the navigation filtering process. The first version of this model was matured under the DARPA individual Precision Inertial Navigation System (iPINS) program. The iPINS system used an IMU, GPS, barometer, and motion classification to estimate a person’s position in both indoor and outdoor environments. In this system, IMU signal characteristics (e.g., peaks and valleys in the accelerations induced by walking) were exploited to differentiate between walking and running. Honeywell recently expanded the human-motion model to identify more specific motion types using a new wavelet motion classification method.
System Description
Figure 1 displays the hardware architecture of the ePINS, a small battery-powered, highly integrated electronic system. The ePINS processing platform is an ARM11-based, i.MX31 system-on-module, paired with support electronics. In addition to the processing platform, the ePINS assembly includes a MEMS IMU, a barometric pressure sensor, a digital magnetometer, and a GPS receiver.
Figure 1. ePINS hardware architecture.
The MEMS IMU provides inertial measurements for strapdown navigation. The IMU’s small package size, light weight, low power consumption, and impressive performance make it attractive for use in the ePINS system. The device is less than 5 cubic inches and weighs less than 0.35 pounds. It consumes about 3 watts of power with a typical current draw of 600mA at 5V.
The ePINS software system is shown in Figure 2. The navigation software runs within Honeywell’s Embedded Computing Toolbox and Operating System (ECTOS IIc), which provides a layered, customizable, and reusable software architecture for implementing navigation, guidance, and control software. A Honeywell-developed simulation tool for offline analysis and development of ECTOS-based software was also used in ePINS development and testing.
The ePINS demonstration device can achieve path performance of less 2 percent distance traveled for walking motion after 1 hour of operation, independent of the magnetic environment. Current performance, packaging characteristics, and interfaces are summarized in Table 1.
Table 1. ePINS performance objectives and physical specifications.
Algorithm Description
Figure 3 depicts the overall sensor integration and data processing scheme used in the ePINS device.
Figure 3. Sensor integration using the ECTOS extended Kalman filter.
Extended Kalman Filter (EKF). The EKF estimates the navigation and sensor errors and computes the resets applied to the strapdown navigation solution to increase its accuracy. Error models for the navigation sensors (IMU, barometric altimeter, magnetometer, GPS, and motion classification) are contained in the EKF. For the ePINS device, the virtual measurements from the step-length model and the strapdown navigation solution are fused by the EKF to assist in bounding the time dependent error growth of the strapdown navigator, which in turn helps maintain calibration of the inertial sensors. A key output of the EKF is the navigation confidence, which is an estimate of the accuracy of the navigation solution.
An important aspect of the EKF and step-length modeling is the residual test that the EKF supports. This test provides a reasonableness comparison between the step-length model estimate and the distance predicted by the strapdown navigation system. This capability significantly increases the robustness of the navigation solution, especially when the user is engaged in motions not recognized during motion classification.
Human-Motion Model. The human-motion model includes two components: wavelet motion classification and step-length model estimation. The wavelet motion classification identifies the type of motion the user is performing, and the step-length model acts as a virtual sensor that quantifies the motion as a distance-traveled estimate.
Wavelet Motion Classification. Human motions are very diverse and highly irregular. Determining what motion is being performed is a challenging problem of classification. Honeywell’s solution is based on wavelet transformation of IMU data. Predefined, or known, characteristics of a variety of motions (such as walking, running, crawling, etc.) are cataloged and stored to a device’s memory. Estimates of those same characteristics for a user are then computed in real time and compared to the catalog of stored information to find the best match.
Generating the catalog of stored information is an offline task that begins by “segmenting” recorded IMU time domain data into individual steps. An example of the output of the segmentation process is shown in Figure 4.
Figure 4. Segmentation of the IMU data using the y-axis accelerometer signal.
Figure 5 displays the segmentation results for two different walking styles (in red and blue) across approximately 15 example steps. As is evident from the graph, walking has characteristics that are common across users, for example, the sharp peaks in the z-axis acceleration caused by foot-ground impacts. Once the data has been segmented, a wavelet transformation on each data channel is performed. Wavelet transformation for many users over many different motion types takes place offline. Subsequently, a wavelet descriptor is built for each motion type based on the transformations into the wavelet domain. With this method, a wide variety of information (that is, descriptors) suitable for input to a classifier is captured about each motion. These descriptors are then cataloged and stored in memory on the ePINS device.
Figure 5. Sample steps for two subjects (red) and (blue).
Finally, for the online phase, the wavelet descriptor of the incoming IMU data is calculated by performing a wavelet transformation on each data channel. This descriptor is then compared to the pre-computed and stored descriptors to classify the motion. FIGURE 7 shows an example of the motion classifier output, where a running motion was used as an input. The classifier successfully determined the motion type (blue field), frequency and phase of the input motion, depicted by the tallest rectangle in the figure.
Figure 7. Classification results from a query of running at a certain frequency and phase (depicted by the dark sphere).
Step-Length Modeling. Once the current motion is identified, a step-length model specific to that motion is used to aid the navigation algorithms. The model for each motion type is obtained by first collecting data that measures step length and step frequency. From this data, the step-length models can be computed by performing a regression analysis of the step-length vs. step-frequency data. Since the step-length models act as a virtual sensor, the models must be as accurate as possible to achieve better system performance. To attain model accuracy, an accurate data collection method is needed.
For ePINS development, step-length models for multiple users have been identified from step-length and timing information using a precise GPS truth reference system. Step-length regression calculations then determine the step length as a function of step frequency (that is, inverse of the step time period). An example of GPS truth data and the corresponding regression model are shown in FIGURE 6 for walking motions.
Figure 6. Step length versus frequency for the walking of subject.
Although basic step-length models are created offline, online calibration of the step-length model can be performed by the EKF if GPS is available during operation. Online calibration tends to increase the overall position accuracy, as variations in the step-length models are likely due to slight variations in biometric differences across humans, terrain features, and even mission plans and duration.
Heading Determination. Heading initialization is one of the key concerns during system start up. In its current operational use, the ePINS device may perform a dynamic or a static initialization of heading. The static method requires the user to survey the system’s initial heading to an accuracy value that is usually specified by mission performance objectives; the absolute position accuracy is dependent upon the accuracy of the initial heading.
The dynamic method is a general method for heading initialization; it is performed without input from the user, but is possible only when GPS is available. This method of heading initialization does not use any a priori information about heading and requires an EKF implementation with a large-azimuth error model. This method requires an additional period of time in which the heading error uncertainty converges.
User Interface. During a mission, the user can interact with the navigation system and monitor its output on a display. The current ePINS prototype offers two-way communication via a serial connection. The serial communication is made wireless by the addition of a Bluetooth interface. Users can use this link to monitor the status of the navigation solution and to send commands to the device.
Honeywell has developed an application for the Android platform for this purpose. One of the key features of the interface design is that the navigation system outputs data in a standard NEMA format. Thus, publically available Android applications, not just proprietary applications, can also receive and display the navigation solution output by the ePINS device.
Honeywell’s personal navigation application displays the user’s traveled trajectory in real-time. The application can be adapted to include building floor plans as well as other navigation information.
Results
The ePINS prototype has been evaluated both in simulations and indoor/outdoor experiments. The navigation results presented here were obtained in February 2012 at a Honeywell facility (FIGURE 8). First, the user completed the heading calibration, and then online step parameter estimation in the presence of GPS was performed. Once calibration and training was completed, the GPS was disabled to simulate a GPS-denied environment outdoors. The user than transitioned to indoors (with GPS still disabled), and walked a course inside that included walking up and down stairs (FIGURE 9) and ended in a conference room (FIGURE 10).
Figure 8. Course for the Honeywell facility demonstration.Figure 9. The user walking up stairs.Figure 10. The user at the end of the demo.
Over these conditions, the ePINS system performed robustly and within performance specifications. Live demonstrations and testing showing similar levels of performance were performed at the 2012 Joint Navigation Conference (JNC) and at military test sites in California and Indiana.
Summary
The technical approach of the ePINS solution to the problem of personnel navigation in GPS-denied environments is based on a strapdown navigation solution maintained using a mid-grade IMU and advanced motion-classification algorithms. We integrated an array of sensors and software into a system that provides accurate position information and is suitable for use by first responders, soldiers, and other personnel where GPS is unavailable. ePINS works well for a variety of pedestrian motion types, including walking, running, crawling, walking upstairs, walking downstairs, sidestepping, and walking backwards. The motion classification and modeling method is extensible to other motion types.
We tested the ePINS system in indoor and outdoor environments. FIGURE 11 depicts the future ePINS concept, and TABLE 2 presents its future physical characteristics.
Figure 11. Future ePINS concept and mounting position.Table 2. Packaging characteristics of the future ePINS.
Acknowledgments
This article is based on a presentation made at ION GNSS 2012.
Manufacturers
The ePINS processing platform uses Honeywell Agile Navigation and Guidance Integrated Electronics support electronics. It includes a Honeywell HG1930 MEMS IMU, a Bosch Sensortec BMP085 barometric pressure sensor, a Honeywell HMC6343 digital magnetometer, and a NovAtel OEMStar GPS receiver.
Yunqian Ma is a principal scientist at Honeywell Aerospace. He received his Ph.D. degree in electrical engineering from the University of Minnesota, Twin Cities. He is currently the program manager of the GPS-denied navigation program and the next-generation personal navigation program.
Wayne Soehren is a senior technical manager at Honeywell Aerospace. He was the program manager for the development of Honeywell’s first MEMS-based GPS/INS, which developed the core capability now used in Honeywell’s IGS-2XX family of MEMS-based GPS/INS products. He holds an MSEE from the University of Minnesota.
Wes Hawkinson is an engineering fellow at Honeywell Aerospace. He holds a BSEE/CE from the University of Wisconsin–Madison. Justin Syrstad is a guidance and navigation scientist. He received a master’s degree in aerospace engineering from the University of Minnesota.
The U.S. Military relies on GPS to aid air, land and sea navigation. Like the GPS units in many automobiles today, a simple receiver and some processing power is all that is needed for accurate navigation. But, what if the GPS satellites suddenly became unavailable due to malfunction, enemy action or simple interference, such as driving into a tunnel? Unavailability of GPS would be inconvenient for drivers on the road, but could be disastrous for military missions. DARPA is working to protect against such a scenario, and an emerging solution is much smaller than the navigation instruments in today’s defense systems.
DARPA researchers at the University of Michigan report that they have made significant progress with a timing and inertial measurement unit (TIMU) that contains everything needed to aid navigation when GPS is temporarily unavailable. The single-chip TIMU prototype contains a six-axis IMU (three gyroscopes and three accelerometers) and integrates a highly-accurate master clock into a single miniature system, smaller than the size of a penny. This chip integrates devices (clocks, gyroscopes and accelerometers), materials and designs from DARPA’s Micro-Technology for Positioning, Navigation and Timing (Micro-PNT) program.
Three pieces of information are needed to navigate between known points A and B with precision: orientation, acceleration and time. This new chip integrates tiny devices that can measure all three simultaneously. The design is accomplished through new fabrication processes in high-quality materials for multi-layered, packaged inertial sensors and a timing unit, in a tiny 10-millimeter-square package. Each of the six microfabricated layers of the TIMU is only 50 microns thick, approximately the thickness of a human hair. Each layer has a different function, akin to floors in a building.
“Both the structural layer of the sensors and the integrated package are made of silica,” said Andrei Shkel, DARPA program manager. “The hardness and the high-performance material properties of silica make it the material of choice for integrating all of these devices into a miniature package. The resulting TIMU is small enough and should be robust enough for applications (when GPS is unavailable or limited for a short period of time) such as personnel tracking, handheld navigation, small diameter munitions and small airborne platforms.”
The goal of the Micro-Technology for Positioning, Navigation and Timing (Micro-PNT) program is to develop technology for self-contained, chip-scale inertial navigation and precision guidance. Other recent breakthroughs from Micro-PNT include new microfabrication methods and materials for inertial sensors.
NovAtel Inc., supplier of OEM GNSS components and subsystems, has announced the addition of a new commercially exportable MEMS IMU to its line of SPAN GNSS/INS products. Available for immediate shipping, this custom Analogue Devices MEMS inertial sensor is exclusive to NovAtel, and can be paired with an OEM6 receiver card to provide continuously available position, velocity and attitude (roll, pitch, yaw) in a small, single-unit form factor.
SPAN tightly couples NovAtel’s precise GNSS technology with highly accurate inertial measurement technology to provide a robust, stable and continuous 3D navigation. The new OEM-ADIS-16488 sensor is designed to be coupled with NovAtel’s OEM6 receivers via the MEMS Interface Card (MIC), providing integrators with a compact, powerful GNSS/INS engine, NovAtel said.
The OEM-ADIS-16488 features low noise gyros and accelerometers in a small, lightweight form factor. This IMU enables precision measurements for applications that require low cost, high performance and rugged durability. Tight-coupling of the two technologies enables continuous robust positioning in difficult environments where satellite signals are unreliable or unavailable for short periods of time.
The OEM-ADIS-16488 is now available for order and immediate shipment.
Unmanned Innovation, a provider of Development Platforms for unmanned aircraft systems (UAS), announced that it has partnered with VectorNav Technologies to integrate VectorNav’s VN-100 inertial measurement unit (IMU) into its os-Series Autopilots. Unmanned Innovation’s os-Series Autopilots offer a customizable solution that enables rapid prototyping and cost-effective production of fixed-wing, helicopter, multi-rotor, and custom configuration UAS. Unmanned Innovation has integrated VectorNav’s VN-100 miniature, calibrated MEMS-based, surface-mount IMU to provide customers the option of a fully calibrated and thoroughly tested IMU.
Unmanned Innovation’s os-Series Autopilots, made commercially available for the first time in November 2012, combine modular hardware with an open architecture, making each autopilot a development platform.
The os-Series Autopilots are offered in multiple form factors with features tailored for various vehicles, payloads, and applications. Each os-Series Autopilot is a complete integrated solution and contains an INS/GPS with air data incorporating the VectorNav VN-100, a datalink radio, payload interfaces, and a Linux computer within one miniature package, starting at 32 grams. The os-Series Autopilots come with professionally written flight control and mission software that Unmanned Innovation provides under a royalty-free license that allows for easy modification, extension, and inclusion in proprietary products.
The partnership between the two companies began during AUVSI’s Unmanned Systems North America 2012 conference in August, where Unmanned Innovation was introduced to VectorNav’s VN-100 and recognized it as an attractive alternative to its existing inertial measurement sensors due to its small form factor, low-cost, and high-precision calibration. Unmanned Innovation’s flexible architecture allowed for quick integration of the VN-100 and VectorNav provided custom firmware with a faster update rate to make the IMU compatible with Unmanned Innovation’s requirements.
The VN-100 IMU, calibrated for bias, scale factor and misalignment errors at room temperature or over the entire thermal operating range of the sensor increased the accuracy of the os-Series Autopilot navigation solution. After a short development cycle, testing and verification, VectorNav’s VN-100 IMUs are now fully integrated within Unmanned Innovation’s os-Series Autopilots. The complete os-Series product line is shipping to customers in the USA and abroad and is free of ITAR restrictions.
“We are very pleased to be working with Unmanned Innovation on their os-Series Autopilot, which we find to be a very unique and high-value product that fills a significant gap in this market,” said John Brashear, VectorNav’s president. “We hope that the VN-100 adds to this value by allowing Unmanned Innovation to focus on its strengths improving the os-Series while securing a long-term, dependable sensing solution and partnership with our company.”
Improving Navigation Continuity Using Parallel Cascade Identification
By Umar Iqbal, Jacques Georgy, Michael J. Korenberg, and Aboelmagd Noureldin
To reliably navigate with fewer than four satellites, GPS pseudoranges needs to be augmented with measurements from other sensors, such as a reduced inertial sensor system or RISS. What is the best way to combine the RISS measurements with the GPS measurements? The classic approach is to integrate the measurements in a conventional tightly coupled Kalman filter. But in this month’s column, we look at how a mathematical procedure called parallel case identification can improve the Kalman filter’s job, when navigating with three, two, one, or even no GPS satellites.
INNOVATION INSIGHTS by Richard Langley
THREE, TWO, ONE, ZERO! Can you still navigate with just a GPS receiver when the number of tracked GPS satellites drops from four to none? As we know, pseu- doranges from a minimum of four satellites, preferably well spaced out in the sky, are required for three-dimensional positioning. That’s because there are four unknowns to estimate: the three position coordinates (latitude, longitude, and height) and the offset of the receiver clock from GPS System Time. If we had a stable clock in the receiver, then we could hold the clock offset constant and have 3D navigation with just three satellites. But for every 3 nanoseconds of clock drift, we will have about 1 meter of pseudorange error, which will lead to several meters of position error depend- ing on the receiver-satellite geometry. On the other hand, we could hold the height coor- dinate constant (viable for navigation over slowly changing topography or at sea) and estimate the horizontal coordinates and the receiver clock offset. So far, so good.
What if the number of tracked satellites then drops to two? We can now only esti- mate two unknowns. They could be the two horizontal coordinates, if we hold both the receiver clock offset and the height fixed. Any errors in those fixed values will propagate into the estimated horizontal coordinates but the resulting position errors might still be acceptable. Instead of holding the clock offset
fixed, we could assume a constant heading and compute the position along the assumed trajectory. However, navigation will rapidly deteriorate as soon as we make the first turn. And one satellite? We would have to make assumptions about the vehicle trajectory, the height, and the clock offset, with likely very poor results. And with no satellites? We might be able to navigate over a short period of time by “dead reckoning,” assuming a constant trajectory and speed, but the resulting positions will be educated guesses at best.
Clearly, if we want to reliably navigate with fewer than four satellites we need to augment the GPS pseudoranges with measurements from some other sensors. A common approach is to use inertial measurement units or IMUs. A complete IMU consists of three accelerometers and three gyroscopes, and small, cost-effective microelectromechanical IMUs are readily available. For land navigation, however, it can be advantageous to use a reduced inertial sensor system or RISS, consisting of one single-axis gyroscope, two accelerometers, and the vehicle speedometer. We can also make use of GPS pseudorange rates along with the pseudoranges.
But what is the best way to combine the RISS measurements with the GPS measurements? The classic approach is to integrate the measurements in a conventional tightly coupled Kalman filter. But in this month’s column, we look at how a mathematical procedure called parallel cascade identification can improve the Kalman filter’s job, when navigating with three, two, or even one GPS satellite.
The Global Positioning System meets the requirements for numerous navigational applications when there is direct line-of-sight (LOS) to four or more GPS satellites. Vehicular navigation systems and personal positioning systems may suffer from satellite signal blockage as LOS to at least four satellites may not be readily available when operating in urban landscapes with high buildings, underpasses, and tunnels, or in the countryside with thick forested areas. In such environments, a typical GPS receiver will have difficulties attaining and maintaining signal tracking, which causes GPS outages resulting in degraded or non-existent positioning information. Due to these well-known limitations of GPS, multi-sensor system integration is often employed. By integrating GPS with complementary motion sensors, a solution can be obtained that is often more accurate than that of GPS alone.
Microelectromechanical systems (MEMS) inertial sensors have enabled production of lower-cost and smaller-size inertial measurement units (IMUs) with little power consumption. A complete IMU is composed of three accelerometers and three gyroscopes. These MEMS sensors have composite error characteristics that are stochastic in nature and difficult to model. In traditional low-cost MEMS-based IMU/GPS integration, a few minutes of degraded GPS signals can cause position errors, which can reach several hundred meters. For full 3D land vehicle navigation, we had earlier proposed a low-cost MEMS-based reduced inertial sensor system (RISS) based on only one single-axis gyroscope, two accelerometers, and the vehicle odometer, and we have integrated this system with GPS. RISS mitigates several error sources in the full-IMU solution; moreover, RISS reduces system cost further due to the use of fewer sensors. Another enhancement can be achieved by using tightly coupled integration, which can provide GPS aiding for a navigation solution when the number of visible satellites is three or lower, removing the foremost requirement of loosely coupled integration, which is visibility of at least four satellites. GPS aiding during limited GPS satellite availability can improve the operation of the navigation system for tightly coupled systems. Thus, in our reseach, a Kalman filter (KF) is used to integrate low-cost MEMS-based RISS with GPS in a tightly coupled scheme.
The KF employed in tightly coupled RISS/GPS integration utilizes pseudoranges and pseudorange rates measured by the GPS receiver. The accuracy of the position estimates is highly dependent on the accuracy of the range measurements. The tightly coupled solutions presented in the literature assume that the pseudorange measurement, after correcting for ionospheric and tropospheric delays, satellite clock errors, and ephemeris errors, only have errors due to the receiver clock and white noise. These latter two are the only errors modeled inside the measurement model in the tightly coupled solutions presented in the literature. Experimental investigation of the GPS pseudoranges for vehicle trajectories in different areas and for different scenarios showed that, in addition, there are residual correlated errors. Since it has been experimentally proven that there are residual correlated errors in addition to white noise and receiver clock errors, we have proposed using a nonlinear system identification technique called parallel cascade identification (PCI) to model such correlated errors in pseudorange measurements.
We propose that the PCI model for the residual pseudorange errors be cascaded with a KF since this nonlinear model cannot be included inside the KF measurement model. The normal operation of a KF is as follows: the difference between the measured pseudorange and pseudorange rate from the mth GPS satellite and the corresponding RISS-predicted estimates of pseudorange and pseudorange rate are used as a measurement update for the KF integration, which computes the estimated RISS errors and corrects the mechanization output. We propose the use of a PCI module, where the role of PCI is to model the pseudorange residual errors. When GPS is available, estimated full 3D position, velocity, and attitude are obtained by integrating the MEMS-based RISS with GPS. In parallel, as a background routine, a PCI model is built for each visible satellite to model its pseudorange error. The actual pseudorange of each visible satellite is used as the input to the model; the target output is the error between the corrected pseudorange (calculated based on corrected receiver position from the integrated solution) and the actual pseudorange. This target output provides the reference output to build the PCI model of the pseudorange residual error. Dynamic characteristics between system input and output help to identify a nonlinear PCI model and the algorithm can then achieve a residual pseudorange error model.
When fewer than four satellites are visible, the identified parallel cascades for the remaining visible satellites will be used to predict the pseudorange errors for these satellites and correct the pseudorange values to be provided to the KF. This improvement of pseudorange measurements will result in a more accurate aiding for RISS, and thus more accurate estimates of position and velocities.
We examined the performance of the proposed technique by conducting road tests with real-life trajectories using a low-cost MEMS-based RISS. The ultimate check for the proposed system’s accuracy is during GPS signal degradation and blockage. This work presents both downtown scenarios with natural GPS degradation and scenarios with simulated GPS outages where the number of visible satellites was varied from three to zero. The results are examined and compared with KF-only tightly coupled RISS/GPS integration without pseudorange correction using a PCI module. This comparison clearly demonstrates the advantage of using a PCI module for correcting pseudoranges for tightly coupled integration.
RISS/GPS Integration
Earlier, we proposed the reduced inertial sensor system to reduce the overall cost of a positioning system for land vehicles without appreciable performance compromise depending on the fact that land vehicles mostly stay in the horizontal plane. It is the gyroscope technology that contributes the most both to the overall cost of an IMU and to the performance of the INS. In RISS mechanization, the heading (azimuth) angle is obtained by integrating the gyroscope measurement, ωz. Since this measurement includes the component of the Earth’s rotation as well as rotation of the local level frame on the Earth’s curved surface, these quantities are removed from the measurement before integration. Assuming relatively small pitch and roll angles for land vehicle applications, we can write the rate of change of the azimuth angle directly in the local level frame as: (1)
where ωe is the Earth’s rotation rate, φ is the latitude, ve is the east velocity of the vehicle, h is the altitude of the vehicle and RN is the normal (prime vertical) radius of curvature of the vehicle’s position on the reference ellipsoid.
The two horizontal accelerometers can be employed for obtaining the pitch and roll angles of the vehicle. Thus, a 3D navigation solution can be achieved to boost the performance of the solution. When the vehicle is moving, the forward accelerometer measures the forward vehicle acceleration as well as the component due to gravity, g. To calculate the pitch angle, the vehicle acceleration derived from the odometer measurements, aod, is removed from the forward accelerometer measurements, fy. Consequently, the pitch angle is computed as:
(2)
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, fx, must be compensated for the normal component of acceleration. The roll angle is then given by:
(3)
The computed azimuth and pitch 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 (ve, vn, and vu respectively) as follows: (4)
where is the rotation matrix that transforms velocities in the vehicle body frame to the navigation frame. The east and north velocities are transformed and integrated to obtain position in geodetic coordinates (latitude, φ, and longitude, λ). The vertical component of velocity is integrated to obtain altitude, h. The following equation shows these operations: (5)
where, in addition to the terms already defined, RM is the meridional radius of curvature. We have used the KF as the estimation technique for tightly coupled RISS/GPS integration in our approach. KF is an optimal estimation tool that provides a sequential recursive algorithm for the optimal least mean variance (LMV) estimation of the system states. In addition to its benefits as an optimal estimator, the KF provides real-time statistical data related to the estimation accuracy of the system states, which is very useful for quantitative error analysis. The filter generates its own error analysis with the computation of the error covariance matrix, which gives an indication of the estimation accuracy.
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, such as atmospheric errors, as well as synchronization errors between the satellite and receiver clocks.
After correcting for the satellite clock error and the ionospheric and tropospheric errors, we can obtain a corrected pseudorange. The receiver clock error and the residual errors remaining in the corrected pseudorange, assumed as white Gaussian noise, are the only errors modeled inside the measurement model in the tightly coupled solutions presented in the literature. Experimental investigation of the GPS pseudoranges in trajectories in different areas and under different scenarios showed that the residual errors are not just white noise as assumed in the literature, but, in fact, are correlated errors. As the GPS observables are used to update the KF, a technique must be developed to adequately model these errors to improve the overall performance of the KF. We propose using PCI to model these correlated errors. A PCI module models these errors, and then provides corrections prior to sending the GPS pseudoranges to aid the KF during periods of GPS partial outages (when the number of visible satellites drops below four).
Parallel Cascade Identification
What is PCI? System identification is a procedure for inferring 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 since nonlinear techniques make analysis much more complicated and difficult to implement than for the linear case. However, for more realistic dynamic characterization nonlinear techniques are suggested. PCI is a nonlinear system identification technique proposed by one of us [MJK]. This 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 shown in Figure 1 can be built up one cascade at a time.
Figure 1. Block diagram of parallel cascade identification.
It has been proven 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. [A Volterra series, named after the Italian mathematician and physicist Vito Volterra, is similar to the more familiar infinite Taylor series expansion of a function used, for example, in systems analysis, but the Volterra series can include system “memory” effects.] It has been shown 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. 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.
The method begins by approximating the nonlinear system by a first such cascade. The residual (that is, 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 kth cascade, with yo(n) = y(n). Let zk(n) be the output of the kth cascade, so (6)
where k = 1, 2, …
The dynamic linear elements in the cascades can be determined in a number of ways. The method we have employed uses cross correlations of the input with the current residual. Best fitting of the current residuals was used to find the polynomial coefficients of the static nonlinearities. These resulting cascades are such that they drive the cross-correlations of the input with the residuals to zero. However, a few basic parameters have to be specified in order to identify a parallel cascade model, including the memory length of the dynamic linear element that begins each cascade, the degree of the polynomial static nonlinearity that follows the linear element (this polynomial is best fit to minimize the mean-square error (MSE) of the approximation of the residual), the maximum number of cascades allowable in the model, and a threshold based on a standard correlation test for determining whether a cascade’s reduction of the MSE justifies its addition to the model.
Augmented Kalman Filter
In the previous section, 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, we propose the use of a KF-PCI module, where the role of PCI is to model the residual errors of GPS pseudoranges.
In full GPS coverage when four or more satellites are available to the GPS receiver, the KF integrated solution provides an adequate position benefiting from both GPS and RISS readings, and the PCI builds the model for the pseudorange errors for each visible satellite. The input of each PCI module is the pseudorange of the visible mth GPS satellite, and the reference output is the difference between the observed pseudorange and the estimated pseudorange from the corrected navigation solution.
The reference output has no corrections during full GPS coverage. It is only used to build the PCI model. Dynamic characteristics between system input and output help to achieve a residual pseudorange error model as shown in the Figure 2.
Figure 2. Block diagram of augmented KF-PCI module for pseudoranges during GPS availability.
During partial GPS coverage, when there are fewer than four satellites available, the PCI modules for all satellites cease training, and the available PCI model for each visible satellite is used to predict the corresponding residual pseudorange errors, as shown in Figure 3. The KF 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 to obtain a correction. Thus, the corrected pseudorange can then be obtained.
During a full GPS outage, when no satellites are available, the KF operates in prediction mode and the PCI modules neither provide corrections nor operate in training mode.
FIGURE 3 Block diagram of augmented KF-PCI module for pseudoranges during limited availability of GPS.
Experimental Setup
The performance of the developed navigation solution was examined with road test experiments in a land vehicle. The experimental data collection was carried out using a full-size passenger van carrying a suite of measurement equipment that included inertial sensors, GPS receivers, antennae, and computers to control the instruments and acquire the data as shown in the Figure 4. The inertial sensors used in our tests are packaged in a MEMS-grade IMU. The specifications of the IMU are listed in Table 1.
Table 1. IMU specifications.
The vehicle’s forward speed readings were obtained from vehicle built-in sensors through the On-Board Diagnostics version II (OBD II) interface. The sample rate for the collection of speed readings was 1 Hz. The GPS receiver used in our integrated system was a high-end dual-frequency unit. Our results were evaluated with respect to a reference solution determined by a system consisting of another receiver of the same type, integrated with a tactical grade IMU.
This system provided the reference solution to validate the proposed method and to examine the overall performance during simulated GPS outages.
Several road test trajectories were carried out using the setup described above. The road test trajectory considered for this article was performed in the city of Kingston, Ontario, Canada, and is shown in Figure 5. This road test was performed for nearly 48 minutes of continuous vehicle navigation and a distance of around 22 kilometers. Ten simulated GPS outages of 60 seconds each were introduced in post-processing (they are shown as blue circles overlaid on the map in Figure 5) during good GPS availability. The trajectory was run four times with the simulated partial outages having three, two, one, and zero visible satellites, respectively. The case with no satellites seen is like a scenario that would occur in loosely coupled integration. The errors estimated by KF-PCI and KF-only solutions were evaluated with respect to the reference solution.
Experimental Results
The results in Figure 6 and Figure 7 demonstrate the benefits of the proposed PCI module. The main benefit of using PCI for pseudorange correction is the modeling capability, which enables correction of the raw GPS measurements. The benefit of more satellite availability can also be seen from these results. Figures 6 and 7 clearly show that both the average maximum position error and the average root-mean-square (RMS) position error are lower with the KF-PCI approach compared to the conventional KF, even when data from only one satellite is available.
Figure 6. Bar graph showing average maximum positional errors for all outages.Figure 7. Bar graph for RMS positional errors for all outages.
To gain more insight about the performance of the proposed technique to enhance the aiding of the KF by correcting the pseudoranges, we can look at the results of KF-PCI and KF approaches with different numbers of satellites visible to the receiver for one of the artificial outages. Figure 8 shows a map featuring the different compared solutions during outage number 8. Eight solutions are presented for the cases of three, two, one, and zero satellites observed for the standard KF and KF with PCI. To get some idea of the vehicle dynamics during this outage, we can examine Figure 9, which depicts the forward speed of the vehicle as well as its azimuth angle as obtained from the reference solution. There is a significant variation in speed, with only a small variation in azimuth.
Figure 8. Performance of tightly coupled 3D-RISS during outage #8.Figure 9. Vehicle dynamics (speed and azimuth) during GPS outage #8.
Figure 10 illustrates the performance differences between the KF-PCI and KF-only solutions for different numbers of satellites for this outage. Similar to Figure 7, Figure 10 shows the average RMS position differences between the KF-PCI and KF-only solutions and the reference solution (without the artificial outages). While the differences increase as the number of available satellites decreases, the accuracies may still be acceptable for many navigation purposes.
And while the differences between the KF-PCI and KF-only approaches for this particular outage are small, the KF-PCI approach consistently provides better accuracy.
Figure 10. Performance of PCI-KF (shades of blue for different number of satellites) and KF (shades of green for different number of satellites) of tightly coupled 3D-RISS during outage #8.
Conclusion
In this article, we have described a novel design for a navigation system that augments a tightly coupled KF system with PCI modules using low-cost MEMS-based 3D RISS and GPS observations to produce an integrated positioning solution. A PCI module is built for each satellite during good signal availability where the integrated solution presents a good position estimate. The output of each PCI module provides corrections to the GPS pseudoranges of the corresponding visible satellite during GPS partial outages, thereby decreasing residual errors in the GPS observations. This KF-PCI module was tested with real road-test trajectories and compared to a KF-only approach and was shown to improve the overall maximum position error during GPS partial outages.
Future work with PCI for modeling the residual pseudorange errors will consider replacing the KF with a particle filter to provide more robust integration and a consistent level of accuracy.
Acknowledgments
The research discussed in this article was supported, in part, by grants from the Natural Sciences and Engineering Research Council of Canada, the Geomatics for Informed Decisions (GEOIDE) Network of Centres of Excellence, and Defence Research and Development Canada. The equipment was acquired by research funds from the Directorate of Technical Airworthiness and Engineering Support, the Canada Foundation for Innovation, the Ontario Innovation Trust, and the Royal Military College of Canada. The article is based on the paper “Modeling Residual Errors of GPS Pseudoranges by Augmenting Kalman Filter with PCI for Tightly Coupled RISS/GPS Integration” presented at ION GNSS 2010, the 23rd International Technical Meeting of the Satellite Division of The Institute of Navigation held in Portland, Oregon, September 21–24, 2010.
Manufacturers
The test discussed in this article used a NovAtel Inc. OEM4 dual-frequency GPS receiver and a Crossbow Technology Inc., now Moog Crossbow IMU300CC-100 MEMS-grade IMU. The On-Board Diagnostics data was accessed with a Davis Instruments CarChip Pro data logger. The reference solutions were provided by a NovAtel G2 Pro-Pack SPAN unit, interfacing a NovAtel OEM4 receiver with a Honeywell HG1700 tactical grade IMU.
Umar Iqbal is a doctoral candidate at Queen’s University, Kingston, Ontario, Canada. He received a master’s of electrical engineering degree in integrated positioning and navigation systems from Royal Military College (RMC) of Canada, Kingston, in 2008. He also holds an M.Sc. in electronics engineering from the Ghulam Ishaq Khan Institute of Engineering Sciences and Technology, Topi, Pakistan, and a B.Sc. in electrical engineering from the University of Engineering and Technology, Lahore, Pakistan. His research focuses on the development of enhanced performance navigation and guidance systems that can be used in several applications including car navigation.
Jacques Georgy received his Ph.D. degree in electrical and computer engineering from Queen’s University in 2010. He received B.Sc. and M.Sc. degrees in computer and systems engineering from Ain Shams University, Cairo, Egypt, in 2001 and 2007, respectively. He is working in positioning and navigation systems for vehicular, machinery, and portable applications with Trusted Positioning Inc., Calgary, Alberta, Canada. He is also a member of the Navigation and Instrumentation Research Group at RMC. His research interests include linear and nonlinear state estimation, positioning and navigation by inertial navigation system/global positioning system integration, autonomous mobile robot navigation, and underwater target tracking.
Michael J. Korenberg is a professor in the Department of Electrical and Computer Engineering at Queen’s University. 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.
Aboelmagd Noureldin is a cross-appointment associate professor with the Department of Electrical and Computer Engineering at Queen’s University and the Department of Electrical and Computer Engineering at RMC. He is also the founder and leader of the Navigation and Instrumentation Research Group at RMC. He received the B.Sc. degree in electrical engineering and the M.Sc. degree in engineering physics from Cairo University, Giza, Egypt, in 1993 and 1997, respectively, and the Ph.D. degree in electrical and computer engineering from The University of Calgary, Calgary, Alberta, Canada, in 2002. His research is related to artificial intelligence, digital signal processing, spectral estimation and de-noising, wavelet multiresolution analysis, and adaptive filtering, with emphasis on their applications in mobile multisensor system integration for navigation and positioning technologies.
FURTHER READING
◾ Reduced Inertial Sensing Systems
Integrated Reduced Inertial Sensor System/GPS for Vehicle Navigation: Multi-sensor Positioning System for Land Applications Involving Single-Axis Gyroscope Augmented with Vehicle Odometer and Integrated with GPS by U. Iqbal and A. Noureldin, published by VDM Verlag Dr. Müller, Saarbrucken, Germany, 2010.
“A Tightly-Coupled Reduced Multi- Sensor System for Urban Navigation” by T.B. Karamat, J. Georgy, U. Iqbal, and A. Noureldin in Proceedings of ION GNSS 2009, the 22nd International Technical Meeting of the Satellite Division of The Institute of Navigation, Savannah, Georgia, September 22–25, 2009, pp. 582–592.
“An Integrated Reduced Inertial Sensor System – RISS/GPS for Land Vehicle” by U. Iqbal, A.F. Okou, and A. Noureldin, in Proceedings of PLANS 2008, IEEE/ION Position Location and Navigation Symposium, Monterey, California, May 5–8, 2008, pp. 912– 922, doi: 0.1109/PLANS.2008.4570075.
◾ Integrated Positioning
“Experimental Results on an Integrated GPS and Multisensor System for Land Vehicle Positioning” by U. Iqbal, T.B. Karamat, A.F. Okou, and A. Noureldin in International Journal of Navigation and Observation, Hindawi Publishing Corporation, Vol. 2009, Article ID 765010, 18 pp., doi: 10.1155/2009/765010.
“Performance Enhancement of MEMS Based INS/GPS Integration for Low Cost Navigation Applications” by A. Noureldin, T.B. Karamat, M.D. Eberts, and A. El-Shafie in IEEE Transactions on Vehicular Technology, Vol. 58, No. 3, March 2009, pp. 1077–1096, doi: 10.1109/TVT.2008.926076.
Aided Navigation: GPS with High Rate Sensors by J.A. Farrell, published by McGraw-Hill, New York, 2008.
Global Positioning Systems, Inertial Navigation, and Integration by M.S. Grewal, L.R. Weill, and A.P. Andrews, 2nd ed., published by Wiley- Interscience, Hoboken, New Jersey, 2007.
“Continuous Navigation: Combining GPS with Sensor-based Dead Reckoning by G. zur Bonsen, D. Ammann, M. Ammann, E. Favey, and P. Flammant in GPS World, Vol. 16, No. 4, April 2005, pp. 47–54.
Applied Optimal Estimation by the Technical Staff, Analytic Sciences Corp., ed. A. Gelb, published by The MIT Press, Cambridge, Massachusetts, 1974.
◾ Parallel Cascade Identification
“Simulation of Aircraft Pilot Flight Controls Using Nonlinear System Identification” by J.M. Eklund and M.J. Korenberg in Simulation, Vol. 75, No. 2, August 2000, pp.72–81, doi: 10.1177/003754970007500201.
“Parallel Cascade Identification and Kernel Estimation for Nonlinear Systems” by M.J. Korenberg in Annals of Biomedical Engineering, Vol. 19, 1991, pp. 429–455, doi: 10.1007/ BF02584319.
“Statistical Identification of Parallel Cascades of Linear and Nonlinear Systems” by M.J. Korenberg in Proceedings of the Sixth International Federation of Automatic Control Symposium on Identification and System Parameter Estimation, Washington, D.C., June 7–11, 1982, Vol. 1, pp. 580–585.
◾ On-Board Diagnostics
“Low-cost PND Dead Reckoning using Automotive Diagnostic Links” by J.L. Wilson in Proceedings of ION GNSS 2007, the 20th International Technical Meeting of the Satellite Division of The Institute of Navigation, Fort Worth, Texas, September 25–28, 2007, pp. 2066–2074.
