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

A single low cost inertial measurement unit (IMU) is often used in conjunction with GPS to increase the accuracy and improve the availability of the navigation solution for a pedestrian navigation system. This paper develops several fusion algorithms for using multiple IMUs to enhance performance. In particular, this research seeks to understand the benefits and detriments of each fusion method in the context of pedestrian navigation. Three fusion methods are proposed. First, all raw IMU measurements are mapped onto a common frame (

As GPS markets continue to expand and new applications are found every day, any new application must abide by a key requirement, namely direct line-of-sight between the satellites and the receiver. So stringent is this requirement that the simple occlusion of satellites renders many navigation systems useless or highly degraded. As users travel in urban canyons, parkades, indoors or in high foliage areas, the ability for GPS to provide a navigation solution is compromised. Although High Sensitivity GPS (HSGPS) receivers can track weak signals through fading, this renders them susceptible to high noise and multipath errors [

Inertial measurement units (IMU) are a common complement to GPS, although it is technically more correct to state that GPS augments an inertial navigation system (INS). The advantage being that together the GPS and inertial sensors can provide a continuous navigation solution, where GPS alone cannot. As competitive consumer markets drive the price of mobile navigation devices lower, an increasingly common choice for IMUs is micro electro-mechanical systems (MEMS). Their size, cost, weight and low power consumption make them an attractive grade of IMU; however their in-run biases, scale factors and high noise require an effective integration scheme to mitigate these errors [

While existing INS research has involved one IMU, the purpose of this paper is to investigate the use of multiple IMUs in tandem with GPS. In particular, this paper will investigate various approaches to integrate multiple IMUs with several filter architectures and constraints that can be used to further improve the accuracy and availability of the navigation solution, with emphasis on pedestrian navigation.

The objectives of this paper, which is based on [

Discuss the implementation and test results of the following techniques to utilize multiple IMUs and GPS observations for pedestrian navigation:

Virtual IMU observation fusion

Centralized filter design

Federated filter design

Assess fault detection capability on the IMU and GPS measurements, discussing any limitations.

Analyze and compare the performance of the different estimation architectures selected and the number of IMUs used.

Analyze the performance of each architecture in residential and indoor conditions.

Potential pedestrian navigation users include: first responders (e.g., emergency search and rescue), cellular phone users (E911 and navigation), health and activity monitoring, recreational users (e.g., hikers, climbers, skiers), self-guided tourists, athletes and athletic trainers, consensual tracking (e.g., elderly, parolees, employees), navigation for the visually impaired and police/military forces.

A key to the success of many INS pedestrian navigation applications is the placement of the IMU on a foot (e.g., [

This configuration reduces the necessity for magnetometers, although these can be used to aid with attitude determination as in [

GPS and IMUs have been successfully integrated since the formal introduction of GPS. More recently, attention has been placed on integration with MEMS IMUs to reduce cost, but still provide robust navigation solutions. A natural progression is to use more IMU sensors and thus capitalize on the decreasing cost of MEMS sensors in order to improve overall accuracy. As such, researchers commonly fuse multiple IMU measurements in the raw observation (

Numerous studies have taken an observation domain approach to redundant IMU (RIMU) integration whereby the observations of several IMUs are fused, generating a single virtual IMU measurement [

In the development of VIMU theory, optimizing the configuration of the IMU sensor axes is an important consideration. Pejsa mathematically determined the optimal configuration for sensor axes; with sensors in a skewed formation rather than an orthogonal one (although the ideal 3-axis sensor is orthogonal) [

The prominent method of RIMU fusion fuses raw IMU observations using least squares estimation, mapping each IMU observation to a virtual IMU frame (which requires

Often, the purpose of virtual IMU integration is not to improve the accuracy (although this is a desirable outcome), but rather to facilitate the detection and exclusion of faulty observations [

Another benefit of the virtual IMU scenario is a direct real time estimate of the VIMU process noise, as derived from each IMU [

Averaging of IMUs’ observations is simple and the least computationally burdensome method of forming a VIMU, however because each IMU is located at a different point on the body, the IMUs measure different specific forces relative to the location of the VIMU origin. Consequently, the fusion must be performed in the same reference frame and the transformation of each gyro and accelerometer observation set into this frame must be performed. The transformation is assumed to be known ^{th} IMU in its body frame,
^{th} IMU (known a priori) and

The specific force, as derived from a VIMU relative to a rigidly attached body, is given as [^{th} IMU,
^{th} IMU and VIMU origins within the VIMU body frame.

To the authors’ knowledge, the second and third term on the right hand side of

In the VIMU least-squares model, the unknown parameters are the angular velocity, angular acceleration and specific force vectors of the VIMU. As a result of the cross products within _{v} ×][_{v} × _{3×1} × b_{3×1} = [a ×]_{3×3}b [

The nine parameter least-squares estimation operates in a standard fashion. It uses all gyro and accelerometer measurements as observations and provides an estimation of the virtual IMU accelerometer and gyro measurements. If five IMUs are used, then the system has 30 observations and operates at the same frequency as the incoming observations. Measurements were weighted equally because the IMUs are all the same brand and model, although this is not a requirement.

This nine-parameter least-squares model has a unique characteristic when two IMUs are used. When using two IMUs, the design matrix will only ever have a maximum rank of eight, indicating that only eight of the nine parameters are actually solvable. Conceptually, the linear dependency arises due to the fact that any angular acceleration about the vector between the two IMUs (

The angular acceleration is the time derivative of the angular velocity and therefore a differential equation exists that relates these states. This forms the basis of a VIMU Kalman filter. A VIMU Kalman filter further reduces noise and can enhance navigation performance. The differential equations of the nine states are as follows:
_{f}_{α}

Determining the optimal values for _{f}_{α}

The filter predicts and updates at the same frequency as the incoming measurements (

The VIMU filter must operate with IMUs which are time synchronized. The adaptive Kalman filter could still function if the IMUs are synchronized but output observations at different data rates or if the observations had different time stamps. The required time synchronization is related to the angular dynamics, specifically the angular acceleration, and will incorrectly determine the specific force at the VIMU location.

This section will demonstrate that FDE is not always a viable option for MEMS IMUs with large biases, scale factors and acceleration-based gyroscope errors, in particular when IMUs experience significant accelerations and angular velocities. Fault detection works on the premise that the misclosure or innovation sequence is zero mean. As the biases and scale factors of each IMU have not been estimated, and therefore not removed from the observations, the observation model is not zero mean and therefore FDE effectiveness is compromised.

Residuals computed from a nine-state least-squares estimation of each sensor axis are shown in ^{2}, which corresponds to the highest acceleration within the gait cycle. Large gyro residuals of nearly 20 °/s are also observed and also correspond to high dynamics. During the stance phase of the gait, the residuals are much smaller, often in the range of the biases. Therefore, the magnitude of the residuals is clearly correlated to high dynamics.

Because the magnitude of the residuals is a function of dynamics rather than sensor errors, the input covariance matrix must accommodate these large variations, otherwise faults will be detected during every gait cycle (or whenever the IMU experiences high dynamics). With a VIMU architecture, each IMUs sensor error cannot be modeled individually. Thus, if FDE was to be performed, the input covariance matrix would not be a function of sensor noise, but rather have to contain an increased amount of error to account for uncorrected sensor errors. Therefore it is a recommendation herein that FDE

This multi-IMU approach uses a centralized filter that is composed of several individual block filters (e.g., [

The centralized filter proposed in this paper is referred to as a stacked filter, consisting of several individual INS filters. In this manner several “block” filters (

The stacked filter contains parameters for position, velocity, attitude, accelerometer and gyro biases and accelerometer and gyro scale factors for each IMU. If five IMUs are used, then there are five 21-states filters contained within one centralized 105 state filter. Each block filter can be updated at the same time or individually, but the entire filter prediction cycle must be synchronized (to avoid different block times, within the stacked filter). An advantageous characteristic of the stacked filter (and federated filters) is that each block filter could contain additional or different IMU error states, thus facilitating varying types and qualities of IMUs and error state models, which the VIMU architecture does not. Since the IMUs are all the same brand and model, the block filters are identical with slightly varied input process noise parameters for each IMU. The block form of the stacked filter is:
^{th} block filter transition matrix,
^{th} block filter states (21 state model),
^{th} block filter of the observations,
^{th} block filter, and
^{th} block filter.

The stacked transition matrix of

During a GPS update, each block filter requires its own misclosure vector, derived from the GPS observations. However, if each block requires its own misclosure vector, the GPS observations must be repeatedly used for each IMU, thereby directly violating fundamental Kalman filter theory [_{k}^{th} block state vector of the k^{th} epoch.

Because the stacked filter contains multiple position, velocity and attitude states, one for each IMU, the filter can be updated with relative position, velocity and attitude (PVA) information that is known

The inter-IMU vector is measured in one of the IMU’s body frame and is computed by differencing the lever arms (^{1} is the estimated position vector of the 1^{st} block filter and ^{2,1} is the

It is important to note that by differencing the lever arms to generate the inter-IMU vector, the lever arms must be in the same frame and not their respective body frames. Since the Earth Centered Earth Fixed (ECEF) frame was used as the navigation frame, the inter-IMU vector must be rotated into that frame. Consequently, there is an inherent relationship between the efficacy of the relative position update (RUPT) and the error in the orientation of the body frame relative to the ECEF frame.

The update is applied periodically to facilitate a convergence of the block INS filter, reduces numerical computations and limits the inter-block correlation accumulation. Using experimental filter tuning, a periodicity of 6 s and a standard deviation of 1 cm (a diagonal matrix) provided the best performance.

The relative velocity of a point on a moving rigid body is given by Marion & Thornton [^{2,1} is the relative velocity between the IMUs 2 and 1, _{1} is the angular velocity vector measured by IMU 1, and ^{2,1} is the vector between IMUs 1 and 2.

The vector between the IMUs is assumed to be known ^{1} is the velocity vector of the 1^{st} block filter and ^{2} is the velocity vector of the 2^{nd} block filter.

As with the relative position update, the relative velocity observation is derived in the body frame and must be rotated into the navigation frame, thus creating a similar relationship between the error of the rotation and the RVUPT. The standard deviation used for RVUPTs was 2 cm/s and was derived using the propagation of variances of

The relative attitude update follows a similar procedure to the relative position update. The misclosure vector is formed using the difference in estimated Euler angles of each IMU and the pre-surveyed Euler angles describing the rotation between them. In this research the IMUs are fixed on the same platform and mounted on adjacent faces thereby allowing simple Euler angle identification. The relative attitude observation equation is given by:
^{B}^{1}, ^{B}^{1}, ^{B}^{1} is the roll, pitch and yaw of the first IMU, respectively. The standard deviation of this observation is 0.1 rad (

Since GPS observations are repeated within the stacked filter, the FDE process is slightly modified for GPS observations. The modification eliminates the possibility that GPS observations may be rejected for one block filter and accepted for another, while at the same time improving the reliability of the fault detection scheme. The effect of the blunder vector and its mapping matrix on the observation vector can be described as:
_{k} is the blunder mapping matrix and ∇_{k} the vector of known blunders.

It is in this equation that the FDE algorithm will be modified to test a series of observations (corresponding to a single GPS measurement) rather than elements of the innovation sequence. The M matrix is generated based on the GPS observations and number of IMUs used. For example, the M matrix with three pseudoranges, repeated for two block filters in a stacked filter, with a single fault in the first observation will be ^{T} . The test statistic is then computed from with direct reference to the GPS observations as [

The test statistic is a chi-squared distribution. The null and alternate hypotheses are:
_{0} is the non-centrality parameter. With these hypotheses, the test is conducted by rejecting the null hypothesis if

The MDB of the stacked filter can then be determined as:

Assuming that the innovation covariance matrix is equivalent between block filters, the improvement in the MDB

To the authors’ knowledge, there has been no published work in the domain of decentralized filters incorporating multiple IMUs. Federated filters were introduced in the late 1980s and early 90s for GPS and INS integration (e.g., [

Federated filtering is defined herein as a decentralized filter that incorporates information sharing between local and master filters. References [

The federated filters discussed herein contain common states. Specifically, the shared states are position (

The reference data of the local filters can be formed by one of two methods. The first method is to use GPS observations, whereby each local filter operates in a tightly coupled manner (

The FNR filter is fundamentally equivalent to running each IMU through an INS filter and combining the final results of each solution via least squares. The master fusion is performed via least squares with each local filter’s PVA providing the observations.

Thus, if there are five IMUs, the master estimator contains 45 observations and correspondingly, a 45 × 45 observation covariance matrix. The master’s input observation covariance matrix is block diagonal, however the internal PVA correlation remains within the off diagonal elements (_{n}_{9}_{x}_{9} is not diagonal). The PVA of the local filters is in reality correlated as a result of using the same GPS observations and moreover by potentially similar dynamics if the IMUs are rigidly mounted together. Therefore the input observation covariance matrix is scaled by n^{−1} to reduce the weight of each correlated observation.

The FFR filter has a similar structure to the FNR filter, but the master filter parameters (and its corresponding covariance matrix) are shared with the local filters. The information factor for each local INS filter is n^{−1} because the IMUs are all the same brand and model. The input to the master fusion is the same as the FNR filter. Furthermore, since the states of the INS extended Kalman filter are zero, the PVA of the master fusion replaces the PVA used to provide the expansion point, rather than the actual values in the state vector. The covariance information of the local filters, however, is replaced with the actual values from the local and master filters. Additionally, because correlation develops within the local filter PVA states and IMU error states, these intra filter correlations must be set to zero, otherwise the filter will diverge. Further, the covariance replacement of the i^{th} local filter with the master state covariance matrix is as follows, the first nine states representing the PVA having been replaced:
_{i}^{th}_{M}_{i12x12} remains unmodified during the covariance replacement because it contains the bias and scale factors of the i^{th}

Tuning the filters presented a significant (and time consuming) problem. There are five tunable parameters for each sensor (

Therefore, a generic set of tuning parameters was used for each data set for all IMUs. Only minor modifications to the spectral densities were allowed to accommodate each sensor noise range. Consequently, the same parameters used in the single IMU solution were used in every other multi-IMU solution. Although the solutions may be somewhat sub-optimal, the methodology facilitates better filter performance comparisons, rather than tuning performance comparisons.

Data was collected in two environments: a typical North American residential home and inside the Olympic Oval of the University of Calgary. The residential home, as shown in

The Olympic Oval, shown in

To collect the data, the test subject carried a rigid aluminum backpack to house a tactical grade reference INS, two laptops to collect the GPS and IMU data, and batteries to power the equipment. A NovAtel SPAN system was used to provide the reference solution. It consists of a Honeywell HG1700 AG58 IMU and a NovAtel OEM4 GPS receiver. The data in this case was differentially post-processed with a nearby (<1 km) reference station to provide a reference trajectory. The data was processed with NovAtel’s Inertial Explorer software in forward and reverse directions, smoothed using RTS smoothing [

The high sensitivity GPS receiver used was a u-blox Antaris 4 Precision Timing AEK-4T evaluation kit with firmware 5.0. The antenna was a u-blox ANN-MS, designed and manufactured by Allis Communications Co Ltd as antenna M827B [

Although the lever arm is time variant, the variation is symmetric about the fixed lever arm. It is under this assumption that solutions can be compared to within a decimetre error envelope.

The VIMU solutions contain more noise as a result of the decreased spectral densities used within the filter. This effect was amplified when GPS measurements were stronger (

The cumulative densities (CDs) of the horizontal and vertical errors are shown in

The stacked filter, FNR and FFR filter’s horizontal errors are shown in

Since the GPS signal strength is still reasonable in this environment, the additional information contained within the relative updates did not further improve the accuracy of the final solution. This indicates that the filter’s biases and scale factors had been resolved and other unmodeled error sources begin to dominate the solution’s accuracy. The FNR (INS) performed 6.3% worse than the FNR (GPS), which indicates that using the raw ranges of the GPS receiver as input to each local filter is superior.

The stacked filter showed the largest percent increase with two IMUs, but then decreased with the addition of the third and fourth IMU. The third and fourth IMUs were among the least accurate SINS solutions. Thus, when the filter combined the block filter solutions, the final solution was degraded. This contradicts the hypothesis that the relative updates would have provided additional information to improve the accuracy of each block filter. This contradiction is refuted with the data set from the Olympic Oval, which shows that in the absence of reasonable GPS observability, the relative updates significantly improve the navigation solution.

The FNR (GPS) results followed a similar trend to that of the block filter, again suggesting that the relative updates were providing little improvement to navigation solutions in this case. The FFR (INS) filter performance plateaued at the third IMU and had similar results with three to five IMUs, only increasing 0.1% per additional IMU. The FNR (INS) percentage improvement was minute with only 0.3, 0.4 and 1.2% for each additional IMU.

Consistent with the results of the VIMU architecture in Section 6.1, the addition of the second IMU had the largest percentage increase, even more so than the third, fourth or fifth IMU. This suggests that if two IMUs are used, the stacked, FNR (GPS) or VIMU AKF all show similar performance. However, when using more than two IMUs, the solution accuracy improves at a lower rate.

The Olympic Oval presents a different approach to that of Section 6 as in this environment, GPS will not provide acceptable performance for most applications and an integrated system is needed.

The VIMU horizontal errors are shown in

The VIMU tends to diverge much more slowly when entering the indoors and converges much more quickly when exiting, compared to the SINS solution. That said, at time 185 s, the solution very quickly diverged from a 6 m error to nearly a 40 m error. This was a direct result of a strong multipath signal that had a high C/N_{o}. The filter consequently overweighed the pseudorange and the VIMU filters were unable to reject this information. This effect has been seen in all the filters during this research and presents a problem that could not be solved without manual intervention of the observation covariance matrix.

The SINS and the FFR (INS) error profiles in

To compare the results of each filter,

For the Oval data, the user entered and exited the track at the same point and therefore provided an interesting metric to compare the solutions. The FNR (GPS) filter only deviated by 2.5 m, the SINS difference was 13.5 m and the standalone GPS solution had a 49.3 m difference. The same check of the reference system yielded a 5.1 m difference.

The accuracy of each architecture as a function of the number of IMUs is shown in

The stacked filter had a linear improvement for each additional IMU of about 3 to 7% per IMU added. This again indicates the value of the relative updates, as each additional IMU provided additional relative information to improve the accuracy of the solution and the error states within the block filters. The FNR (INS) and the FFR (INS) results did not increase linearly, but plateaued similarly to the results of two IMUs. The FNR (GPS) slightly decreased with each additional IMU in excess of two.

The FNR (INS) and FFR (INS) results were very similar to the residential data set with very moderate improvements as each IMU was added. The FNR (GPS) also had similar results between data sets with a slight decrease in performance with more IMUs. The two data sets confirm that the federated filter architecture did not increase the accuracy, but merely processed the data in a similar manner to that of the centralized version.

There is a large difference in the computer processing speed of each architecture and for the number of IMUs used. An exact comparison of the computational load is beyond the scope of this paper, but

All data was processed on an Intel Core 2 Quad CPU with 3.25 GB of RAM. This analysis is merely intended to be comparative, since there are numerous factors that determine processing speed. The slowest architecture was the stacked filter. This was mostly due to the inversion required for the gain matrix computation, which has n times m rows and columns (n is the number of IMUs and m is the number of GPS observations); propagating the filter forward was also a burden. This was the only filter that could not run in real time. For those interested in operating a stacked filter in real time, several processing enhancements could be made to reduce the computational load. These include processing observations sequentially, using integer based data types, reducing the IMU data rates, propagating the filter for longer intervals rather than shorter more frequent ones or using factorization methods such as Cholesky decomposition. Readers are referred to [

Three architectures were proposed for which multi-IMU data can be fused to provide improved navigation performance. The filters proposed specifically assess the integration schemes within the scope of pedestrian navigation. The objective was to compare the results of three architectures and provide insight into the advantages and disadvantages of each, providing a better understanding of the accuracy and availability for each filter.

The stacked filter provided better results compared to its federated reset free counterpart, which showcases the use of relative updates and a better fault detection algorithm. Although the improvement was minor in the residential data set, the filter was already operating at a high performance level with the use of only moderately attenuated GPS signals. In the Olympic Oval data set, the stacked filter performed 9% better with five IMUs than the federated reset free filter. The multi-IMU federated filters accuracy reached a maximum with two IMUs, whereas the stacked filter accuracy linearly increases 3 to 7% with each additional IMU. This suggests that the relative updates provide a linear relationship with the number of IMUs, at least up to five units.

When GPS measurements were used as the reference information for the local filters of the federated filter, the performance was 15% better than when a single INS solution was used as the reference for the federated filter. The time correlation of the output of the INS solution resulted in a dramatic decrease in performance of the local filters.

Within the VIMU scope, FDE is not practical unless the systematic errors have been removed prior to testing for faults. Performance within the FDE is severely hindered by the dynamics of the IMU and the magnitude of the scale factors, biases and acceleration based gyroscope errors. There is also no evidence within this research to suggest that FDE on IMU measurements would increase navigation accuracy or availability; the primary interests of pedestrian navigation.

Processing times of the filters differ, but the stacked filter requires the most processing time, followed by the federated filters, VIMU AKF, VIMU LSQ and VIMU average.

Virtual IMU Observation Fusion Architecture.

Specific Force Residuals from a Virtual IMU Computed from Least-Squares Estimation.

Federated Filter Architecture of Multiple IMUs.

Residential House used for Data Collection.

Olympic Oval (Left: roof top with trajectory in red, Right: inside showing track and ice level).

Rigidly Mounted IMUs on the Foot.

VIMU Horizontal Errors (Five IMUs Used in Residential Data Set).

CD of Horizontal and Vertical Errors (Residential Data Set).

Stacked and Federated Filter Horizontal Errors (Residential Data Set).

CD of Horizontal and Vertical Errors for Stacked and Federated (Residential Data Set).

VIMU Accuracy as a Function of IMUs Used (Residential Data Set).

Average C/N_{o} and HDOP (Olympic Oval Data Set).

VIMU Horizontal Errors (Five IMUs Used in Olympic Oval Data Set).

CD of VIMU Horizontal and Vertical Errors (Five IMUs Used in Olympic Oval Data Set).

Horizontal Error of Stacked and Federated Filters (Five IMUs Used in Olympic Oval Data Set).

CD of Horizontal and Vertical Errors for Stacked and Federated Filters (Olympic Oval Data Set).

Loop 2 (Clock Wise) Map View of Best Performing Filters—Truth Solution (

VIMU Accuracy Improvement as a Function of IMUs Used (Olympic Oval Data Set).

Processing Speed of Various Architectures.

Comparison of the Various Architectures.

Enhanced GPS Observation FDE | No | Yes | No |

IMU Observation FDE | Not Recommended | No | No |

Reduced Noise at Mechanization Input | Yes | No | No |

Constrains Estimator using Relative PVA | No | Yes | No |

Estimates Each IMUs Bias and Scale Factor | No | Yes | Yes |

IMU Time Synchronization Not Required | No | Yes | Yes |

Reference and MEMS Grade IMU Maximum Errors.

| |||
---|---|---|---|

Accelerometer | In Run Bias (mg) | 1 | 51 |

Turn on Bias (mg) | - | 30 | |

Scale Factor (PPM) | 300 | 10,000 | |

Random Walk (g/√Hz) | 2.16 × 10^{−6} |
370 × 10^{−6} | |

| |||

Gyro | In Run Bias (°/h) | 1 | 2,160 |

Turn on Bias (°/h) | - | 5,400 | |

Scale Factor (PPM) | 150 | 10,000 | |

Random Walk (°/h/√Hz) | .5 | 226.8 |