^{*}

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/).

Most portable systems like smart-phones are equipped with low cost consumer grade sensors, making them useful as Pedestrian Navigation Systems (PNS). Measurements of these sensors are severely contaminated by errors caused due to instrumentation and environmental issues rendering the unaided navigation solution with these sensors of limited use. The overall navigation error budget associated with pedestrian navigation can be categorized into position/displacement errors and attitude/orientation errors. Most of the research is conducted for tackling and reducing the displacement errors, which either utilize Pedestrian Dead Reckoning (PDR) or special constraints like Zero velocity UPdaTes (ZUPT) and Zero Angular Rate Updates (ZARU). This article targets the orientation/attitude errors encountered in pedestrian navigation and develops a novel sensor fusion technique to utilize the Earth’s magnetic field, even perturbed, for attitude and rate gyroscope error estimation in pedestrian navigation environments where it is assumed that Global Navigation Satellite System (GNSS) navigation is denied. As the Earth’s magnetic field undergoes severe degradations in pedestrian navigation environments, a novel Quasi-Static magnetic Field (QSF) based attitude and angular rate error estimation technique is developed to effectively use magnetic measurements in highly perturbed environments. The QSF scheme is then used for generating the desired measurements for the proposed Extended Kalman Filter (EKF) based attitude estimator. Results indicate that the QSF measurements are capable of effectively estimating attitude and gyroscope errors, reducing the overall navigation error budget by over 80% in urban canyon environment.

The provision to pedestrians of location and orientation information, which in some way can be used for simplifying the task of reaching a particular destination, is called pedestrian navigation. The versatility of environments in which the pedestrian navigation system has to work, differentiates it from land vehicle, sea vessel and aircraft navigation. Of all the possible environments in which the PNS has to work, the urban and indoor environments are the most challenging ones. These challenges arise from the amount and reliability of information available for estimating the navigation parameters. This diversity of environment for pedestrian navigation imposes the use of self-contained sensors for navigation that can provide users with navigation parameters irrespective of the availability of external aids.

Pedestrian navigation can be either accomplished using some man-made information source or by measuring the planetary/universal forces. Radio Frequency (RF) signals are the most widely used man-made information sources for pedestrian navigation [

The use of sensors additional to gyroscopes and accelerometers constitutes the first category. For example, with magnetometers, the measure of the Earth’s magnetic field can assist the estimation of the direction of motion, which can further be used for estimating errors associated with gyroscopes [

The second category comprises the use of special constraints dictated by the biomechanical description of the user’s dynamics [

As is evident from the above discussion, a number of approaches can be taken to target pedestrian navigation. Only one approach seems feasible for seamless navigation in all the pedestrian navigation environments, namely the PDR approach, which utilizes the self-contained navigation systems. Assuming successful detection of gait events, the main limitation of PDR is the orientation estimation. A novel idea has emerged for mitigating the gyroscope errors and estimating the orientation parameters using the Earth’s magnetic field even when it is perturbed by man-made infrastructure [

Section 2 introduces the Earth’s magnetic field, its usefulness for orientation estimation, and the effects of indoor environments on it. Section 3 describes different approaches feasible for pedestrians’ attitude estimation, whereas, Section 4 details an Extended Kalman Filter (EKF) based estimator required for mitigation of attitude and sensor errors. The novel technique utilizing perturbed magnetic field for estimating attitude and sensor errors is introduced in Section 5 and Section 6 details the measurement error models required for the EKF. Section 7 addresses the statistical analysis of the proposed mitigation technique. Finally, Section 8 is dedicated to the experimental assessment of the proposed algorithm in a real world environment.

The Earth’s magnetic field is a naturally occurring planetary phenomenon, which can be modeled as a dipole and follows the basic laws of magnetic fields summarized and corrected by Maxwell [

With the advancements in sensor technology, this field can now be precisely measured with the help of a sensor commonly known as the magnetometer. With the proper transformation of the field components to the horizontal plane and knowing the declination angle specific to the measurement area and time, a simple trigonometric operation estimates the geographic heading. The magnetic field vector is elaborated in _{x}_{v}_{z}

The orthogonal components of the horizontal magnetic field are used for estimating the heading. Thus the field components must first be transformed to a local level in order to find the horizontal field component of the measured Earth’s magnetic field. For estimating the heading with respect to true North instead of the magnetic North, the declination angle also needs to be predicted using one of the Earth’s magnetic field models [

Modeling of the Earth’s magnetic field is possible in an indoor environment in the presence of magnetic dipoles known as magnetic perturbations [

For better understanding the nature of artificial magnetic perturbations and their influence on Earth’s magnetic field measurements, an exhaustive assessment of the magnetic field was conducted by surveying different indoor and outdoor environments [

The digital implementation of the mathematical equations governing the attitude of a body with respect to a reference frame is known as an attitude computer [

using two or more vector measurements,

propagating the attitude using angular rate measurements.

Using vector measurements for attitude estimation requires finding a common transformation matrix that maps all of the vector measurements in body frame to those in a navigation frame by minimizing the cost function known as Wahba’s problem [^{th}^{th}

As the vector measurements are not available for estimating the attitude at every epoch for pedestrian navigation, propagation of the attitude in time is necessary. For this purpose, an inertial sensor providing angular rate measurements, namely the rate gyroscope, can be used [_{1}, _{2}, _{3} and _{4} are the four elements describing the quaternion

Attitude and its error estimation constitute a non linear problem [

As only the attitude/orientation estimation for pedestrian navigation is targeted herein, the main states to be estimated are the three attitude angles. The attitude is primarily determined using the angular rates obtained using the rate gyroscopes. The deterministic errors associated with this sensor can be compensated for using the calibration parameters [

Here, _{ωx}, _{ωy}_{ωz}

The perturbed state vector representing errors in _{ω}

From

The matrix

Substituting

In _{ω}_{ω}_{ω}_{ω}_{ω}

The time varying gyroscope bias is modeled as an exponentially correlated noise term, resulting in the derivative of perturbations in the time varying gyroscope bias δ_{ω}_{bω} is the correlation time and _{bω} is the noise vector for the stochastic modeling of the time varying gyroscopes’ biases.

The measurements used for compensating the errors associated with the attitude vector in general and gyroscope biases in particular are herein solely based on the magnetic field vector measurements. In order to utilize the clean as well as perturbed magnetic field measurements, a novel technique for estimating the sensor frame’s angular rates using a tri-axis magnetometer was developed and is detailed in the following section.

The novel idea for estimating attitude and gyroscope errors using magnetic field measurements for pedestrian navigation environments mainly involves detecting quasi-static total magnetic field periods during pedestrian motion and utilizing them as measurements for estimating attitude and gyroscope errors. Indeed when the local magnetic field is quasi-static, the rate of change of the magnetic field is combined with the rotational rate of change of the inertial device generating an estimated gyroscope error, which can be further used to correct for time-varying inherent gyroscope errors [

The Earth’s magnetic field, though a good source of information for estimating heading outdoor, suffers severe degradations in the indoors caused by magnetic field perturbations [_{k}_{k−1}

Let the hypothesis for a non-static field be _{0}_{1}

The rate of change of the total magnetic field is also contaminated by white Gaussian noise _{k}, which, when modeled with the measurements, gives:
_{k}_{0}_{1}_{0}_{k}_{k}_{n} = {l ∈ N:

As the complete knowledge about ||_{k}_{0}

Let the Maximum Likelihood Estimator (MLE) for the unknown parameter in case of _{0}

Now the PDF for _{0} becomes:

For hypothesis _{1}, the rate of change of the total magnetic field is known (it will be zero), therefore the PDF in this case becomes:

The Generalized Likelihood Ratio Test (GLRT) for detecting a quasi-static field is given by:

Substituting for the PDFs in

Taking the natural log on both sides and simplifying yields:

Once the QSF periods are detected during pedestrian motion, the next step is to utilize the magnetic field information during such periods for estimating the attitude angles and the gyroscope errors. This section derives the equations required for using QSF magnetic field measurements, even perturbed, in the EKF developed in Section 4. It principally consists of a measurement error model that is used for updating the state vector of the navigation filter.

Let the magnetic field measurement in the sensor frame at the start, ^{th} epoch, of the quasi-static field period be given by:

Considering the attitude at the start of quasi-static period as the reference for the measurement model, the magnetic field measurement can be transformed to the navigation frame using:

At the ^{th} epoch,

Ideally, as the magnetic field during QSF period is locally static (magnetic field vector not changing its magnitude or direction), _{B}

Simplifying (37) to get a relationship between measurements and states, one obtains:

Because the last term in

From

During the quasi-static field periods, the rate of change of the reference magnetic field is zero. Using this information as a measurement, one gets:

Taking the derivative of

During user motion, the magnetic field components in the body frame will encounter changes, which can be modeled by

The first two terms give the rate of change of the reference magnetic field, which, during QSF periods, is zero.

Combining

In order to quantify the performance of the proposed quasi-static magnetic field detector, statistical analysis was conducted. From

This factor corresponds to the variance of the total field gradient when the field itself is not changing. This gives a measure of the gradient noise that is encountered during quasi-static field periods. More than seven hours of data in five different environments that are common for pedestrian navigation have been collected and analyzed for evaluating this parameter [^{2}.

The Receiver Operating Characteristics (ROC) curve allows one to select the test statistics acceptance threshold based on the required probability of detection _{d}_{f}_{d}_{d}_{d}_{d}

In order to assess the performance of the proposed QSF based orientation/attitude estimation algorithm, test data was collected in a real world environment, which is most common for pedestrian navigation applications: downtown. The test setup used for analyzing the impact of the proposed algorithm on attitude estimation comprised a Multiple Sensor Platform (MSP) and an optical wheel encoder developed by the author [^{−3} m/s. This walking speed is later resolved into North and East components using the estimated attitude to compute the position, the latter being obtained by integrating the velocity components. As this article focuses only on attitude estimation, the wheel encoder provides accurate speed measurements, which are necessary for de-correlating the velocity error budget from the attitude one, allowing the assessment of attitude accuracies only.

In an actual portable device such as a smart-phone, the walking speed would be measured by accelerometers. Although smart-phones of today are equipped with accelerometers, performing gait analysis with a handheld device is a challenging task and constitutes a research topic in itself. Thus with the use of wheel encoder, the experimental assessments of the proposed attitude estimator in the position domain as described herein, can be considered free of errors induced by speed sensors (accelerometers), providing a better insight into attitude accuracy. However the MSP developed for this research is hosting a tri-axis of accelerometers, which can be used in future for investigating different methods to estimate stride length and speed. This work targets the implementation of a complete pedestrian navigation system.

The MSP was rigidly mounted on a plastic plate, which can be easily carried in a hand. The wheel encoder is mounted on a pole that can be held by a pedestrian and pushed along the ground for measuring the walking speed.

In order to assess the impact of the proposed algorithm on attitude estimation for pedestrian navigation, the solution repeatability criterion is chosen. Multiple paths of the same trajectory were followed in different environments keeping the same starting and ending point to assess the performance. The paths were followed so as to keep the separation between them within one metre if possible. This is achieved by following prominent patterns on the ground (tiles boundaries, pavement markings/intersections

In order to assess the impact of the proposed algorithms on attitude estimation for pedestrian applications, a urban canyon was selected.

Urban canyons can be considered as one of the regions where pedestrian navigation applications have a lot of commercial significance. Also before moving indoor, one often ends up being in an urban canyon for some time. Hence detailed analysis of the proposed algorithm in this environment is very important.

The observability of gyroscope biases using QSF measurements is presented graphically in

The maximum trajectory error obtained using raw magnetic heading based on measurement models for estimating continuous trajectory, whose length was longer than 1 km (for the three loops) is approximately 87 m whereas that for the QSF measurement model is approximately 16 m. Thus the trajectory errors are reduced by 80% by utilizing QSF measurements for constraining the attitude error growth and estimating the gyroscopes’ errors.

This article investigated the use of handheld devices (smart-phones) equipped with low cost consumer grade sensors for pedestrian navigation. As the attitude/orientation errors play a major role in the overall navigation error budget, the focus was on improving the attitude estimates in environments where GPS is denied. For this purpose, the use of the Earth’s magnetic field as a measurement source for estimating the errors associated with low cost inertial sensors was investigated.

A novel method for utilizing the Quasi-Static magnetic Field (QSF), regardless of perturbation, to mitigate gyroscope errors has been developed and the corresponding equations have been presented. The novel estimation technique proved to deliver a high level of performance, reducing the trajectory errors by 80% for a distance of more than 1 km. This method detected the QSF periods during pedestrian’s motion and related the changes in the magnetic field components during these periods with the angular rates of the sensor block, thus providing measurements for directly assessing the errors associated with the rate gyroscopes. Development of an Extended Kalman Filter (EKF) for modeling the attitude and gyroscope errors as well as relating these to the magnetic field measurements have been conducted. They gave an insight into the interdependence of different parameters, which proved beneficial in identifying the limitations of the proposed model.

Selection of an urban canyon (magnetically disturbed outdoor field) for the experimental assessments provided data sets for a realistic and detailed analysis of the performance of the proposed algorithm. Analyzing the results in the position domain with the sensor platform carried in a hand gave detailed insight into the impact of the attitude estimator on the position error budget. A high accuracy wheel encoder made it possible to isolate the attitude errors from the position ones.

Even with an 80% error reduction in the positioning domain as compared with classical integration of magnetometers and gyroscopes in attitude estimation filter, the use of QSF measurements to successfully estimate the gyroscope errors resulted in constant orientation errors. This showed that this scheme is not sufficient for observing the absolute attitude errors. The position errors reached 16 m for a traversed trajectory of more than 1 km.

The use of accelerometers and pressure sensors for estimating pedestrian’s position as well as speed is necessary to completely assess the impact of this research in real world navigation scenarios. Both of these sensors are already incorporated in the MSP developed for this research and will form the basis for the above suggested research.

Research into the resolution of the ambiguity between sensor and body frames, which was constrained to be zero for the experiment herein, is needed. This can be achieved by using the accelerometers, but requires detailed modeling of the pedestrian’s walk related to arm swing or hip joint motion. The algorithm developed herein is self-contained and assumed a fully denied GNSS environment. However, GNSS is partly available in urban canyons and in numerous indoor environments. Hence research into the integration of the two approaches to maximize availability and accuracy is in order.

The financial support of Research In Motion (RIM), the Natural Science and Engineering Research Council of Canada, Alberta Advanced Education and Technology and Western Economic Diversification Canada is acknowledged.

Earth’s magnetic field in Cartesian coordinate system.

Heading estimates from clean and perturbed magnetic field.

Test setup for the magnetic field survey including the Bartington fluxgate and the SPAN CPT HG1700 INS System from NovAtel.

Probability Density Function (PDF) of total field gradient during constant field periods.

ROC for different window sizes.

Continuous QSF periods and their occurrence.

Duration of gaps between QSF periods.

Test data collection setup.

Downtown Calgary data collection environment.

Data collection in downtown Calgary.

Total field and QSF detections for similar paths in urban canyon.

Raw heading measurements versus QSF measurements.

Estimated

Estimated

Trajectories obtained using QSF in urban canyon.

Continuous trajectory in urban canyon using QSF.

Parameters selected for the QSF detector.

Window Size (N) | 3 |

Probability of detection (P_{d}) |
0.82 |

Probability of false alarm (P_{f}) |
0.30 |

Threshold (γ) | 0.14 |