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IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 52, NO.1, FEBRUARY 2003
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A Method for Dead Reckoning Parameter Correction in Pedestrian Navigation System Rommanee Jirawimut, Piotr Ptasinski, Vanja Garaj, Franjo Cecelja, and Wamadeva Balachandran, Senior Member, IEEE
Abstract—This paper presents a method for correcting dead reckoning parameters, which are heading and step size, for a pedestrian navigation system. In this method, the compass bias error and the step size error can be estimated during the period that the global positioning system (GPS) signal is available. The errors are used for correcting those parameters to improve the accuracy of position determination using only the dead reckoning system when the GPS signal is not available. The results show that the parameters can be estimated with reasonable accuracy. Moreover, the method also helps to increase the positioning accuracy when the GPS signal is available. Index Terms—Dead reckoning (DR), global positioning system (GPS), Kalman filter, parameter correction, pedestrian navigation.
I. INTRODUCTION
I
N a pedestrian navigation system, it is necessary to locate the position of the user in any environment. For this reason, a self-contained navigation system based on a dead reckoning (DR) principle is of interest. To locate the position of the user, distance and heading from a known origin have to be measured at an acceptable level of accuracy. In a pedestrian navigation system, an electronic pedometer can be used to count the number of steps, which can be combined with the step size for obtaining the distance traveled. In addition, a terrestrial magnetic compass can be used as a heading sensor. The dead reckoning system is dependent upon the accuracy of measured distance and heading and the accuracy of the known origin. Apart from sensor measurement noise the main factors, which affect the accuracy, are compass bias error and step size error. The compass bias is a result of several possible causes such as magnetic declination and body offset. In a clean environment, the total bias can be changed slowly over a long period and may need to be recalibrated occasionally. The step size error can be defined as the difference between the actual step size and the predetermined step size entered by the user. Although the exact step size is not necessary for the distance calculation, the average step size over a short period has to be measured. The reason for this is that the step size of the user may vary according to the environment; for example, the step size of the user is shorter when the user is walking in a crowded area. Hence, the
predetermined step size cannot be used effectively for the distance measurement. The global positioning system (GPS) is a satellite-based radio-navigation system, which provides three-dimensional absolute position, velocity, and time information. The horizontal positioning accuracy of GPS is currently about 10 m and can be improved to about 5 m under a differential mode. In a signal blocked environment, the “availability” is significantly deteriorated. The key point of a dead reckoning algorithm for a pedestrian navigation system is the step size of the user, which is used as a scale factor in an ordinary dead reckoning algorithm. Unlike a scale factor of an odometer for a car navigation system [7], the step size is a time-varying process. Moreover, the step size is dependent on several factors. The main ones are the velocity and step frequency of the user. In 1999, Levi and Judd [1] suggested that the step size could be estimated online based on a linear relationship between measured step frequency and the step size. A real-time step calibration algorithm was proposed in [2] using a Kalman filter with GPS positioning measurement and three estimated parameters: frequency, amplitude, and mean. These parameters are obtained from another extended Kalman filter, the input of which is the measured -axis acceleration. On the other hand, Jee et al. [8] proposed a system, which calculated walking velocity from a pre-determined step size together with step frequency (pulse/second) measured by an electronic pedometer. A Kalman filter, implemented as an indirect filter, was used to estimate the error of the walking velocity with the velocity measurement from a GPS receiver. The corrected walking velocity was then fed into a dead reckoning equation for determining the position of the user. In this paper, we propose an integrated system utilizing the measured time interval between two successive steps, which is a fundamental measurement of an electronic pedometer. The inputs of an extended Kalman filter (EKF) are measured step period and heading, as well as position and velocity from a GPS receiver. When the GPS signal is available, the outputs of the filter are the position and velocity, together with the step size error and the compass bias error at each epoch of the GPS update. II. DEAD RECKONING SENSORS
Manuscript received May 29, 2001; revised August 19, 2002. This work was supported by the Electronic Systems and Information Technology Research Group, Department of Systems Engineering, Brunel University, Middlesex, U.K. The authors are with the Brunel University, Uxbridge, Middlesex, U.K. (e-mail: [email protected]). Digital Object Identifier 10.1109/TIM.2002.807986
In a DR system, the displacement from the previous known position is calculated from the measured heading and single integral of measured velocity. The walking velocity is an estimated step size divided by the measured step period. The step size or step length is defined as the distance between the same
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Fig. 1. E-position versus N-position from GPS position (GPS) and Kalman-filtered GPS position (KF-GPS). The numbers 1 and 2 refer to the first and second round, respectively.
point on each foot, usually the heel, during double-limb support [4]. Several methods have been suggested to detect step occurrences. One such method is to detect the peaks of vertical acceleration [1], which correspond to the step occurrences. In this study, a solid-state orientation sensor module, comprised of orthogonal arrays of magnetometers and accelerometers, was used for collecting the three-axis accelerometer signals and calculated body angles, which are roll, pitch and yaw, at 20 Hz sampling frequency. In the sensor module, the roll and pitch angles were measured by two inclinometers, formed by the orthogonal accelerometers. The yaw angle or heading was measured by the magnetometers. The sensor was placed rigidly, around the center of the waist, on the back of three walkers. Using Fourier analysis, the fundamental frequencies of roll and yaw angles were determined and found to be identical and equal to half of the pitch fundamental frequency. This is similar to the results published in [3]. Furthermore, we found that the Fourier analysis of the fundamental frequency of the measured vertical acceleration was also identical to that of pitch. In this paper the pitch measurement was used effectively to detect the step occurrence. In the experiment, the signals from the sensor and GPS receiver were collected and processed offline on a personal computer. For the DR, the pitch signal was fed into a digital low-pass filter for canceling double peaks; then a peak-detection algorithm was applied to detect single peaks. The time between two successive peaks was used as step time or step period for the DR. The yaw signal was fed into a low-pass filter and was sampled at the update rate of the GPS receiver, which was set to 1 Hz. The sampled yaw signal was used as the heading measurement for the DR.
III. KALMAN FILTER A. Process Model The state equation of the extended Kalman filter is written as follows [5]: (1) , is considered as a zero-mean The process noise vector, white process. The position and velocity for pedestrian navigation are in local East-North-Up (ENU) coordinates. In this experiment, we considered only two-dimensional (2D) position and velocity, i.e., east (E) and north (N). Other parameters of interest are step size error and compass bias error together with position and velocity for navigating the pedestrian. Hence, the state vector for the EKF can now be written as (2) E-position, E-velocity, N-position, where N-velocity, step size error, compass bias error, and denotes the vector and matrix transposition. Denoting the time index by , the relationships between the true step size and step size error can be written as (3) step size used for DR velocity calculation, where true step size, and step size error. The step size error was modeled as a first-order Gauss-Markov process. The reason behind this is that the previous step sizes affect the current step
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Fig. 2. E-position versus N-position from GPS position (GPS) and integrated positioning solution (Integration). The numbers 1 and 2 refer to the first and second round, respectively. For the integration in both rounds, complete GPS position data and DR measurements were fed into the integration model.
size in order to maintain the walking velocity. The effect of previous steps is controlled by the time constant of the process. The compass bias is a result of body offset and magnetic declination. Without magnetic field disturbance, the measured heading can be written as (4) measured heading, true heading, where true compass bias, and additive zero-mean Gaussian noise . The relationships between the true compass with variance , and the compass bias error, , can be written as bias, (5) compass bias used for correcting heading measurewhere ment in the DR velocity calculation. The compass bias error, , is also correlated with the previous errors. Thus, it is modeled as a first-order Gauss-Markov process. The velocity and position are modeled as random walk and integrated random walk processes, respectively. Hence, the state transition matrix, , can be written as
B. Measurement Model The general measurement model can be written as follows. (9) , is assumed to be a The measurement noise vector, zero-mean white Gaussian noise process with co-variance , and uncorrelated with the process noise vector, . The measurement vector is written as (10) measured E-position from the GPS receiver, where measured N-position from the GPS receiver, measured measured N-velocity E-velocity from the GPS receiver, measured step period, and from the GPS receiver, measured compass heading. is calcuThe measurement matrix or observation matrix lated using the following equations; (11)
(6) (7) (8) update rate of DGPS and GPS, time conwhere time constant of the comstant of the step size error, and pass bias error.
(12) To give an accurate solution from a Kalman filter, the error budgets of the measurement-noise covariance matrix should be as close to the real measurement noise statistics as possible. Hence, we take into account a method to adapt and tune the variance of the position and velocity measurement noises, according to the horizontal dilution of precision (HDOP) of the GPS satellite
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Fig. 3. Step sizes of Fig. 2: the step size calculated from GPS velocity (GPS) and the corrected step size from the integration model (Integration: Corrected).
constellation [6]. Let and denote the variances of E-position, N-position, E-velocity, and N-velocity, respectively. The variances can be calculated at each GPS update as follows: (13) (14) where
one-sigma pseudo-range error of GPS, and one-sigma delta-range error of GPS.
C. Parameter Correction When the GPS signal was available, the step size and compass bias errors were estimated by the EKF. The estimated errors were fed into the DR system to correct the step size and compass bias in (3) and (5), respectively. This is an indirect feedback structure. When the GPS signal was unavailable, the step size and the compass bias were calculated from the step size average and compass bias average of the last 100 seconds. Over the period that the GPS signal was unavailable, the E- and N-velocities were calculated from the step size average and the heading measurement corrected by the compass bias average. The calculated E- and N-velocities were then fed into an ordinary Kalman filter (KF) to estimate the position and velocity for the navigation. IV. RESULTS AND DISCUSSIONS An experiment to test the method was conducted in a field of size 47.4 m 34 m. A man was asked to walk for two rounds with a set of equipment. The equipment is comprised of the orientation sensor module for the roll, pitch and yaw measurements
sampled at 20 Hz, a GPS receiver, and a computer notebook for collecting data. In order to specify the start and stop samples of each round, the walker was asked to press time stamps when walking past those points. As shown in Fig. 1, 3, and 6, the walking path started about 30 m from the lower left corner. The path was straight with those 90-degree turns at the corners, and the walker walked clockwise. In the figures, the “GPS” refers to the raw position from the GPS receiver, and the “KF-GPS” refers to the Kalman filtered GPS position using a position-velocity model [5]. On the other hand, the “Integration” refers to the GPS and DR data processed by the method described in Section III. The number after these captions refers to the number of times the person walked around the field. It can be seen from Fig. 1 that at the beginning and the end of the second round, the GPS position data deviate from the actual path. The Kalman filtered-GPS data the other hand, in Fig. 2, the positioning solution from the integrated system using the integration model in Section III is closer to the actual path than those of the GPS and KF-GPS. This points out the fact that without magnetic field disturbance, the magnetic compass with the bias correction from the integration model gives the correct heading, which leads to higher positioning accuracy. Therefore, even when the GPS signal is available, the heading measured by the magnetic compass can help to increase the positioning accuracy. The step size and heading of the walking in Fig. 2 are shown in Fig. 3 and 4, respectively. The vertical lines in the figures represent the end of each round. It should be noted that the initial step size and initial compass bias were set to zero. Figs. 3 and 4 show that the step size and heading are corrected effectively by the integration model without the necessity of initializing the true step size and true compass bias. In addition, the step size and compass bias could be estimated within the first few
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Fig. 4. Headings of Fig. 2: the heading calculated from GPS velocity (GPS), the heading measured by the magnetic compass (Compass heading), the heading bias from the integration model (Integration: Bias) and the corrected heading from the integration model (Integration: Corrected).
Fig. 5. E-position versus N-position from GPS position (GPS) and Kalman-filtered GPS position (KF-GPS). The numbers 1 and 2 refer to the first and second round, respectively. On round 2, the GPS position data were not fed into the integration model.
seconds. In Fig. 4, the compass bias at about radians is the result of the body offset since the orientation sensor module was placed on the back of the user. The time constants for step size error and compass bias error processes in (8) were assigned to large values and were not changed for the whole experiment. As large time constants bring
about slow changes of step size error and compass bias error estimates, and vice versa, this assumption is adequate for our experiment in which the subject was walking in the same environment. However, some improvements can be made to this system for more robust estimation. One possible approach is to use a Kalman filter bank with different levels of Gauss-Markov
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(a)
(b) Fig. 6. State variance of (a) E-position and (b) N-position. KF-GPS refers to the Kalman-filtered GPS position as plotted in Fig. 1. Integration-Complete data refers to the position from the integration model with complete GPS and DR data as plotted in Fig. 2. Integration-Incomplete data refers to the position from the integration model without GPS data but DR data only in the second round as plotted in Fig. 2.
process parameters including the time constant. The result from each filter in the bank can be combined by probabilistic approaches such as in the Interacting multiple model (IMM), generalized pseudo-Bayesian approach of first order (GPB1), and generalized pseudo-Bayesian approach of second order (GPB2) [9], [10]. By this method, the system can cope with the step size and compass error changes due to the changing environment. However, the computational complexity will be considerably increased. To analyze the system when the GPS signal is unavailable, in Fig. 5, the GPS data of the second round was not fed into the integration model. It can be seen that on the right side of the second round, the unexpected compass heading error due to magnetic field disturbances brings about the positioning error, and the error of the estimated step size causes a shorter path. However, the shape of the path is still acceptable and may be fitted to a digital map (map matching), if available. To compare the positioning error, the state variances of E-position and N-position are plotted in Fig. 6. It can be seen that the variances of the integrated positioning solution with GPS data (shown as “Complete data”) are much lower than the variance of the KF-GPS. This supports the result in Fig. 2. When the GPS signal was unavailable (the second round of the “Incomplete data”), the state variances of the integrated positioning solution increase with time and almost reach the same level as the state variances of the KF-GPS.
V. CONCLUSION This paper describes a method to estimate the errors of the DR parameters, which are step size and heading measurement,
in a pedestrian navigation system. The estimated step size and compass bias errors are used for correcting the DR parameters. With this method, the averaged step size can be estimated online, so that the user does not need to specify his/her step size. This also benefits when the user is not walking with the usual step size, e.g., walking in a crowded area. In addition, the magnetic compass can be placed anywhere on the body since the compass bias can be estimated effectively. Moreover, this method also helps to increase the positioning accuracy when the GPS signal is available.
REFERENCES [1] R. W. Levi and T. Judd, “Dead reckoning navigational system using accelerometer to measure foot Impacts,”, U.S. Patent US5583776, Dec. 1999. [2] V. Gabaglio and B. Merminod, “Real-time calibration of length of steps with GPS and accelerometers,” in GNSS, 1999, pp. 599–605. [3] N. Molton, S. Se, J. M. Brady, D. Lee, and P. Probert, “A stereo visionbased aid for the visually impaired,” Image Vis. Comput., vol. 16, no. 4, pp. 251–263, Apr., 3 1998. [4] J. Rose and J. G. Gamble, Human Walking. Baltimore, MD: Williams & Wilkins, 1993. [5] R. G. Brown and P. Y. C. Hwang, Introduction to Random Signals and Applied Kalman Filtering. New York: Wiley, 1997. [6] B. W. Parkinson and J. J. Spilker, Global Positioning System: Theory and Application: AIAA, 1996, vol. 2. [7] J. Mar and J.-H. Leu, “Simulations of the positioning accuracy of integrated vehicular navigation systems,” in Proc. Inst. Elect. Eng. Radar, Sonar Navigation, vol. 143, Apr. 1996, pp. 121–128. [8] G.-I. Jee, J.-S. Hong, S.-I. Yoon, Y. J. Lee, and C. G. Park, “Portable positioning system for mobile communication network performance test,” in ION GPS, Sept. 1999, pp. 1091–1098. [9] Y. Bar-Shalom and X.-R. Li, Estimation and Tracking : Principles, Techniques and Software. Norwood, MA: Artech House, 1993. [10] S. Blackman and P. Robert, Design and Analysis of Modern Tracking Systems. Norwood, MA: Artech House Radar Library, 1999.
JIRAWIMUT et al.: METHOD FOR DEAD RECKONING PARAMETER CORRECTION
Rommanee Jirawimut received the B.Eng. degree from King Mongkut’s University of Technology Thonburi, Thailand, in 1998 and the M.Sc. degree in communication and signal processing from Imperial College of Science, Technology, and Medicine, London, U.K., in 1999. She is currently pursuing the Ph.D. degree at the Department of Systems Engineering, Brunel University, Uxbridge, Middlesex, U.K.
Piotr Ptasinski received the Eng. and M.Sc. degrees in electronics and telecommunication in 1998 from the University of Mining and Metallurgy, Krakow, Poland. He is currently pursuing the Ph.D. degree at the Electronic Systems and Information Technology Group in the Systems Engineering Department, Brunel University, Uxbridge, Middlesex, U.K.
Vanja Garaj received the B.Sc. degree in product design from the Department of Design, Faculty of Architecture, University of Zagreb, Zagreb, Croatia, in 1998. She is currently purusing the Ph.D. degree at the Department of Systems Engineering, Faculty of Technology and Information Systems, Brunel University, Uxbridge, Middlesex, U.K.
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Franjo Cecelja was born in Krizevci, Croatia, on February 21, 1955. He received the Dipl.Eng. degree in electronics from the University in Zagreb, Zagreb, Croatia, the M.Sc. degree in control from Cranfield Institute of Technology, Cranfield, U.K., and the Ph.D. degree in electrical engineering from Brunel University of West London, Middlesex, U.K. From 1978 to 1992, he pursued research in control and sensor technology. In 1992, he joined the University of Surrey, Surrey, U.K., where he was engaged in research on optical sensors and EM interaction with biological systems. He is currently a Lecturer at Brunel University, with research interests in EM radiation, near-field measurements, and general signal processing, which resulted in more than 80 scientific papers.
Wamadeva Balachandran (M’91–SM’96) received the B.Sc. degree from the University of Ceylon, Colombo, Sri Lanka, in 1970, and the M.Sc. and Ph.D. degrees from the University of Bradford, Bradford, U.K., in 1975 and 1979, respectively. He is Professor of electronic systems and Head of the Department of Systems Engineering, Brunel University of West London, Middlesex, U.K. He is actively pursuing research in electromagnetic sensors, measurement systems, medical electronics, DGPS navigation systems, electrohydrodynamics, and charge particle dynamics. He has published over 200 papers in these fields and filed eight patent applications. He is an editorial board member of the International Journal of Atomization and Sprays and the International Journal of Particle Science and Technology. Prof. Balachandran is the past Chairman of the Static Electrification Committee of the Institute of Physics, London, International President of the Institute of Liquid Atomization and Spray Systems, and a member of the IEEE Electrostatic Process Committee. He is a Fellow of the IEE, Institute of Measurement and Control, Institute of Physics, and the Royal Society of Arts, U.K.
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A Method for Dead Reckoning Parameter Correction in Pedestrian Navigation System Rommanee Jirawimut, Piotr Ptasinski, Vanja Garaj, Franjo Cecelja, and Wamadeva Balachandran, Senior Member, IEEE
Abstract—This paper presents a method for correcting dead reckoning parameters, which are heading and step size, for a pedestrian navigation system. In this method, the compass bias error and the step size error can be estimated during the period that the global positioning system (GPS) signal is available. The errors are used for correcting those parameters to improve the accuracy of position determination using only the dead reckoning system when the GPS signal is not available. The results show that the parameters can be estimated with reasonable accuracy. Moreover, the method also helps to increase the positioning accuracy when the GPS signal is available. Index Terms—Dead reckoning (DR), global positioning system (GPS), Kalman filter, parameter correction, pedestrian navigation.
I. INTRODUCTION
I
N a pedestrian navigation system, it is necessary to locate the position of the user in any environment. For this reason, a self-contained navigation system based on a dead reckoning (DR) principle is of interest. To locate the position of the user, distance and heading from a known origin have to be measured at an acceptable level of accuracy. In a pedestrian navigation system, an electronic pedometer can be used to count the number of steps, which can be combined with the step size for obtaining the distance traveled. In addition, a terrestrial magnetic compass can be used as a heading sensor. The dead reckoning system is dependent upon the accuracy of measured distance and heading and the accuracy of the known origin. Apart from sensor measurement noise the main factors, which affect the accuracy, are compass bias error and step size error. The compass bias is a result of several possible causes such as magnetic declination and body offset. In a clean environment, the total bias can be changed slowly over a long period and may need to be recalibrated occasionally. The step size error can be defined as the difference between the actual step size and the predetermined step size entered by the user. Although the exact step size is not necessary for the distance calculation, the average step size over a short period has to be measured. The reason for this is that the step size of the user may vary according to the environment; for example, the step size of the user is shorter when the user is walking in a crowded area. Hence, the
predetermined step size cannot be used effectively for the distance measurement. The global positioning system (GPS) is a satellite-based radio-navigation system, which provides three-dimensional absolute position, velocity, and time information. The horizontal positioning accuracy of GPS is currently about 10 m and can be improved to about 5 m under a differential mode. In a signal blocked environment, the “availability” is significantly deteriorated. The key point of a dead reckoning algorithm for a pedestrian navigation system is the step size of the user, which is used as a scale factor in an ordinary dead reckoning algorithm. Unlike a scale factor of an odometer for a car navigation system [7], the step size is a time-varying process. Moreover, the step size is dependent on several factors. The main ones are the velocity and step frequency of the user. In 1999, Levi and Judd [1] suggested that the step size could be estimated online based on a linear relationship between measured step frequency and the step size. A real-time step calibration algorithm was proposed in [2] using a Kalman filter with GPS positioning measurement and three estimated parameters: frequency, amplitude, and mean. These parameters are obtained from another extended Kalman filter, the input of which is the measured -axis acceleration. On the other hand, Jee et al. [8] proposed a system, which calculated walking velocity from a pre-determined step size together with step frequency (pulse/second) measured by an electronic pedometer. A Kalman filter, implemented as an indirect filter, was used to estimate the error of the walking velocity with the velocity measurement from a GPS receiver. The corrected walking velocity was then fed into a dead reckoning equation for determining the position of the user. In this paper, we propose an integrated system utilizing the measured time interval between two successive steps, which is a fundamental measurement of an electronic pedometer. The inputs of an extended Kalman filter (EKF) are measured step period and heading, as well as position and velocity from a GPS receiver. When the GPS signal is available, the outputs of the filter are the position and velocity, together with the step size error and the compass bias error at each epoch of the GPS update. II. DEAD RECKONING SENSORS
Manuscript received May 29, 2001; revised August 19, 2002. This work was supported by the Electronic Systems and Information Technology Research Group, Department of Systems Engineering, Brunel University, Middlesex, U.K. The authors are with the Brunel University, Uxbridge, Middlesex, U.K. (e-mail: [email protected]). Digital Object Identifier 10.1109/TIM.2002.807986
In a DR system, the displacement from the previous known position is calculated from the measured heading and single integral of measured velocity. The walking velocity is an estimated step size divided by the measured step period. The step size or step length is defined as the distance between the same
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Fig. 1. E-position versus N-position from GPS position (GPS) and Kalman-filtered GPS position (KF-GPS). The numbers 1 and 2 refer to the first and second round, respectively.
point on each foot, usually the heel, during double-limb support [4]. Several methods have been suggested to detect step occurrences. One such method is to detect the peaks of vertical acceleration [1], which correspond to the step occurrences. In this study, a solid-state orientation sensor module, comprised of orthogonal arrays of magnetometers and accelerometers, was used for collecting the three-axis accelerometer signals and calculated body angles, which are roll, pitch and yaw, at 20 Hz sampling frequency. In the sensor module, the roll and pitch angles were measured by two inclinometers, formed by the orthogonal accelerometers. The yaw angle or heading was measured by the magnetometers. The sensor was placed rigidly, around the center of the waist, on the back of three walkers. Using Fourier analysis, the fundamental frequencies of roll and yaw angles were determined and found to be identical and equal to half of the pitch fundamental frequency. This is similar to the results published in [3]. Furthermore, we found that the Fourier analysis of the fundamental frequency of the measured vertical acceleration was also identical to that of pitch. In this paper the pitch measurement was used effectively to detect the step occurrence. In the experiment, the signals from the sensor and GPS receiver were collected and processed offline on a personal computer. For the DR, the pitch signal was fed into a digital low-pass filter for canceling double peaks; then a peak-detection algorithm was applied to detect single peaks. The time between two successive peaks was used as step time or step period for the DR. The yaw signal was fed into a low-pass filter and was sampled at the update rate of the GPS receiver, which was set to 1 Hz. The sampled yaw signal was used as the heading measurement for the DR.
III. KALMAN FILTER A. Process Model The state equation of the extended Kalman filter is written as follows [5]: (1) , is considered as a zero-mean The process noise vector, white process. The position and velocity for pedestrian navigation are in local East-North-Up (ENU) coordinates. In this experiment, we considered only two-dimensional (2D) position and velocity, i.e., east (E) and north (N). Other parameters of interest are step size error and compass bias error together with position and velocity for navigating the pedestrian. Hence, the state vector for the EKF can now be written as (2) E-position, E-velocity, N-position, where N-velocity, step size error, compass bias error, and denotes the vector and matrix transposition. Denoting the time index by , the relationships between the true step size and step size error can be written as (3) step size used for DR velocity calculation, where true step size, and step size error. The step size error was modeled as a first-order Gauss-Markov process. The reason behind this is that the previous step sizes affect the current step
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Fig. 2. E-position versus N-position from GPS position (GPS) and integrated positioning solution (Integration). The numbers 1 and 2 refer to the first and second round, respectively. For the integration in both rounds, complete GPS position data and DR measurements were fed into the integration model.
size in order to maintain the walking velocity. The effect of previous steps is controlled by the time constant of the process. The compass bias is a result of body offset and magnetic declination. Without magnetic field disturbance, the measured heading can be written as (4) measured heading, true heading, where true compass bias, and additive zero-mean Gaussian noise . The relationships between the true compass with variance , and the compass bias error, , can be written as bias, (5) compass bias used for correcting heading measurewhere ment in the DR velocity calculation. The compass bias error, , is also correlated with the previous errors. Thus, it is modeled as a first-order Gauss-Markov process. The velocity and position are modeled as random walk and integrated random walk processes, respectively. Hence, the state transition matrix, , can be written as
B. Measurement Model The general measurement model can be written as follows. (9) , is assumed to be a The measurement noise vector, zero-mean white Gaussian noise process with co-variance , and uncorrelated with the process noise vector, . The measurement vector is written as (10) measured E-position from the GPS receiver, where measured N-position from the GPS receiver, measured measured N-velocity E-velocity from the GPS receiver, measured step period, and from the GPS receiver, measured compass heading. is calcuThe measurement matrix or observation matrix lated using the following equations; (11)
(6) (7) (8) update rate of DGPS and GPS, time conwhere time constant of the comstant of the step size error, and pass bias error.
(12) To give an accurate solution from a Kalman filter, the error budgets of the measurement-noise covariance matrix should be as close to the real measurement noise statistics as possible. Hence, we take into account a method to adapt and tune the variance of the position and velocity measurement noises, according to the horizontal dilution of precision (HDOP) of the GPS satellite
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Fig. 3. Step sizes of Fig. 2: the step size calculated from GPS velocity (GPS) and the corrected step size from the integration model (Integration: Corrected).
constellation [6]. Let and denote the variances of E-position, N-position, E-velocity, and N-velocity, respectively. The variances can be calculated at each GPS update as follows: (13) (14) where
one-sigma pseudo-range error of GPS, and one-sigma delta-range error of GPS.
C. Parameter Correction When the GPS signal was available, the step size and compass bias errors were estimated by the EKF. The estimated errors were fed into the DR system to correct the step size and compass bias in (3) and (5), respectively. This is an indirect feedback structure. When the GPS signal was unavailable, the step size and the compass bias were calculated from the step size average and compass bias average of the last 100 seconds. Over the period that the GPS signal was unavailable, the E- and N-velocities were calculated from the step size average and the heading measurement corrected by the compass bias average. The calculated E- and N-velocities were then fed into an ordinary Kalman filter (KF) to estimate the position and velocity for the navigation. IV. RESULTS AND DISCUSSIONS An experiment to test the method was conducted in a field of size 47.4 m 34 m. A man was asked to walk for two rounds with a set of equipment. The equipment is comprised of the orientation sensor module for the roll, pitch and yaw measurements
sampled at 20 Hz, a GPS receiver, and a computer notebook for collecting data. In order to specify the start and stop samples of each round, the walker was asked to press time stamps when walking past those points. As shown in Fig. 1, 3, and 6, the walking path started about 30 m from the lower left corner. The path was straight with those 90-degree turns at the corners, and the walker walked clockwise. In the figures, the “GPS” refers to the raw position from the GPS receiver, and the “KF-GPS” refers to the Kalman filtered GPS position using a position-velocity model [5]. On the other hand, the “Integration” refers to the GPS and DR data processed by the method described in Section III. The number after these captions refers to the number of times the person walked around the field. It can be seen from Fig. 1 that at the beginning and the end of the second round, the GPS position data deviate from the actual path. The Kalman filtered-GPS data the other hand, in Fig. 2, the positioning solution from the integrated system using the integration model in Section III is closer to the actual path than those of the GPS and KF-GPS. This points out the fact that without magnetic field disturbance, the magnetic compass with the bias correction from the integration model gives the correct heading, which leads to higher positioning accuracy. Therefore, even when the GPS signal is available, the heading measured by the magnetic compass can help to increase the positioning accuracy. The step size and heading of the walking in Fig. 2 are shown in Fig. 3 and 4, respectively. The vertical lines in the figures represent the end of each round. It should be noted that the initial step size and initial compass bias were set to zero. Figs. 3 and 4 show that the step size and heading are corrected effectively by the integration model without the necessity of initializing the true step size and true compass bias. In addition, the step size and compass bias could be estimated within the first few
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Fig. 4. Headings of Fig. 2: the heading calculated from GPS velocity (GPS), the heading measured by the magnetic compass (Compass heading), the heading bias from the integration model (Integration: Bias) and the corrected heading from the integration model (Integration: Corrected).
Fig. 5. E-position versus N-position from GPS position (GPS) and Kalman-filtered GPS position (KF-GPS). The numbers 1 and 2 refer to the first and second round, respectively. On round 2, the GPS position data were not fed into the integration model.
seconds. In Fig. 4, the compass bias at about radians is the result of the body offset since the orientation sensor module was placed on the back of the user. The time constants for step size error and compass bias error processes in (8) were assigned to large values and were not changed for the whole experiment. As large time constants bring
about slow changes of step size error and compass bias error estimates, and vice versa, this assumption is adequate for our experiment in which the subject was walking in the same environment. However, some improvements can be made to this system for more robust estimation. One possible approach is to use a Kalman filter bank with different levels of Gauss-Markov
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(a)
(b) Fig. 6. State variance of (a) E-position and (b) N-position. KF-GPS refers to the Kalman-filtered GPS position as plotted in Fig. 1. Integration-Complete data refers to the position from the integration model with complete GPS and DR data as plotted in Fig. 2. Integration-Incomplete data refers to the position from the integration model without GPS data but DR data only in the second round as plotted in Fig. 2.
process parameters including the time constant. The result from each filter in the bank can be combined by probabilistic approaches such as in the Interacting multiple model (IMM), generalized pseudo-Bayesian approach of first order (GPB1), and generalized pseudo-Bayesian approach of second order (GPB2) [9], [10]. By this method, the system can cope with the step size and compass error changes due to the changing environment. However, the computational complexity will be considerably increased. To analyze the system when the GPS signal is unavailable, in Fig. 5, the GPS data of the second round was not fed into the integration model. It can be seen that on the right side of the second round, the unexpected compass heading error due to magnetic field disturbances brings about the positioning error, and the error of the estimated step size causes a shorter path. However, the shape of the path is still acceptable and may be fitted to a digital map (map matching), if available. To compare the positioning error, the state variances of E-position and N-position are plotted in Fig. 6. It can be seen that the variances of the integrated positioning solution with GPS data (shown as “Complete data”) are much lower than the variance of the KF-GPS. This supports the result in Fig. 2. When the GPS signal was unavailable (the second round of the “Incomplete data”), the state variances of the integrated positioning solution increase with time and almost reach the same level as the state variances of the KF-GPS.
V. CONCLUSION This paper describes a method to estimate the errors of the DR parameters, which are step size and heading measurement,
in a pedestrian navigation system. The estimated step size and compass bias errors are used for correcting the DR parameters. With this method, the averaged step size can be estimated online, so that the user does not need to specify his/her step size. This also benefits when the user is not walking with the usual step size, e.g., walking in a crowded area. In addition, the magnetic compass can be placed anywhere on the body since the compass bias can be estimated effectively. Moreover, this method also helps to increase the positioning accuracy when the GPS signal is available.
REFERENCES [1] R. W. Levi and T. Judd, “Dead reckoning navigational system using accelerometer to measure foot Impacts,”, U.S. Patent US5583776, Dec. 1999. [2] V. Gabaglio and B. Merminod, “Real-time calibration of length of steps with GPS and accelerometers,” in GNSS, 1999, pp. 599–605. [3] N. Molton, S. Se, J. M. Brady, D. Lee, and P. Probert, “A stereo visionbased aid for the visually impaired,” Image Vis. Comput., vol. 16, no. 4, pp. 251–263, Apr., 3 1998. [4] J. Rose and J. G. Gamble, Human Walking. Baltimore, MD: Williams & Wilkins, 1993. [5] R. G. Brown and P. Y. C. Hwang, Introduction to Random Signals and Applied Kalman Filtering. New York: Wiley, 1997. [6] B. W. Parkinson and J. J. Spilker, Global Positioning System: Theory and Application: AIAA, 1996, vol. 2. [7] J. Mar and J.-H. Leu, “Simulations of the positioning accuracy of integrated vehicular navigation systems,” in Proc. Inst. Elect. Eng. Radar, Sonar Navigation, vol. 143, Apr. 1996, pp. 121–128. [8] G.-I. Jee, J.-S. Hong, S.-I. Yoon, Y. J. Lee, and C. G. Park, “Portable positioning system for mobile communication network performance test,” in ION GPS, Sept. 1999, pp. 1091–1098. [9] Y. Bar-Shalom and X.-R. Li, Estimation and Tracking : Principles, Techniques and Software. Norwood, MA: Artech House, 1993. [10] S. Blackman and P. Robert, Design and Analysis of Modern Tracking Systems. Norwood, MA: Artech House Radar Library, 1999.
JIRAWIMUT et al.: METHOD FOR DEAD RECKONING PARAMETER CORRECTION
Rommanee Jirawimut received the B.Eng. degree from King Mongkut’s University of Technology Thonburi, Thailand, in 1998 and the M.Sc. degree in communication and signal processing from Imperial College of Science, Technology, and Medicine, London, U.K., in 1999. She is currently pursuing the Ph.D. degree at the Department of Systems Engineering, Brunel University, Uxbridge, Middlesex, U.K.
Piotr Ptasinski received the Eng. and M.Sc. degrees in electronics and telecommunication in 1998 from the University of Mining and Metallurgy, Krakow, Poland. He is currently pursuing the Ph.D. degree at the Electronic Systems and Information Technology Group in the Systems Engineering Department, Brunel University, Uxbridge, Middlesex, U.K.
Vanja Garaj received the B.Sc. degree in product design from the Department of Design, Faculty of Architecture, University of Zagreb, Zagreb, Croatia, in 1998. She is currently purusing the Ph.D. degree at the Department of Systems Engineering, Faculty of Technology and Information Systems, Brunel University, Uxbridge, Middlesex, U.K.
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Franjo Cecelja was born in Krizevci, Croatia, on February 21, 1955. He received the Dipl.Eng. degree in electronics from the University in Zagreb, Zagreb, Croatia, the M.Sc. degree in control from Cranfield Institute of Technology, Cranfield, U.K., and the Ph.D. degree in electrical engineering from Brunel University of West London, Middlesex, U.K. From 1978 to 1992, he pursued research in control and sensor technology. In 1992, he joined the University of Surrey, Surrey, U.K., where he was engaged in research on optical sensors and EM interaction with biological systems. He is currently a Lecturer at Brunel University, with research interests in EM radiation, near-field measurements, and general signal processing, which resulted in more than 80 scientific papers.
Wamadeva Balachandran (M’91–SM’96) received the B.Sc. degree from the University of Ceylon, Colombo, Sri Lanka, in 1970, and the M.Sc. and Ph.D. degrees from the University of Bradford, Bradford, U.K., in 1975 and 1979, respectively. He is Professor of electronic systems and Head of the Department of Systems Engineering, Brunel University of West London, Middlesex, U.K. He is actively pursuing research in electromagnetic sensors, measurement systems, medical electronics, DGPS navigation systems, electrohydrodynamics, and charge particle dynamics. He has published over 200 papers in these fields and filed eight patent applications. He is an editorial board member of the International Journal of Atomization and Sprays and the International Journal of Particle Science and Technology. Prof. Balachandran is the past Chairman of the Static Electrification Committee of the Institute of Physics, London, International President of the Institute of Liquid Atomization and Spray Systems, and a member of the IEEE Electrostatic Process Committee. He is a Fellow of the IEE, Institute of Measurement and Control, Institute of Physics, and the Royal Society of Arts, U.K.
