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Theoretical And Empirical Analysis Of Path Magnetic Lane Tracking For The Intelligent Vehicle Highway System

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https://escholarship.org/uc/item/5g49r59q Authors

Andrews, Angus

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1992

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INSTlTUTE OF TRANSPORTATION STUDIES UNIVERSITY OF CALIFORNIA, BERKELEY

Theoretical and Empirical Analysis of PATH Magnetic Lane Tracking for the Intelligent Vehicle Highway System

Dr. Angus Andrews Rockwell International Science Center

PATH Research Report UCB-ITS-PRR-92-9

This work was performed as part of the California PATH Program of the University of California, in cooperation with the State of California, Business, Transportation, and Housing Agency, Department of Transportation, and theUnited States Department of Transportation, Federal Highway

Administration.

The contents of this report reflect the views of the author who is responsible for the facts and the accuracy of the data presented herein. The contents do not necessarily reflect the official views or policies of the State of California.

This report does not constitute a standard, specification, or regulation.

August 1992 ISSN 10551425

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THEORETICAL AND EMPIRICAL ANALYSIS OF PATH MAGNETIC LANE TRACKING FOR THE INTELLIGENT VEHICLE HIGHWAY SYSTEM

FINAL REPORT

January 22,1992 through August 14,1992

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CONTRACT NO. PPBOO2162- Prepared for:

University of California at Berkeley PATH Program

Building 452 Richmond Field Station

1301 S. 46th Street Richmond, CA 94804 Attn: Prof. J. Walrand

Dr. Steve Shladover Prepared by:

Dr. Angus Andrews AUGUST 1992

Approved for public release; distribution unlimited

Rockwell International

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IVHS ANALYSIS AND DESIGN in Support of the PATH Program

Task 2

Analysis of Magnetic Dipoles for

Road-Vehicle Communication

Final Report August, 1992 Prepared for:

The University of California at Berkeley under

Contract Number: PPB002162

Prepared by:

Rockwell International Science Center 1049 Camino DOS Rios

Thousand Oaks, CA 91360

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Preface

Purpose. This report documents the principal technical findings of studies performed under Contract PPB002152 for the PATH program office at the University of California at Berkeley by the Science Center of Rockwell International.

IVHS and the PATH Project. Partners for Advanced Transit and Highways (PATH) is a research center for vehicular technologies, located at the University of California field station in Richmond, California, and funded by CalTrans. The PATH Program is also a center for research for the Intelligent Vehicle Highway System (IVHS), a national program to improve the transportation infrastructure of the United States by developing and implementing application-specific technology

“Magnetic nails” lateral steering concept. The PATH program team had developed an autonomous lateral control (steering) system using “magnetic nails” to mark the center of the lane.

These are small magnets imbedded in the pavement along the centerline of the lane. Magnetic sensors mounted underneath the vehicles in the lane can sense the resulting disturbances in the background magnetic field, and use this information to estimate the lateral position error as the vehicles navigate down the lane. This approach has been demonstrated successfully on a test track with imbedded magnetic nails, using a vehicle with four magnetic sensors coupled to an automatic steering system. The signal processing method used in PATH demonstrations is based on a rather elegant and reliable signal classification approach. The information in the signals can be used to classify the location of the vehicle as it passes each magnet into discrete ranges of the lateral position error.

Prior work at the Science Center. Under commercial research and development funding (CR&D) at the Science Center [4] in FY 1991, a signal processing method for determining the instantaneous position of a vehicle within the highway lanes from processing of magnetic sensor signals was reduced to practice. This work was based, in part, on work performed a decade earlier under IR&D funding from the Autonetics Marine Systems Division of Rockwell in Anaheim.

The author had developed a statistical estimation method based on extended Kalman filtering, Bayesian inferencing, discrete adaptive estimation methods, and similar physical principles for the magnetic airborne detection and location of submarines, and had applied this method on a government contract a year later. A solution of the matrix Riccati differential equation for this estimation problem was obtained in closed form under Science Center IR&D Project 839 in FY1988.

This signal processing method was extended in FY1991 to include a method for determining the orientation of the magnetic dipoles of the magnetic nails. This allows the orientations to be used as a code to communicate information to the vehicle control system-such as changes in routing, or curves in the lane ahead. A classification method based on Bayesian inferencing was derived for this purpose and reduced to practice on simulated signal data.

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Scope. This report covers Task 2 of the statement of work set forth in Proposal SC611332T [7].

The principal objectives of Task 2 are the following:

1. Characterize the statistical performance of the magnetic nails concept for estimating the positions of vehicles within their marked lanes. The performance is to be characterized in terms of the mean-squared position estimation error and its dependence on the sizes and orientations of the magnets used, the mean-squared noise in the sensor outputs (due to sensor noise and the ambient magnetic noise), and the relative spacing between the sensor and the magnets in operation.

2. Develop a mathematical model for sensor noise due to the distortions of the earth field by regular patterns of ferrous reinforcing bars in the pavement.

Organization of results. This report includes summaries of the principal findings organized by objectives in chapters, with more detailed technical results presented in appendices. The first, introductory chapter includes an executive summary of the principal findings. It also contains a glossary of symbols used throughout the text, and background information on the general problem area. The second and third chapters contain the principal findings of the two subtasks of Task 2, with more technical details contained in the appendices. The first subtask (in chapter 2) addresses the issue of characterizing the theoretical performance of the lateral position estimator, and the dependence of performance on various design parameters. The methodology for design of a distributed sensor system is derived, and results are plotted. The second subtask (in chapter 3) addresses the effects of reinforcing bars in the pavement, and the characteristics of the noise that they contribute to the measurements.

Acknowledgments. The author gratefully acknowledges the assistance of the following individuals in preparing the material for this report.

Weibin Zhang from the PATH Project for providing the signal data recorded on the test track at the Richmond Field Station, and a sample mangetic nail of the type used on the test track.

Ira Goldberg at the Science Center for setting up the field mapping experiment to test the validity of the dipole field model for the magnetic nails.

Jane Hanamoto at the Science Center for measuring and recording the magnetic field over a grid of locations from a centimeter to half a meter from a magnetic nail.

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E Signal and Noise Segments of Data Recorded at 30 MPH F Analysis of Noisy Data at 30 MPH

F.l , Power Spectral Densities of Noise Data . . . . F.2 Estimated RMS Sensor Noise . . . .

G Fourier Analysis of Signal Data at 30 MPH

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G.l Power Spectral Densities . . .

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G.2 Estimating the Sensor Scale Factor . . .

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Chapter ₁ Introduction

1.1 Executive Summary

All technical objectives of the contract statement of work were met, additional analyses of test data were performed, and the results generally indicate very good performance characteristics for the PATH magnetic lane tracking approach.

The following principal results of the investigation are summarized in the subsections that follow :

1. The theoretical limit of lateral position estimation accuracy due to noise and the statistical structure of the estimation problem is about one millimeter rms uncertainty for the current PATH sensor/magnet configuration. This result is based on statistical modeling and noise analysis of data recorded on the test track. It is not known whether this noise is representative of likely IVHS system environments.

2. Signal processing implementation equations for distributed (i.e., non-collocated) magnetic field sensors were derived and used for assessing location accuracy performance for these configurations.

3. The lateral position estimation problem can be solved with only one single-axis sensor.

In fact, the rms lateral position estimation error is the same whether one, two, or three axes of the field are measured.

4. Background magnetic noise due to regular patterns of steel reinforcing bars in the pavement was modeled using the Born approximation. The power spectral density of the resulting noise contribution was modeled by convolving the resulting signal with the sampling function (due to rebar spacing) to obtain a closed-form solution. The resulting formula is a parametric model for background noise PSD as a function of

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(a) Magnetic susceptibility of the reinforcing bar material.

(b) Thickness of the reinforcing bars.

(c) Horizontal spacing of the reinforcing bars.

(d) Vertical placement of the reinforcing bars relative to the nominal placement of the magnetic sensor.

(e) Magnitude and direction (relative to the highway) of the background “permanent”

magnetic field.

(f) Vehicle velocity.

5. The input magnetic field deformations due to reinforcing bars in the pavement do not couple into the signal axis that affects lateral position error. As a consequence-to first-order effects-reinforcing bars do not influence steering errors. (This result does not take into account the effects of the reinforcing bars on the field patterns of the magnetic nails, however.)

6. The magnets currently used in the test track at the Richmond Field Station are about 10 centimeters in length. This results in a magnetic field pattern with about 4% peak difference from the field of an idealized dipole, measured at the separation distances between the magnet and the sensor in the current test configuration. A first-order model for the field of an elongate magnet was derived and verified with field measurements of a sample magnet. (It is not known what effect this has on the accuracy of estimating the lateral position error, however.)

7. The rms noise in samples of signals taken at the test track is estimated to be equivalent to input magnetic field noise of about 0.01 Gauss.

1.1.1 Statistical Performance Analysis

Details of the theoretical analysis are presented in Chapter 2 and the Appendices.

Theoretical Performance Limits

Figure 1.1 shows the theoretical functional dependence of rms lateral position estimation error on the background noise (assumed white) and the height of the sensor above the pavement. This result represents the theoretical limits of performance for the current magnetic nail design and sampling interval (3 millisecond).

This result assumes

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RMS Lateral Position Uncertainty

RMS Noise from PATH Test Data

I 10 cm --

Sensor Height

I

o.bl ^0.2 1.0

RMS Sensor Noise in Gauss

Figure 1 .l: RMS Lateral Position Uncertainty versus Sensor RMS Noise and Sensor Height

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l A “magnetic nail” is a cylindrical ferrite magnet 4 inches long and 1 inch in diameter (R5 1.9 x lo-5Tm3), with its top end recessed l/4 inch below the surface.

l An extended Kalman filter is used for estimating position.

l No assumed a priori position information used in the estimate.

l The sensor noise is white within the signal frequency band.

The rms noise estimated from the data analysis of test results on the PATH test track at the Richmond Field Station (see Appendix F) is approximately 0.01 Gauss’.

For the current sensor height (x lOcm), this results in a rms lateral position uncertainty of about 1 millimeter at the time that the sensor is nearest the magnet. For a sensor height of one foot (x 3Ocm), the corresponding rms lateral position uncertainty would be about 4 centimeters.

Performance Analysis of Distributed Sensor Systems

A previous study [4] had characterized the performance of a vehicular position estimation method using a single 3-axis sensor to measure the magnetic field disturbance due to a magnet embedded in the pavement.

This study extended those results to include single-axis sensors that are spatially distributed, as are the sensors used on the instrumented test track of the PATH program at the Richmond Field Station of the University of California at Berkeley.

The principal findings of this study are:

1. Performance can be characterized in two ways:

(a) By the rank of the observability Gramian, which determines whether or not the relative vehicle location can be estimated with the given sensor configuration.

This is a qualitative characterization of observability of position from the data.

(b) By the steady-state covariance matrix of location uncertainty from a single en- counter with a magnet. This is a quantitative characterization of how well the relative location of the vehicle can be estimated from the magnetic signals.

2. A two-axis sensor is capable of estimating all three components of the relative location of the sensor (i.e., of the vehicle) with respect to the magnets.

‘This estimate is considered conservative. The test data show considerable harmonic noise that would appear to be due to pickup from power supplies, etc., and not actually magnetic noise. This type of noise can be effectively eliminated by careful design of the sensor system, and/or by added filtering.

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Science Center SC7701 1 JR 5 3. A sensor design measuring only one component (the lateral component) of the field has a singular observability Gramian, which means that the Scomponent location of the sensor relative to the magnet is not observable from the data.

4. If,the estimation problem is restricted to estimating only the lateral position of the vehicle, and the magnets are oriented as they are on the test track (dipole axes vertical), then the mean-squared lateral position uncertainty is the same with ^asingle- axis sensor as it is with a-axis and S-axis sensors. The dependence of lateral position uncertainty on all design parameters (including vehicle speed, sensor height, sensor noise, and magnet size) is then the same for single-axis sensors as with multi- axis sensors. This implies that steering system performance does not benefit from having more than a single-axis sensor capability.

1.1.2 Background Noise due to Reinforcing Bars

Ferrous metal reinforcing bars laid laterally and longitudinally in the pavement perturb the background terrestrial magnetic field. A first-order analysis of this effect shows that the lateral component of the magnetic field is not, perturbed. Because this field component produces the signal used in estimating the lateral position error, the perturbation does not, contribute to the lateral position error.

The power spectral density shapes are plotted for representative ranges of the spacings of reinforcing bars and the height of the sensor above the reinforcing bars.

1.1.3 Analysis of Test Data

Two empirical studies were included as additional sub-tasks:

1. Data taken on the test track at the Richmond Field Station was used to separate the empirical signals and noise, and to estimate the sensor scale factors and rms noise.

2. The magnetic field within one half meter of a magnetic nail was measured at a regular grid of locations, and compared to the theoretical magnetic field model. An altema- tive magnetic field model (for an elongate source) was derived and compared to the empirical field, also.

The principal findings of these studies are the following:

1. The PSD of the track test data shows the same general shape as the theoretical model, but with more apparent sharpening of the magnet spacing-rate peaks. These peaks

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occur at multiples of the frequency at which the sensors pass individual magnets in the test track.

2. The magnetic field measured near a magnetic nail shows about 4% peak distortion from the simple dipole field model at test-track conditions (sensors 4-5 inches above the magnet). Most of this is due to the elongation of the source, which is not insignificant when the magnet dimensions are comparable to the minimum separation distance between the sensor and the magnet. The anomalous field distortion is significantly greater > 20% within a centimeter or two of the magnet, which may be a result of inhomogeniety of the remnant flux density in the magnetic material.

3. The elongated magnet model gives a better fit to the data than the dipole model at distances comparable to the magnet dimensions (in this case, 4-5 inches). However, it complicates the signal processing equations considerably.

1.2 General Background Information

IVHS and the PATH Program. The Intelligent Vehicle Highway System (IVHS) is a national program to improve the transportation infrastructure of the United States by applications of appropriate technologies. Partners for Advanced Transit and Highways (PATH) at the University of California at Berkeley is a center for coordinated university and industrial research, development, testing, and evaluation of these technologies.

1.2.1 Technical Issues for Steering Using Magnetic Nails

One component of the PATH project is the use of Advanced Vehicle Control Systems (AVCS) to substantially improve the capacity and delays of freeway systems. One of the subsystems of AVCS is a lateral steering system for autonomous lane-holding. The lateral steering sub- system uses “magnetic nails” (magnets) buried in the pavement to mark the lanes. Onboard magnetic field sensors are used to detect the resulting disturbances in the background magnetic field. Onboard signal processors can use these signals to estimate the position of the vehicle relative to the lane centerline (marked by magnets). The resulting lateral position information is used in for steering in autonomous lane-holding.

A problem of interest for IVHS development is to characterize how the performance of this lateral positioning system depends upon its design attributes, such as

l The spacing between magnets.

l The size of the magnets.

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l The orientation of the magnetization of each magnet.

l The types of magnetic sensors used.

‘- Search coils.

- Hall-effect sensors.

- Magnetoresistive sensors.

- Magnetodiodes.

- Magnetotransistors.

- Magnetorestrictive-clad fiber-optic sensors.

- Magneto-optical sensors (Faraday cells).

- Fluxgate magnetometers.

- Zeeman effect (optically pumped) magnetometers.

- Nuclear precession (“total field”) magnetometers.

- Superconducting quantum interference devices (SQUIDS).

l The number of sensors used.

l The orientations of the input axes of the sensors.

l The locations of the sensors on the vehicle.

Performance will also depend on other problem attributes that are not under the control of the designer, such as

l The speed of the vehicle.

l The level of background magnetic noise from the vehicle.

l Sensor input variations due to the rather large (x .3 Gauss) natural field and - Attitude variations from vehicle dynamics.

- Attitude variations from vibrations.

- Distortions from passing cars.

- Distortions from inhomogeneous permeability, due to distributions of ferrous materials such as

* Reinforcing bars in the pavement.

* Structural steel in bridges.

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* Guardrails and sign posts.

* Debris of magnetic materials (or magnets)

l Distortion of the magnetic field from the magnet, due to such factors as.

- Ferrous materials in the pavement (e.g., reinforcing bars).

- Ferrous materials in the vehicle This report addresses only a few of these issues.

Other Considerations

Minimum clearance considerations. The current clearance of about 5-6 inches between the sensor and the pavement probably requires some sort of protection of the sensor from contact with debris in the road. Commercial trucks typically have a foot or more of clearance under the bumpers and chassis, which would require mounting the magnetic sensor higher above the pavement or risk damage from debris on the road. The problem is that the rms lateral position estimation uncertainty falls off as the seventh power of sensor height.

Signal bandwidth and sampling rate requirements. The usable bandwidth of the signal is approximately 160 Hertz2. This would imply that sampling rates should be at least twice that, or 320 samples per second. (Although economic considerations may favor sampling rates consistent with low-cost voiceband codecs with 7000 samples per second, these have prefilters with a passband of about 300-3000 Hertz that effectively eliminate the magnetic signals of interest.)

1.2.2 Added Studies of Empirical Test Data

These studies were not proposed in the original technical proposal, in part because the test track data was not expected to be available within the proposed contract period (late 1990 or early 1991). The contract award was delayed much more than the test program, however, and the test data were available at the start of the contract period. Therefore, it was possible to verify some attributes of the theoretical models using actual test data.

The test track at its Richmond Field Station includes ceramic magnets buried at intervals of about one meter along the centerline of a lane. The cylindrical magnets imbedded in the test track are approximately 2.2 centimeters in diameter and 10 centimeters long, with their

‘This assumes a vehicle speed of about 110 kilometers per hour, about 1 foot minimum separation between the sensor and the magnet, and 50dB of usable signal range.

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Science Center SC7701 1. FR 9 dipole axes parallel to the cylindrical axis. They are oriented with their dipole and cylindrical axes vertical, with their top ends about one-half centimeter below the pavement surface. Test data have been collected using four Hall-effect magnetic field sensors mounted underneath the test vehicle, approximately lo-15 centimeters (4-6 inches) above the pavement. This distance is lower (by a factor of 2 or 3) than assumed in the original analysis, and is close to the longest magnet dimension. Therefore, it was suspected that the actual magnetic field might differ significantly from the idealized dipole field used in the analysis. The field pattern near a sample “magnetic nail” provided by Weibin Zhang was measured and analyzed to test the validity of the model.

1.3 Historical Background

The modeling and analysis of magnetic fields goes back a long way-at least as far back as Carl Friedrich Gauss (1777-1855), who performed a spherical harmonic analysis of the magnetic potential of the “constant” natural background field. The analytical treatment of signals from magnetic sensors on moving platforms passing dipole sources goes back at least to the 194Os, when the problem of interest was the location of a submarine relative to a surveillance airplane. This antisubmarine warfare problem is mathematically the same as the lateral steering problem for IVHS. The signal interpretation problem for both applications is to determine the relative location of the sensor and the magnet. The historical roots of the common problem are examined in the following subsections.

1.3.1 John Anderson and Anderson Functions

An early use of magnetic sensors was for detecting the presence of submarines from airborne sensor platforms. The earliest submarines were made of iron or steel that was usually magnetized during the manufacturing process, or acquired a remnant field during fabrication or from diving stresses. This magnetic signature could be reduced by “de-Gaussing” the entire submarine to remove the remnant magnetization, but ferrous materials still produce distortions in the natural magnetic field that are detectable in the “magnetically quiet” oceans.

(Other sources of magnetic fields around submarines come from corrosion currents resulting from the use of sacrificial anodes to prevent corrosion in sea water, and from magnetization induced by diving stresses.)

The dominant magnetic field components at distances large compared to the diameter of the source (in this case, a submarine) are the dipole components. They tend to fall off inversely as the cube of distance from the source. For World War II-vintage submarines, the detectable distances were in the order of kilometers.

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The problem of characterizing the signals coming from magnetic field sensors as they are passed by a magnetic dipole was solved in 1949 by John E. Anderson [2], who showed that all such signals are linear combinations of three parametric basis functions. He also calculated the Fourier transforms of these functions, which characterize their frequency bandwidths and power spectral densities. Since that time, these basis functions have been called “Anderson functions.” They were originally used to characterize the signals used in magnetic anomaly detection for antisubmarine warfare, using the dipole component of the field disturbance as the distinguishing characteristic of the submarine. The same basis functions occur in the modeling of pressure-induced absorption of electromagnetic radiation by gasses [6], and in the present application for autonomous control in the Intelligent Vehicle Highway System[l7].

1.3.2 Statistical Methods for Signal Processing and Analysis

Methods using Anderson functions. Once the Anderson functions are detected in the sensor signals, their amplitudes and phases can be estimated. From these, it is possible to estimate the position of the source. The phasing of Anderson functions is defined with respect to the “time of closest approach” between the dipole and the sensor. This time is not known beforehand, of course, and finding it by searching is relatively computationally intensive. Once the time of closest approach is known, the Anderson function components can be found by least-squares fitting or its statistical variants.

Classification methods. A simpler and faster method was developed by Zhang [17] for the IVHS application. It takes advantage of the relative invariance of the ratio of the vertical to lateral field components to the distance of the sensor from the magnet to derive a lateral position error classification method that is robust against inital position errors. It uses a pattern classification approach to identify how the patterns of vertical and lateral magnetic field components are related to the lateral position offset of the sensor from the magnet.

Adaptive extended Kalman filtering and Schweppe likelihood-ratio detection.

The estimation problem was solved using extended Kalman filtering in 1979 by the author.

This approach has the further advantage that it is compatible with Schweppe likelihood-ratio detection[l5], a detection method based on decision theory and Bayesian statistics. The nonlinearity for the submarine location problem is too great for straightforward extended Kalman filtering, however. It was solved satisfactorily by carrying several potential solutions in parallel, and using a generalization of Schweppe likelihood-ratio detection to resolve which of the potential solutions is the best. Another advantage of the Kalman filtering approach is that it includes its own performance characterization-something that is not possible with the other methods. With it, one can assess the performance limits as a function of the design variables.

Statistical performance characterization. The matrix Riccati differential equation char-

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Science Center SC7701 1. FR 11 acterizes the evolution over time of the covariance matrix of position estimation uncertainty from Kalman filtering. This equation includes the effects of random processes (e.g., vehicle dynamics and sensor noise) as well as the effects of using measurements. The random processes tend to increase uncertainty, and the measurements tend to decrease uncertainty. The Riccati equation is most often solved numerically, because analytical solutions are unknown except in trivial cases. Fortunately, the dipole location problem is one of those cases-or, at least, its Rccati equation can be solved analytically. This is especially useful for design trade-offs, because system performance can be expressed as a function of the design variables by a simple algebraic formula.

1.4 Technical Conventions

1.4.1 Mathematical Notation

Vectors and matrices. Where convention permits3, matrices will be denoted by upper case letters, and vectors will be denoted by lower case letters in boldface type. The su- perscript T will denote the transpose of a matrix or vector. The elements of vectors and matrices will usually be denoted by the corresponding “un-bold” lower case letters-unless it might conflict with other usage. The z‘th element of the vector v will then be denoted by vi, and the element in the ith row and jth column of A will be denoted by aij. The corresponding element of a matrix expression will be denoted by enclosing the entire expression in delimiters and subscripting the result, so that {AB),j will denote the element in the ith row and jth column of the matrix product AB. The matrix inverse of A, if it exists, will be denoted by A-‘.

3For example, B will be used as the (conventional) symbol for magnetic field, a vector quantity.

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1.4.2 Symbols Used

A

a

ai B d

di

m

r

X

PO

<ijk

7r

a3x3matrix

a column 3-vector representing the sensitivity of a sensor output to the three components of the magnetic field where it is located. The magnitude [al of a equals the sensitivity of the sensor in appropriate units (e.g., volts per Tesla), and the direction of ^ais the direction of the sensor input axis.

an element of vector ^a

magnetic field (in units of Teslas)

relative displacement vector of the location of a sensor with respect to the origin of the vehicle coordinate axes

components of d

the dipole moment vector of a magnet (Tesla-meter3)

= x - d, the relative location vector from the location of a magnetic dipole to the position where its magnetic field is measured (or vice versa)

position vector of a magnetic dipole in vehicle coordinates

= 47r x 10s7, a constant for the SI system of magnetic units.

intermediate scalar variables.

= 3.14159.. .

1.4.3 Default Vehicle Coordinates

Where possible, derived formulas will not depend on the choice of coordinates. Otherwise, the coordinate system shown in Figure 1.2 will be used for the location and input axes of the sensors. It has the following attributes:

l The origin of the vehicle-fixed coordinates is at its nominal center of mass. The exact location of the true center of mass will depend on fuel and passenger loading, but some standard definition for control purposes is assumed.

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Second axis .

Magnet location ,‘u

Figure 1.2: Vehicle-Fixed Coordinate System

l The first coordinate axis is nominally horizontal and parallel to the longitudinal axis of the vehicle, and its positive direction is forward.

l The second coordinate axis is nominally horizontal and its positive direction is to the right side of the vehicle.

l The third coordinate axis is nominally downward.

l The coordinate axes are mutually orthogonal.

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Chapter 2 Statistical Performance Analysis

2.1 Objectives and Approach

The objective of this sub-task is to characterize the performance of the position estimates derived from the magnetic signals and its dependence on certain “design parameters.” These design parameters include some which we are free to choose in the design process, and some which are forced upon us by nature:

1. The size and material composition of the magnetic nails.

2. The orientation of the principal (dipole) moment of each magnet with respect to “highway coordinates” (down-lane, cross-lane, and vertical).

3. The locations of magnetic sensors on the vehicle, including their heights above the pavement and their relative lateral offsets from the of magnets.

4. The orientations of the sensitive axes of the sensors.

5. The speed of the vehicle.

6. The “noise floor” of the sensor, due to internal noise.

7. The background magnetic field noise due to external sources, such as the magnetic field of the earth and the electrical system of the vehicle.

The two noise sources have the same effect, but the first (sensor noise) can be influenced by the choice of sensor type. (The background magnetic noise is effectively a design constraint, not a design parameter. It is assigned a value by nature, not by the designer.)

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The approach uses the mean-squared uncertainty in the estimate of position as a measure of performance of the system. A mathematical model for the mean-squared position uncertainty can be derived in terms of stochastic differential equations. The result is a degenerate form of the matrix Riccati differential equation. The differential equation can be transformed to simpler form by solving for the matrix inverse of the covariance matrix of estimation uncertainty. The resulting matrix differential can be solved in closed form for certain sensor geometries. The solution can be re-inverted in closed form to obtain the complete solution for the covariance of position estimation uncertainty as a function of the design parameters listed above.

2.2 Formulas for Distributed Sensors

The functional dependence of the instantaneous value of a signal from a single-axis magnetic sensor on its location and orientation relative to a magnetic dipole is derived here in terms of the instantaneous location and orientation of the magnetic dipole, and the direction and scale factor of the sensor input axis. These formulas are used in the derivation of a signal processing algorithm for estimating the vehicle position relative to a magnet, given a sequence of sensor measurements at known times and with known (or estimated) vehicle velocity.

2.2.1 Field Dependence on Dipole Moment and Location

The magnetic field B at a location r with respect to a magnetic dipole of moment m is given by the formula

B ( r ) = g$ {3rrT - lr121} m,

(2.1)

where I is a 3 x 3 identity matrix and ~0 = 47r x 10s7 for SI units. Note that this formula is independent of the sign of r. That is, r can be taken as a vector from the location where the field is measured to the location of the dipole center, or vice versa. It is more natural for the problem at hand to assume that r is measured from the location where the field is measured to the location of the dipole source.

2.2.2 A Matrix Inversion Formula

It can easily be shown that the matrix inverse

{& [3rrr - jr[21])-’ = jr1 [ir rT - jr121] ,

(2.2)

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m=- - [r/2I] B.

(2.3)

is the inverted formula for m, given B and r. That is, m can be determined from a single 3-axis measurement of B if r is also known.

2.2.3 Formulas for Distributed Single-Axis Sensors

“Vehicle coordinates.” Let the location x of a magnetic dipole be represented with respect to a vehicle-fixed coordinate system. The origin of this coordinate system is a point on the vehicle that remains fixed with respect to the magnetic sensors. (It should be the reference point for the steering control system, which may be the “nominal” center of mass. The center of mass depends on the fuel and passenger loading, which are not always the same.) Location and orientation of single-axis sensors. Let d be the displacement of a single- axis sensor from the origin of that coordinate system, and let a be the input sensitivity of that sensor. That is, the scale factor of the sensor is /aI and its input axis is in the direction of the unit vector a/la/. The two vectors d and a characterize the sensor.

Location of a dipole with respect to a sensor. The formula

r = x - d (2.4

relates the location r of the dipole with respect to the sensor to the location x of the dipole with respect to the vehicle.

Field at a sensor. This depends on the relative location r:

B = B(r) (2-5)

= B (x- d) (2.6)

= 4~1 (x? d) I5 (3 (x - d) (x - d)T - I (x - d) I’I} m.

(2-7)

Sensor outputs. The output of the sensor from a dipole of moment m at location x will be the “dot product”

y (x, m) = aTB (2*8)

= 4+iyd)/5aT

⁽³ (x - d) (x - d)T - I (x - d) [‘I} m. P-9)

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18 Science Center SC7701 1. FR

2.2.4 Measurement Sensitivity Matrices

In the terminology of extended Kalman filtering, the measurement sensitivity matrix is the matrix of partial derivatives of the sensor outputs with respect to the state variable. In this application, the state variable is the S-component position vector x specifying the instantaneous position of the magnet with respect to the vehicle.

General Formulas for Dipole Sources

The resulting formulas can be simplified computationally by taking advantage of known bilinearity. That is, the field B is independently linear in both a and m-but not both simultaneously. As a consequence, its partial derivatives with respect to x will also be linear with respect to both (i.e., b&ear). In particular, the result can be expressed in vector form with elements represented by bilinear forms aTAm:

aY 3Po

-= ax

47~ Ix - d/z

where, for

(

aTA[‘]m>

(aTA121m>

(aTA[“]m>

(2.10)

rl = x1 - dl

f2 = x2 - d2

r3 = x3-d3,

the matrices Afil of the bilinear forms are symmetric, with elements A[‘] = 7-l (27-f -37-i

At:l - r&l1

- 37-i)

12 -

t411 = (4rf - ri - ri) All1 = r3J411

Ailj = A/

A!$ = rl5141

5^{141 =} (-7-f +47-i - ri) A!$ = 5rlrg3

A!! = A!;

(2.11) (2.12) (2.13)

(2.14) (2.15) (2.16) (2.17) (2.18) (2.19) (2.20) (2.21) (2.22)

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Science Center SC7701 1 .FR 19

t114 = (-T; - T; + 4T;)

AI21 = -Atj ,$ = /#

T2 (3T; - 2T; + 3T;)

-T3&41

2.2.5 Simplifications for Special Situations

Sensor Axes along Coordinate Axes

In the case that _{a = Ial}

[

⁰₀

1 1

^{, or}

[

⁰

a = Ial ₀1

1

^{7 or}

(2.23) (2.24) (2.25) (2.26) (2.27) (2.28) (2.29) (2.30) (2.31) (2.32) (2.33) (2.34) (2.35) (2.36) (2.37) (2.38) (2.39) (2.40) (2.41) (2.42) (2.43)

(2.44)

(2.45)

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20

m Rockwell International

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0

a = Ial 0

[ 1

¹ ^,

m = b-4 k

[ 1 > or

⁰

m = /ml

[

ⁱ₀

1

^{, or}

m = Im/

i

⁰⁰¹

1

^,

(2.46)

only the rows of the Ali corresponding to the non-zero row of a need be used.

Dipole Axis along a Coordinate Axis In the case that

(2.47)

(2.48)

(2.49)

only the column of the AIil corresponding to the non-zero row of m need be used.

Dipole and Sensitivity Axes along Coordinate Axes

If both the dipole moment vector m and the sensitivity vector ^aare along coordinate axes, only the corresponding element of each AIil needs to be considered.

For example, consider the cases that

m = Jml 0

[ 1

0 (dipole axis vertical), and 1

0

a = Ial 0

[ 1

(input axis vertical), or 1

0

a = Ial 1

[ 1

(input axis lateral).

0

(2.50)

(2.51)

(2.52)

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Rockwell International

Science Center SC7701 1. FR 21 In the first case (input axis vertical),

dY 3 I4 PO b-4

ax=

47~ Ix - dlz ^-T2[114 (2.53)

T3 (3T7 + 3T; - 2T;)

In the second case (input axis lateral),

8Y

3 lal PO Iml dx=

47r Ix - dli

5TlV3

-7-3&41

-

^(2.54)

T25114

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Science Center SC7701 1 .FR

Chapter 3 Noise from Reinforcing Bars

It is expected that the major small-scale magnetic anomalies immediately above the surfaces of highways will come from reinforcing bars (here called “rebars”) of steel imbedded in the concrete. These tend to distort the background magnetic field pattern, which is otherwise approximately uniform. A magnetic sensor moving parallel to the surface will pick up a signal from this “ripple” in the magnetic field. This noise will tend to interfere with the signal from a magnet buried in the pavement, which is to be used for lateral steering.

The derivations of the resulting field perturbation patterns are in Appendix C. The perturbed field is modeled by what is called the “Born approximation” in physics (described below).

The model derivation results in formulas for signal perturbations and the power spectral density of the “rebar noise,” in terms of the magnitude and direction of the natural magnetic field relative to the highway, the spacing of the reinforcing bars, and the vehicle velocity.

3.1 Born Approximation Model

The Born approximation is a first-order approximation method due to Max Born (1882- 1970). The essential idea behind it is that field perturbations due to minor anomalies in the medium do not influence the “illuminating” field1 on other anomalies. This approximation allows one to sum up (or integrate) the field contributions from individual parts of a distributed anomaly, ignoring the second-order interactions. In this case, the distributed anomaly is a pattern of ferrous reinforcing bars in the pavement.

The presence of a reinforcing bar in the pavement is an anomalous pattern of magnetic

‘The illuminating field is the field that would be there if the anomalies were not present. In essence, this approximation assumes that the perturbations from one anomaly do not change the effects of another anomaly on the field.

23

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susceptibility. Each infinitesimal volume element of this pattern makes an infinitesimal perturbation in the otherwise uniform background field. This perturbation can be represented by an infinitesimal magnetic dipole aligned with the illuminating field. The Born approximation is invoked to represent the resulting field as the integral of its infinitesimal contributions-ignoring the second-order interactions. The technical details of this derivation are in Appendix C.

3.1.1 Rebar Noise Differs Spectrally from the Signal

The resulting field pattern causes a signal disturbance that is not functionally equivalent to the signal coming from a pass-by of a single magnetic dipole source. The signals from passing by a dipole source can be represented by linear combinations of three basis functions called Anderson functions. The corresponding basis functions for the “rebar model” are called “rebar functions.” The Fourier transform of these functions is derived in Appendix C.

3.1.2 Rebar Noise Does Not Couple into Lateral Position Error

The other piece of good news is that-to first-order effects2-the lateral component of the field perturbation is zero. This means that the rebar noise does not corrupt the signal component that determines the lateral position error, and that reinforcing bars do not influence steering errors. The formulas supporting this result are derived in Appendix C.

3.2 Convolution Model in Frequency Space

The signal resulting from a pattern of reinforcing bars in the pavement can be represented by a convolution integral of the rebar function with a sampling function. This can be represented more efficiently in the frequency domain by a pointwise product of the Fourier transforms of the two functions.

3.2.1 PSDs for Representative Geometries

Representative shapes of the noise spectra are shown in Figures 3.1, 3.2, and 3.3 for sensor heights of 10, 20, and 30 centimeters (roughly, 4, 8, and 12 inches) above the reinforcing

%econd-order effects that could couple this ripple into the lateral channel include the field distortions due to ferrous metal in the vehicle. These can couple vertical or longitudinal field variations into lateral field variations.

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Bars 20 cm apart

Science Center SC7701 1 .FR 25

Bars 10 cm apart

\I\/ . uu

50 100 150 200

Frequency in Hertz

Figure 3.1: Rebar Noise PSD Shapes at 10 Centimeter Sensor Height and 100 km/hr bars. For each sensor height, the spectra shapes are plotted for spacings between the bars of 10, 20, and 30 centimeters (roughly, 4, 8, and 12 inches.

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26

Rockwell International

Science Center &‘7011. FR

Bars 10 cm apart I A

Bars 20 cm apart

Bars 30 cm apart

Frequency in Hertz

Figure 3.2: Rebar Noise PSD Shapes at 20 Centimeter Sensor Height and 100 km/hr

-Bars 10 cm apart

Bars 20 cm apart

Bars 30 cm apart

1 n n I E;n 7 n n

J” Freque~&f in Hertz* J” L ” ”

Figure 3.3: Rebar Noise PSD Shapes at 30 Centimeter Sensor Height and 100 km/hr

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Science Center SC7701 l.FR

Bibliography

[l] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, U. S. Department of Commerce, National Bureau of Standards, Applied Mathematics Series, Publication No. 55, 1964.

[2] J. E. Anderson, Magnetic Anomaly Detection Frequency Responses, Report No. ADC- EC-47-51, Naval Air Development Center, Warminster, Pennsylvania, October 1949.

[3] A. P. Andrews, Closed-form Evaluation of Uncertainties in Detecting and Locating Mug- netic Dipoles, with Applications to Submarine Surveillance, Science Center Technical Report SCTR 88-4, September 1988.

[4] A. Andrews, Design Analysis of a Passive Magnetic Guidance Method for the Intelli- gent Vehicle Highway System, Rockwell International, Science Center Technical Report SCTR92-1, September 1991.

[5] A. P. Andrews, “The accuracy of navigation using magnetic dipole beacons,” Navigation, the Journal of the Institute of Navigation, Vol. 38, pp. 367-397, Winter 1991-1992.

[6] George Birnbaum and E. Richard Cohen, “Theory of line shape in pressure-induced absorption,” Canadian Journal of Physics, Vol. 54, pp. 593-602, 1978.

[7] A. Firstenberg, Specific Tasks of IVHS Analysis and Design in Support of the PATH Program, Rockwell International Science Center, Proposal SC61332T, to The University of California at Berkeley, October 1990.

[8] Arthur Gelb, Joseph F. Kasper, Jr., Raymond A. Nash, Jr., Charles F. Price, and Arthur A. Sutherland, Jr., Applied Optimal Estimation , MIT Press, Cambridge, Massachusetts,

1974.

[9] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, NY, 1980.

[lo] Thomas Kailath, “An innovations approach to least squares estimation, Part I: Linear filt>ering in additive white noise,” IEEE Transactions on Automatic Control, Vol. AC-13, pp. 646-655,1968.

27

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Science Center SC7701 1. FR

[ll] R. E. Kalman, “A new approach to linear filtering and prediction problems,” ASME Journal of Basic Engineering, Vol. 82, pp. 34-45,196O.

[12] R. E. Kalman and R. S. Bucy, “New results in linear filtering and prediction theory,”

ASME Jozirnal of Basic Engineering, Series D, Vol. 83, pp. 95-108, 1961.

[13] J. E. Lenz, “A review of magnetic sensors,” IEEE Proceedings, Vol. 78, pp. 973-989, 1990.

[14] S. E. Schladover, C. A. Desoer, J. K. Hedrick, M. Tomizuka, J. Walrand, W.-B. Zhang, D. H. McMahon, H. Peng, S. Sheikholeslam, and N. McKeown, “Automatic vehicle control developments in the PATH program,” IEEE Trans. Veh. Tech., Vol. 40, pp.

114-130, 1991.

[15] F. C. Schweppe, “Evaluation of likelihood functions for Gaussian signals,” IEEE Trans.

Info. Theory, Vol. IT-11, pp. 61-70, 1965.

[16] Stephen Wolfram, MathematicaTM, A System for Doing Mathematics by Computer, Addison-Wesley Pub., NY, 1988.

[17] W.-B. Zhang and R. E. Parsons, “An intelligent roadway reference system for vehicle lat- eral guidance/control,” ASCE 1st International Conference on Applications of Advanced Technologies in Transportation Engineering, American Society of Civil Engineers, pp.

281-286, 1989.

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m Rockwell International

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APDendix A

A A

Elongate Magnet Model

Signals recorded on the test track at the Richmond Field Station exhibit patterns that differ somewhat from idealized dipole field patterns. (This should be expected when the field is measured at ranges comparable to the dimensions of the magnet.) If the field distortion due to these conditions will affect better model of the magnetic field may be needed for the signal processing. The derivation and evaluation of a slightly more sophisticated model is the subject of this appendix. It includes only the effects of the finite vertical extent of the magnet, and evaluates those effects in the region 10 centimeters or more above the pavement surface.

The following notation and units are observed for this derivation.

A cross-section area of magnet, square meters a, b, c intermediate variables, defined where used B magnetic field (a vector), Tesla

Bi , BZ, Ba the components of B, Tesla

B, remnant flux density in magnetized material, Tesla d diameter of magnet, meters

F (rj, L, A, B,) vector function representing the dependence of the magnetic field from ^a magnet on the cross-section area A, length L, remnant flux density B,, and the relative location r at which the field is measured.

Pi, F’, F. the components of F I 3 x 3 identity matrix

29

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Rockwell Intqational

Science Center SC7701 1. FR L length of magnet, meters

m effective dipole moment vector of a magnet. This is usually expressed in units of Ampere per meter, but can also be given in units of Tesla cubic meter. The latter is preferable for permanent magnets, the magnetic moments of which equal the product of the magnet volume (in czlbic meters) and the remnant flux density (in Tesh). (These units differ by a constant factor of ~0 = 47r x 10m7.)

m the magnitude of m

r = b-1, 7-2, r31T, the position vector at which B is measured, meters ri, r-2, r3 components of r, meters

7-12, T+, T- intermediate variables with units of meters T superscript denoting vector transposition

x a variable of integration Q, ,f3, y, 6 intermediate variables

~0 471. x 10m7, a constant for SI magnetic units, Tesla-meter per ampere IT 3.14159.. .

A.1 A One-dimensional Model

The model for an elongated magnet is a finite extended distribution of infinitesimal dipole sources in one dimension. This dimension is along the dipole axis, which we will take to the third axis of a Cartesian coordinate system, as depicted in Figure A-1. The magnet has length L, diameter d, and remnant flux density B,. Its total magnetic moment is therefore’

0

m=f_7rcP

[ 1

^B,LA⁰ ^(A4

A = 4,

where A is the cross section area. The units will be Tesh cubic meter if B, is expressed in Tesla, A is expressed in square meters, and L is expressed in meters.

‘The sign (zk) on the moment vector can vary from magnet to magnet, and the modulation of this sign can be used to encode information for the vehicle control systemsuch as lead information about the curves in the lane, route and location information, or local driver services available.

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Cross Section AreaA = 1r&/4

Science Center SC7701 1 .FR 31

k7-3

(dipole axis)

‘;T . ..I L

^I^I

d - - i I+--- Figure A. 1: Coordinates

A . l . l Idealized Dipole Field (Zero-Order Model)

The formula for the dipole field B measured at location r relative to an infinitesimal magnetic dipole with finite magnetic moment m is

B = 1

~ { 3rrT - lx-l2 I} m 47r lr15

0 m=f

[ 1

^B,LA^{0 .}

Consequently, for m as given by Equation A.1,

lBol =

^B,ALdm

47r jr\”

A.1.2 Differential Model for a Distributed Dipole Source

(A.3) (A-4)

(A4 (A-6)

As a rough rule of thumb, the field from a magnet with finite dimensions differs substantially from the idealized dipole field at distances that are “small” relative to the magnet dimensions.

In the case of “magnetic nails,” the largest dimension is the length L. The magnitude of L is in the order of 10 centimeters. The closest approach to the magnets on the test track at the Richmond field station is in the order of I5 centimeters (about half a foot). The other

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Rockwell lnt~rnational

Science Center SC7701 1. FR

Figure A.2: Cross Section Field Contribution

magnet dimensions are in the order of a centimeter, which is about an order of magnitude smaller than the distance between the sensor and the magnet at closest approach. Therefore, a simplified model for the departure of the field from the idealized dipole field considers just the longest magnet dimension-along its length.

The first-order model considers the infinitesimal field contribution dB from a cross-section disk of the magnet of infinitesimal thickness dz, where -L/2 5 z 5 L/2, as shown in Figure A.2. Consider the infinitesimal field contribution dB at the location r from the infinitesimal dipole along the longitudinal axis of the magnet with dipole moment

0

dm =

[ 1

^fB,A⁰ ^a?

^(A.7)

where -L/2 5 x < L/2 and dz is the infinitesimal length of a cross section of the magnet at z. The location of r relative to this (assumed2) infinitesimal dipole source is

0 r’ = r-

[

⁰

1

^z

= [ n

r2

1 2

7-3 - z

‘i.e., neglecting the finite diameter of the cross-section.

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Science Center SC7701 1.FR 33 and the field contribution aB at r from that infinitesimal source at z will be

{ 3r’ (r’)’ - lr’12 I} dm (A.lO)

f &A

=

x 1

^(A-11)

The total field contribution from the distributed source will the integral

B = (A.12)

A . l . 3 Field Equations for Elongate Magnet

The result of carrying out the integration above is the following set of formulas for the magnetic field components as a function of the position at which they are measured.

B = Bi = FI = F2 = s =

[ 1 Bl

^B2^B3

4 (r, L, A, 8)

l<i<3 ST1

ST2

(A.13) (A.14)

(A.15) (A.16)

a =

r- = r+ = r-T2 = F3 = b =

16AB,Lrsa 7rr?r$ (r- + r+)

/(L” + 4 IrI’)’ - 16L2rz + 2L2 + 8rf, + Srz

(A.17)

(A.18) (A.19) (A.20) (A.21) (A.22) L - 2r3)2 + 47-T,

d(

L + 2r3)2 + 47-T, r; + 7-s

48AB,Lrib 7rr3r~ (r- + r,) (L” + 4 IrI’)’

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Rockwell Interndonal

Science Center S(X’7011. FR

- 8L2ri - (r” + rt)” (A.23)

Note that the functional forms Fi all include the common factor

the dipole moment magnitude. The factors B, and A occur only in this common factor.

That is, the Pi are linear functions of the product AB,. The Fi are algebraic functions of the length L (and the position components rj), however. This relationship can be used for estimating B,, given A, L, and r.

A.2 Relative Field Difference due to Elongation

The relative &#erence between the zero-order field model and the first-order field model is defined as the difference between the magnetic field predicted by the two models, divided by the magnitude of the field for the zero-order model:

B - B.

Pal * (A.25)

This statistic is independent of the magnet cross-section area and magnetic material, but does depend on the length of the magnet and the location at which the field is measured.

This figure of merit for the idealized field model is plotted for an assumed magnet length of 10 centimeters and two representative heights above the end of the magnet in Figure A.3.

The representative heights are 0.15 meters (about half a foot) and 0.3 meters (about a foot).

The plot shows the relative approximation errors in the vertical and horizontal components of the field for these two heights, plotted as functions of the distance from “ground zero”

(the closest spot to the magnet). These represent the relative errors in a pass directly over the magnet, from half a meter of the closest approach to the magnet in both directions. The vertical components are symmetric functions, and the horizontal components are antisym- metric functions. The larger relative errors correspond to the shorter height of the sensor above the pavement.

The relative errors peak at about 4% for a sensor height of 0.15 meters (the approximate sensor height used in the first tests at the Richmond Field Station), and at about 1% for a sensor height of 0.3 meters (about 1 foot, which is a more conservative sensor height for IVHS). Over this range of heights, it appears to vary roughly as the inverse square of the sensor height.

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Science Center SC7701 1. FR 35

Figure A.3: Relative Field Approximation Error versus Distance from Closest Approach

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36

Rockwell International

Science Center SC7701 1.

FR

A.3 Least Squares Estimation of BT

The magnetic field pattern of a “magnetic nail” is characterized by its length L, cross- section area A; and remnant magnetic field B,. The parameters L and A can be measured with a ruler. The value of B, must be determined by using a magnetic field sensor. The following is a derivation of the least-squares solution for determining B, from multiple field measurements.

A.3.1 Measurement Model for Elongate Magnet in Uniform Back- ground Field

Measured field contribution from the magnet. The elongate magnet model in Equa- tions A.15 through A.23 includes B, as a linear factor. This model can be expressed in the general functional form

Bij = Fi (rj, L , A , B,) (A.26)

= B,fii (rj, L, A) (A.27)

I$ = Fi/Br, (A.28)

where Bij is the value of the ith component of the magnetic field B measured at the location specified by the vector rj. The functions Fi are specified by Equations A.15 through A.23.

These depend on the location rj where the field is measured, and on the known parameters L and A.

Added uniform background field. If the background magnetic field Bb is uniform with components

Bb =

then the measured field B, with components Bmij will be Bmij = Bij + Bh.

(A.29)

(A.30)

A.3.2 Representation as a Linear System of Equations

If the field is measured at N discrete locations specified by the set {rj 11 < j 5 N}, then the system of 3N equations

6 h L, A) Br + & = Bmu

UC Berkeley Research Reports

UC Berkeley

Research Reports

This paper has been mechanically scanned. Some

errors may have been inadvertently introduced.

Theoretical and Empirical Analysis of PATH Magnetic Lane Tracking for the Intelligent Vehicle Highway System

THEORETICAL AND EMPIRICAL ANALYSIS OF PATH MAGNETIC LANE TRACKING FOR THE INTELLIGENT VEHICLE HIGHWAY SYSTEM

FINAL REPORT

January 22,1992 through August 14,1992

- _

Rockwell International

Science Center

, .

Rockwell lnternartronal

Preface

Rockwell Interntional

Contents

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.FR

83

83

83

77

77

77

Chapter 1 Introduction

1.1 Executive Summary

Rockwell lnternatronal

1.1.1 Statistical Performance Analysis

!!!!I!9 Rockwell Interntional

Rockwell Internallonal

1.1.2 Background Noise due to Reinforcing Bars

1.1.3 Analysis of Test Data

Rockwell International

1.2 General Background Information

1.2.1 Technical Issues for Steering Using Magnetic Nails

Rockwell lntqrntional

1.2.2 Added Studies of Empirical Test Data

1.3 Historical Background

1.3.1 John Anderson and Anderson Functions

Rockwell Int~rntional

1.3.2 Statistical Methods for Signal Processing and Analysis

1.4 Technical Conventions

1.4.1 Mathematical Notation

Rockwell InternatIonal

1.4.2 Symbols Used

1.4.3 Default Vehicle Coordinates

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Chapter 2

Statistical Performance Analysis

2.1 Objectives and Approach

2.2 Formulas for Distributed Sensors

2.2.1 Field Dependence on Dipole Moment and Location

(2.1)

2.2.2 A Matrix Inversion Formula

(2.2)

Rockwell International

(2.3)

2.2.3 Formulas for Distributed Single-Axis Sensors

(2-7)

= 4+iyd)/5aT

2.2.4 Measurement Sensitivity Matrices

aY 3Po

-= ax

(

Rockwell lntermtional

2.2.5 Simplifications for Special Situations

[

1 1

[

1

m Rockwell International

[ 1

[ 1 > or

[

1

i

1

[ 1

[ 1

Chapter ₁ Introduction

^(A.7)