calibration and enhancement of inertial measurement units frank schubert

8/11/2019 Calibration and Enhancement of Inertial Measurement Units Frank Schubert

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University of KarlsruheDelft Universityof Technology

Institute of Theory and Systems Optimizationin Electrical Engineering

Faculty of Electrical and Information Engineering

Control and Simulation Division

Faculty of Aerospace Engineering

Calibration and Enhancement

of Inertial Measurement Units

cand. el. Frank M. Schubert

Report of a student

research project

Studienarbeit

Delft, July 11 2005th



Declaration of AuthorshipI hereby declare that this report contains no material which has been ac-

cepted for the award of any other degree or diploma at any university orequivalent institution and that, to the best of my knowledge and belief, thisthesis contains no material previously published or written by another per-son, except where due reference is made in the text of the thesis. A completelist of references is included.

iii



Recipients

• Delft University of TechnologyControl and Simulation Department,Faculty of Aerospace EngineeringAnthony Fokkerweg 12629 HC Delft, The Netherlands

– Prof. Dr. Ir. J. A. Mulder

– Dr. Q. P. Chu

– Ph. D. Candidate José Lorga

• University of KarlsruheInstitute of Theory and Systems Optimizationin Electrical Engineering

Kaiserstr. 1276128 Karlsruhe, Germany

– Prof. Dr.-Ing. Gert Trommer

– Dipl.-Ing. Armin Teltschik

– Dipl.-Ing. Jürgen Metzger

iv



Acknowledgements

I would like to thank Prof. Dr.-Ing. G. Trommer and Prof. Dr. Ir. J.A. Mulderto make it possible for me to do a short research project ("Studienarbeit") at theDelft University of Technology (TU Delft). I would also like to thank Dr. Q.P.Chu for his help on theoretical and mathematical topics and Ph.D. Candidate J. Lorga for his strong support and his inspiring ideas as my supervisor herein Delft. Furthermore, I am very thankful for the received help and assistanceof Dipl.-Ing. A. Teltschik and Dipl.-Ing. J. Metzger. Additionally, I am verythankful for the excellent work of Kees Woerkam on the preparation of theexperiments and O. Stroosma for establishing the first contact at the TU Delft.Besides, I would like to thank Ph. D. Candidate J. Oliveira for compiling theflight plan and making it available for this document. For the support in alladministrative questions, A. Muis, C. Dam, P. Kraan, A.M. Markus and H.Lindenburg deserve my gratefulness.

Doing this research project among the frontier of electrical and aerospaceengineering in a really international environment was an extraordinary expe-rience for me and I appreciate very much the help and advise I received fromall people involved. Especially participating in flight tests is a rare experienceand I am very grateful that the opportunity was given to me.

Delft, July 11th, 2005Frank M. Schubert

v



Acronyms

This is an overview of acronyms used in this document:

ADC Analog-to-Digital Converter

DC Direct Current

DMS Differential Measurement System

GNSS Global Navigation Satellite System

GPS Global Positioning System

GRSE Ground-Referenced Single-Ended Measurement System

IMU Inertial Measurement Unit

MTBF Mean Time Between Failure

MTI Moving Target Indicator

NRSE Non-Referenced Single-Ended Measurement System

PCMCIA Personal Computer Memory Card International Association

RTK Real Time Kinematics

SAR Synthetic Aperture Radar

SIMONA International Center for Research in Simulation, Motion and Nav-igation Technologies

TNO Toegepast Natuurwetenschappelijk Onderzoek - Applied research in science

TU Delft Delft University of Technology

vi



Contents

1 Introduction 11.1 The MiniSAR Project . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Calibration of an Inertial Measurement Unit . . . . . . . . . . . 2

2 The MiniSAR Navigation System 32.1 Hardware . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2.1 Acqisition Program . . . . . . . . . . . . . . . . . . . . . . 4

3 Analog-to-Digital Conversion 93.1 Ground-Referenced Single-Ended Measurement System . . . . 93.2 Non-Referenced Single-Ended Measurement System . . . . . . . 103.3 Differential Measurement System . . . . . . . . . . . . . . . . . 103.4 Classification of Measurement Systems . . . . . . . . . . . . . . 123.5 Configuration of the MiniSAR system . . . . . . . . . . . . . . . 12

4 Calibration 154.1 Regression Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.1.1 Linear Regression . . . . . . . . . . . . . . . . . . . . . . . 154.1.2 The Least-Squares Method . . . . . . . . . . . . . . . . . 194.1.3 Polynomial Regression . . . . . . . . . . . . . . . . . . . . 214.1.4 Computational Tools . . . . . . . . . . . . . . . . . . . . . 214.1.5 The Calibration Problem . . . . . . . . . . . . . . . . . . . 21

4.2 Building the model . . . . . . . . . . . . . . . . . . . . . . . . . . 234.2.1 Evaluation of models . . . . . . . . . . . . . . . . . . . . . 234.2.2 Stepwise regression methods . . . . . . . . . . . . . . . . 24

4.3 Accelerometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.3.1 The Tilt Table . . . . . . . . . . . . . . . . . . . . . . . . . 244.3.2 Data Acquisition . . . . . . . . . . . . . . . . . . . . . . . 284.3.3 Possible sources of error . . . . . . . . . . . . . . . . . . . 304.3.4 Experiments with Analog-to-Digital Converter Parame-

ters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.3.5 Adopting the model . . . . . . . . . . . . . . . . . . . . . 33

4.4 Gyroscopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.4.1 Rotation Table . . . . . . . . . . . . . . . . . . . . . . . . . 364.4.2 SIMONA Flight Simulator IMU . . . . . . . . . . . . . . . 364.4.3 Evaluating Time Facilities . . . . . . . . . . . . . . . . . . 404.4.4 Final Gyroscope Calibration . . . . . . . . . . . . . . . . . 40

4.5 Thermal Calibration . . . . . . . . . . . . . . . . . . . . . . . . . 414.6 Magnetometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

vii



4.6.1 Description . . . . . . . . . . . . . . . . . . . . . . . . . . 424.6.2 Swinging . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.7 A new algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.8 Estimation of the Time Need . . . . . . . . . . . . . . . . . . . . . 47

5 Test Flight 495.1 Preparations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.2 Maneuvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

6 Conclusion and Future Perspective 55

A Datasheets 59A.1 Accelerometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59A.2 Gyroscopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

B Calibration Results of the SIMONA IMU Calibration 65

C MATLAB listings 67C.1 Calibration program for three accelerometers . . . . . . . . . . . 67

C.1.1 Calibration Module . . . . . . . . . . . . . . . . . . . . . . 67C.1.2 File Information Module . . . . . . . . . . . . . . . . . . . 69C.1.3 Module for loading binary data . . . . . . . . . . . . . . . 70C.1.4 Plotting Module . . . . . . . . . . . . . . . . . . . . . . . . 71

D Publication 73

viii



List of Figures

2.1 The opened IMU box, containing three accelerometers to theleft and three gyroscopes to the right. . . . . . . . . . . . . . . . 4

2.2 Overview of the hardware parts and the model of the fusion of the sensors’ datas. . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.3 The schematic diagram of the IMU box. . . . . . . . . . . . . . . 52.4 The schematic diagram of the junction box. . . . . . . . . . . . . 6

2.5 Internal circuit diagram of the accelerometer (source: QA700 Q-Flex Accelerometer, Construction and Principle of Operation Manual). 6

2.6 The main screen of the acquisition software for selecting thechannels to be recorded. . . . . . . . . . . . . . . . . . . . . . . . 7

2.7 The visualization module of the acquisition software. . . . . . . 72.8 Recorded data of a rotation experiment has been visualized. . . 8

3.1 Non-grounded Signal Source. . . . . . . . . . . . . . . . . . . . . 9

3.2 Ground-referenced measurement system for use with non-referencedsignal sources. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.3 Ground-referenced Signal Source. . . . . . . . . . . . . . . . . . . 10

3.4 Non-referenced single-ended measurement system for use withreferenced signal sources. . . . . . . . . . . . . . . . . . . . . . . 11

3.5 Differential measurement system for use with signal sourceswhere each signal can refer to a different reference. . . . . . . . . 11

3.6 Ground-loop caused by measuring ground-referenced signalswith a ground-referenced single-ended measurement system. . 12

3.7 All possible configurations with two types of signals and threekinds of measurement systems. . . . . . . . . . . . . . . . . . . . 13

3.8 Measurements taken of a temperature sensor. The left signalhas been taken before the design changes of the schematic dia-gram were made, the right signal after the change. . . . . . . . . 14

4.1 Distances and the times needed of several swimming laps. . . . 164.2 Straight line relationship between swum distance and time. . . 17

4.3 How observations are generated in a linear regression. . . . . . 18

4.4 Difference of the least-squares and the total least-squares ap-proach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4.5 Ideal characteristic curve (red) of a sensor and possible realisticcurve (green) of a sensor. . . . . . . . . . . . . . . . . . . . . . . . 22

4.6 The tilt table and the angle θ. . . . . . . . . . . . . . . . . . . . . 25

4.7 The directions of the sensitive axes of the accelerometers andthe ϕ measurement angle. . . . . . . . . . . . . . . . . . . . . . . 26

ix



4.8 Using a non-perfect inclinometer to align a plain horizontally.Letter a describes the shorter side of the non-perfect inclinome-

ters, b is the longer side. . . . . . . . . . . . . . . . . . . . . . . . 274.9 References of the expected accelerometer measurements (theso-called ’true data’). The samples 0-12 show the true datafor an inclination of θ=0, then it goes up in 10steps amongthe next sample segments until the samples 108-120 where θreached 90. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.10 Raw accelerometer measurements at θ=40of the three acceler-ation data channels. . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.11 Results of a least-squares fit with previously measured accelerom-eter data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.12 Result of the cabling test. The signals show peaks in the chan-nels while moving the cables, especially for the y-channel. . . . 31

4.13 Raw data of an x-channel measurement while the hydraulic

pump in the basement was running. . . . . . . . . . . . . . . . . 324.14 Residual of the accelerometers’ calibration with the computed

parameters of Tab. 4.6. . . . . . . . . . . . . . . . . . . . . . . . . 354.15 Genisco C181 Turn Table (source: Genisco C181 Operations Man-

ual). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.16 Signal of a temperature sensor (1000 Hz sampling frequency,

100/s rotation rate). It is noteable that the noise is dependenton the position and thus on the sliprings of the rotation table. . 37

4.17 The rate changer of the turn table. The sketch shows two arbi-trary positions, left with a low rotation rate, right with a higherrotation rate (source: Genisco C181 Operations Manual). . . . . . . 37

4.18 The three gyroscopes of the SIMONA IMU are in the center of

the rotation table. On the right the power supply can be seen. . 384.19 Calibration for the International Center for Research in Simu-

lation, Motion and Navigation Technologies (SIMONA) IMU:The upper plot shows the estimation of the actual calibrationand the calibration provided by the manufacturer, both are lin-ear models. The plot in the middle presents the residuals of the actual, the bottom plot shows the residuals of the manufac-turer’s calibration. . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.20 Results of the calibration for the r-gyroscope. The top plot showsthe different rotation rates. The second plotshows the residualsfor the calibration with the ADC used as the stop watch. Forplot three the GPS board was used and the bottom plot pointsout the result when a dedicated timer was used. . . . . . . . . . 39

4.21 Residuals of the final gyroscope calibration. . . . . . . . . . . . . 414.22 Raw output of the temperature sensor during temperature cal-

ibration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.23 Calibration results of the P gyroscope temperature calibration. . 434.24 Impact of errors on magnetic field measurements of a measured

circle in two dimensions. . . . . . . . . . . . . . . . . . . . . . . . 46

5.1 The scheduled route for the test flight. . . . . . . . . . . . . . . . 505.2 The rack with the mounted equipment on it. . . . . . . . . . . . 505.3 The rack with the mounted equipment on it. . . . . . . . . . . . 51

x



List of Tables

2.1 The used hardware. . . . . . . . . . . . . . . . . . . . . . . . . . . 3

4.1 Distances a swimmer swam and the corresponding measured

times. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.2 Comparison between a calibration done on raw data and after

computing the medians first. Former σ is the standard devi-ation of the residuals of the original program, actual σ is theoutput of the modified program. . . . . . . . . . . . . . . . . . . 30

4.3 The row Ground indicates, if a grounded channel was acquiredin between data channels. The used multiplexing frequency if given through F Mx . Standard deviations of the residuals of allchannels were calculated. The unit of all values is m/s2. . . . . . 33

4.4 The first three models that were examined. Parameters markedwith a * represent the scale factors of the three accelerometers.β 0 is the bias of each model. . . . . . . . . . . . . . . . . . . . . . 34

4.5 The second attempt in finding best fitted models for the ac-celerometers. β 4, β 5 and β 6 are the cross-coupling coefficients. . 34

4.6 The final models of the accelerometers’ calibration. . . . . . . . . 354.7 The approximate expenditure of time for calibrating the accelerom-

eters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

A.1 Technical data of the used gyroscopes. . . . . . . . . . . . . . . . 63A.2 The pin assignment of the used gyroscopes. . . . . . . . . . . . . 63

B.1 Results of the SIMONA IMU clockwise calibration. . . . . . . . 66

xi



1 Introduction

1.1 The MiniSAR Project

Many fields like autonomous vehicles, autopilots, identification of aerody-namic models in aircraft design and Synthetic Aperture Radar (SAR) requirean extremely precise navigation system. These applications can demand anaccuracy at centimeter level in position, millimeter per second level in veloc-ity and miliradian level in attitude.

This project develops a new navigation device with the goal of process-ing SAR images in real time within the MiniSAR project. The final resultsof SAR image processing are highly dependent on the autofocus algorithmwhich requires again a high-accuracy navigation solution. MiniSAR is devel-oped in cooperation with Toegepast Natuurwetenschappelijk Onderzoek - Applied

research in science (TNO), a physics and electronics laboratory in The Nether-lands.

MiniSar is a miniature, lightweight and scalable SAR/Moving Target Indicator(MTI) system. The homepage1 gives a short overview of MiniSAR. MiniSARwill be introduced into the market in 2006-2007 and will contain 24 transmit-ters/receivers elements in a row. The radar will operate in the X-band, witha center frequency of about 10 GH z. The resolution will be better than 50 cm.The optional MTI mode will be able to detect moving targets at speeds fasterthan 3 km/h.

The navigation system presented here uses commercial off-the-shelf com-ponents wherever possible. The Inertial Measurement Unit (IMU) containsthree accelerometers and three gyroscopes, a fluxgate magnetometer and ahigh-end dual-frequency Global Navigation Satellite System (GNSS) receiver board. The GNSS receiver is responsible for the absolute position of the air-craft, but these datas are for example available only once in a second withan accuracy of approximately 1 m (with the use of a two frequency GPS re-ceiver and differential GPS). As stated above, SAR image processing has muchhigher requirements. Therefore, a GNSS receiver is combined with an IMUwhich can provide more accurate position updates with higher update fre-quencies. The SAR system needs a navigation solution that provides a highaccuracy during straight level flight to reconstruct the SAR images correctly.

1http://www.cs.lr.tudelft.nl/index.php?page=projects&id=25

1

http://www.cs.lr.tudelft.nl/index.php?page=projects&id=25

http://www.cs.lr.tudelft.nl/index.php?page=projects&id=25



Introduction

1.2 Calibration of an Inertial Measurement Unit

The reacheable accuracy of an IMU is strongly correlated with the inital cal-ibration of the used sensors. Indeed, the manufacturers of the componentsprovide data like scale factors and coefficients for the compensation of sensorsystem errors, but errors like misalignments between the different axes do notoccur until the system is assembled. Thus, a calibration has to be performedthat regards the navigation system as a whole.

First, the process of data acquisition using a Personal Computer Mem-ory Card International Association (PCMCIA) Analog-to-Digital Converter(ADC) with the possibility of using different setups to minimize noise on mea-sured signals is illuminated. This part is followed by a short introduction intothe theory of regression analysis and the least-squares parameter estimationmethod. After an introduction on how to use the tilt table as a calibration de-vice, the calibration for the accelerometers is described. The next part explains

how a well-fitted model for the acceleration sensors can be developed. Duringthat part emphasis is laid on the observance of environmental influences, e.g.vibrations or other disturbances within the building. These remarks follow adescription of the rotation table and the calibration of gyroscopes. Two sec-tions depict the calibration which is necessary to avoid thermal dependanceand a new calibration method for magnetometers. Another chapter is devotedto the preparation and performance of test flights in a Cessna Citation to col-lect flight path data that took place in June 2005.

A look-out gives ideas for future research and areas that need further im-provement.

Again, a sophisticated calibration for the IMU is needed to ensure a highperformance of the IMU and therefore for the whole navigation solution. This

is needed to meet the demanding requirements of the SAR image processing.

2



2 The MiniSAR NavigationSystem

2.1 Hardware

One of the main characteristics of the developed navigation system is flexibility. The adaptability to different kinds of operations is possible due to thedesign of the IMU. It consists of three Q-Flex accelerometers, three fiberopticgyroscopes (one for every axe) and a three axes fluxgate magnetometer. Theaccelerometers and the magnetometer are delivered with temperature sensorsinside their housing for a temperature calibration. Temperature informationfor the gyroscopes is obtained by a sensor that is placed next to the gyroscopes.A GPS board with dual-frequency Real Time Kinematics (RTK) capabilitiesserves as the GNSS part of the navigation system.

For the increase of the system’s flexibility, the analog-to-digital conversionis done outside the IMU box. This allows a post-processing of the sampleddata with different filter designs and cut-off frequencies after the digitaliza-tion. A 16 bit ADC with a maximum sampling frequency of 1000 H z is usedfor the digitalization process.

Fig. 2.1 gives an insight in the IMU assembly, a prototype of the to be de-veloped MiniSAR IMU. The left half of the photograph shows the accelerom-eters, in the right half the gyroscopes can be seen.

Tab. 2.1 lists the models and the manufacturers of the particular hardwareparts. Detailed datasheets are given in appendix A.

The accelerometers contain already temperature sensors in their enclo-sures. A separate temperature was mounted next to the gyroscopes to com-pensate temperature influences on measurements (as shown in section 4.5).

Additionally, there is a junction box that provides four DC/DC (Direct

Type Model ManufacturerAccelerometers Q-Flex 700-20 Honeywell (Plymouth, USA)Gyroscopes VG941-3AM Fizoptika (Moscow, Russia)Magnetometer APS-536 Wuntronic (Munich, Germany)AD-Converter DAQ-6036E National Instruments (Austin, USA)GPS Board Euro4-RTL L1/L2 NovAtel (Calgary, Canada)GPS antenna 512C GPS NovAtel (Calgary, Canada)

Table 2.1: The used hardware.

3



The MiniSAR Navigation System

Figure 2.1: The opened IMU box, containing three accelerometers to the left and threegyroscopes to the right.

Current) converters as power supplies for the sensor groups. Moreover, the junction box connects the PCMCIA ADC card to the IMU box with two dif-ferent cables. A 12V switched mains adaptor connects to the junction box asa power source. The schematic diagram for the IMU and the junction box arepresented in Fig. 2.3 and Fig. 2.4 respectively. The schematic diagram for theaccelerometers is shown in Fig. 2.5.

2.2 Software

2.2.1 Acqisition Program

The acquisition program was developed by J. Lorga in Visual C++. It hasdifferent modules for settings, data acquiring, visualization and convertingthe readings to a binary format.

The settings module e.g. allows to assign the sampling frequency, the mul-tiplexing frequency of the ADC, the channels to be recorded, the duration andthe signal gains (see Fig. screenshot-main). After saving the data it can beviewed at once using the visualization module. Here, all of the channels or

selected channels can be plotted and a first evaluation of the recorded datacan be done on-site, an example show Fig. 2.7 and Fig. 2.8.

This binary format can be read in by MATLAB routines that use a digitallow-pass filter to process the oversampled data to avoid the aliasing problem.After this step, the data is downsample to the wished sampling frequency.

4



2.2 Software

I/EKF

inputs

Position coordinates

Attitude angles

Velocity coordinates

magnetic fieldcomponents

accelerations androtation rates

Magnetometer

GNSSobservables

GNSS Receiver

IMU

PolinomialCalibration

PolinomialCalibration IMU error model

parameters

Figure 2.2: Overview of the hardware parts and the model of the fusion of the sensors’datas.

Temp

1 3

Accelerometer X1 2 3 4 5 6 7 8 9 10

Accelerometer Y1 2 3 4 5 6 7 8 9 10

Accelerometer Z1 2 3 4 5 6 7 8 9 10

Gyro p1 2 3 4

Gyro q1 2 3 4

Gyro r1 2 3 4

11 12

1314 15 16 17 18 19201 2 3 4 5 6 7 8 9 10 21 22 23 2425

IMUP1Power/Data

25 Pins Serial Plug

R7R3R1 R5

IMU

R2 R4 R6

+ 5 V

Digital GroundGround

+ 15 V

- 15 V

Temperature Data

Accelerations Data

Rotational Rate Data

Figure 2.3: The schematic diagram of the IMU box.

5




1 2 3 4 5 6 7 8 9

JunP3

MagnetometerPower/DataSerial 9 Pins

Socket

1

2

C

1

2

3

2 8 V D C - 1 5 V D C

5 0 mA / 1 . 5 W

JunP6

AD 68 Pins Plug

1 2 3 4 5 6 7 8 9 10 1112 13141516 17 18 19 20 2122 23 24 2526 27282930 31 323334 35 36 37 38 3940 4142 43 44 4546 4748495051 525354 5556 575859 60 6162 6364 65 6667 68

1

2

C

1

2

3

2 8 V D C - 5 V D C

1 1 0 mA / 3 W

1

2

C

1

2

3

2 8 V D C - 1 5 V D C

6 0 mA / 2 W

1

2

C

1

2

3

2 8 V D C - 5 V D C

1 A / 5 W

A B C

JunP1

External PowerMS3120E12-3P

JunP4

MagnetometerTemperatureBNC Socket

A B C 5 6 7 8 101 2 3 4 9

Socket PlugJunP5

GPSStrobes Power

Lemo 2x 5 Pins

11 1213 14 15 16 17 1819201 2 3 4 5 6 7 8 9 10 21 222324 25

JunP2

IMU Power/DataSerial 25 Pins Socket

Junction Box

DC-DC4

DC-DC3

DC-DC2

DC-DC1

AB

B

B

B

A

A

A

+15V0V

-15V

0V

+5V0V-5V

+15V0V

-15V

+5V0V

-5V9-36V

0V9-36V

+ 5 V Digital Ground

+15/28 V

- 15 V

Temperature Data

Accelerations Data

Rotational Rate Data

Magnetic Fields Data

Strobes

Analog Ground

Figure 2.4: The schematic diagram of the junction box.

Figure 2.5: Internal circuit diagram of the accelerometer (source: QA700 Q-Flex Ac-celerometer, Construction and Principle of Operation Manual).

6



2.2 Software

Figure 2.6: The main screen of the acquisition software for selecting the channels to berecorded.

Figure 2.7: The visualization module of the acquisition software.

7




Figure 2.8: Recorded data of a rotation experiment has been visualized.

8



3 Analog-to-DigitalConversion

There are two different kinds of signals to be measured: grounded signals andnon-grounded signals.

Also, there are three ways of usage of an ADC: As a referenced single-ended measurement system, as a non-referenced single-ended measurementsystem or as a differential measurement system (assumimg that the ADC cardprovides these possibilities).

Following, an introduction to the different ways of acquiring data is given.Precautions that are to be met and more detailed explanations can be found in[Sha01] and [DAQ02].

3.1 Ground-Referenced Single-Ended Measurement

System

An Ground-Referenced Single-Ended Measurement System (GRSE) can beused with an non-referenced or a floating signal (ungrounded signal) source.These are for example devices with a battery as power source or special in-struments that explicitly float its output signal (Fig. 3.11).

In that case the ground2 of the measured system can be connected directlyto the ground of the ADC. The ADC supplies then a common ground for thewhole system (Fig. 3.2).

1source of the figures 3.1, 3.2, 3.3, 3.4, 3.5, 3.6, and 3.7 is [DAQ02].2The ground of the signal in Fig. 3.1 is the V S “-” connector. “Ground” in 3.1 describes the

case or the shielding.

Figure 3.1: Non-grounded Signal Source.

9



Analog-to-Digital Conversion

Figure 3.2: Ground-referenced measurement system for use with non-referenced sig-nal sources.

Figure 3.3: Ground-referenced Signal Source.

3.2 Non-Referenced Single-Ended Measurement System

A Non-Referenced Single-Ended Measurement System (NRSE) can be usedwith ground-referenced signals (grounded signals) that provide their ownground (see Fig. 3.3).

All devices that plug into power outlets belong to these types of signals, because the system’s ground and the ground or shielding of the case is con-nected to the electricity network. It is necessary to use the ADC as a NRSE

(Fig. 3.4) in order to measure signals where the signal and the ADC can haveits own grounds (references).

3.3 Differential Measurement System

In a Differential Measurement System (DMS) environment each signal canhave its own ground that might be different from the other signals’ references.However, then it is necessary to use a separate input for each signal’s ref-erence, which would cut the amount of available channels in half. Fig. 3.5shows the plan of such a system.

10



3.3 Differential Measurement System

Figure 3.4: Non-referenced single-ended measurement system for use with referencedsignal sources.

Figure 3.5: Differential measurement system for use with signal sources where eachsignal can refer to a different reference.

11




Figure 3.6: Ground-loop caused by measuring ground-referenced signals with aground-referenced single-ended measurement system.

3.4 Classification of Measurement Systems

The easiest way to use the ADC is the GRSE, because there is only one groundand all signals are measured in respect to that single ground. But the problemthat appears with connecting a ground-referenced signal source to a GRSEinstead of a non-referenced source is the so-called ground-loop losses (see Fig.3.6).

Because both system connect to the same ground (in the electricity system)the measured voltage is the sum of the signal’s voltage V S and the difference between the two grounds ∆V g . In most cases there will be some noise in ∆V g,

e.g. influences of the 50 H z power-line frequency.However, if the signal voltage level is high and the signal and the connec-

tions to the AD converter have a low impedance, it is still possible to takereadings with such a configuration, because the degradations caused by theground-loop are tolerable. However, it is not recommended.

Fig. 3.7 shows an overview of all possible configurations.

3.5 Configuration of the MiniSAR system

In the case of the MiniSAR system, the notebook with the PCMCIA card isconnected to the power outlet as is the navigation system with the 12V power

supply. The signals for the accelerometers and the gyros are typically be-low 1V during slow or zero motion. All that disqualifies a GRSE configu-ration for the given navigation system as given in Fig. 2.3 and 2.4. This factwas discovered during this project and therefore the design of the IMU waschanged to a non-referenced singel-ended measurement system. The best so-lution would have been a differential measurement system because the gyrosand the accelerometers work with different grounds. But for that configura-tion the amount of channels is not sufficient.

Regarding the accelerometers, another point was improved. The accelerom-eter detects accelerations with the help of a quartz proof-mass that is mountedrelative to a stationary ring. Motion displaces the proof-mass slightly. A ca-pacitive null-position detector sends now current through an actor that com-pensates this displacement. The current that is required to balance the proof-

12



3.5 Configuration of the MiniSAR system

Figure 3.7: All possible configurations with two types of signals and three kinds of measurement systems.

13




0 500 1000 15002.97

2.975

2.98

2.985

2.99

2.995

3

3.005

3.01

3.015

3.02

0 500 1000 15002.97

2.975

2.98

2.985

2.99

2.995

3

3.005

3.01

3.015

3.02

V o l t

V o l t

Samples Samples

Figure 3.8: Measurements taken of a temperature sensor. The left signal has been taken before the design changes of the schematic diagram were made, the right signal afterthe change.

mass is directly proportional to the input acceleration.This principle reveals that the resistor RL in Fig. 2.5 must be connected

directly between pin 1 (Signal) and pin 8 (Signal Ground) of the accelerometer.After the modifications were made, this precondition was met, too.

Fig. 3.8 shows 15 seconds of measurements of a temperature sensor beforeand after the changes. The decrease in the noise level can clearly be seen.

14



4 Calibration

This chapter will give an introduction on how to calibrate components of an IMU: Accelerometers, gyroscopes, and the magnetometer. This enfolds

a short theoretical introduction to regression analysis before the calibrationfacilities and their usage are presented and explained. Then the calibration of accelerometers, gyroscopes and an algorithm for calibrating magnetometersare shown. In addition environmental influences that can disturb the mea-surements are pointed out.

4.1 Regression Analysis

Regression Analysis is one of the most widely used methods for analyzingmany different kinds of data. Universally, it uses a process to find a rela-tionship between a set of independent variables and a variable of interest, the

response or dependent variable. This chapter describes the construction of themodel (how the variables relate to each other) and the estimation of the un-known parameters by using the least-squares method. After a general intro-duction, the problem is illuminated from the side of the calibration problem.

It is not the intention of this chapter to cover the vast theory of regressionanalysis or the least-squares method in total. Instead, it will rather give a shortintroduction to the subjects. References like [Dra98], [Raw98], and [Mon01]cover the topic in a general theoretical way on how to use regression analysis.This chapter is based on the remarks of those references.

4.1.1 Linear Regression

As an example on how to use regression analysis, imagine a swimmer wantsto predict the time he needs for doing his daily exercise in a swimming pool.For that, he would measure the time he needs for different distances like100m, 200m, 400m and 500m during some of his trainings. Tab. 4.1 showsa list of possible values. If he will plot the data in a diagramm he would prob-ably get something such as Fig. 4.1.

Such a plot is called a scatter diagram. This diagram clearly shows a linearrelationship between the distance and the time needed for that. Because of thislinearity the best approximation for the points is a straight line relationship.If x represents the distance of a lap and y the used swimming time, then theequation for a straight line would be

y = β 0 + β 1 x (4.1)

15



Calibration

Swum distance Time neededin meters in seconds50 5550 25

100 45100 35100 39150 70150 60200 88200 88250 90250 120

250 115300 150300 180350 157400 155450 180450 165500 190500 240500 205

Table 4.1: Distances a swimmer swam and the corresponding measured times.

Figure 4.1: Distances and the times needed of several swimming laps.

16




Figure 4.2: Straight line relationship between swum distance and time.

where β 0 is called intersept and β 1 is called slope. Since the data points donot fall exactly on a straight line, an error is to be introduced:

y = β 0 + β 1 x + (4.2)

The error could contain measurement errors regarding the clock facility orthe swimmer having to share the lane with other swimmers which will resultin a delay.

Equation 4.2 is a linear regression model. The variable x is called the in-dependent variable, predictor or regressor, y is called the dependent variableor response. The error is the difference of the straight line and the mea-surements. If our model is perfect, covers all errors related to the measuredvalues y , what could be measurement errors etc. If we consider the mean of is 0 and the variance is σ2 the mean response of any value of the regressorvariable is

E (y|x) = µy|x = E (β 0 + β 1 x + ) = β 0 + β 1 x (4.3)

Furthermore, the variance of y under the assumption of a noise-free x is

V ar(y|x) = σ2y|x = V ar(β 0 + β 1 x + ) = σ2 (4.4)

Fig. 4.3 clarifies the relationship of the variance σ2 and the values of y.These values of y are samples of normal distributions of any given point onthe straight line in the simple linear regression case. The variance σ2 is anindicator for variability or noise of the values y . If σ2 is small, the observedvalues will fall close to the line, if it is large, the values can deviate sizeablyfrom the line.

The parameters β 0 and β 1 are unknown and have to be estimated. Here theleast-squares method is used to get such an estimation. Another option would be the total least-squares method. Fig. 4.4 shows the different approaches:least-squares minimizes the difference dLS of the measurements in respect tothe straight line, total least-squares minimizes the difference dTLS which isorthogonal to the straight line.

17



Calibration

N(0, )2

observed values y ( )are samples fromthese distributions

•

N(0, )2

y

x

Figure 4.3: How observations are generated in a linear regression.

dTLS

dLS

Figure 4.4: Difference of the least-squares and the total least-squares approach.

18




The least-squares method only allows the dependent variable y to be noisy,not the independent variable x. A method that allows the y-values to be noisy

as well is total least-squares. The difficulty in the total-least squares methodlies in the knowledge of the noise levels of the measurement errors that areneeded. An introduction to total-least squares can be found in [Huf91] and[But99].

4.1.2 The Least-Squares Method

The least-squares method is used to estimate the parameters β 0 and β 1 withthe gathered sample data:

yi = β 0 + β 1xi + i, i = 1,2,...,n (4.5)

These equations can easily be written in matrix notation:

y0y1...

yn

=

1 x0

1 x1

......

1 xn

β 0β 1...

β k

+

01...

n

(4.6)

Or in a more compact display

y = X β + (4.7)

where

y =

y0y1...

yn

, X =

1 x01 x1

......

1 xn

, β =

β 0β 1...

β k

, =

01...

n

(4.8)

The goal is to find the vector β that minimizes

S (β) =

ni=1

2i = = (y − X β)(y − X β)1 (4.9)

The criterion S (β) can be transformed to

S (β) = y y − β X y − y X β + β X X β (4.10)

= y y − 2 β X y + β

X X β (4.11)

because β X y is a scalar and its transpose (β X y) = y X β is the samescalar.

To find the minimums of S , the least-squares estimator must satisfy

∂S

∂ β

β

= −2X y + 2X X β = 0 (4.12)

This can be simplified into

1whereX describes the transpose of a vectorX.

19



Calibration

X

X ˆ

β = X

y (4.13)β = (X X)−1 X y (4.14)

If the inverse matrix (X X)−1 exists, there is a solution β as the least-squares estimator of β. That is true, if the independent variables, the regres-sors, are always linearly independent.

The vector of fitted values yi of the observed values yi is

y = X β (4.15)

The differences between the measured values yi and the fitted values yi arecalled residuals and they are to be calculated as follows:

e = y − y (4.16)

If the data of the swimming experiment is used and the least-squares esti-mation is calculated, the following solution is found:

β 0 = 11.2166

β 1 = 0.3958

These values represent the straight line in Fig. 4.2. With the formula

y = 11.2166 + 0.3958 x (4.17)

the swimmer can calculate the time needed for a given length he wouldlike to swim. This means in general that an estimated value y for a givenindependent variable x can be calculated through

y = x β. (4.18)

The matrix notation allows to add more parameters easily. In the givenexample, the number of fellow swimmers on the lane could be modelled as anew parameter to describe the distraction of the swimmer and the delay in hismeasurements. This multiple linear regression model would be

y = β 0 + β 1 x1 + β 2 x2 + (4.19)

with x1 as the distance input and x2 as the number of people on the samelane. A geometrical interpretation of this formula is a plane in the three-dimensional space. Other examples could include curved planes or even hy-perplanes in spaces of higher degrees than three dimensions.

So far, the preconditions for using the least-squares method are

• The relationship between dependent variable and independent variablesis (at least approximately) linear.

• The error term has zero mean and a constant variance σ2.

• The errors are uncorrelated and normally distributed.

20




4.1.3 Polynomial Regression

The next step is to use polynomials to build the models. If for example a scatterplot gives the impression of a quadratic relationship between independentand dependent variables, a model like

y = β 0 + β 1 x1 + β 2 x21 + (4.20)

could be suitable. Such a model is called a second-order model (or quadraticmodel) in one variable. Imaginable are also models of higher degrees and theinclusion of combined regressors to cover dependances on a relationship be-tween two independant variables, for example the misalignment of the axesof two acceleration sensors to each other.

4.1.4 Computational Tools

The program MATLAB offers many statistical functions and a huge toolbox of different operations. It is easy to calculate the least-squares coefficient itself by using Eqn. 4.14 but a shorter possibility is to use the command lsqr. Thesyntax x = lsqr(A, B) provides the least-squares estimation of a system of equations like A ∗ x = B. Written in the nomenclature used in this chapter,

A ∗ x = B refers to X β = y .Another command regarding least-squares is polyfit, which calculates

the coefficients of a polynomial of a certain degree in a least-squares manner.The syntax is P = polyfit(x,Y,n), where x refers to the known X, Y refers to

y and P are the returned coefficients β of a polynomial with degree n. Thecoefficients are sorted from the highest order to the lowest.

4.1.5 The Calibration Problem

Most regression problems want to gain knowledge of a value y to a givenvalue x like the swimmer who wants to estimate times for future distances.But a calibration procedure deals with the inverse problem: First, a set of known states is measured, in the example of a newly developed thermometerthese could be measurements at several temperatures, say from 10to 100in10steps. This discovers maybe a quadratic dependence or a dependece of an even higher degree of the new thermometer in respect to the temperature.Probably, a model is developed of the third or fourth order probably. Now, if a measurement is taken with the new thermometer, how could it be correctedusing the developed model? In the case of a linear model like

y = β 0 + β 1 x + (4.21)

this is very easy:

x = y − β 0

β 1(4.22)

with y as the measured temperature as input and x as the estimated (cor-rected in a calibration way) value. But in the case of models of higher degrees,for example

y = β 0 + β 1 x1 + β 2 x21 + β 3 x3

1 + β 4 x41 + (4.23)

21



Calibration

10 2 3 54

1

2

3

5

4

Input, x

Output, y

0: bias

1: scale factor

2: non-linearity

Figure 4.5: Ideal characteristic curve (red) of a sensor and possible realistic curve(green) of a sensor.

it is not trivial to find a solution. In a multi-regressor environment withindependent variables x1, x2, and x3 in the second, third or even higher or-der, this task seems too complex and the possible calculation through numeric

solution of equations of the fourth order is not suitable in a productive envi-ronment. A solution could be to give up precision and to develop a linearizedmodel of the high-degree model. This is not an option in most cases becausethrough that the gained precision of a high-degree model is lost.

The practical way is to use a e.g. model

x = α0 + α1 y + α2 y2 + α3 y3 (4.24)

where x can now be estimated by y directly. This approach is called theinverse estimator. It assumes that x is a random variable. In most cases, x willnot be a random variable because it is controlled. But this is still a possibilityto perform calibrations even if they are done in a less correct mathematicalway regarding the preconditions of the variables.

An ideal characteristic curve of a measurement device should look like thered curve in Fig. 4.5. If the input x to a sensor, for example a thermometer is5C the output y of the sensor should result in 5C as well.

In fact, most sensors show several irregularities: The parameter β 0 (theso-called bias) describes the output of a sensor even if there is no excitationpresent at all. β 1 is the scale factor of a characteristic curve; the scale factortakes a different slope than 1 of the gradient into account. If the gradient isnot even a straight line, but a curve, these anomalies can be modeled by us-ing non-linearities, e.g. here illustrated by parameter β 2 as a second-ordercoefficient. Thinkable are also parameters of even greater order. By using anappropriate sensor model and by applying the least-squares method, these pa-rameters, parameters with dependances of higher degrees and cross-couplingparameters can be determined in an optimal way for each sensor.

22



4.2 Building the model

4.2 Building the model

In order to design a suitable model for a calibration problem, tests have to beintroduced on how to evaluate models to be able to answer the question whichmodel describes the actual context of measurements and references better thanthe other. This is a vast statistical field, with many possible tests to perform.However, the extent of the work on hand only allows to present few. [Pik97]presents more different approaches more detailed.

Second, there are stepwise regression methods to build a model. Here, theforward selection and the backward elimination are explained.

4.2.1 Evaluation of models

Significance of Regression

The test for significance of regression examines, if there is a linear relation-ship between the dependent variable y and any of the independent variablesx1, x2, . . . , xk. The test is true, if at least one of the parameters β 0, β 1, . . . , β k ,does not equal 0. It is trivial in a logical sense, that if a coefficient β k is 0 it doesnot have any impact on the response. Furthermore very low coefficients β k inrespect to the other coefficients can be discarded to achieve a more compactmodel. In the following process of evaluating models, tables will be givenwith the values of the calculated parameters to be compared.

Residual Mean Square

The residual sum of squares SS Res is given by

SS Res =

ni=1

e2i =

ni=1

(yi − yi)2 = e e (4.25)

with the substitution e = y − X β this can be transformed to

SS Res = y y − β

X y (4.26)

as stated in [Mon01]. The residual mean square M S Res is

M S Res = SS Res

n − p (4.27)

where n is the number of measurements taken and p is the number of co-efficients β.

Since MS Res can be seen as an unbiased estimator of the variance (likeshown in [Mon01]), it is desired to get a low value for the residual meansquare.

Coefficient of Determination

First, the definition of the total sum of squares:

SS T = y y − (n

i=1 yi)2

n (4.28)

With this, the coefficient of determination is determined by

23



Calibration

R = 1 − S S Res

SS T (4.29)

Its value lies between 0 and 1. The higher the value, the better the qualtiyof fit. The coefficient of determination is maybe the first value a model builderwould look at to determine a quality of fit parameter for the used model, asdone in [Lan04]. The sole use of R should be balanced carefully, because ingeneral, R always increases when a term is added to the model. However, thequestion if this term adds an important contribution to the model lies in the judgment of the modeller. The definition of

RAdj =

1 −

S S Res/(n − p)

SS T /(n − 1) (4.30)

provides an adjusted version of R. It provides a penalty for adding termsthat are actually unnecessary and thus prevents the overfitting of the model.

Residuals

Like the view on the coefficients, one should never forget to observe the resid-uals yi − yi as well. For a calibration it is wished to get the residuals as faras possible to zero, what promises an interpolation of future measurementsof high quality. A criterion for this is the standard deviation. The lower thestandard deviation, the closer is the residual error to zero.

4.2.2 Stepwise regression methods

The forward selection method starts with few or only one regressor and addsone regressor at a time. For each modification the model is evaluated us-ing techniques like the (adjusted) coefficient of determination or the residualmean square. Once the desired quality of fit is reached, the iteration termi-nates.

The backward elimination method however starts with all possible regres-sors included in the model. Using the evaluation methods mentioned above,regressors that do not contribute significantly to the accuracy of the model arediscarded.

The method used for the present calibration problem is to give a reasonablemodel first with all terms included and then terms of higher order are addedto improve the performance of the model.

4.3 Accelerometers

In order to calibrate the accelerometers, they have to be set to a reference con-dition. Between -1 g and +1g this can be achieved through mounting the IMUon a tilting table and setting this table to a certain angle. Thus, the earth’sgravity field is used to exposure the accelerometers to dedicated accelerations.

4.3.1 The Tilt Table

With the tilt table it is possible to adjust the attitude of the IMU respectiveto the earth’s gravity vector in a very accurate way. An already mentioned

24



4.3 Accelerometers

Figure 4.6: The tilt table and the angle θ.

disadvantage is, that it allows to measure the accelerometer sensors in a range

from -1 g to +1 g only. The tilt table used has two degrees of freedom: Thetilt angle θ that adjusts the z-axis of the IMU relative to the verticality (seefigure 4.6) and the rotation angle ϕ that rotates the xy-plane orthogonally tothe z-axis. Both angles can be set within an accuracy of 0,25 arcseconds.

Fig. 4.7 shows the IMU mounted on the tilt table platform with the ϕ angle.Before starting with the calibration, the attitude of the table has to be aligned

horizontally to the local gravitational field. This can be done by using an in-clinometer. However, to reach an exact horizontal alignment of the tilt tablerespective to the horizontal, some precautions have to be taken care of:

• The z-axis is supposed to be parallel to the gravity vector at a tilt an-gle θ of 0. The inclinometer is placed on top of the IMU and θ is setto 0. The inclinometer shows a difference from the optimal horizontal

position. But since the inclinometer itself does not give a perfect output,the measurement has to be repeated with the inclinometer now turned by 180. The inclinometer shows another difference in the opposite di-rection2. Now, through the adjuster on the table, θ is adjusted so thathalf of the difference is counterbalanced. The inclinometer is turned an-other time. When the inclinometer shows the same differences from theequality in both ways, the table is aligned correctly. Figure 4.8 clarifiesthis procedure. It is crucial that the adjusting wheels of the tilt table areonly turned in one direction before taking readings of the angle scale.Otherwise, the clearance ratio of the gear is falsifying the scale readings.

2There are two unknows: the error angles of the table and of the inclinometer. Therefore, therehave to be done two measurements to get knowledge of both of them

25



Calibration

z

x y

Figure 4.7: The directions of the sensitive axes of the accelerometers and the ϕ mea-surement angle.

If one went too far during the adjustment, it is possible to go back (a bit

over the wished point) and then try to set the desired angle again.This error of θ was typically around 01’, it is a table error and has to betaken into account during every measurement.

• Secondly, the alignment of the table in the y direction has to be verified.It is very difficult to compensate this error because the table construc-tion itself would have to be adjusted. Fortunately, the table is alignedvery well in this direction and the error lays below the accuracy of theinclinometer.

• Now, the table is set to θ=90. The same procedure as for the z-axis hasto be done for the ϕ angle to align the xy-plain vertically to the gravity

vector. This error ranges between 120’ and 145’, dependent on theposition in that the IMU is mounted on the tilt table.

After gaining knowledge of the table’s errors the measurement procedurecan begin. Measurements in 10steps in θ direction and with 30steps in the ϕdirection were taken. That means that there were 12 measurements for everytilt angle. The first run was done with the z-axis in -g direction (the IMUwas assembled upside-down on the tilt table), the secon run was done in + gdirection.

The nature of readings of the tilt table leads to sinusoidal or cosinusoidalresults, as the following calculations show. A vector of given accelerations inright-handed cartesian coordinates may have the form

26



4.3 Accelerometers

Perfect inclinometer

The same non-perfect inclinometer in two different positions, inclined plane

The same non-perfect inclinometer in two different positions, horizontal plane

Non-perfect inclinometer

baa a

b

a a b

b a a b

Figure 4.8: Using a non-perfect inclinometer to align a plain horizontally. Letter adescribes the shorter side of the non-perfect inclinometers, b is the longer side.

27



Calibration

a = ax

ayaz

(4.31)

During calibration with the tilt table, only the gravity vector in z-direction(down) is effective on the IMU:

g =

0

0g0

(4.32)

Transformation around the y- and z-axis are given through3

C y(α) = cos(α) 0 − sin(α)

0 1 0

sin(α) 0 cos(α) , (4.33)

C z(α) =

cos(α) sin(α) 0

− sin(α) cos(α) 00 0 1

(4.34)

The tilting table rotates around the y-axis with the angle θ and around thez-axis with the angle ϕ. The complete rotation in those two axes is

A = C y(θ) C z(ϕ) g =

−g0 cos(ϕ) sin(θ)

g0 sin(ϕ) sin(θ)g0 cos(θ)

(4.35)

Thus, the equations that transform the given angles to the respective accel-

erations are

Ax = −g0 cos(ϕ) sin(θ) (4.36)

Ay = g0 sin(ϕ) sin(θ) (4.37)

Ay = g0 cos(ϕ) (4.38)

Fig. 4.9 shows the reference data for the chosen angles. For example, thedata for θ=40can be found as samples 48 to 60 in the plots. The raw readingsof the accelerometers directly taken from the ADC card voltage output can beseen in Fig. 4.10 for θ=40.

The constant g0 describes the exact strength of the local gravitational field.

It is dependent on the position on earth (i.e. g0 is a function dependent onlatitude and longitude). It was determined exactly by the geodetic instituteto g0 = 9.81242231 what was entered as reference in the MATLAB calibrationprogram.

4.3.2 Data Acquisition

First, the calibration program and data taken by a former student were ana-lyzed. These measurements cover IMU outputs at angles from θ=0to θ=90in10steps. The least-squares fit in Fig. 4.11 shows an absolutely unsatisfyingresult. The main task was to improve this calibration.

3see [Moo97] for nearer explanations on transformation of coordinates

28



4.3 Accelerometers

Figure 4.9: References of the expected accelerometer measurements (the so-called ’truedata’). The samples 0-12 show the true data for an inclination of θ=0, thenit goes upin10steps among the next sample segments until the samples 108-120 where θ reached90.

29



Calibration

0 200 400 600 800 1000 1200 1400 1600 1800−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

o l t s

Samples

Accx

AccyAcc

z

Figure 4.10: Raw accelerometer measurements at θ=40of the three acceleration datachannels.

0 36 72 108 144 180 216 252 288 324 360−10

0

10

20

0 36 72 108 144 180 216 252 288 324 360−400

−200

0

200

400

0 36 72 108 144 180 216 252 288 324 360−1

−0.5

0

0.5

1

Samples

x − a x i s , m / s ²

y − a x i s , m / s ²

z − a x i s , m / s ²

Figure 4.11: Results of a least-squares fit with previously measured accelerometer data.

30



4.3 Accelerometers

Former σ Actual σx 2,374e−002 3,300e−003

y 3,685e−001 3,695e−001z 2,471e−002 2,990e−003

Table 4.2: Comparison between a calibration done on raw data and after computingthe medians first. Former σ is the standard deviation of the residuals of the originalprogram, actual σ is the output of the modified program.

All calculations mentioned below were done in Matlab. The data of theIMU was acquired with a 1000 H z sampling frequency and is downsampledto 100 H z after applying a digital low-pass filter to avoid aliasing caused bynoise with frequency parts greater than 500 H z. For every position, 15 sec of data was saved.

The program computed coefficients for a polynomial using the least-squaresmethod. Inputs were the raw output of the 15 sec sensor data. To terminatethe noise and to lower the impact of peaks in the raw data, the medians of the channels were calculated first. See Tab. 4.2 for a comparison of the per-formance of the original program and the modified version. It can clearly beseen that with calculating the medians of the gathered data first, the standarddeviation of the residuals is greatly decreasing.

4.3.3 Possible sources of error

After taking measurements of the accelerometers, irregular behaviour was ob-served. Sometimes the results were quite good, the next day they were com-

pletely out of the line although the same angles and parameters were used.After testing the cabeling, considering the building and the environment, anddouble-checking the measured data, three causes were found for the prob-lems:

The cabling

A test was done where the cables and connector where touched one after an-other while data was gathered of the IMU for 30 seconds. The IMU itself wasfixed to a certain position. A separate plot for every cable and connector wasused. In the plot of the connectors going to the junction box, a peak in channeltwo could be seen, which is the channel of the output of the y-accelerometer

(Fig. 4.12). The differences are too high to be caused by movements of thecable. Because the IMU was mounted on the tilt table and according to thedata it would have to move in y-direction (the movement is the area bounded by 0 and the y-accelerometer curve) which is impossible.

To solve the problem, the junction box was disassembled, the solderingswere checked and the serial cable going from the junction box to the IMUwas exchanged. The tests were repeated and the results did not show peaksanymore.

The Hydraulic Pump

Sometimes, measurements taken during the day showed a very good accu-racy, but the next day, the same measurements taken with the same setting,

31



Calibration

0 50 100 150−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

V o l t s

Samples

X

Y

Z

Figure 4.12: Result of the cabling test. The signals show peaks in the channels whilemoving the cables, especially for the y-channel.

were completely inexact. The cause of this behaviour was found in a hydraulicpump that is installed in the basement. It is directly mounted to the wall be-neath the calibration facility. The pump is used for the hydraulic system of aflight simulator that operates in the same building. Moreover, in the basementare a machine for transforming voltages for equipment to be used in aircrafts.Fig. 4.13 shows raw data while the pump was running. A comparison withfigure 4.10 shows the high impact of the building’s vibrations caused by thepump.

The pump has an impact on the noise of the measurements and is there-fore not leveled out by the calculation of the median over the sampling time, because the vibrations vary in time and directions and are not the same for all

of the three accelerometers.The examined IMU is very sensitive: Even the closing of doors or traces

of people moving chairs in the laboratory appear in the plots. Hence, thefinal measurements were done between 6pm and 10pm, where less staff waspresent, the pump, the transfomer, and the air conditioning were shut off.

Software

A software bug was found in the acquisition software. The module, that cal-culated binary data which can be processed by the M ATLAB program, waserroneous. The mistake was fixed.

32



4.3 Accelerometers

0 200 400 600 800 1000 1200 1400 1600 1800−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

V o l t s

Accx

Accy

Accz

Figure 4.13: Raw data of an x-channel measurement while the hydraulic pump in the basement was running.

4.3.4 Experiments with Analog-to-Digital Converter Parame-ters

After excluding all evironmental impacts, there was still a bad performance of the y-channel calibration compared to the x- and z-measurements. The cou-pling of an ADC-channel to the next one caused by the ADC multiplexing wascalled to account for this problem.

To lower the influence of this effect, the multiplexing frequency of the ADCcard was adjusted. Firstly, tests with different multiplexing frequencies (200kH z, 50 kH z, and 20 kHz ) were done. Secondly, a channel that was directlyconnected to the ground of the ADC card was sampled in between two datachannels. The results of this test can be seen in table 4.3. For this test, 12measurements were taken for 15 s at 10, 50, and 90respectively with 1000Hz sampling frequency.

The hopping to a ground channel did not lead to a significant improve-ment of the standard deviation but the decreasing of the multiplexing fre-quency improved the calibration results enormously. The final calibration wasdone with an even lower multiplexing frequency of 20 kH z. With this multi-plexing frequency, the time between the first channel to be sampled and thelast is 16 · (20000Hz)−1 = 0,0008s. Both gives the multiplexer enough timeto recover from a previous measurement and at the same time an acceptablesynchrony of the readings is given for the design of the integrated navigationsystem.

33



Calibration

Ground no no no yes yesf Mx 200 kHz 50 kHz 20 kHz 200 kHz 50 kHz

σx 0,00204 0,00177 0,00197 0,00213 0,00194σy 0,00622 0,00176 0,00214 0,00407 0,00164σz 0,00187 0,00163 0,00136 0,00156 0,00178

Table 4.3: The row Ground indicates, if a grounded channel was acquired in betweendata channels. The used multiplexing frequency if given through F Mx . Standard devi-ations of the residuals of all channels were calculated. The unit of all values is m/s2.

A|x,y,z = β 0 + β 1 ax + β 2 ay + β 3 azi β i|ax β i|ay β i|az0 0,01000107 −0,00190695 0,003526321 1,00348456* −0,00370471 −0,00208968

2 0,00553735 0,99006634* 0,000194133 −0,00106205 −0,00055637 1,00818215*

Ax Ay Az

σres 0,00603819 0,00600338 0,00776800M res 0,00121367 0,00281355 0,00334097

R 0,99997504 0,99994366 0,99996532Radj 0,99997472 0,99994295 0,99996488

Table 4.4: The first three models that were examined. Parameters marked with a *represent the scale factors of the three accelerometers. β 0 is the bias of each model.

4.3.5 Adopting the model

As stated in section 4.1.5, the calibration problem is considered as a reversedleast-squares estimation. In the models that are surveyed in this section, Arepresents the reference acceleration data that the IMU was aligned to onthe tilting table and ax, ay, and az are the measured accelerations of the ac-

celerometers. The computation of the parameters β for first three modelsA|x,y,z = β 0 + β 1 ax + β 2 ay + β 3 az are shown in Tab. 4.4. Obviously the scalefactors (represented by the parameters β 1 for model Ax, β 2 for model Ay, andβ 3 for model Az) have the greatest influence on their respective model, be-cause they determine the sensitive axes of the respective accelerometers. Butthis model takes already misalignments in account, e.g. β 2 = 0.00553735 of Ax

is the contingent of the y-accelerometer that influences the output in directionof the sensitive axis of the x-accelerometer.The models that are shown in Tab. 4.5 regard also cross terms like ax az .

This part of the model take cross-channel influences into account, that can becaused by interference of tightly mounted cables that are unshielded and/ortoo close to each. This leads to an improvement (= decreasing) of the residuals’standard deviation. But the value of some of these parameters is not verysignificant, so only ay az for Ax and ax az for Ay and Az is kept for the nextstep.

More experiments with polynomials of higher degrees were performed, but there was no improvement in the standard deviation of the residuals withpolynomials higher than quadratic terms except for the z-axis. Therefore, onlythis term was kept in the models. Tab. 4.6 shows the final result.

34



4.3 Accelerometers

A|x,y,x = β 0 + β 1 ax + β 2 ay + β 3 az++ β 4 ax ay + β 5 ax az + β 6 ay az

0 0,01000381 −0,00192272 0,003540021 1,00348452 −0,00370406 −0,002090262 0,00553675 0,99006641 0,000194123 −0,00106176 −0,00055505 1,008180934 0,00000134 −0,00000268 0,000000145 0,00000336 0,00015533 −0,000139716 −0,00013689 0,00000910 0,00000928

Ax Ay Az

σres 0,00515873 0,00486450 0,00709341M res 0,00123939 0,00286247 0,00339427

R 0,99997483 0,99994341 0,99996521Radj 0,99997418 0,99994196 0,99996432

Table 4.5: The second attempt in finding best fitted models for the accelerometers. β 4,β 5 and β 6 are the cross-coupling coefficients.

A|x

= β 0 + β 1 ax + β 2 a2x + β 3 ay + β 4 a2y + β 5 az + β 6 a2z++ β 7 ay azA|y = β 0 + β 1 ax + β 2 a2x + β 3 ay + β 4 a2y + β 5 az + β 6 a2z+

+ β 7 ax azA|z = β 0 + β 1 ax + β 2 a2x + β 3 ay + β 4 a2y + β 5 az + β 6 a2z++ β 7 ax az

β i β i|Axβ i|Ay

β i|Az

0 0,00011348 0,41206041 0,000065491 1,00348508 −0,00374945 −0,002089712 0,00007102 −0,00431949 0,000046713 0,00553646 0,99008258 0,000193904 0,00007265 −0,00422615 0,000047065 −0,00106102 −0,00058151 1,00818112

6 0,00013531 −0,00436952 0,000025387 −0,00013701 0,00018011 −0,00013989

Ax Ay Az

σres 0,00467461 0,00482296 0,00705226M res 0,00124964 0,00287522 0,00340950

R 0,99997473 0,99994340 0,99996521Radj 0,99997397 0,99994170 0,99996416

Table 4.6: The final models of the accelerometers’ calibration.

35



Calibration

0 50 100 150 200

−0.01

0

0.01

X E r r o r , m / s 2

0 50 100 150 200

−0.01

0

0.01

Y E r r o r . m / s 2

0 50 100 150 200

−0.01

0

0.01

Z E r r o r , m / s 2

Samples

Figure 4.14: Residual of the accelerometers’ calibration with the computed parametersof Tab. 4.6.

Fig. 4.14 shows the residuals of the calibration of Tab. 4.6. 240 measure-ments were taken for the calibration. Compared to Fig. 4.11 the calibrationhas greatly improved and is now usable.

4.4 Gyroscopes

4.4.1 Rotation Table

All of the calibrations for the gyroscopes were done with a Gensico, Inc. C181turn table (Fig. 4.15).

The turn table is able to carry a 50 kg capacity and adjust the infinitely vari-

able rotation rate from 0.010

/s to 1200

/s. Though the table was developedin the US, it operates with 110 V , 60 H z power requirements. It comprises asynchronous motor that requires to multiply the readings of the rotation rate by 5

6 due to its operation in Europe with a 50 H z frequency. An optical pulse

generator was added to measure the time of one turn with high precision.The table offers 16 conductors to connect the load with devices off the table

through slip rings. Fig. 4.16 shows a temperature signal of an accelerometer’stemperature sensor. A periodicity can be seen which can be traced back to thefact, that the sliprings behave differently in various different positions.

The ability of the turn table to infinitely vary the rotation rate is achieved by a movable ball carriage arranged in a way that it incorporates two setsof two transfer balls between three integrator disks. Fig. 4.17 points up themechanism. This technique allows precise adjustments and eliminates all ex-

36



4.4 Gyroscopes

Figure 4.15: Genisco C181 Turn Table (source: Genisco C181 Operations Manual).

4 5 6 7 8 9 10

x 104

3.02

3.03

3.04

3.05

3.06

3.07

3.08

Figure 4.16: Signal of a temperature sensor (1000 Hz sampling frequency, 100/s ro-tation rate). It is noteable that the noise is dependent on the position and thus on thesliprings of the rotation table.

37



Calibration

Figure 4.17: The rate changer of the turn table. The sketch shows two arbitrary po-sitions, left with a low rotation rate, right with a higher rotation rate (source: GeniscoC181 Operations Manual).

tra movements except the linear rate changing movement desired. The rota-tion is controlled by a handwheel that changes the position of the ball carriagerelative to the integrator disc.

4.4.2 SIMONA Flight Simulator IMU

Besides the main calibration for the MiniSAR project, a rough calibration forthe gyroscope components of the International Center for Research in Simula-tion, Motion and Navigation Technologies (SIMONA) Flight Simulator4 wasdone. SIMONA is a joint venture of several institutes of the TU Delft andpartners from the industry and the government. It is a six degrees-of-freedomadvanced flight simulator designed to simulate present and future aircrafts.See f or more information. Fig. 4.18 shows the build up for the first calibrationof a new SIMONA IMU.

The three gyros are mounted together on the rotation table which acceler-ates the readings. But the misalignments of the gyroscopes’ axes in the finalIMU are not covered with this setup of course.

Only an approximate calibration has been done in that way, that the ro-

tation table was set to a specific rotation rate and the voltage output of thegyros were just read off a voltage meter. The rotation table includes an opticalsensor that gives an output pulse every time a complete round is done. Thispulse was connected to a stop watch that measured the time for one roundwith microsecond precision.

The measurement results of this calibration can be seen in appendix B.

The reachable accuracy of the calibration is highly dependent on the avail-able time facility: For example, if the rotation table rotates at a speed of 500/s and the clock has a resolution of 1e − 6s, then the measurement error isabout ±0,0005 per round. If the clock has only a resolution of 1ms, the errorwould grow to 0,5 at 500 /s!

4http://www.simona.tudelft.nl

38

http://www.simona.tudelft.nl/

http://www.simona.tudelft.nl/



4.4 Gyroscopes

Figure 4.18: The three gyroscopes of the SIMONA IMU are in the center of the rotationtable. On the right the power supply can be seen.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.450

20

40

60

80

P−Gyro Output / V

R o t a t i o n R a t e

P −

G y r o / ° / s

ActualManufacturer

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45−0.2

−0.1

0

0.1

0.2

R e s i d u a l s a c t u a l

c a l i b r a t i o n / ° / s

P−Gyro Output / V

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.450

2

4

6

8

R e s i d u a l s m a n u f a c t u r e r

c a l i b r a t i o n / ° / s

P−Gyro Output / V

Figure 4.19: Calibration for the SIMONA IMU: The upper plot shows the estimation of the actual calibration and the calibration provided by the manufacturer, both are linearmodels. The plot in the middle presents the residuals of the actual, the bottom plotshows the residuals of the manufacturer’s calibration.

39



Calibration

0 2 4 6 8 10 12 14 160

200

400

600

R o

t a t i o n R a t e ,

° / s

Raw Data

True

Calibration

0 2 4 6 8 10 12 14 16−0.1

−0.05

0

0.05

0.1

R e s i d u a l s ,

° / s

A D C

a s t i m e r

Rotation Rate

0 2 4 6 8 10 12 14 16−0.06

0

0.06

0 2 4 6 8 10 12 14 16−0.03

0

0.03

R e s i d u a l s ,

° / s

G P S b o a r d a s t i m e r

R e s i d u a l s , ° / s

d e d i c a t e d c h r o n o m e t e r

Figure 4.20: Results of the calibration for the r-gyroscope. The top plot shows thedifferent rotation rates. The second plotshows the residuals for the calibration with theADC used as the stop watch. For plot three the GPS board was used and the bottomplot points out the result when a dedicated timer was used.

Measurements in clockwise and counter-clockwise directions were made.Unfortunately, the voltage meter was set to autoscale. It changed the measure-

ment range for the last nine measurements meaning, that the fitted curve dif-fers slightly from the part with the measurements that were taken previously.This does not matter in the presented situation because only the data between0 and 5 /s is needed for the actual experiment. A least-squares estimation of a linear model for the calibration was calculated and compared to the factorprovided by the manufacturer in Fig. 4.19. Even this rough calibration is waymore accurate than the data provided by the manufacturer.

4.4.3 Evaluating Time Facilities

For the gyroscopes’ calibration, the calibration program was modified to han-dle measurements of different durations, because a round at 1/s lasts 360s =

6min. To acquire data for 6min for several rotation speeds would be too timeconsuming.In an experiment, the transmitter that generates a pulse after each com-

pleted round was connected to a channel of the ADC to measure the timefor one round in a convenient way. An algorithm was developed to extractthe pulse data and calculate the rotation rate for every measurement. But, asshown in section 4.4.2, the reachable accuracy is dependant on the accurateknowledge of the rotation rate. And since the ADC has a maximum samplingrate of 1000Hz, the resolution for the pulse measurement is 1ms minimum.Readings were taken for 1, 2, 4, 8, 10, 20, 40, 60, 80, 100, 150, 200, 300, 400 and500 /s. Hence, the plot of the residuals of a calibration of the r-Gyro showsgreat discrepancies starting from 10/s up to 500/s, see Fig. 4.20, second plotfrom top.

40



4.4 Gyroscopes

0 10 20 30 40 50 60 70−0.1

0

0.1

0.2

0.3

0 10 20 30 40 50 60 70−0.2

−0.1

0

0.15

0 10 20 30 40 50 60 70−0.1

−0.05

0

0.05

0.1

Rotation Rate

R e s i d u a l P −

G y r o ° / s

R e s i d u a l Q −

G y r o ° / s

R e s i d u a l R −

G y r o ° / s

Figure 4.21: Residuals of the final gyroscope calibration.

The third plot from the top in the same figure shows the residuals for a cal-culation where the Global Positioning System (GPS) board was used to gen-erate a log output every time a pulse is received on a special mark input. TheGPS board internal clock is corrected by the GPS system continuously andreaches a higher accuracy as the time measurement with the ADC card.

Finally, a dedicated timer (a Hewlett-Packard 6700 Measurement System)was used to measure the time per round for the calibration. This option offersthe best results, expressed in the lowest residuals, as seen in the lowest plot inFig. 4.20.

4.4.4 Final Gyroscope Calibration

The final calibration for the gyroscopes was done at the rotation rates (1, 2, 4,8, 10, 20, 40, 80, 100) · 56

/s.5 As a timer, the dedicated Hewlett-Packard stopwatch was used. Fig. 4.21 shows the resulting residuals.

The model for the gyroscopes was developed in the same way as section4.3.5 describes, the final versions are

Ω|( p) = β 0 + β 1 ω p + β 2 ω2 p + β 3 ωq + β 4 ω2

q + β 5 ωr +

+ β 6 ω2r + β 7 ω pωq + β 8 ω pωr + β 9 ωqωr, (4.39)

5for an explanation of the factor 56

see section 4.4.1

41



Calibration

0 1 2 3 4 5 6x 10

4

2.3

2.4

2.5

2.6

2.7

2.8

2.9

3

3.1

3.2

T e m p e r a t u r e / V

Samples

Figure 4.22: Raw output of the temperature sensor during temperature calibration.

Ω|(q) = β 0 + β 1 ωq + β 2 ω2q + β 3 ω p + β 4 ω2

p + β 5 ωr +

+ β 6 ω2r + β 7 ω pωq + β 8 ω pωr + β 9 ωqωr, (4.40)

Ω|(r) = β 0 + β 1 ωr + β 2 ω2r + β 3 ω p + β 4 ω2 p + β 5 ωq +

+ β 6 ω2q + β 7 ω pωq + β 8 ω pωr + β 9 ωqωr, (4.41)

where ω p, ωq, ωr are the measured rotation rates in V and Ω the referencerotation rate, determined by the stop timer. The biases are represented by β 0,the scale factors are modeled by β 1, and the second-order coefficients are β 2for each model. Misalignments are given through β 3, β 5 (first order) and β 4,β 6 (second order). The cross-coupling terms are β 7, β 8, and β 9.

4.5 Thermal Calibration

For the thermal calibration a Associate Testing Laboratories, Inc. (Wayne,New Jersey, USA) Model SW-5101 temperature testing facility was used. TheIMU was placed into the SW-5101 and the temperature was adjusted from -50C to +50C in some 10C steps in two cycles. Figure 4.22 shows the rawoutput of the IMU’s temperature sensor.

The data of the accelerometers where collected, each of them own its sep-arate temperature sensor. The gyroscopes use a common temperature sensorthat was assembled in the IMU next to them.

Fig. 4.23 shows the result for the P-gyroscope. The blue circles in the upperplot are the raw data of the gyroscopes rotation rate output and they indicatea linear dependance on the temperature (the two temperature cycles of thecalibration manifest themselves in more than one measruement for a certaintemperature). The MATLAB command polyfit was used to fit the measured

42



4.6 Magnetometer

2.3 2.4 2.5 2.6 2.7 2.8 2.9 3 3.1 3.20.012

0.013

0.014

0.015

0.016

0.017

0.018

0.019

M e a n P G y r o s c o p e / V

Temperature / V

P Gyroscope

2.3 2.4 2.5 2.6 2.7 2.8 2.9 3 3.1 3.2−1

−0.5

0

0.5

1x 10

−3

E r r o r P G y r o s c o p e / V

Temperature / V

Raw DataCalibrated

Figure 4.23: Calibration results of the P gyroscope temperature calibration.

values to the refence with a third order polynomial). The polyfit commandfits the data in a least-squares sense and returned in the given case two coeffi-cients. The red crosses in the plot represent the fitted values. This proceduredetermined the temperature dependance of the bias. To get knowledge of the temperature’s impact on other errors like the scale factor and misalign-ment errors for the sensors, the IMU has to be set to different attitudes in thetemperature-controlled chamber. For a detailed procedure, see section Tem- perature model calibration and temperature hysteresis in [IEEE1]. The correctionof the temperature influences was done prior to the least-squares correctionof the accelerometers and gyroscopes respectively. Hence, the temperature in-fluence is not modelled in the final models in Tab. 4.6 and in Eqns. 4.39, 4.40,and 4.41.

4.6 Magnetometer

[Elk01] describes a sophisticated method for calibrating solid-state magne-tometers. Unlike the traditional method where an aircraft has to performan inconvenient procedure called swinging for calibrating the magnetometer,this new method allows a much easier calibration. Different from the originalapproach the method presented here is also usable for applications other thanheading determination. After an introduction to magnetometers and their de-scription in general, an introduction is given to the old swinging procedure

43



4.7 A new algorithm

Finally, the misalignment errors Cm in the axes are considered very low if care is taken during the installation and the misalignment of the magnetome-

ter triad and the aircraft’s axes. For the following analysis, these errors areconsidered 0 and Cm is an identity matrix.The presented algorithm deals with the hard iron biases and scale factor

errors.

4.6.2 Swinging

The well-known calibration in the heading domain is based on perturbationsof the basic heading Eqn. 4.42 and a substitution of the error Eqns. 4.43. Moredetails on the exact approach can be found in [ Elk01].

Emerging problems using this kind of calibration are the time invariantsoft iron bias, that are heading dependent and the inaccuracy of the referencemeasurements of the heading. For the reference, the standard practice is to usea compass rose painted on the tarmac and set the vehicle to e.g. 12 predefinedheadings. This procedure is called swinging.

Moreover, the earth’s magnetic field changes its strength in respect to dif-ferent locations on the earth. To circumvene these dependances, several cal-ibrations at multiple locations have to be performed to allow the aircraft totravel over a large geographic area. This would lead to different sets of pa-rameters A to E that could be used respectively to each actual location.

If the system consists of more than two magnetometers, this procedurecannot be used.

4.7 A new algorithm

The new approach is based on the fact, that the locus of an error-free measure-ment of two magnetometers that are mounted orthogonally is a circle, as thefollowing equation shows:

Bbx

2+ Bb

y

2= BH

2 (cos ϕ)2 + BH 2 (sin ϕ)2 = BH

2, (4.44)

where BH is the strength of the horizontal component of the magnetic fieldof the earth. Eqn. 4.44 describes a circle with the center at the origin and witha radius of the magnitude of the earth’s magentic field. The variation of themagnetic field vector of the earth can be described with the 1999 InternationalGeomagnetic Reference Field Model.

The previously discussed errors alter the shape and the position of the

circle described by Eqn. 4.44. E.g. a hard iron bias would change the origin of the circle:

Bbx = δ Bx + Bb

x (4.45)

Bby = δ By + Bb

y (4.46)

where Bbx and Bb

y are the measured magnetic field strengths. These twoequations transform Eqn. 4.44 to

( Bbx − δ Bx0)2 + ( Bb

y − δ By0)2 = BH 2 (4.47)

with its new center at (Bx0,By0).

45



Calibration

BH

B +H BH

Scale factor andsoft iron effects

Misalignments andsoft iron effects

Figure 4.24: Impact of errors on magnetic field measurements of a measured circle intwo dimensions.

Furthermore, the alteration of the circle caused by scale factor errors isinteresting. Those errors cause different measurements of the magnetometerseven for the same field, as stated above which can be expressed as

Bbx = (1 + sf x) BH cos ϕ (4.48)

Bby = −(1 + sf y) BH sin ϕ (4.49)

which can be rearranged to Bb

x

1 + sf x

2

+

Bb

y

1 + sf y

2

= B2H (4.50)

which is the expression of an ellipse with different minor and major axes. If both hard iron biases δBx0, δBy0 and scale factors sf x, sf y errors are applied,the final ellipse is given by

Bbx − δBx0

1 + sf x

2

+

Bby − δBy0

1 + sf y

2

= B2H . (4.51)

Fig. 4.24 shows the effects graphically on the ellipse that is caused by theerrors. Misalignments are not considered in the presented algorithm but theycan be described as a rotation of the minor and major axes as can be seen inthe figure.

The goal is now to estimate the parameters of the ellipse in a least-squaressense, which will give knowledge to the unknown hard iron errors and scalefactor errors. For that, the one ellipse is calculated that fits best to the mea-sured data using a least-squares estimator. In the three-dimensional case witha triad of magnetometers, the best fitting ellipsoid is calculated.

Two ways of achieving this are presented in [Elk01]. First, a classical ap-proach using an Extended Kalman Filter to estimate the parameters of a non-linear system of equations is described and upcoming problems such as therequirement of a close guess of the initial scale factor and hard iron errors are

46



4.8 Estimation of the Time Need

Warming up the IMU 15 minAcquiring time 15 s

Time for 12 positions in xy-plane (ϕ-plane) 10 minTime for 10 different tilting angles (θ-plane) 120 minTime for 20 different tilting angles (complete calibration) 280 min

Table 4.7: The approximate expenditure of time for calibrating the accelerometers.

discussed. The second approach is a non-linear estimater that breaks up thecalculation in two steps, a linear and a non-linear step. The mentioned refer-ence shows further on simulation and verifications of the presented algorithm.And last but not least it works well for the MiniSAR project. For the magne-tormeter’s calibration the initial circle maneuver on the tarmac is performed

prior to every test flight, see section 5.2.

4.8 Estimation of the Time Need

For comparison and as a future reference, tables are provided to describe theapproximate time needed for calibrating the accelerometers and gyroscopes.Tab. 4.7 shows the time need of partial calibration procedures and a completecalibration for the accelerometers.

It is important to take some time for testing and altering the calibration build up into account. Moreover, letting the IMU adopt to the temperatureand continuously checking the acquisitioned data during a calibration are cru-cial to detect possible failures.

For the final gyroscope calibration 16 measurements per gyro were taken.The time needed was approximately 60 min per gyro inclusive assembly andchecking of the datas.

47



5 Test Flight

Test flights for the MiniSAR navigation system and two other navigation sys-tem research groups took place on May 31st and June 26th, 2005. Main goals

of these flights for the MiniSAR project were tests of the data acquisition soft-ware and the performance under the use of the new calibration. The flightswhere performed in the faculty’s Cessna Citation aircraft (Fig. 5.3) over thenorthern part of The Netherlands and the North Sea (Fig. 5.1).

The other projects were interested e.g. in validating of mass and enginemodels of the aircraft. A set of manoeuvres were prepared to met those objec-tives.

5.1 Preparations

Before the flight, following preparations were made:

• An engineer was consulted to find a suitable way for assembling theequipment on a special rack so that it can be mounted in the aircraft.

• The system has to be tested while the aircraft is still grounded. For ex-ample, the notebook on the rack where the IMU was mounted on tohad to be relocated to a place more central because the lid could not beopened completely. A test acquisition was done to check the correctnessof cabeling and software, see Fig. 5.2.

• Just before the flight, an Interference Test was performed: All experi-mental devices in the airplane were powered on and the emissions weremeasured by the flight supervisor with an interferometer. The result

showed that the aircraft’s systems were not disturbed by the experi-ments and the flight could start as scheduled.

5.2 Maneuvers

The planned route started at Schiphol International Airport, lead to Delft wherea synchronisation of the GPS receiver with a Differential GPS Station on the building of the Geodesy Institute was possible. After that the test maneuverswere performed over the dutch islands and the Northern Sea.

In a briefing before the flight, the planned maneuvers were discussed withthe two test pilots. Important for the MiniSAR project is to test the reachable

49



Test Flight

Figure 5.1: The scheduled route for the test flight.

Figure 5.2: The rack with the mounted equipment on it.

50



5.2 Maneuvers

Figure 5.3: The rack with the mounted equipment on it.

accuracy of the navigation system during a straight flight with only few al-

terations in altitude and attitude. These conditions are typical during SARoperation. The MiniSAR group requested a certain time of steady flight toexaminate the performance of the navigation system in this typical operationsituation. The pilots granted these en-route requirements on the way fromDelft to the area over the Northern Sea were the other tests took place.

During the flight, the flight coordinater was constantly in contact with thepilots and gave feedback about the flown maneuvers. Some parameters werechanged during the flight, e.g. enabling or disabling the auto-pilot in somesituations because it did not behave as expected.

En route, the teams maintained a log in which they entered start time, endtime and the quality of the maneuvers.

The following list describes a selection of the flown maneuvers.

Ground Maneuver

In order to calibrate sensors such as the magnetometer, a turn was made whilethe aircraft was still on the ground. This is necessary to apply the algorithmfor the calibration of the magnetometer. Since the aircraft’s roll and pitch aremaintained approximately constant to zero the driven circle on the groundserves as an input to the calibration technique as shown in section 4.6.

3-2-1-1 Elevator

Objective: To measure the lateral/directional response characteristics of theaircraft due to elevator input.

51



Test Flight

Pilot test procedures:

1. Trim the aircraft in straight and level flight at the nominal conditions,

2. Control the elevator by means of chirp aft column movements, proceed by pulling the column with the adequate displacement D,

3. hold for three seconds,

4. push the control until it reaches the same displacement on the oppositedirection, −D,

5. hold for two seconds,

6. pull the column until displacement D is obtained,

7. hold for one second,

8. push the column until displacement −D is reached,

9. return column to initial position (free hands position, although this timethe pilot keeps it fixed),

10. allow the aircraft to naturally damp out.

Short period dynamics

Objective: To measure the aircraft’s response characteristics due to a step ele-vator input.


1. Trim the aircraft in straight and level flight at the specified initial condi-tions.

2. Begin recording, then hold straight and level flight for five seconds.

3. Apply a double elevator doublet with two seconds period for each doublet. Then leave the control column force free.

4. Allow the aircraft to naturally damp out.

5. Continue the experiment for 10 seconds after control column is released.

6. Establish straight and level flight, and hold for five seconds.

7. The test ends when step six is completed.

Alpha-dot response with S-turn and stepped bank angles fea-turing elevator doublets

Objective: To gain information about the alpha-dot response of the aircraft byapplying elevator doublets at steep bank angles to the left and to the right.


1. Trim the aircraft in straight and level flight at the specified initial condi-tions.

2. Apply a control wheel displacement until a bank angle of 20is reached.Maintain altitude using the control column.

52



5.3 Results

3. Apply a sharp elevator doublet of 2 seconds duration while maintainingroll angle.

4. Repeat steps two and three for 30, 45and 60,

5. Roll the aircraft to the opposite direction until a bank angle of 15isreached.

6. Apply a sharp elevator doublet of two seconds duration while maintain-ing roll angle.

7. Repeat steps two and three for 30, 45and 60,

8. Turn the aircraft to straight and level flight and hold for 5 seconds.

5.3 ResultsThe saved data of the perfomed flight was processed after the duration of thepresent project, unfortunately. Hence, no results can be given here. However,a first glimpse on the data promised that it is of satisfying quality. Now thedeveloped post-processing filters of the MiniSAR project can be evaluated us-ing the data sets of the test fligth. And, moreover, they can be compared to the board-mounted IMU and to the recorded data of the other projects that wereon board.

53



6 Conclusion and FuturePerspective

This Short Research Project explained the necessary steps needed for calibrat-ing an Inertial Measurement Unit. The basic principles of analog-to-digitalconversion were illuminated and the theoretical fundamental of the least-squares calibration problem were discussed. Using this technique and its var-ious valuation factors, a well-fitting model of the accelerometers and gyro-scopes, were found. An instruction for calibration facilities such as the tilt table, rotation table and the thermal facility were given. Different ways in usingvarious units (e.g. GPS receiver or stop watch) as timers were evaluated. Aninteresting approach for the on-line calibration of a solid-state magnetometerwas depicted. The description of the test flight gave an insight in practicalrequirements of developing navigation solutions for the aerospace.

Many difficulties with the system were discovered and eliminated withinthe project duration. The Analog-to-Digital Converter system was changedfrom a Ground-Referenced Single-Ended Measurement System to a non-re-ferenced single-ended measurement system to avoid ground-loop losses. Adifferential measurement system was able to perform even better, but sincethe amount of channels were limited, this configuration was not possible.

Furthermore, the cabling was checked and defective parts were discoveredand replaced. The author would like to highlight the importance of an almost(in the meaning of possible) disturbance-free calibrating environment. An hy-draulic pump in the basement confounded often usable calibration runs dur-ing the day. To repeat the readings after 6pm increased the expenditure of timethat was needed for complete calibrations remarkably. Also, factors that wereunimportant at first sight had to be taken into account: vibrations caused bythe air conditioning for example or even ships which are driving past close- by canals were able to be revealed as a source of errors as the experience of the staff at the TU Delft showed. Special precautions, especially a dedicatedcalibration location with sophisticated constructions for the buildung and thefundament could significantly minimize the impacts described above.

Further concern is necessary for the calibration equipment. Up-to-date fa-cilities (regarding e.g. gaging of timers) assure high-quality calibration results.The measurement disturbances caused by the sliprings of the rotation tablecould have been avoided. There are existing plans to replace the rotation tablesoon.

The accelerometers’ calibration on the tilt table allowed a measurementrange from -1g to +1g. To take readings with accelerations greater than 1g it is

55



Conclusion and Future Perspective

necessary to mount the IMU on an arm onto the rotation table. This could bedone to expand the range of the accelerometer calibration.

Difficulties that were inherent to the system are for example the influenceof one channel of the ADC to the other. They could have been avoided byusing a different build-up, e.g. impedance converters as input buffer to everychannel. Of course, a separate ADC for every single channel would be the op-timal solutions and indeed, such plans exist already. These recommendationswill be incorporated into the next generation of the IMU, where a completelynew ADC design is developed solely for the MiniSAR navigation system.

56



Bibliography

[Sha01] Syed Jaffar Shah, Field Wiring and Noise Considerations for Analog Signals, 2001, National Instruments Developer Zone,http://zone.ni.com/devzone%5Cconceptd.nsf/

webmain/01F147E156A1BE15862568650057DF15?opendocument&node=dz00000_us

[DAQ02] DAQ NI 6034E/6035E/6036E User Man-ual, 2002, National Instruments Corporation,http://www.ni.com/pdf/manuals/322339d.pdf

[Dra98] Norman R. Draper, Harry Smith, Applied Regression Analysis, ThirdEdition, 1998, John Wiley & Sons, Inc.

[Mon01] Douglas C. Montgomery, Elizabeth A. Peck, G. Geoffrey Vining, In-troduction to Linear Regression Analysis, Third Edition, 2001, John Wi-ley & Sons, Inc.

[Raw98] John O. Rawlings, Sastry G. Pantula, David A. Dickey, Applied Re- gression Analysis - A Research Tool, Second Edition, 1998, SpringerVerlag New York, Inc.

[Huf91] Sabine van Huffel, Joos Vandewalle, The Total Least Squares Problem- Computational Aspects and Analysis, 1991, Society for Industrial andApplied Mathematics

[Lan04] J. K. Langendoen, Realtime Navigation - Design and Implementation forGuidance and Control, 2004, Master Thesis, Delft University of Tech-nology

[But99] T. Butarbutar, Sensor Model Development using Advanced Regression

Methods, Master Thesis, 1999, Delft University of Technology[Pik97] A. J. Pikaar, Regression Analysis applied to Experimental Data and Simu-

lated Flight Data, Master Thesis, 1997, Delft University of Technology

[Moo97] E. Mooij, The Motion of a Vehicle in a Planetary Atmosphere, 1997, Fac-ulty of Aerospace Engineering, Delft University of Technology

[Elk01] D. Gebre-Egziabher, G. H. Elkaimy, J. D. Powellzand B. W. Parkin-son, 2001, A non-linear two-step Estimation Algorithm for calibratingsolid-state Strapdown Magnetometers, Department of Aeronautics andAstronautics, Stanford University,http://waas.stanford.edu/ wwu/papers/gps/PDF/de-mozins201.pdf

57

http://zone.ni.com/devzone%5Cconceptd.nsf/webmain/01F147E156A1BE15862568650057DF15?opendocument&node=dz00000_us



http://www.ni.com/pdf/manuals/322339d.pdf

http://waas.stanford.edu/~wwu/papers/gps/PDF/demozins201.pdf




http://www.ni.com/pdf/manuals/322339d.pdf






BIBLIOGRAPHY

[IEEE1] IEEE Standard Specification Format Guide and Test Procedure for Linear,Single-Axis, Nongyroscopic Accelerometers

58



A Datasheets

A.1 Accelerometers

This specification applies to the QA700 Model 979-0700-001 QFlex accelerom-eter, product of AlliedSignal Instrument Systems.

The accelerometer consists of a pendulous proof mass with quartz flex-ures. Acceleration of the proof mass is sensed by a capacitive pickoff system,generating the error term to a highgain servo loop. The self-contained servoelectronics drive a permanent magnet and torquer coil system to bring theproof mass back to the null position. Output current is a linear function of the input acceleration. Accelerometer temperature is monitored by an inter-nal sensor, however, modeling coefficients are not supplied with this unit. Amore detailed description is contained in Technical Note TN- 103, Q-Flex Ac-celerometer, Construction and Principle of Operation. Information for the use of this accelerometer as a highly accurate inclinometer is contained in the Appli-

cation Note, Leveling and Small Angle Measurement.The load resistor, RL, connected between the signal output pin and the

isolated return pin, is selected by the user for the desired full-scale range. Aschematic diagram of the accelerometer is given in Fig. 2.5.

Performance

Input Limits

The input limits for the QA700 are + and -30g. The maximum full-scale rangewithout output voltage limiting is given by:

gmax = + (V in-2.5)/[(0.180(1+DT *0.0039)+0.043+RL)K 1]V in = absolute value of the lesser of the two input voltages (volts)

DT = operating temperature (C ) minus 25 C RL = load resistance (kΩ)K 1 = scale factor (mA/g)gmax = full-scale range (g)

Bias

Absolute bias K0 is less than 8 mg at 25C .

Bias Errors

Temperature sensitivity over the operating temperature: +70 µg/C (maxi-mum)

59



Datasheets

Temperature hysteresis over the operating temperature: +1000 µg (maxi-mum)

Input voltage sensitivity: +10 µg/V (maximum).One-year stability (over constant environment): +500 µg (maximum) .One-year repeatability: +1200 µg (maximum).

Scale Factor

The output currrent scale factor K1 is nominally 1.33 mA/g + 10%. Actualvalue at 25C is furnished on each unit’s data sheet.

Scale Factor Errors

Temperature sensitivity over the operating temperature: +200 ppm/C (max-

imum)Temperature hysteresis over the operating temperature: +1000 ppm (max-imum)

Input voltage sensitivity: +10 ppm/V (maximum).Nonlinearity: 48 µg/g2 (maximum) One-year stability (over constant envi-

ronment): +500 ppm (maximum)One-year repeatability: +1200 ppm (maximum)

Input-Axis Misalignment

The absolute value of the input-axis misalignment with respect to the mount-ing flange is less than or equal to 2 mrad.

Warm-up Time

The output of the accelerometer is within 100 µg of its steady-state value in nomore than 4 minutes after turn on.

Threshold and Resolution

Less than 1 µg.

Frequency Response

The output current as a function of input acceleration is flat within 0.1% from0 Hz to 10 Hz, within 5% from 10 H z to 200 Hz, and within 10% from 200 Hzto 300 H z, with less than 5 dB peaking beyond 300 Hz.

Natural Frequency

Greater than 300 H z.

Vibration Rectification

50 µg/g squared rms (maximum) from 50 to 200 H z. 100 µg/g squared rms(maximum) from 200 to 750 H z. 150 µg/g squared rms (maximum) from 750to 2000 H z.

60



A.1 Accelerometers

Mechanical

Life

The unit is designed for a service life of 5 years, and a storage life of 10 years.

Weight

The weight of the unit is less than 46 g.

Seal

The unit has a welded hermetic seal with a measured helium leak rate of lessthan 1x10-6 ATM cc/s with the unit fill gas being 10% He/90%N2.

Electrical

Input Power

Input voltage, ± 13 to ± 18 V (Operation up to ± 28 V is possible. Consultmarketing for specific details.)

Quiescent current (each supply): 16 mA, maximum. Full-scale current: 60mA (maximum).

Output Noise

From 0 to 10 H z, less than 10 nA rms. From 10 to 500 H z, less than 100 nA

rms. From 500 H z to 10 kH z, less than 2 µA rms.

Temperature Sensor Output

Nominal Current output (at 25C , 298.2 K ): 298.2 µA. Nominal Tempera-ture Coefficient:1 µA/K . Absolute Error over operating temperature (aftercalibration at 25rC): + 3.0rC (maximum).

Environmental Requirements

Operating/Non-Operating Environment

The accelerometer will continue to meet this specification after any reasonablecombination of the following environments. It will not provide output to thisspecification during the vibration and shock environments.

Temperature: -55 to 96C at 5C/min maximum rate of change.Vibration: 25 g peak sine, 20 to 2000 Hz. Shock: 250 g, 6 ms, half-sine pulse.

385 g, 1.5 ms, half-sine pulse.Storage Temperature: -65 to 125C .

Reliability

The unit has a Mean Time Between Failure (MTBF) greater than 620000 h ascalculated in accordance with MIL-HDBK-217E, for a benign ground environ-ment at 71C .

61



A.2 Gyroscopes

Weight 80 gSize (35 x 35 x 60) mm (without output

pins)Power consumption 1 W typicalPower source + 5V Activation time 0.1 s

Bias repeatability (1 σ) 0.003 /sBias variation (steady state, 1 σ) 0.001 /sScale factor nominal (SF) 6 mV //sSF repeatability 0.10%

SF stability (steady state) 0.03%Scale factor variation (operatingtemperature range)

5%

Random walk PSD = 0.0015 /s/sqrtHzFrequency range 0...500 H zInput range 500 /s

Temperature -30 C to +71 C Vibration 6g, 20 H z to 2000 H zShock, acceleration 90g

Table A.1: Technical data of the used gyroscopes.

Type Pin number Pin name DESCRIPTIONPower 1 +5V Power input, +5 V DC regulated.

Ripple 50 mV max.

4 GND Power Ground (DC return line for+5 V )

SIGNAL 3 OUTPUT OUTPUT SIGNAL (Rate propor-tional voltage)

2 AGND Analog Ground (Signal return)

Table A.2: The pin assignment of the used gyroscopes.

63



Datasheets

64



B Calibration Results of theSIMONA IMU Calibration

The following table gives the measurements for the

SIMONA calibration as described in section 4.4.2.

65



Calibration Results of the SIMONA IMU Calibration



E x p

e c t e d D u r a t i o n

M e a s u r e d D u r a

t i o n

V o l t a g e O u t p u t



a s s e

e n o n t a b l e

* 5 / 6

o f o n e r o u n d

o f o n e R o u n d

P - G y r o

Q- G y r o

R - G y r o

0 , 0

0 , 0 0

0 , 0 0

0 , 0 0

0 , 0 0 0 0 0

0 , 0 0 0 0 0

0 , 0 0 0 0 0

0 , 5

0 , 4 2

8 6 4 , 0 0

9 3 7 , 9 5

0 , 0 0 2 0 7

0 , 0 0 2 0 9

0 , 0 0 2 1 6

1 , 0

0 , 8 3

4 3 2 , 0 0

4 5 3 , 7 2

0 , 0 0 4 3 0

0 , 0 0 4 3 1

0 , 0 0 4 3 9

1 , 5

1 , 2 5

2 8 8 , 0 0

2 9 8 , 3 7

0 , 0 0 6 5 6

0 , 0 0 6 5 7

0 , 0 0 6 6 5

2 , 0

1 , 6 7

2 1 6 , 0 0

2 2 4 , 6 5

0 , 0 0 8 7 2

0 , 0 0 8 7 3

0 , 0 0 8 8 0

2 , 5

2 , 0 8

1 7 2 , 8 0

1 8 0 , 7 2

0 , 0 1 0 8 6

0 , 0 1 0 8 8

0 , 0 1 0 9 3

3 , 0

2 , 5 0

1 4 4 , 0 0

1 4 9 , 1 4

0 , 0 1 3 1 8

0 , 0 1 3 1 8

0 , 0 1 3 2 4

3 , 5

2 , 9 2

1 2 3 , 4 3

1 2 8 , 4 6

0 , 0 1 5 2 9

0 , 0 1 5 3 0

0 , 0 1 5 3 4

4 , 0

3 , 3 3

1 0 8 , 0 0

1 1 1 , 6 7

0 , 0 1 7 6 0

0 , 0 1 7 6 0

0 , 0 1 7 6 5

4 , 5

3 , 7 5

9 6 , 0 0

9 8 , 7 2

0 , 0 1 9 9 4

0 , 0 1 9 9 3

0 , 0 1 9 9 5

5 , 0

4 , 1 7

8 6 , 4 0

8 8 , 9 9

0 , 0 2 2 0 9

0 , 0 2 2 1 1

0 , 0 2 2 1 1

6 , 0

5 , 0 0

7 2 , 0 0

7 4 , 5 4

0 , 0 2 6 3 8

0 , 0 2 6 3 8

0 , 0 2 6 3 9

7 , 0

5 , 8 3

6 1 , 7 1

6 3 , 9 4

0 , 0 3 0 7 5

0 , 0 3 0 7 6

0 , 0 3 0 7 2

8 , 0

6 , 6 7

5 4 , 0 0

5 5 , 2 0

0 , 0 3 5 6 2

0 , 0 3 5 6 2

0 , 0 3 5 5 6

9 , 0

7 , 5 0

4 8 , 0 0

4 8 , 9 9

0 , 0 4 0 1 5

0 , 0 4 0 1 6

0 , 0 4 0 0 9

1 0 , 0

8 , 3 3

4 3 , 2 0

4 4 , 0 4

0 , 0 4 4 6 9

0 , 0 4 4 6 8

0 , 0 4 4 5 8

2 0 , 0

1 6 , 6 7

2 1 , 6 0

2 2 , 0 8

0 , 0 8 9 1 0

0 , 0 8 8 9 8

0 , 0 8 8 8 5

3 0 , 0

2 5 , 0 0

1 4 , 4 0

1 4 , 6 3

0 , 1 3 4 5 0

0 , 1 3 4 3 5

0 , 1 3 3 9 8

4 0 , 0

3 3 , 3 3

1 0 , 8 0

1 0 , 9 4

0 , 1 7 9 3 0

0 , 1 7 9 2 0

0 , 1 7 8 7 0

5 0 , 0

4 1 , 6 7

8 , 6 4

8 , 7 4

0 , 2 2 4 2 0

0 , 2 2 3 9 0

0 , 2 2 3 2 0

6 0 , 0

5 0 , 0 0

7 , 2 0

7 , 3 0

0 , 2 6 8 5 0

0 , 2 6 8 3 0

0 , 2 6 7 4 0

7 0 , 0

5 8 , 3 3

6 , 1 7

6 , 2 6

0 , 3 1 2 7 0

0 , 3 1 2 6 0

0 , 3 1 1 4 0

8 0 , 0

6 6 , 6 7

5 , 4 0

5 , 4 7

0 , 3 5 8 2 0

0 , 3 5 7 8 0

0 , 3 5 6 7 0

9 0 , 0

7 5 , 0 0

4 , 8 0

4 , 8 7

0 , 4 0 1 8 0

0 , 4 0 1 3 0

0 , 4 0 0 0 0

1 0 0 , 0

8 3 , 3 3

4 , 3 2

4 , 3 7

0 , 4 4 7 2 0

0 , 4 4 6 8 0

0 , 4 4 5 0 0

T a b l e B . 1 : R e s u l t s o f t h e S I M O N A

I M U c l o c k w i s e c a l i b r a t i o n .

66



C MATLAB listings

C.1 Calibration program for three accelerometers

C.1.1 Calibration Module

1%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Function: Tilt Table Calibration for Accelerometers

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Copyright 2005 by Control and Simulation Division - TU-Delft

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

6% W ri tt en b y: F ra nk M . S ch ub er t, E ma il : fs @c hi li 23 .d e

% based on a program by Jose’ F. M. Lorga Email: [email protected]

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

close all ;11clear all ;

% load path and files information

DataFilesInfo

16% common informations

NTAngles = 2 0 ; % Number of tilt angles that are analized

AcqFreq = 1 0 0 0 ; % in Hz

21FilesPerAngle = 1 2 ; % 12 files per tilt angleFilesDuration = 1 5 ; % in seconds

DownSample = 1 ; % 1 = downsample, 0 = do not downsample

ReadBinData = 0 ; % 1 = read from .bef, 0 = read .mat file

26if DownSample

NewFreq = 1 0 0 ;else

NewFreq = AcqFreq ;end ;

31% A different downsample rate requires to read binary data (.bef files) again!

disp ( ’’ ) ;disp ( ’=============================================’)

36disp ( sprintf( ’Data Base Dir1: %s’ , DataBaseDir1 ) ) ;disp ( sprintf( ’Data Base Dir2: %s’ , DataBaseDir2 ) ) ;disp ( sprintf( ’Number of Tilt Angles: %d’ , NTAngles ) ) ;disp ( sprintf( ’Acquisition Frequency: %d’ , AcqFreq ) ) ;disp ( sprintf( ’Files per Angle: %d’ , FilesPerAngle ) ) ;

41disp ( sprintf( ’Duration (sec): %d’ , FilesDuration ) ) ;disp ( sprintf( ’Downsample? %d’ , DownSample ) ) ;disp ( sprintf( ’Read Bin Data? %d’ , ReadBinData ) ) ;disp ( ’=============================================’)

46% parameters for downsampling

FiltData . flag = 0 ;FiltData . cutoff_freq = 5 0 ; % in Hz, should be below half the sampling frequency

FiltData . n = 5 ; % filter order

51FiltData . Wn= FiltData . cutoff_freq/( AcqFreq/2) ;

TiltVertical = [ 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 9 0 8 0 7 0 6 0 5 0 4 0 3 0 2 0 1 0 0 ] . /1 8 0∗pi ;RotHorizontal = ( 0 : 3 0 : 3 5 0 ) . / 18 0∗pi ;

56% g0

g0 = 9 . 8 1 2 4 2 2 3 1 ;

for iFile =1 : NTAngles

61disp ( sprintf( ’Started File %d...’ , iFile) ) ;

RawData iFile = LoadBinData ( AllFilesInfo iFile , AcqFreq , NewFreq , ReadBinData , DownSample , FiltData ) ;

% calculate true values

66NSamplesPartFile = NewFreq∗AllFilesInfo iFile . Duration ;

for jangle =1 :FilesPerAngle

if iFile < 11

67



MATLAB listings

71TrueData iFile ( jangle , 1 ) = ( sin ( TiltVertical( iFile ) ) ∗ cos ( RotHorizontal( jangle) )∗g0 ) ; % ax true

acceleration

TrueData iFile ( jangle , 2 ) = ( sin ( TiltVertical( iFile ) ) ∗ sin ( RotHorizontal( jangle) )∗g0 ) ; % ax true

acceleration

TrueData iFile ( jangle , 3 ) = −

(cos ( TiltVertical( iFile) ) ∗

g0 ) ; % ax true accelerationelse

TrueData iFile ( jangle , 1 ) = ( sin ( TiltVertical( iFile ) ) ∗ cos ( RotHorizontal( jangle) )∗g0 ) ; % ax true

acceleration

76TrueData iFile ( jangle , 2 ) = −(sin ( TiltVertical( iFile) ) ∗ sin ( RotHorizontal( jangle) )∗g0 ) ; % ax true

acceleration

TrueData iFile ( jangle , 3 ) = ( cos ( TiltVertical( iFile ) ) ∗ g0 ) ; % ay true acceleration

end ;end ;disp ( sprintf( ’Ended File %d.’ ,iFile ) ) ;

81disp ( ’ ’ ) ;end ;

% optionally print some raw data channels:

86% for i=1:NTAngles

% figure

% plot(RawDatai(:,1), ’b’);

% hold on

% plot(RawDatai(:,2), ’m’);

91% plot(RawDatai(:,3), ’g’);

% legend(’X’, ’Y’, ’Z’);

% grid on;

% ylabel(’Volts’);

% Xlabel(’Samples’);

96% end;

% first calculate medians from measurements:

for iFile =1 :NTAngles

101NSamplesPartFile = NewFreq∗AllFilesInfo iFile . Duration ;for jangle=1 : FilesPerAngle

MedianRawData iFile ( jangle , 1 ) = median ( RawData iFile ( ( jangle−1)∗NSamplesPartFile+1 :jangle∗

NSamplesPartFile, 1) ) ;MedianRawData iFile ( jangle , 2 ) = median ( RawData iFile ( ( jangle−1)∗NSamplesPartFile+1 :jangle∗

NSamplesPartFile, 2) ) ;MedianRawData iFile ( jangle , 3 ) = median ( RawData iFile ( ( jangle−1)∗NSamplesPartFile+1 :jangle∗

NSamplesPartFile, 3) ) ;106end ;

end ;

Y_xtm = [ ] ;X_xtm = [ ] ;

111Y_ytm = [ ] ;X_ytm = [ ] ;Y_ztm = [ ] ;X_ztm = [ ] ;

116for iFile =1 :NTAngles

%f or the x axisY_xtm = vertcat ( Y_xtm , TrueData iFile ( : , 1 ) ) ;X_xtm = vertcat ( X_xtm , horzcat ( ones ( FilesPerAngle, 1) , . . .

121MedianRawData iFile ( : , 1 ) , . . .MedianRawData iFile ( : , 1 ) . ^ 2 , . . .MedianRawData iFile ( : , 2 ) , . . .MedianRawData iFile ( : , 2 ) . ^ 2 , . . .MedianRawData iFile ( : , 3 ) , . . .

126MedianRawData iFile ( : , 3 ) . ^ 2 , . . .MedianRawData iFile ( : , 3 ) . ^ 3 , . . .MedianRawData iFile ( : , 2 ) .∗MedianRawData iFile ( : , 3 ) . . .) ) ;

131% for the y axis

Y_ytm = vertcat ( Y_ytm , TrueData iFile ( : , 2 ) ) ;X_ytm = vertcat ( X_ytm , horzcat ( ones ( FilesPerAngle, 1 ) , . . .

MedianRawData iFile ( : , 1 ) , . . .136MedianRawData iFile ( : , 1 ) . ^ 2 , . . .

MedianRawData iFile ( : , 2 ) , . . .MedianRawData iFile ( : , 2 ) . ^ 2 , . . .MedianRawData iFile ( : , 3 ) , . . .MedianRawData iFile ( : , 3 ) . ^ 2 , . . .

141MedianRawData iFile ( : , 3 ) . ^ 3 , . . .MedianRawData iFile ( : , 1 ) .∗MedianRawData iFile ( : , 3 ) . . .

) ) ;

%for the z axis

146Y_ztm = vertcat ( Y_ztm , TrueData iFile ( : , 3 ) ) ;X_ztm = vertcat ( X_ztm , horzcat ( ones ( FilesPerAngle, 1 ) , . . .

MedianRawData iFile ( : , 1 ) , . . .MedianRawData iFile ( : , 1 ) . ^ 2 , . . .MedianRawData iFile ( : , 2 ) , . . .

151MedianRawData iFile ( : , 2 ) . ^ 2 , . . .MedianRawData iFile ( : , 3 ) , . . .MedianRawData iFile ( : , 3 ) . ^ 2 , . . .MedianRawData iFile ( : , 3 ) . ^ 3 , . . .MedianRawData iFile ( : , 1 ) .∗MedianRawData iFile ( : , 3 ) . . .

156) ) ;

end ;

disp ( ’lsqr...’)161

disp ( ’X:’)Coefm . xt = lsqr ( X_xtm , Y_xtm , 1e−6, 10 ) ;

disp ( ’Y:’)166Coefm . yt = lsqr ( X_ytm , Y_ytm , 1e−6, 10 ) ;

68




disp ( ’Z:’ )Coefm . zt = lsqr ( X_ztm , Y_ztm , 1e−6, 10) ;

171

% apply calibration

176for iFile =1 : NTAngles

ComplexCaliData iFile ( : , 1 ) = X_xtm ( ( iFile−1)∗FilesPerAngle+ 1 : ( iFile)∗FilesPerAngle , : )∗Coefm . xt ;ComplexCaliData iFile ( : , 2 ) = X_ytm ( ( iFile−1)∗FilesPerAngle+ 1 : ( iFile)∗FilesPerAngle , : )∗Coefm . yt ;ComplexCaliData iFile ( : , 3 ) = X_ztm ( ( iFile−1)∗FilesPerAngle+ 1 : ( iFile)∗FilesPerAngle , : )∗Coefm . zt ;

end ;181

% calculate residuals

for iChan =1: 3 %AllFilesInfoNTests.NChanels

ResidualsComplex iChan = [ ] ;186end ;

for iFile =1 : NTAngles

for iChan =1: 3 %AllFilesInfoNTests.NChanels

ResidualsComplex iChan = vertcat ( ResidualsComplex iChan , ComplexCaliData iFile ( : , iChan )−TrueDataiFile ( : , iChan ) ) ;

191end ;end ;

for iChan =1: 3

MeanResidualComplex = mean ( ResidualsComplex iChan ( : , 1 ) ) ;196StdResidualComplex = std ( ResidualsComplex iChan ( : , 1 ) ) ;

disp ( sprintf( ’Channel %i residual mean is: %d’ , iChan , MeanResidualComplex ) ) ;disp ( sprintf( ’Channel %i residual std is: %d\r’ , iChan , StdResidualComplex ) ) ;

end ;

201% creat mean data for plotting:

MeanTrueData = [ ]MeanComplexCaliData = [ ] ;MeanMedianRawData = [ ] ;

206for iFile =1 : NTAngles

MeanTrueData = vertcat ( MeanTrueData , TrueData iFile ) ;MeanComplexCaliData = vertcat ( MeanComplexCaliData, ComplexCaliData iFile ) ;MeanMedianRawData = vertcat ( MeanMedianRawData, MedianRawData iFile ) ;

211end ;

% calculate (adjusted) coefficient of determination

for i = [ 1 2 3 ]216

SSres = sum ( ( MeanMedianRawData( : , i ) − MeanComplexCaliData ( : , i ) ) .^2) ;n = size ( MeanMedianRawData( : , i ) , 1 ) ;SST = MeanMedianRawData ( : , i ) ’ ∗ MeanMedianRawData ( : , i ) − ( ( sum ( MeanMedianRawData( : , i ) ) ) . ^ 2 ) / n ;

R ( i ) = sqrt (1 − SSres / SST ) ;221p = size ( Coefm . xt , 1 ) ;

MSres( i ) = SSres / ( n − p ) ;

Radj ( i ) = sqrt (1 − ( SSres / ( n − p ) ) / ( SST / ( n − 1 ) ) ) ;disp ( sprintf( ’channel %d: sig: %f \t MSres: %f \t R: %f \t Radj: %f \t \n’ , i , std ( ResidualsComplex i ( : , 1 ) ) ,

MSres ( i ) , R ( i ) , Radj ( i ) ) ) ;226

end ;

isp ( ’formatted output for Latex:’ ) ;

231disp ( ’\hline’) ;disp ( sprintf( ’\\multicolumn4|l|Models: $\\left. A\\right|_x,y,z = $ \\\\’ ) ) ;disp ( ’\hline’) ;disp ( sprintf( ’$\\beta_i$ & \\multicolumn1c|$\\beta_i|_A_x$ & \\multicolumn1c|$ \\beta_i|_A_y$ &

\\multicolumn1c|$\\beta_i|_A_z$ \\\\’) ) ;236disp ( ’\hline’) ;

for i = 1 : p

disp ( sprintf( ’%d & %.8f & %.8f & %.8f \\\\’ , i−1, Coefm . xt ( i ) , Coefm . yt ( i ) , Coefm . zt ( i ) ) ) ;end ;disp ( ’\hline’) ;

241disp ( ’\hline’) ;disp ( sprintf( ’& \\multicolumn1c|X & \\multicolumn1c|Y & \\multicolumn1c|Z\\\\’) ) ;

disp ( ’\hline’) ;disp ( sprintf( ’$\\sigma_res$ & %.8f & %.8f & %.8f\\\\ ’ , std ( ResidualsComplex 1 ( : , 1 ) ) , std ( ResidualsComplex

2 ( : , 1 ) ) , std ( ResidualsComplex 3 ( : , 1 ) ) ) ) ;disp ( sprintf( ’$M_res$ & %.8f & %.8f & %.8f \\\\’ , MSres ( 1 ) , MSres ( 2 ) ,MSres (3) ) ) ;

246disp ( sprintf( ’$R$ & %.8f & %.8f & %.8f \\\\’ , R ( 1 ) , R ( 2 ) ,R (3) ) ) ;disp ( sprintf( ’$R_adj$ & %.8f & %.8f & %.8f \\\\’ , Radj ( 1 ) , Radj ( 2 ) , Radj (3) ) ) ;disp ( ’\hline’) ;

% plot data

251PlotMeanData( MeanTrueData, MeanComplexCaliData , MeanMedianRawData , 1 , NTAngles ) ;

% Save Coefficients:

save ( ’accel_coef.mat’ , ’Coefm’ ) ;

C.1.2 File Information Module

1%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Function: Load data files information

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Copyright 2005 by Control and Simulation Division - TU-Delft

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

6% W ri tt en b y: F ra nk M . S ch ub er t, E ma il : fs @c hi li 23 .d e

69



MATLAB listings


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

AllFilesInfo = 1 : 1 ;

11DataBaseDir1 = ’C:\Documents and Settings\fschubert\My Documents\IMU\CalibrationData\Frank-NRSE\complete run1\’ ;DataBaseDir2 = ’C:\Documents and Settings\fschubert\My Documents\IMU\CalibrationData\Frank-NRSE\complete -run1\’ ;

FileInfo . DataPath = strcat ( DataBaseDir1, ’0\’ ) ; %Data Path

16FileInfo . Name = ’’ ;FileInfo . FileExt = ’.bef’ ;FileInfo . FirstFile = 1;FileInfo . NFiles = 1 2 ;FileInfo . NChanels = 1 0 ;

21FileInfo . Duration = 1 5 ; %in seconds

%save data

AllFilesInfo 1 = FileInfo ;




%save data





%save data





%save data




61FileInfo . Duration = 1 5 ; %in seconds%save data


% file was truncated, it contains tilting angels 0, 10, ..., 90 for calibration

66% of z = g and 0, 10, ..., 90 for calibration of z = -g

C.1.3 Module for loading binary data

Various versions of this modules were developed within this project, e.g. fordetecting pulse information on the temperature sensor input and handlingcalibration data with different acquisition times.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Function: Load data files information

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

4% Copyright 2005 by Control and Simulation Division - TU-Delft

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% W ri tt en b y: F ra nk M . S ch ub er t, E ma il : fs @c hi li 23 .d e


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%9

AllFilesInfo = 1 : 1 ;

DataBaseDir1 = ’C:\Documents and Settings\fschubert\My Documents\IMU\CalibrationData\Frank-NRSE\complete run1\’ ;DataBaseDir2 = ’C:\Documents and Settings\fschubert\My Documents\IMU\CalibrationData\Frank-NRSE\complete -run1\’ ;

14FileInfo . DataPath = strcat ( DataBaseDir1, ’0\’ ) ; %Data Path

FileInfo . Name = ’’ ;FileInfo . FileExt = ’.bef’ ;FileInfo . FirstFile = 1;

19FileInfo . NFiles = 1 2 ;FileInfo . NChanels = 1 0 ;FileInfo . Duration = 1 5 ; %in seconds

%save data

AllFilesInfo 1 = FileInfo ;24


FileInfo . Name = ’’ ;FileInfo . FileExt = ’.bef’ ;FileInfo . FirstFile = 1;


%save data

70






FileInfo . Name = ’’ ;

FileInfo . FileExt = ’.bef’ ;FileInfo . FirstFile = 1 ;


%save data



FileInfo . Name = ’’ ;FileInfo . FileExt = ’.bef’ ;FileInfo . FirstFile = 1 ;


%save data



FileInfo . Name = ’’ ;FileInfo . FileExt = ’.bef’ ;FileInfo . FirstFile = 1 ;


%save dataAllFilesInfo 5 = FileInfo ;

64% file was truncated, it contains tilting angels 0, 10, ..., 90 for calibration

% of z = g and 0, 10, ..., 90 for calibration of z = -g

C.1.4 Plotting Module

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Function: Plot Calibration Results

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

4% Copyright 2005 by Control and Simulation Division - TU-Delft

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% W ri tt en b y: F ra nk M . S ch ub er t, E ma il : fs @c hi li 23 .d e


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

9function PlotData ( TrueData , ComplexCalibratedData , MeanUnCali , FirstFile , LastFile )

% plot Data

14FigureSize = get (0 , ’ScreenSize’) ;FigureSize = FigureSize + [0 −40 0 0 ] ;

% X

Handle_Pos = figure ( ’Visible’ , ’off’) ;19set ( gcf , ’position’ , FigureSize ) ;

orient landscape ;

subplot( 2 1 1 )plot ( MeanUnCali( : , 1 ) , ’co-’ , ’MarkerFaceColor’ , [. 4 9 1 . 6 3 ] ) ;

24hold on

plot ( TrueData ( : , 1 ) , ’r-d’ , ’MarkerFaceColor’ , ’r’ ) ;plot ( ComplexCalibratedData( : , 1 ) , ’k+’ ) ;legend( ’Raw Data’ , ’Reference Data’ , ’Calibration’ ) ;grid on ;

29ylabel( ’Mean X Acceleration, m/s^2’ )Xlabel( ’Samples’)

subplot( 2 1 2 )plot ( ComplexCalibratedData( : , 1 )−TrueData ( : , 1 ) , ’k.’ ) ;

34hold on ;ylabel( ’Difference X Acceleration, m/s^2’ )Xlabel( ’Samples’)

set ( Handle_Pos , ’Visible’ , ’on’ ) ;

39% Y

Handle_Pos = figure ( ’Visible’ , ’off’) ;set ( gcf , ’position’ , FigureSize ) ;orient landscape ;subplot( 211) ;

44plot ( MeanUnCali( : , 2 ) , ’co-’ , ’MarkerFaceColor’ , [. 4 9 1 . 6 3 ] ) ;hold on

plot ( TrueData ( : , 2 ) , ’r-d’ , ’MarkerFaceColor’ , ’r’ ) ;plot ( ComplexCalibratedData( : , 2 ) , ’k+’ ) ;legend( ’Raw Data’ , ’Reference Data’ , ’Calibration’ ) ;

49ylabel( ’Mean Y Acceleration, m/s^2’ )Xlabel( ’Samples’)grid on ;

subplot( 2 1 2 )54hold on

plot ( ComplexCalibratedData( : , 2 )−TrueData ( : , 2 ) , ’k.’ ) ;ylabel( ’Error Y Acceleration, m/s 2’ )Xlabel( ’Samples’)set ( Handle_Pos , ’Visible’ , ’on’ ) ;

59

% Z

71



MATLAB listings

Handle_Pos = figure ( ’Visible’ , ’off’) ;set ( gcf , ’position’ , FigureSize ) ;

64orient landscape

subplot( 2 1 1 )

plot ( MeanUnCali( : , 3 ) , ’co-’ , ’MarkerFaceColor’ , [. 4 9 1 . 6 3 ] ) ;hold on ;plot ( TrueData ( : , 3 ) , ’r-d’ , ’MarkerFaceColor’ , ’r’ ) ;

69plot ( ComplexCalibratedData( : , 3 ) , ’k+’ ) ;legend( ’Raw Data’ , ’Reference Data’ , ’Calibration’ ) ;ylabel( ’Mean Acceleration, m/s 2’)Xlabel( ’Samples’)grid on ’

74

subplot( 2 1 2 )plot ( ComplexCalibratedData( : , 3 )−TrueData ( : , 3 ) , ’k.’ ) ;hold on ;

79ylabel( ’Error Z Acceleration, m/s 2’ )Xlabel( ’Samples’)set ( Handle_Pos , ’Visible’ , ’on’ ) ;

figure ;84subplot( 311) ;

hold on ;plot ( ComplexCalibratedData( : , 1 )−TrueData ( : , 1 ) , ’k.’ ) ;ylabel( ’X Error, m/s^2’ )Xlabel( ’Samples’)

89axis ( [ 0 2 4 0 −. 01 5 . 0 1 5 ] ) ;grid on ;

subplot( 312) ;hold on ;

94plot ( ComplexCalibratedData( : , 2 )−TrueData ( : , 2 ) , ’k.’ ) ;ylabel( ’Y Error. m/s^2’ ) ;Xlabel( ’Samples’) ;axis ( [ 0 2 4 0 −. 01 5 . 0 1 5 ] ) ;grid on ;

99subplot( 313) ;plot ( ComplexCalibratedData( : , 3 )−TrueData ( : , 3 ) , ’k.’ ) ;hold on ;ylabel( ’Z Error, m/s^2’ )

104Xlabel( ’Samples’)axis ( [ 0 2 4 0 −. 01 5 . 0 1 5 ] ) ;grid on ;

72



D Publication

The following paper was prepared for the European Conference for AerospaceSciences (EUCASS) that took place July 4th-7th 2005 in Moscow. More informa-

tion about the conference can be found athttp://www.tsagi.ru/eng/news/eucass.

73

http://www.tsagi.ru/eng/news/eucass

http://www.tsagi.ru/eng/news/eucass



EUROPEAN CONFERENCE FOR AEROSPACE SCIENCES (EUCASS)

ENHANCING THE PERFORMANCE OF AN INTEGRATED

NAVIGATION SYSTEM USING ON-LINE AND OFF-LINE

CALIBRATION PROCEDURES

J. F. M. Lorga, F. Schubert , Q. P. Chu, and J. A. Mulder Delft University of Technology

Delft, The Netherlands

Abstract

This paper analyzes and compares on-line

and off-line calibration methods applied to an

integrated navigation system. The influences

in the global performances of using bothmethods separately or combined are shown.

Ground data, collected in dynamic and static

environments, is processed and incorporated

using various Global Positioning System (GPS)

observables to demonstrate the improved

performances combining both calibration

topologies.

1. Introduction

Terrestrial, maritime and aerial navigation

applications respectively requiring centimeter,

millimeter per second, and tenths of radian

error level in position, velocity, and attitude

have increased in the last few years and are

no longer difficult to find. The challenging

objective was to completely design and develop

an integrated navigation system that achieves

these performances.

The navigation system development started

in the Fall 2002. The first results obtained by

simulations (see [1]) proved that it was possible

to achieve the desired precisions. The design

and implementation of the complete hardware

and software is described in [2]. The first

successful flight and ground tests are also

presented in [2] and [3].

The present status of the navigation system

is well illustrated by Fig. 1 in terms of software

and hardware through the flowchart and pic-

tures of the several components, respectively.

The hardware main components are an Inertial

Measurement Unit (IMU) composed of three

fiber optic gyroscopes, three accelerometers

and temperature sensors, a high-end Global

I/EKF

inputs

Position coordinates

Attitude angles

Velocity coordinates

magnetic fieldcomponents

accelerations androtation rates

Magnetometer

GNSSobservables

GNSS Receiver

IMU

PolinomialCalibration

PolinomialCalibration IMU error model

parameters

Fig. 1. Present status of the integrated navigation system.

J. F. M. Lorga, F. Schubert , Q. P. Chu, and J. A. Mulder , ENHANCING THE PERFORMANCE OF AN INTEGRATED NAVIGATION SYSTEMUSING ON-LINE AND OFF-LINE CALIBRATION PROCEDURES

Positionin S stem GPS receiver and a The third art describes the on-line calibration




celerometers the model is:

axmaymazm

=

bxbybz

+

wx

wy

wz

+ (1)

sx mxy mxz

myx sy myz

mzx mzy sz

axayaz

,

where axm, aym, and azm are the specific

forces measured by the accelerometers in

the body reference frame; ax, ay, and az are

the real values of the specific forces also in

body reference frame; sx, sy, and sz are the

scale factors; bx, by, and bz are the biases (theaccelerometer bias derivative is equal to the

drift); mxy, mxz, myx, myz , mzx, and mzy

are the axis misalignments; and wx, wy, and

wz are the three body axes specific forces

measurements noise. Since the accelerometers

measurements are given in the body reference

frame and all the other used equations were

written in the ECEF reference frame, a coor-

dinates transformation matrix from body toECEF is necessary. The navigation system

under development should be able to estimate

besides the position, velocity, and attitude, the

described parameters from the error models of

the inertial sensors. The estimation of these

parameters is obviously dependent in their

observability which is discussed in Section 4.

3. Off-Line Calibration

The goal of the off-line calibration is to find the

coefficients of the polynomial that relates the

raw output of the sensors − measured in Volt −

to the respective physical values. The next sub-

sections describe the implemented off-line cali-

bration procedures to the inertial and magnetic

sensors.

3.1. Inertial

The inertial off-line calibration was initiatedby placing the inertial sensors in a tempera-

ture controlled environment and several consec-

utive temperature cycles ranging from −40oC

to +40oC were applied. The cycle repetitions

were done to detect the temperature dependent

hysteresis of the components. Fig. 2 shows the

changes in the outputs of the Z axis accelerom-

eter and r gyroscope with temperature varia-

tions. For the accelerometers and gyroscopes

a third and first order polynomial were used, re-

spectively, to model the temperature influences.

The second part of the inertial calibration

was done by using a tilt table to determine

the accelerometers calibration polynomial co-

efficients for accelerations inferior to the grav-

ity. This test consists in changing the attitude

of the IMU box between exactly known posi-tions imposing different accelerations to the ac-

celerometers. The attitude angles imposed to

the IMU can be known with an accuracy of

0.25 arc minute.

The used error model for the X axis ac-

celerometer is:

ax = C 0 +C 1ax +C 2a2x +C 3a

3x +C 4a

4x

+C 5ay +C 6a2y +C 7az +C 8a

2z+

C 9axay +C 10axaz +C 11ayaz,

(2)

where ai are the accelerometers output and the

coefficients C i are determined using the least-

squares method. The models for the Y and Z

axis accelerometers are similar. The residuals

between the true and the calibrated output are

2.5 2.6 2.7 2.8 2.9 3 3.1 3.2

1.28

1.282

1.284

1.286

1.288

1.29

Z Accelerometer Temperature Sensor Output, V

z a c c e l e r o m e t e r , V

2.3 2.4 2.5 2.6 2.7 2.8 2.9 3 3.1 3.26

6.5

7

7.5

8

8.5

x 10−3

IMU Box Temperature Sensor Output, V

r g y r o s c o p e ,

V

Fig. 2. Z accelerometer and r gyroscope temperature

dependencies.

3




shown in Fig. 3. We can also see that our cali-

bration outperforms the one supplied by the ac-

celerometers manufacturer. This was expected

because our calibration takes into account the

axis misalignments and Analog-Digital Con-

verter (ADC) card cross-coupling and change

injection phenomenons. These two phenome-

nas are mainly caused by the ADC card minia-

turization and the fact that a single ADC has to

be multiplexed for the sixteen available chan-

nels.

The third calibration step of the inertial

sensor was the calibration of the gyroscopes

using a turn table. The used table can rotate

around its vertical axis in both clockwise andcounter-clockwise direction. The rotation speed

can be adjusted from 0.01o/s to 1200o/s.

The calibration procedure consisted in ro-

tating the IMU around its three axes in both

directions from 0o/s to 300o/s. The collected

data was processed and a polynomial calibra-

tion was applied to the three gyroscopes. The

used p gyroscope the error model is:

p = C 0 +C 1 p+C 2 p2+C 3 p

3+C 5q

C 7r +C 9 pq +C 10 pr +C 11qr, (3)

where p, q , and r are the gyroscopes outputs and

coefficients C i are determined using the least-

squares method. The models for the q and rgyroscopes are similar. The residuals between

0 40 80 120−0.2

−0.1

0

0.1

0.2

Y , m / s 2

0 40 80 120−0.1

−0.05

0

0.05

0.1

X , m / s 2

Manufacturer’s

0 40 80 120−0.01

−0.005

0

0.005

0.01

Our

0 40 80 120−0.01

−0.005

0

0.005

0.01

0 40 80 120−0.1

−0.05

0

0.05

0.1

Samples

Z , m / s 2

0 40 80 120−0.01

−0.005

0

0.005

0.01

Samples

Fig. 3. Accelerometers calibration residuals.

the true and the calibrated output are shown in

Fig. 4. We can also see that our calibration out-

performs the simple one supplied by the gyro-

scopes manufacturer including only scale fac-

tors. This was expected because of misalign-

ments estimation and ADC problems described

previously.

3.2. Magnetic

Further improvements of the navigation system

performance were achieved by applying the

off-line magnetic calibration method described

in [6]. This method does not require any

external reference. Furthermore, unlike the

usual calibration methods, it performs thecalibration in the magnetic field domain and

not in the heading domain. The results of the

magnetic calibration are presented in Fig. 5.

The system was rotated at constant a altitude.

Therefore, the two horizontal components of

the measured magnetic field, X and Y , have

a sinusoidal behavior. When one of them is

approximately zero, the other should point

north and measure approximately 19018

nT ,which is the horizontal intensity of the Earth’s

magnetic field at the tests location using the

International Geomagnetic Reference Field

model. The vertical axis Z and the norm

of the magnetic filed should have a constant

value. The norm of the total magnetic field was

0 16 32 48−30

−20

−10

0

10

0 16 32 48−1

0

1

0 16 32 48−30

−20

−10

0

10

0 16 32 48−1

−0.5

0

0.5

1

0 16 32 48−30

−20

−10

0

10

0 16 32 48−1

−0.5

0

0.5

1

Manufacturer’s Our

Samples Samples

p ,

° / s

r , ° / s

q ,

° / s

Fig. 4. Gyroscopes calibration residuals.

4




0 1 2 3 4 5 6

x 104

−2

−1

0

1

2

x 104

X M a g .

F i e l d , n T

0 1 2 3 4 5 6

x 104

−2

−1

0

1

2

x 104

Y M a g .

F i e l d ,

n T

0 1 2 3 4 5 6

x 104

4.4

4.6

4.8

5x 10

4

Samples

M a g .

F i e l d N o r m , n T

ManufacturerOur

Fig. 5. Measured X , Y , and Earth magnetic field norm

with and without our calibration.

approximately 48723 nT . The improvements

achieved with the implemented calibration

method are well visible in the top of the two

plots of Fig. 5 showing the two horizontal axes.

As expected, in these plots the sinusoid is

centered around zero after the calibration. In

the bottom plot, a constant magnetic field norm

was achieved.

4. On-Line Calibration

In on-line calibration the inputs’ error models

parameters of the accelerometer and gyroscopes

(see Eq. 1) are included as states in the EKF.

Before all these parameters were introduced as

states, an observability analysis was done.

4.1. States ObservabilityThe summary of the observability analysis us-ing the so-called Grammian matrix is presentedin Table 1. This matrix is defined as:

G =N

k=0

ΦT (k + 1|0)H T (k)H (k)Φ(k + 1|0) (4)

where Φ is the system transition matrix and H

the observation matrix. The system is com-

pletely observable if the matrix G is positive

definite, or in other words, if all singular val-ues are positive (non-zero) real values. The in-

clusion of the magnetometer was necessary to

make observable the r gyroscope bias and the

heading angle under certain conditions. The

misalignments only became observable when

some maneuvers producing sufficient accelera-

tions in all axes were performed. Furthermore,

the mechanical misalignments of the system are

constant, so once known they are always the

same. Their inclusion as filter states would only

add complexity to the filter and more computa-

tional power would be needed. Similar reasons

led us also to exclude accelerometers and gyro-

scopes scale factors from the filter states. Ad-

ditionally, as seen in the off-line calibration, a

simple scale factor would not be sufficient for

both type of sensors. Therefore, the polyno-

mial coefficients from the inertial sensors errormodel (see Eq. 2) would also have to be added,

raising new observability problems.

4.2. Static case

A LC filter was set up with 18 states: 3 position

and 3 velocity coordinates, 3 attitude angles, 3

accelerometer and 3 gyroscopes biases, and 3

accelerometer temperatures. This configuration

was used to process the data collected in

the static and dynamic (see next subsection)

environments.

The static test consisted in placing the

navigation system with different attitudes. The

selected roll and pitch angles were approxi-

mately 0o and 0o , −30o and 0o , and 0o and

−30o, respectively. Their estimated values are

presented in Fig. 6. As can be seen in all cases,

the introduction of the off-line calibrations

improved the quality of the final results. Asseen in Fig. 7, the off-line calibrations not

only influence position, velocity, and attitude

estimation, but also the on-line calibration

parameters, accelerometer and gyroscope

Table 1. Gyroscopes and accelerometers observable pa-

rameters from the input error models.

Is observable? Gyroscopes Accelerometers

Biases Yes Yes

Scale Factors Yes Yes

Misalignments Not Always Not Always

5




5 25 45 65 85

−0.05

0

0.05

R o l l , o

5 25 45 65 85−0.5

0

0.5

P i t c h ,

o

5 25 45 65 85−0.2

−0.1

0

0.1

0.2

R o l l , o

5 25 45 65 85−31

−30

−29

−28

P i t c h ,

o

10 30 50 70 90−1

−0.5

0

0.5

1

Time, s

P i t c h ,

o

10 30 50 70 90

−30.2

−30.1

−30

−29.9

−29.8

Time, s

R o l l , o

BeforeAfter

PitchRoll

True≈ 0o

True≈ 0o

True≈ 0o

True≈ 0o

True ≈ −30o

True≈ −30o

Fig. 6. Estimated roll and pitch in different situations.

0 50 100 0 50 100−0.01

0

0.01

0.02

0.03

X A c c .

B i a s , m / s 2

0 50 100 0 50 100−1

0

1

2

3x 10

−3

p G y r o B i a s ,

o / s

0 50 100 0 50 100−0.06

−0.04

−0.02

0

0.02

Y A c c .

B i a s , m / s 2

0 50 100 0 50 100−2

0

2

4x 10

−3

q G y r o B i a s ,

o / s

0 50 100 0 50 100−0.05

0

0.05

0.1

0.15

Time, s

Z A c c .

B i a s , m / s 2

0 50 100 0 50 100−2

0

2

4

6x 10

−3

Time, s

r G y r o B i a s ,

o / s

Before CalibrationAfter Calibration

Fig. 7. Accelerometers and gyroscopes biases.

biases. The EKF includes in these biases

the imprecisions and non-linearities of the

used models, for instance, the Earth’s gravity

and magnetic models. Therefore, it was ex-pected that after the calibration these states had

a smaller value, but still are different from zero.

4.3. Dynamic case

A set of ground tests were done to collect

data in a dynamic environment. The GPS

antenna was placed on the top of a van, the

magnetometer on the tip of a 5 m aluminum

boom sticking out of the van back, and theremaining components in the interior. These

dynamic tests consisted in driving the van to

an open field through a series of rectilinear

trajectories and perform circular maneuvers.

The altitude changes were always inferior to

5m during the complete runs. The GPS board

estimated heading angle − angle between body

center line and true north − for one of the

several completed runs can be seen in the top

of Fig. 8.

The ground tests have the advantage of

being done in a more controllable environment

when compared to flight tests. Vertical and

lateral body velocities are almost non existent,

and the track angle − angle between ground

speed and true north − is coincident with the

van heading angle. The van velocity vector

is coincident with the ground velocity vec-tor.Therefore, the estimated heading angle of

the LC filter can be compared with the track

angle estimated by the GPS board, as it is also

shown in Fig. 8. As expected, the moments the

van is stopped the GPS board does not estimate

correctly the heading angle. This can be seen in

the top plot of Fig. 8 where the estimated GPS

heading varied randomly between 0o and 360o.

The comparison of the GPS and LCestimated heading angles in the bottom plot

of Fig. 8 show a 10o bias correction when

the off-line magnetic calibration procedure is

implemented.

The estimated pitch and roll attitude

4.639 4.64 4.641 4.642 4.643 4.644 4.645 4.6457

x 105

0

100

200

300

360

GPS time, s

H e a d i n g ,

d e g

4.64 4.6402 4.6404 4.6406 4.6408 4.641 4.6412

x 105

−20

−15

−10

−5

0

5

10

15

GPS time, s

R e s i d u a l s ,

d e g

GPS HeadingLC ManufacurerLC Our

Fig. 8. GPS and LC using two different calibrations es-

timated van heading angle (top). The residuals between

the GPS and LC heading for a shorter period (bottom).

6




4.64 4.6402 4.6404 4.6406 4.6408 4.641 4.6412

x 105

−5

−4

−3

−2

−1

0

GPS time, s

P i t c h ,

d e g

4.64 4.6402 4.6404 4.6406 4.6408 4.641 4.6412

x 105

−3

−2

−1

0

1

2

3

4

GPS time, s

R o l l , d e g

Fig. 9. EKF roll and pitch angle estimates.

angles, shown in Fig. 9, should be approxi-

mately constant for all the run because it was

performed in almost flat terrain with minor

changes of altitude. No other source was

available for comparison. Around 4.6406 s the

break pedal was pushed hard, sudden changes

of pitch and roll were registered. At the end of

the run, the van was parked for some seconds

in a small ramp, as can be seen by the increasein the pitch angle.

5. Conclusions

Enhances in overall performance of the

integrated navigation system, through the

employment of on-line and off-line calibration

methods to the inertial and magnetic compo-

nents were shown. The implemented off-line

calibrations outperformed largely the ones sup-

plied by the manufacturers because accounted

for misalignments and ADC related problems.

The biggest improvement was the correction

of a 10o bias in the heading angle estimation

due to poor magnetic calibration. Further

improvements through off-line calibration will

be achieved using the turn table and a metal

arm, to extend the rotation radius, to excite the

system with centrifugal accelerations superiorto the gravity.

The best obtained configuration estimated

the magnetic and inertial sensors biases, mis-

alignments and scale factors using off-line

calibration methods, and an extra bias state

in the EKF for each of the accelerometers

and gyroscopes. This extra states improved

the filter performance because imprecisions

and non-linearities of the used models were

’looked’ as a bias by the EKF.

Acknowledgments

The work presented and still under development

is possible due to the financial and intellectual

support of TNO - Defence, Security and Safety, The

Hague, The Netherlands. The authors would also

like to thank Kees van Woerkom for the help in the

preparation of the ground experiments and system

calibration.

References

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[2] Lorga J. F. M., van Rossum W., van Halsema E.,Chu Q. P., and Mulder J. A. The development of asar dedicated navigation system: From scratch to thefirst test flight. In PLANS 2004 - Position Locationand Navigation Symposium Proceedings, pages 249–258, Monterey, California, 26-29 April 2004. IEEEAerospace and Electronic Systems Society.

[3] Lorga J. F. M., Chu Q. P., van Rossum W., and Mul-der J. A. Developing and testing a tightly coupledgps/imu navigation system. In 2nd ESA Workshopon Satellite Navigation User Equipment Technolo-gies Navitec 2004 Workshop Proceedings, Noord-wijk, The Netherlands, 8-10 December 2004. Euro-pean Space Agency - ESTEC.

[4] Chu Q. P., Mulder J. A., and Van Woerkom P. T. L.M. Modified recursive maximum likelihood adap-tive filter for nonlinear aircraft flight-path reconstruc-tion. Journal of Guidance, Control, and Dynamics,19(6):1285–1295, November-December 1996.

[5] Mulder J.A., Chu Q.P., Sridha J.K.and Breeman J.H.,and Laban M. Non-linear aircraft flight path re-construction review and new advances. Progress in Aerospace Sciences, 35:673–726, 1999.

[6] Elkaim G. H., Gebre-Egziabher D., Powell J. D., andParkinson W. B. A non-linear, two-step estimationalgorithm for calibrating solid-state strapdown mag-netometers. In 8th St. Petersburg Conference on Nav-igation Systems (IEEE/AIAA), St. Petersburg, Russia,27-31 May 2001.

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