저 시-비 리- 경 지 2.0 한민

는 아래 조건 르는 경 에 한하여 게

l 저 물 복제, 포, 전송, 전시, 공연 송할 수 습니다.

다 과 같 조건 라야 합니다:

l 하는, 저 물 나 포 경 , 저 물에 적 된 허락조건 명확하게 나타내어야 합니다.

l 저 터 허가를 면 러한 조건들 적 되지 않습니다.

저 에 른 리는 내 에 하여 향 지 않습니다.

것 허락규약(Legal Code) 해하 쉽게 약한 것 니다.

Disclaimer

저 시. 하는 원저 를 시하여야 합니다.

비 리. 하는 저 물 리 목적 할 수 없습니다.

경 지. 하는 저 물 개 , 형 또는 가공할 수 없습니다.

Master's Thesis

DEVELOPMENT OF ORIENTATION

MEASUREMENT SYSTEM WITH MAGNETIC

FIELD AND NEURAL NETWORK

Junguk Kim

Department of Mechanical Engineering

Graduate School of UNIST

2018

DEVELOPMENT OF ORIENTATION

MEASUREMENT SYSTEM WITH MAGNETIC

FIELD AND NEURAL NETWORK

Junguk Kim

Department of Mechanical Engineering

Graduate School of UNIST

1

ABSTRACT

Many objects having multi-degree of freedom(DOF) motion have been appeared such as joystick,

multi-DOF actuator and multi-DOF motion platform. These applications have been widely studied with

their different purposes to apply industry. Orientation measurement system is required to improve

orientation control by sensing the orientation of an object. The motivation of this research was demands

for precise orientation control of these applications.

This thesis presents to develop an orientation measurement system using magnetic field of permanent

magnet and magnetic sensors such as Hall-effect sensor. Using magnetic field of a permanent magnet,

the orientation of an object embedded with this magnetic sources can be tracked in real-time.

Orientation tracing method based on magnetic field has advantages such as contact-free measurement

without any extra mechanics and no requirement of line-of-sight problem. However, due to the

complexities of the relation between magnetic field and position/orientation, it is challenging to

calculate the position/orientation from magnetic field in closed-form. In this thesis, Artificial neural

network has been applied to compensate the difficulties as it can fit any practical function between input

(=magnetic field) and output (=orientation). Using distributed multipole model which can provide a fast

and closed-form to analyze magnetic field, issues related this system are studied include the number of

optimized sensor and position of sensor. Considering these issues, effective orientation measurement

system can be presented.

In both simulation and experiment, two examples are introduced to provide better understanding how

proposed measurement system considering issues can measure orientation and evaluate a measurement

accuracy according to a motion region. This system can be applied to measure orientation of interested

application.

TABLE OF CONTENTS

ABSTRACT .......................................................................................................................................... 1

LIST OF FIGURES ................................................................................................................................ 2

LIST OF TABLES .................................................................................................................................. 4

NOMENCLATURE ............................................................................................................................... 5

CHAPTER 1. INTRODUCTION ........................................................................................................... 8

1.1 Background and Motivation .......................................................................................................... 8

1.2 Prior and Related Works ............................................................................................................... 9

1.3 Research Objective ..................................................................................................................... 11

1.4 Thesis Outline ............................................................................................................................. 11

CHAPTER 2. TWO-DOF ORIENTATION MEASUREMENT SYSTEM ......................................... 12

2.1 Overview ..................................................................................................................................... 12

2.2 Orientation Measurement System based on Magnetic Field and Artificial Neural Network...... 13

2.2.1 Magnetic field of Permanent Magnet Based on DMP ....................................................... 14

2.2.2 Analysis of Bijective Domain ............................................................................................ 17

2.2.3 Investigation of Optimized Sensor’s Position and Orientation .......................................... 19

2.2.4 Measurement Accuracy based on Jacobian matrix ............................................................ 23

2.2.5 Inverse Computation by using Artificial Neural Network ................................................. 23

2.2.6 Multi-Sensor Approach ..................................................................................................... 25

2.2.7 Calibration of Magnetic Field ............................................................................................ 26

2.3 Numerical Simulation and Experiment ....................................................................................... 27

2.3.1 Example 1: Two-DOF Orientation Measurement System with Single Sensor .................. 27

2.3.2 Example 2: Two-DOF Orientation Measurement System with Multi-Sensor ................. 47

2.4 Conclusion .................................................................................................................................. 56

CHAPTER 3. CONCLUSION AND FUTURE WORKS .................................................................. 57

3.1 Accomplishments and Contributions ........................................................................................ 57

3.2 Future Works ............................................................................................................................. 57

REFERENCES ..................................................................................................................................... 59

ACKNOWLEDGEMENTS .................................................................................................................. 62

2

LIST OF FIGURES

Figure 1.1 Various objects having multi-DOF motion .......................................................................... 9

Figure 1.2 Orientation measurement system developed previously ................................................... 10

Figure 2.1 X1Y2Z3 Euler angles .......................................................................................................... 14

Figure 2.2 Flowchart of orientation estimation ................................................................................... 14

Figure 2.3 Coordinate systems ............................................................................................................ 15

Figure 2.4 DMP model for a cylindrical PM ...................................................................................... 16

Figure 2.5 Relation between orientation and MFD ............................................................................. 17

Figure 2.6 Domain of orientation and MFD ......................................................................................... 18

Figure 2.7 Rotating magnetic field shape in the case of LP≠0 .............................................................. 20

Figure 2.8 Two shape of measurable MFD-field ................................................................................ 20

Figure 2.9 Unit sphere indicating sensing performance ...................................................................... 22

Figure 2.10 Structure of ANN ............................................................................................................. 24

Figure 2.11 Example of multi-sensor approach .................................................................................. 25

Figure 2.12 Coordinate systems of example 1 ...................................................................................... 28

Figure 2.13 Magnetic field distribution .............................................................................................. 28

Figure 2.14 Sensing performance in unit sphere ................................................................................. 29

Figure 2.15 Sensing performance in a unit sphere according to Ls ..................................................... 30

Figure 2.16 Sensing performance in a unit sphere according to sensor’s orientation ......................... 31

Figure 2.17 Bijection in orientation domain ....................................................................................... 32

Figure 2.18 Orientation and MFD domain .......................................................................................... 34

Figure 2.19 Magnitude of G according to and ............................................................................... 35

Figure 2.20 Measurement error according to rc by using ANN in simulation ....................................... 36

Figure 2.21 Experimental setup for example 1 ..................................................................................... 38

Figure 2.22 Calibration of sensor’s measuring points .......................................................................... 39

Figure 2.23 Goniometer stage ............................................................................................................. 40

Figure 2.24 Adjustment range of Goniometer stage ........................................................................... 40

Figure 2.25 Weight and bias of each axis according to the number of actual measurements ............. 42

Figure 2.26 MFD calibration of 625 data applying weight and bias with Ns=10 .................................. 44

Figure 2.27 MFD calibration of 625 data applying weight and bias with Ns=20 .................................. 45

Figure 2.28 Measurement error according to rc by using ANN in experiment ..................................... 46

Figure 2.29 Coordinate systems of example 2 .................................................................................... 48

Figure 2.30 Region expansion using a closed square ......................................................................... 49

3

Figure 2.31 Region expansion using multi-sensor with switching conditions .................................... 51

Figure 2.32 Experiment setup for example 2 ........................................................................................ 52

Figure 2.33 MFD calibration of 81 data applying weight and bias of each sensors ........................... 53

Figure 2.34 MFD calibration error of 81 data applying weight and bias of each sensors .................. 54

Figure 2.35 Experiment measurement error of example 2 .................................................................... 55

4

LIST OF TABLES

Table 2.1 Parameters of measurement system and DMP .................................................................... 28

Table 2.2 Hall-effect Sensor specifications ......................................................................................... 38

Table 2.3 Calibration results of the sensor’s measuring points in example 1 ..................................... 39

Table 2.4 Scale mark of front indicating α and β orientation angles in Goniometer stage ................. 41

Table 2.5 Weight, bias and calibration error of each axis ................................................................... 43

Table 2.6 Numerical value of experiment measurement error ............................................................ 47

Table 2.7 Rotation angles of four sensors ........................................................................................... 48

Table 2.8 Expanded range of closed square according to the number of sensors ............................... 49

Table 2.9 The Number of sensor with desired accuracy ..................................................................... 50

Table 2.10 Calibration results of multi-sensor’s measuring points in example 2 ................................. 52

Table 2.11 Weight, bias and calibration error of each axis for multi-sensor ....................................... 53

Table 2.12 Numerical error of experiment in example 2 .................................................................... 55

5

NOMENCLATURE

Upper Case Descriptions

B MFD vector

BC1, sp MFD vector satisfying condition 1 in sphere shape of measurable MFD-

field

BC1, sq MFD vector satisfying condition 1 in square shape of measurable MFD-

field

BC2 MFD vector satisfying condition 2

BDMP MFD vector computed by DMP method

Bvar MFD variation model

B max Radius of outer in sphere shape of measurable MFD-field

B min Radius of inner in sphere shape of measurable MFD-field

Bi,DMP MFD component on the i axis computed by the DMP

Ei,B Error of MFD between the actual measurements and the simulated MFD on

the i axis

G Square root of a sum of Jacobian matrix’s components

J Jacobian matrix

Lp Distance from the origin of global coordinate system to a center of PM

LS Distance from the origin of global coordinate system to a center of sensor

Ns The number of actual measurements

Pi Initial sensor position

Pji Position vector of ith dipole in the jth loop dipole

Ps Sensor position after rotating initial sensor position by s and s

Q Domain of MFD

Rji±- Distance vectors of source and sink from ith dipole in the jth loop to a sensor

position

Lower Case Descriptions

a1, a2 Weight and bias of each axis for LS

ac One side length of closed square

aji± Radius of DMP’s sink and source

h The number of layer in ANN

6

k Index of loop for DMP

l Length of PM

l Length of circular loops in DMP

m2 The number of sensors in use

mj Strengh of jthcircular loop in DMP

n Index of pole in a loop

nb Constant bias of field variation

ni The number of neurons in ith layer

nr Constant of relative percentages in field variation

q Orientation vector in terms of X1Y2Z3 Euler angles

qc Orientation vector of expanded range of closed square

qm Modified desired motion region

qu Orientation vector in unit sphere

q̂ Estimated orientation vector by ANN

sq Sensor’s orientation vector

rc Radius of approximated circle

Greek Descriptions

, , γ X1Y2Z3 Euler angles

s, s Rotation angles of sensor position about the X axis and y’ axis

μ0 Free space permeability

Г XYZ rotational coordinate transformation matrix

Ω Domain of orientation

ΩSi Motion region to measure using ith sensor in Ω

λ Area where the unique solution exists in orientation domain

Λ Area where the unique solution exists in orientation domain for numerical

implementation

ε A value bigger than zero

εi Boundary condition value of ith sensor

εi,j Boundary condition value of ith sensor on j axis

η Sensing performance

ηu Sensing performance using unit sphere

7

Abbreviation Descriptions

DMP Distributed Multi-Pole model

DOF Degree of Freedom

EM(s) Electromagnet(s)

LS Least Squares

LUT Lookup Tables

MFD Magnetic Flux Density

MIMO Multiple Inputs and Outputs

PM(s) Permanent Magnet(s)

PMSM Permanent Magnet Spherical Motor

RMSE Root Mean-Squared Error

8

CHAPTER 1

INTRODUCTION

1.1 Background and Motivations

Recently, various objects having degree of freedom (multi-DOF) motion have been appeared with

various purposes such as joystick, multi-DOF actuator and multi-DOF motion platform [1-5] and [8].

A joystick is an input device consisting of a stick that pivots on a base and reports its angle or direction

to the device [1-2]. It is used to control video games and different types of machines in industry such

as fork lift trucks. According to the different purposes of the joysticks, they can be divided into two

types, digital and analog joystick. As digital joystick has only four switches for four directions (up,

down, left, right), it cannot implement sensitive control. According to this fact, analog joysticks lately

replaced digital ones to improve the control accuracy.

A permanent magnet spherical motor (PMSM), which is capable of three DOF motion with

continuous and smooth in one joint, is typical example of multi-DOF actuator and have been widely

studied [3-5]. It has great potential applications in a variety of area such as the robotic wrist, propellers

for boat and UAV, camera actuators, omnidirectional wheels [6] and so forth. Open-loop control of

PMSM have been published to control the spherical motor [7] and an orientation of rotor has to be

sensed to estimate torque and to improve the motion accuracy by applying closed-loop controller.

Motion platform have been appeared in popularity because it has the advantage of capability to

control multi-DOF motion [8]. Motion simulation with motion platform can be applied in variety of

area such as ride simulation [9], flight simulators [10], medical manipulators [11] and so forth. A

singularity in motion control and a closed-loop control system have been still investigated for better

motion control accurately and smoothly.

These applications have been widely studied with their different purposes to apply industry.

Orientation measurement system is required to improve orientation control by sensing the orientation

of an object.

9

(a) Gimbaled mount type joystick [2] (b) Multi-DOF actuator [5]

(c) Multi-DOF motion platform [8]

Figure 1.1 Various objects having multi-DOF motion

1.2 Prior and Related Works

A number of orientation measurement system for a closed-loop controller of the spherical motor have

been proposed using a variety of methods and techniques. In [12], contact sensing systems have been

developed, which has fast response and high accuracy as shown in Figure 1.2 (a). However, it needs to

design additional extra mechanics, increasing friction and additional moment inertia and resulting in

dynamical imbalance to the system. To overcome these difficulties, non-contact sensing systems such

as optical [13] and vision-base [14] have been developed. However, a specially manufactured surface

needs for using these systems and it requires line-of-sight as shown in Figure 1.2 (b). Recently as the

magnetic field is invariant to environmental factors such as pressure, temperature and can be measured

instantaneously with low costs and footprint size, many methods using magnetic field measurements

from a magnet installed in addition or built in rotor have been developed by several researchers.

10

(a) Contact system [12] (b) Vision based system [14]

Figure 1.2 Orientation measurement system developed previously

Joysticks have been developed with similar trend. As conventional digital joysticks do not have

sensitive control, analog joysticks lately appeared to overcome this difficulty. Analog joysticks using

potentiometer for 2 axis rotation are used. A potentiometer is to translate the stick’s physical position

into an electrical signal. However, it has limitations of durability and reliability due to the wearing of

moving parts. Analog joysticks using three-axis magnetic sensor can extend lifetime of joystick

functionality and more options for future games or industrial implementations of joystick are possible.

However, due to the complexities of the relation between magnetic field and position/orientation, it

is challenging to calculate the position/orientation from magnetic field in closed-form. In order to apply

to human medical applications, a linear algorithm have been proposed for a magnet’s location and

orientation using three-axis magnetic sensors [15]. Lee and Son developed a method to obtain

orientation from magnetic field inversely by using combination of sine and cosine function [16]. As

relation between magnetic field and orientation has periodic patterns, function that defines the relation

can be represented by a combination of sine and cosine. However, it has a measurement error because

a magnetic field, which is desired value for their method, always has uncertainty for several reasons

such as noise. Another approach is using artificial neural network (ANN). ANN have been used to solve

the inverse problem as it can fit any practical function. Son and Guo have been proposed utilizing the

magnetic-field measurements of the moving rotor permanent magnets (PMs) and ANN [17]. Lee and

Foong developed a method utilizing the periodic field of PMs and ANN as a direct mapping for

orientation determination [18] and [19]. In [15-19], the measurement system consists of many magnets

and Hall-effect sensors.

In position/orientation measurement system based on magnetic field, another critical issue is one-to-

one mapping between orientation of PM and magnetic field (=bijective relation). It means the

orientation-field correspondence has to be unique to solve the inverse problem. The method to analyze

the bijection relation between the magnetic fields and the position/orientation of motions systems has

been widely studied by several researchers [19] and [20]. In [16], Lee and Son developed a method to

11

compute inverse function in surjective relation by using combination of sine and cosine function.

1.3 Research Objective

The objective of this research is to develop an orientation measurement system using magnetic field

of PM and magnetic sensors such as Hall-effect sensor. Using magnetic field of a PM, the orientation

of an object embedded with this magnetic sources can be tracked in real-time.

As PM moves, a static magnetic field is established in the motion region and can be calculated by

DMP method. Compared to previous method such as ANSYS or analysis, DMP can provide a fast and

closed-form solution to calculate the magnetic field. Using DMP method, issues related this system are

studied to make system effective. ANN is applied to obtain orientation from measured magnetic field.

ANN requires data for training the relation between input (=magnetic field) and output (=orientation).

Data for training is prepared in advance using DMP and calibration of magnetic field are implemented

with some actual measurements. As a result, this system can prepare ANN with fast calibration using

DMP method even if there is an uncertainty in magnetic field.

According to various purposes, a sensing system requires desired accuracy for measuring orientation.

Considering the issues, the system is simplified and can avoid installing more sensors than necessary.

1.4 Thesis Outline

The remainder of the thesis is outlined as follows.

Chapter 2 presents a two-DOF orientation measurement system using a cylindrical PM and Hall-

effect sensors. Algorithms to obtain orientation from magnetic field with artificial neural network is

introduced and issues related to this system are studied. Two examples are introduced to provide better

understanding how proposed measurement system considering issues can measure orientation in

numerical simulation and experiment. Example 1 consists of single sensor and it is identified how much

accuracy it can be measured in a motion region. Based on the result of example 1 using single sensor,

it is identified that the orientation can be measured accurately using multi-sensor in example 2.

Finally, Chapter 3 summarizes this thesis and a conclusion is presented. The future research about

orientation measurement system based on the magnetic field is proposed.

12

CHAPTER 2

TWO-DOF ORIENTATION MEASUREMENT SYSTEM

2.1 Overview

This chapter presents to develop a measurement system to obtain two orientation angles of an object

using magnetic field and artificial neural network (ANN). A cylindrical permanent magnet (PM) is used

as excitation source and a static magnetic field generated by a PM. Orientation of an object embedded

with PM can be tracked in real-time by using the magnetic field.

Orientation tracing method based on magnetic field has several advantages such as contact-free

measurement without any extra mechanics and no requirement of line-of-sight problem. In addition,

magnetic field can be measured across nonferromagnetic mediums including air and the human body,

and it is invariant to environmental factors such as pressure and temperature. Recently as the

improvements in manufacturing technology, a magnetic sensor (Hall-effect sensor is used in this system)

can measure three dimensional magnetic field with low costs, footprint size, a wide range and high

resolution.

However, due to the complexities of the relation between magnetic field and orientation, it is

challenging to calculate the orientation from magnetic field in closed-form (=inverse problem). ANN

with simulated magnetic field have been used to solve the inverse problem as it can fit any practical

function. However, actually measured magnetic field and simulated magnetic field are not always same

for several reasons such as noise, production error of PM, measurement error, experiment setup error

etc. In order to match the simulated magnetic field to actual magnetic field, a calibration has to

implemented.

In orientation measurement system based on magnetic field, there are some issues: one-to-one

mapping between orientation and magnetic field (=bijective relation), optimized sensor’s position and

orientation, measurement accuracy, multi-sensor approach and calibration of magnetic field between

simulation and actual measurements. Bijective relation means the orientation-magnetic field

correspondence has to be unique to solve the inverse problem. The sensor’s position and orientation

have to be considered as it can affect the system performance. In order to expand a sensing region and

get a high resolution, the multi-sensor approach is required.

The remainder of the chapter is organized as followings:

1) Magnetic flux density (MFD) according to the orientation is calculated by DMP (distributed

13

multi-pole model) method. This method can provide a fast and closed-form solution comparing

other methods such as ANSYS or analysis. The MFD calculated by DMP is used to study the

issues related to this system.

2) The issues related to this system are studied: one-to-one mapping between orientation and

magnetic field, sensor’s position and orientation, measurement accuracy, multi-sensor approach

and calibration of magnetic field between simulation and actual measurements.

3) The algorithm using ANN to measure orientation from MFD is proposed. Two examples are

introduced to validate the proposed method can measure orientation in numerical simulation and

experiment. Example 1 consists of a PM and single sensor. Using only single sensor, it is

identified how much the orientation can be measured in a motion region. Example 2 consists of

a PM and multi-sensor. Based on the result of example 1 using single sensor, the multi-sensor

approach is applied to expand a sensing region and reduce measurement error.

2.2 Orientation Measurement System based on Magnetic Field and Artificial Neural Network

A PM and an electromagnet(EM) are generally used as excitation sources to generate magnetic field

in a position/orientation measurement system. Magnetic field variation according to motions of an

interested object is analyzed to get the position or orientation inversely as magnetic field is varying with

orientation of magnetic sources. Unlike PM generating the field without any power source, the EM

requires and controls currents to detect the motion of object by using phase and magnitude differences

between the excitation signal and measured signal. Thus, the PM could be more attractive excitation

source and has been used in measurement system to generate magnetic field.

X1Y2Z3 Euler angles as shown in Figure 2.1 are used to define the orientation of a PM in global

coordinate system. XYZ is the global coordinate system, the PM rotates along X axis of global

coordinate system for a first, and then rotates along the new Y1 for a , and finally rotates along the

new Z2 for a γ.

14

Figure 2.1 X1Y2Z3 Euler angles

Figure 2.2 shows a flowchart of orientation estimation based on a magnetic field and ANN. First,

MFD in a motion region is analyzed by DMP method in [4], [16] (this method will be further presented

in Section 2.2.1). The calibration of MFD is implemented to match the simulated MFD to actual

measurement and the ANN is trained to calculate the orientation of the PM with the calibrated MFD.

The trained ANN can be applied to solve the inverse problem.

2.2.1 Magnetic Field of Permanent Magnet Based on DMP

Figure 2.3 shows the PM-based orientation measurement system consisting of a cylindrical PM with

single three-dimensional magnetic sensor and the coordinate systems.

MFD

(by 3-axis Hall sensor)Neural Network

(Trained)Orientation

Neural Network

(Initail)

Calibration

of MFD

DMP method

MFD

(by DMP method)

Tˆ [ ] qTP [ ]x y zB B BB

DMP,caliB

Forward prediction

Target

OrientationT[ ] qT

DMP [ ]x y zB B BB

Figure 2.2. Flowchart of orientation estimation

15

Figure 2.3 Coordinate systems

XYZ is the global coordinate system, xyz is the local coordinate system (=X2Y2Z2 Euler angles in

Figure 2.1) and x′y′z′ is the intermediate coordinate system (=X1Y1Z1 Euler angles in Figure 2.1). The

PM is installed at the end of a rod with a constant length Lp from the origin. Similarly, the initial sensor

position Pi = [0 0 Ls]T is located along the Z axis at LS as constant distance between a center of the

sensor and origin of global coordinate system. Ps is the sensor position after rotating Pi by s and s in

Eq (2.1).

cos sin

sin sin

cos

s s

s s s s

s

L

P (2.1)

where s and s.are the rotation angles of sensor position about the X axis and y’ axis.

The orientation of PM can be defined as , and γ, which are the rotation angles about the X axis, y′

axis and z axis. Due to the symmetry of the magnetization of PM along the z axis, only two Euler angles

( and in Figure 2.3) can be measured.

As PM moves, a static magnetic field is established in the motion region. The DMP method is utilized

to analyze magnetic field to calculate orientation. This method offers a closed-form of the field

distribution since it consists of a set of magnetic dipoles as shown in Figure 2.4.

16

Figure 2.4 DMP model for a cylindrical PM

The magnetic dipoles, consisting of magnetic sources and sinks, are located inside of the PM with a

number of circular loops whose length is l . Each jth circular loop has n dipoles parallel to the

magnetization vector at a radius aji+ and aji- of the source and the sink respectively. Rji+ and Rji- are a

distance vectors of the source and the sink from ith dipole in the jth loop to a sensor position PS in the

global coordinate system, where i = 1,2,…,n and j = 0,1,…,k.

The PM rotates by and , its static magnetic field (=magnetic flux density, MFD) is expressed in

the global coordinate system in Eq (2.2).

0

4 2 20 1

k nRji Rji

mj

j i R Rji ji

a aB where

2 3

S

S

Rji ji

Rji

ji

a P P

P P

. (2.2)

where μ0 is the permeability of free space; mj is the strengh of jth circular loop; Pji is the position vector

of ith dipole in the jth loop dipole in the global coordinate system after rotating by and .

After the moving PM by and , Pji± is described in Eq (2.3).

( )ji ji P Γ q p (2.3)

where

cos 0 sin

( ) sin sin cos sin cos

cos sin sin cos cos

Γ q is the coordinate transformation matrix from the

glocal coordinate system to the local coordinate system; q=[ ]T is orientation vector in terms of

X1Y2Z3 Euler angles; pji± is the initial position of each dipoles expressed in the global coordinate system

[16].

17

2.2.2 Analysis of Bijective Domain

The relation between orientation and MFD have to be a one-to-one mapping (=bijective relation) to

solve the inverse problem. This means a unique solution of orientation alwasys exist with measured

MFD. Figure 2.5 shows relation between orientation and MFD.

Figure 2.5 Relation between orientation and MFD

The MFD can be calculated with given posture by using theoretical/analytical field models

(=Forward prediction). The DMP method was used to implement forward prediction in this system. In

this chapter, the method which determines bijective relation between orientation and MFD is presented

based on the forward prediction.

Unlike forward prediction which solve the MFD with given posture, the inverse computation, which

solve the orientation from measured MFD, is much more challenging. ANN have been applied to solve

the inverse problem as it can fit any practical function. Both MFD and orientation must be uniquely

determined without singularity so as to inversely compute the orientation from MFD by using ANN.

MFD is a function of the orientiation of the PM in Eq (2.4):

( )fB q (2.4)

Bijecitve relation can be determined by using simulated MFD and orientation.The continuous domain

of orientation and MFD can be visually illustrated in Figure 2.6 [19].

18

Figure 2.6 Domain of orientation and MFD

As the continuous domain of MFD and orientation are descretized by DMP, the interested points to

be measured are scatterd in the space. Mathematically, it should satify the bijective relation requiring

two conditions in Eq (2.5) and (2.6).

Condition 1: ( ) ( )n Q n (2.5)

Condition 2: { } { }( )i i j j i j B B B B (2.6)

where Q and Ω are domain of MFD and orientation respectively in Figure 2.5; Bi and Bj are the ith and

jth MFD vector (such as [Bx Bz]T or [Bx By Bz]T);

Condition 1 means that the number of Q and Ω are same and condition 2 means the correspondence

between MFD and orientation should be unique to avoid singularity. This method able to confirm the

bijective relation visually. However, it cannot explain the relation exactly in concentrated points.

Bijecitve domain can be determined using another method. In [20] Lee and Bai used the Jacobian

matrix to find an area that satisfies the bijective relation. Jacobian matrix is defined in Eq (2.7).

[ / ],where 1,2,..., , 1,2,...,i jB q i N j M J (2.7)

where N and M equal to the demension of MFD and orientation respectively; i, j=1,2,3 represent x,y,z

axis and ,,γ respectively.

f is bijective if satisfying the condition in Eq (2.8).

{ det([ ]) 0} q J (2.8)

19

Zero determinant means there is no unique solution and λ is the area where the unique solution exists.

As the continuous domain of MFD and orientation are descretized by DMP, the bijective domain can

be determined numerically. For numerical implementation, another condition can be used in Eq (2.9).

{ det([ ]) } q J (2.9)

where ε is a value bigger than zero and is used to avoid error occurring in numerical approximations

If the dimention of MFD and orientation is same, it is easy to calculate the determinant of Jacobian

matrix because the Jacobian matrix is square matrix. However, if the dimention of MFD and orientation

is not same, the determinant cannot be calculated beacause the Jacobian matrix is not square matrix. In

this case, the puedo inverse is required. If the dimention of MFD is much than orientation, the left

inverse have to applied in Eq (2.10) and the determinant of the former term can be calculated as it is

square matrix. Similarly, the dimention of orientation is much than MFD, the right inverse have to

applied in Eq (2.11).

T 1 T( )J J J (2.10)

T T 1( )J JJ (2.11)

2.2.3 Investigation of Optimized Sensor’s Position and Orientation

The sensor’s position and orientation have to be considered in the position/orientation measurement

system based on the magnetic field as it can affect system performance. When the magnetic source is

attached to the rotating object and rotates with respect to the origin, the magnetic field is varying

nonlinearly and can be illustrated with certain pattern. The shape of the rotating mangetic field by

magnetic source located at the origin becomes an ellipsoid. However, the method in [21] using single

dipole model is only valid some shapes because single dipole only can be modeled for some shapes.

Son and Fang eliminated the previous constraint by modeling the magnetic source by using DMP

method in [22]. As DMP method used a set of magnetic dipoles, most magnetic source can be modeled

inclued a cylindrical PM.

The method in [22] approximates the rotating mangetic field shape of a cylindrical PM as ellipsoid.

When the PM is rotating at the orgin without certain distance offset (it defined LP), the rotating mangetic

field shape is almost ellipsoid as the maximum sum of the squared error between excited magnetic field

and approximated ellipsoid is about 10-14 T in an illustrative example. However, when the PM is rotating

with respect to the origin with certain distance offset (LP≠0), the rotating magnetic field shape is not a

20

ellipsoid. Thus, in case of LP≠0, the sum of the of the squared error between excited magnetic field and

approximated ellipsoid is bigger than the case of LP=0.

In this chapter, the optimized sensor’s position and orientation are investigated mathematically not

using approximated ellipsoid. Figure 2.7 shows rotating magnetic field shape when a cylindrical PM is

rotating with certain distance offset.

Figure 2.7 Rotating magnetic field shape in the case of LP≠0

where Ps is an arbitrary sensor position.

The rotating magnetic field shape (it called MFD-field in Figure 2.7) is like heart because the signal

is very week when the PM is far from the sensor. Measurable MFD-field is the range that a three axes

magnetic sensor can measure. The MFD-field is formed according to some parameters such as shape of

a PM, certain distance offset, orientation of PM and the position of sensor. The MFD-field can be formed

by orientation of PM and position of a sensor when assuming that the rest of parameters are determined

based on the system design.

(a) Sphere shape (b) Square shape

Figure 2.8 Two shape of measurable MFD-field

21

In the sensor position, the measurable MFD-field is sphere or square as shown in Figure 2.8. If a

magnetic sensor is isotropic for all axes, the shape of the measurable MFD-field becomes a hollow

sphere, the radius of outer and inner are the B max and B min respectively and B is the magnitude of

MFD 2 2 2( )X Y ZB B B . In the square shape of measurable MFD-field, the measured MFD of each

axis are independent. Unlike the sphere shape, the MFD has the maximum and minimum range of each

axis in square shape, B max,j and B min,j respectively. j=1,2,3 represent x,y,z axis respectively. Usually

B min,j is the resolution of sensor in use.

The MFD-field has to be in the measurable MFD-field to avoid saturation of sensor. In the sphere

shape of measurable MFD-field, the magnitude of MFD is smaller than maximum value and bigger than

the minimum value in Eq (2.12).

1, max min{ (B ) (B )}C sp i i iB B B B (2.12)

where Bi is the ith MFD vector; Bi is the magnitude of Bi.

In the square shape of measurable MFD-field, the absolute MFD is smaller than the maximum value

and bigger than the minimum value of each axis in Eq (2.13). The MFD satisfying this condition for all

axes is in Eq (2.14).

, , 1 , , max, , min,{ ( ) ( )}i j c i j i j j i j jB B B B B B (2.13)

T

, 1 ,1, 1 ,2, 1 ,3, 1[ ]sq C i c i c i cB B BB (2.14)

where j=1,2,3 represent x,y,z axis respectively; Bi,j is the magnitude of ith MFD vector on the j axis.

Eq (2.15) indicates the magnetic field variation according to desired angle resolution must be larger

than the resolution of magnetic sensor in use.

2 { }C i rS B B J (2.15)

where J is Jacobian matrix in Eq (2.7); Sr is the resolution of magnetic sensor.

The sensing performance can be defined as in Eq (2.16)

12( , ) Cs s

Af P

A q (2.16)

where qs is the orientation of sensor; A is all of the MFD-field; AC12 is MFD satisfying the both

22

conditions simultaneously in Eq (2.12)/ (2.14) and (2.15).

The optimized sensor’s position and orientation for the given magnetic sensor can be obtained by

maximizing the sensing performance in Eq (2.16). The sensing performance is determined based on the

sensor’s position and orientation.

Unit sphere as shown in Figure 2.9 is formed by two orientation angles qu={[qu1 qu2]T: -π ≤ qu1 ≤ π

radian and –π/2 ≤ qu2 ≤ π/2 radian}. It can be used to confirm the sensing performance visually instead

of Eq (2.16) if two constraints are satisfied in Eq (2.17) and (2.18).

Constraint 1: 2 1( )R q (2.17)

Constraint 2: uq q (2.18)

The unit sphere can be formed by rotating the unit vector by qu1 and qu2 about the X axis and y’ axis.

Constraint 1 means the unit sphere is formed by using two orientation angles with unique vectors.

Constraint 2 means the two interested orientation angles are in the range qu. It means the all possible

motion region in which orientation can move is the surface of unit sphere. Then the sensing performance

using the unit sphere is defined newly in Eq (2.19)

Figure 2.9 Unit sphere indicating sensing performance

23

12C uu

u

A

A (2.19)

Where Au is the surface of unit sphere; AC12u is the area in unit sphere satisfying the both conditions

simultaneously in Eq (2.12)/ (2.14) and (2.15).

2.2.4 Measurement Accuracy based on Jacobian matrix

The gradient of MFD with respect to orientation affects measurement performance and should be

considered to reduce the measuremnt error. In general, the bigger gradient means higher accuracy in

measurement and results in higher sensitivity. The measured MFD, the input of ANN, has always

measuremet error due to several reason such as noise, production error of PM, measurement error and

experiment setup error etc. If the gradient of MFD with respect to orientation is large, system has high

accuracy when solving the orientation from measured MFD with uncertainty due to the high sensitivity.

However, as the gradient increases, the satisfied region decreases simultaneously. In other words, there

is a tradeoff relationship between gradient and region. The components of Jacobian matrix in (2.7) are

the gradient and square root of a sum of its components expressed in Eq (2.20).

2 2

1 1

( , )N M

i j

G i j

J (2.20)

where, J(i, j) is ith row and jth column components of the Jacobian matrix;

2.2.5 Inverse Computation by using Artificial Neural Network

Once the motion region, which has a unique correspondence between MFD and orientation, is

determined, a method to get orientaion from MFD is required. The simple solution is to linearize Eq

(2.1) and to obtain orientation from MFD inversly. However it is difficult to linearize Eq (2.1) because

the MFD depending on orientation is highly nonlinear and has uncertainties. The ANN is applied to

obtain orientation from the MFD as it can compensate the difficulties.

In our brain, many brain cells (called neurons) form a network and a neuron is conneted to multiple

neurons. The ANN is a mathematical model that mimics this brain. Figure 2.10 shows structure of ANN.

24

where, N1 and N2 are the number of input and output respectively; h is the number of hidden layer; nh

is the number of neurons for hth hidden layer.

The ANN consists of a lot of neurons and connections between neurons are determined by using

weights, bias and transfer function. The ANN with initial weight and bias is calculate the outputs given

inputs and compare result with desired target and then adjust the weight and bias minimizing the

difference between outputs of ANN and desried target. In particular, Levenberg-Marquardt back-

propagation algorithm has been used to adjust weights and bias. Among the data required for training

ANN, 90% of the data are used for training, 5% for validation and 5% for testing. The ANN can be

mathematically represented as in Eq (2.21).

where ANNh,n consists of h hidden layers and n neuron per hidden layer; q̂ is the estimated orientation

by ANNh,n.

The root mean-squared error (RMSE) is the cost function that try to minimize the square root of the

squared error between the outputs of ANN and the desired target in Eq (2.22).

2ˆRMSE ( ) q q (2.22)

The training of ANN is continued when RMSE has a value less than a certain value. The orientation

measurement system is multiple inputs and outputs (MIMO) system. Fortunately, ANN is more suitable

for applying to MIMO system than least squares(LS) and lookup tables(LUT) [19].

Figure 2.10 Structure of ANN

,ˆ ANN ( )

h nq B (2.21)

25

2.2.6 Multi-Sensor Approach

The orientation measuremnet system with single sensor has short sensing region because the signal

becomes weak as the PM moves farther from the sensor. The MFD can not be measured if MFD is

lower than minimum measurable-MFD field of sensor or lower than the sensor resolution (it means the

condition 1 and 2 as mentioned in chapter 2.2.3). The multi-sensor approach is required to expand a

sensing region and get high accuracy. The sensor having individual bijective domain can be connected

to each other. The switching condition is required to distinguish which sensor is used. It can be

illustrated with a simple example. Figure 2.11 shows the example: the orientation measurement system

consists of a PM and two sensors..

(a) Coordinate systems (b) Motion region to measure with each sensor

in orientation domain

Figure 2.11 Example of multi-sensor approach

where Ps1 and Ps2 are the position of sensor 1 and 2; the red and blue dash line are a motion region to

measure with MFD of sensor 1 (=ΩS1) and sensor 2 (=ΩS2) respectively.

Figure 2.8 (b) shows motion region to measure with each sensor in the continuous orientation domain.

In order to selsect the sensor to be used on, the switching condition can be applied and is defined using

MFD.

1 1, 1,

2 2, 2,

: , 1,2,3

: , 1,2,3

S j j

S j j

B j

B j

(2.23)

where j=1,2,3 represent x,y,z axis respectively; B1,j is MFD component of sensor 1 on the j axis; B2,j is

about the sensor 2; ε1,j is the boundary condition value of sensor 1 on j axis; ε2,j is about sensor2.

26

When the motion region of each sensor is connected to each other, overlapped areas exist and there

are many solutions in this area. However every solutions can be applied to obtain orientation from MFD

of each sensor since they are satisfied with bijective relation respectively. In the overlapped areas, the

sensor having a biggest gradient of B with respect to q is used because it has higher accuracy.(as

mentioned chapter 2.2.4)

2.2.7 Calibration of Magnetic Field

The actual MFD measured by magnetic sensor is differ from the simulated MFD due to several

reasons such as noise, production error of PM, measurement error, experiment setup error etc. Therefore

the calibration of MFD has to implimented to match the simulated MFD to the actual measurements

before training ANN with simulated MFD data. In order to match the simulated MFD data made by the

DMP method to actual measurements, the least square (LS) is applied. The calibration error using LS

is defined in Eq (2.24).

where Ei,B is the error of MFD between the actual measurements and the simulated MFD on the i axis;

Bi,S is the actual measurement by magnetic sensor on the i axis; Bi,DMP is the MFD on the i axis computed

by the DMP; ai,1 and ai,2 are the bias and weight of Bi,DMP; Ns is the number of actual measurements.

The weight and bias of each axis, which minimize the calibration error, are determined by using

derivative of Eq (2.24). It requires one or more actual measurements that is not zero value in each axis.

The optimized number of actual measurements is determined by considering the difference of the

weight and bias according to the number of actual measurements. It is possible to avoid unnecessary

actual measurements and can be reducing efforts to measure a number of actual measurements for

training.

For practical simulation, the magnetic field variation and uncertainties are included in the simulated

MFD in Eq (2.25).

var DMP (1 )r bn n B B (2.25)

where nr is a constant of relative percentages in field variation and nb is a constant bias of field variation;

BDMP1 1

( )N

R

is the MFD vector computed by DMP method and Bvar1 1

( )N

R

is the MFD variation by

applying nr and nb to BDMP.

2

, , ,2 ,DMP ,1

1

( ) , 1,2,...,sN

i B i S i i i

j

E B a B a where i N

(2.24)

27

In numerical simulation, the MFD variation model in Eq (2.25) is applied to identify the measurement

accuracy according to a motion region.

2.3 Numerical Simulation and Experiment

The Numerical simulation and experiment are executed to evaluate the measurement performance of

proposed method. It provides a better understanding how the proposed measurement system considering

issues can measure orientation and it is illustrated with the following two examples.

Example 1 is a two-DOF measurement system consisting of a cylindrical PM with single Hall-effect

sensor. PM is rotating with certain distance offset. Example 2 is a two-DOF measurement system

consisting of a cylindrical PM with multi-sensor. Based on the result of example 1 using single sensor,

the multi-sensor approach is applied to expand a sensing region and improve the measurement accuracy.

In both simulation and experiment, it is identified how much accuracy it can be measured in a motion

region considering the issues based on magnetic field and ANN.

2.3.1 Example 1: Two-DOF Orientation Measurement System with Single Sensor

Magnetic field of PM based on DMP

A cylindrical PM is rotating by and with a constant offset Lp. The DMP method is numerically

simulated to compute MFD according to a rotation of the PM as shown in Figure. 2.12. Due to the

symmetry of the magnetization of the PM along the z axis, only two Euler angles ( and in Figure

2.12) can be measured. The range of Euler angles is set as q={[ ]T: -14 ≤ and ≤ 14 degree} and

dicretized with 1 degree interval. The total number of discretization is 841(=29×29) pairs by using the

DMP method. The single sensor is located in the initial sensor position Pi = [0 0 Ls]T. Figure 2.13 shows

the magnetic field distribution Bx, By, Bz, and 2 2 2B( )X Y ZB B B with respect to the orientation (

and ). The detailed parameters of the DMP method and the measurement system, which is used in

simulation and experiment, is summarized in Table 2.1.

28

Figure 2.12 Coordinate systems of example 1

Figure 2.13 Magnetic field distribution

Table 2.1 Parameters of measurement system and DMP

PM specifications and sensor/PM location

a = 6.5mm, l = 8mm, M = 1.19T, LP = 72mm, LS = 83mm

DMP parameters

2

8, 2, 2.8 mm

m 0.2354, 0.0478, 0.5015 T / m 1.0e 4 ( 0,1,2)j

n k l

j

29

Investigation of optimized sensor’s position and orientation

The Hall-effect sensor, the measurable MFD-field shape is square, was used in this system. The

absolute maximum and minimum range of MFD on the each axis are same as

max,1 max,2 max,3 130mTB B B and min,1 min,2 min,3 0.098mTB B B respectively as the Hall-effect

sensor of which the magnetic linear range is ±130mT with 0.098mT resolution was chosen for the

experiment.

The unit sphere in Figure 2.9 can be applied in this system to identify the sensing performance

because the two constraints in Eq (2.17) and (2.18) are satisfied. Figure 2.14 shows result of the sensing

performance using a unit sphere when the sensor position Pi is set as [0 0 Ls=83]T. The area satisfying

the both conditions simultaneously in Eq (2.14) and (2.15), AC12u, is in the unit sphere. The colored area

is the area satisfying the both conditions and black sphere is the unit sphere. The sensing performance

using unit sphere ηu is 3.2185%.

Figure 2.14 Sensing performance in unit sphere

30

(a) Ls = 81 (b) Ls = 82

(c) Ls = 83 (d) Ls = 84

3.2017% ( 81)

3.2017% ( 82)

3.2185% ( 83)

3.2152% ( 84)

3.2093% ( 85)

u s

u s

u s

u s

u s

L

L

L

L

L

(e) Ls = 85 (f) Sensing performance using unit sphere

Figure 2.15 Sensing performance in a unit sphere according to Ls

Figure 2.15 shows sensing performance in a unit sphere according to Ls. The area AC12u is in

orientation domain and the range of Ls is set as 80~85mm (Figure 2.15 (a) to (e)). In Figure 2.15 (a) and

(b), the hollow area is appeared in the center of AC12u because the close distance between PM and sensor

is occurring the sensor saturation. The sensing performance according to Ls is shown in Figure 2.15 (f).

The maximum sensing performance is appeared in Ls=83 because the sensor position not occurring the

sensor saturation is Ls=83 and the portion where the signal is weak increases in the Ls=84 or more. As

a result, the optimized sensor position is Ls=83 maximizing the sensing performance not considering

the sensor’s orientation

31

(a) T[45 45]s q (b)

T[ 45 45]s q

(c) T[45 45]s q (d)

T[ 45 45]s q

Figure 2.16 The sensing performance in a unit sphere according to sensor’s orientation

Figure 2.16 shows sensing performance in a unit sphere according to the sensor’s orientation

T[ ]s s s q . The area AC12u is in orientation domain and the sq is set as four cases (Figure 2.16 (a)

to (d)). The shape is different and it has symmetric shape each other however sensing performance is

same when the sensor rotates according to sq . It means that the sensor’s orientation is not an important

factor affecting the sensing performance. In the measurement system of example 1, the sensor’s

orientation is set as sq = [0 0]T. It means he sensor coordinate system is equal to the global coordinate

system and it has the advantage of easy installation of the Hall-effect sensor in the experiment.

Analysis of bijective domain

Figure 2.17 shows the bijective domain using Jacobian matrix according to MFD components in

orientation domain ((a) is using [Bx Bz]T, (b) is using [By Bz]T, (c) is using [Bx By]T, (d) is using [Bx By

Bz]T and [Bx By B]T).

32

(a) Using [Bx Bz]T

(b) Using [By Bz]T

(c) Using [Bx By]T

33

(d) Using [Bx By Bz]T and [Bx By B]T

Figure 2.17 Bijection in orientation domain

In this system, the orientation domain is a plane because only two orientation angles can be measured.

The Jacobian matrix in Eq (2.7) is square matrix when a two MFD components are used. The solid line

in Figure 2.17 is a contour of the determinant with its value. The area between zero determinants another

is the bijective domain. Since there are two unknown orientation components (q=[ ]T), it requires at

least two MFD components. However the area, which is satisfying the bijective relation using two MFD

components, has singularity in = 0 ([Bx Bz]T) or = 0 ([By Bz]T) and there is a contour of zero

determinant in the region q={[ ]T: -6 ≤ and ≤ 6 degree} ([Bx By]T). In the case using three MFD

components ([Bx By Bz]T or [Bx By B]T in Figure. 2.17 (d)), the Jacobian matrix is non-square matrix as

the dimension of B is bigger than q. The left inverse is considered and the determinant of (JTJ) can be

calculated. The area which is using three MFD components has a simple circle shape of bijective domain

than using two MFD components and has no singularity in the circle. B (=[Bx By B]T) are used for the

ANN since the area satisfying the bijective relation between q and [Bx By B]T is larger than [Bx By Bz]T

as shown in Figure 2.17 (d).

Figure 2.18 shows the MFD domain of B (=[Bx By B]T) and the narrow parts with orienatation domain.

Two conditions in Eq (2.5) and (2.6) are satisfied since B and q are uniquely determined without overlap

as shown in Figure 2.18 (a). However, the results show B is not uniformly distributed unlike the q. As

the PM moves around the conner of the square motion region in the orientation domain, the MFD

becomes weak and the sensor can not measure the MFD in practice since a number of measurement of

MFD are concentrated around the origin. The difficulty in measurement can be overcomed by

considering the error according to the gradient in a motion region.

34

(a) MFD domain of B

(b) Narrow parts in MFD domain

Figure 2.18 Orientation and MFD domain

Measurement accuracy based on Jacobian matrix

The Jacobian matrix consisting of gradient of B(=[Bx By B]T) with respect to q and the square root

of a sum of its components are expressed in Eq (2.26) and (2.27). As the dimension of B is bigger

than q, the Jacobian matrix is non-square matrix.

35

/ /

/ /

B / B /

x x

y y

B B

B B

J (2.26)

3 22 2

1 1

( , )i j

G i j

J (2.27)

Figure 2.19 shows magnitude of G according to and . Each color circle line is the contour of the

same gradient and the value is the magnitude of G. The region inside the contour has higher gradient

than that of the magnitude of G. The gradient in origin is the largest and as the distance from the origin

increases, the gradient becomes small. As the gradient increases, the satisfied region decreases

simultaneously as shown in Figure 2.19. The region with some or more magnitude of G can be

approximated in the closed circle. The boundary condition is the radius (=magnitude of G) of outer

circle.

Figure 2.19 Magnitude of G according to and

Measurement result by using ANN in simulation

The ANN for q is numerically simulated from B made by DMP method. For practical simulation, the

MFD variation model in Eq (2.25), Bvar3 1( )R , is applied to ANN to simulate the accuracy of

measurement system in the approximated closed circle. The nr is set as two value (nr = +5%, +10%)

36

respectively and nb is set as only 0.1mT. The initial number of data is 841 pairs and the orientation is

set as the square region (q={[ ]T: -14 ≤ and ≤ 14 degree}). As the interested region is closed

circle, the edge of region does not have to be taken into account. Therefore, modified total number of

data in closed circle for training the ANN is 625 pairs. Two layers with seven neurons is applied in the

ANN (h=2, n=n1=n2=7) and the transfer function of each layers is given in Eq (2.28).

1 2( ) ( )

n ne ef n f n n ne e

(2.28)

where n is the number of neurons in ANN.

(a) n = +5%

37

(b) n = +10%

Figure 2.20 Measurement error according to rc by using ANN in simulation

Figure 2.20 shows simulated result using ANN with two MFD variation model. The maximum and

mean error of and in each cloesd circle, which is approximated considering gradient in Figure 2.19,

were shown in Figure 2.20, where rc2 2( ) is the radius of each circle. The error tends to

converge to 0 when rc decreases and the error in origin is the smallest since it has the maximum gradient.

The simulation result shows that the measurement is more accurate in the large gradient.

Calibration for experiment

Figure 2.21 shows experimental setup of example 1 consisting of a cylindrical PM and single three-

dimensional Hall-effect sensor (TLV493d). The Hall-effect sensor of which the magnetic linear range

is ±130mT was chosen for the experiment. The Goniometer stage, which realizes rotation of α and β,

was used as a reference. XYZ stage was used to set the position of sensor in a desired place.

38

Figure 2.21 Experimental setup for example 1

The Hall-effect sensor communicates MFD data using i2c. The resolution of 12-bit readout is 98uT

and the sensor specifications related to the MFD are summarized in Table 2.2.

Table 2.2 Hall-effect Sensor specifications

Magnetic linear range ±130mT

Resolution 12-bit readout 98uT/LSB

Magnetic noise 0.1mT

Resolution drift ±20%

Offset drift -1 ~ +1mT

Due to manufacturing errors and imperfection of experimental setup, the positions of the actual

measuring point and the designed measuring point can differ. Considering this issue, the measuring

points of sensor are calibrated by using XYZ stage shown in Figure 2.22 (a) before applying the Hall-

effect sensor to the measurement system. The XYZ stage can move in 3 DOF with ranges of 20+mm in

two planar directions and 10mm in a vertical direction with 10μm travel resolution. Due to the symmetry

of the magnetization of the PM along the z-axis, the actual position of the measuring points can be

found by moving stage. It is possible to find the center of yz plane by looking for that the value of By

and Bz are zero simultaneously as shown in Figure 2.22 (b). Similarly, the center of xz plane can be

found by looking for that the value of Bx are Bz are zero simultaneously as shown in Figure 2.22 (c).

Table 2.3 summarizes the calibration results of the sensor’s measuring points.

39

(a) Setup for the calibration of sensor’s measuring point

(b) Magnetic field and yz plane of sensor (c) Magnetic field and xz plane of sensor

Figure 2.22 Calibration of sensor’s measuring points

Table 2.3 Calibration results of the sensor’s measuring points in example 1 (unit: mm)

Position of Measuring points (desired + discrepancy)

X axis Y axis Z axis

0 + 1.015 0 + 0.31 0 - 0.34

The Goniometer stage, TP 65-W30-W40, was used as a reference of orientation as shown in Figure

2.23 [23]. It can rotate orientation angles with 200mdegree travel resolution and the center of rotation

is located 20mm above the mounting surface. The Goniometer enables a precise adjustment with the

help of precision worm gears and adjustable dovetail guide. However, the actual adjustment by worm

gears can differ from a defined scale mark due to a gear elasticity and manufacturing error. Considering

this issue, the adjustments of Goniometer stage was calibrated to find actual adjustment.

40

Figure 2.23 Goniometer stage

(a) α adjustment range (b) β adjustment range

Figure 2.24 Adjustment range of Goniometer stage

Maximum adjustment ranges of each stage are ±15° and ±20° respectively indicated in product

information as shown in Figure 2.24. However, actually it can move ±17° and ±23° respectively. When

each stages moved from minimum to maximum ranges, the scale mark of front changes. In α adjustment

range, the adjustment range per rev of warm gear is 1.6°. The 21 rev + 0.4° is required to move α stage

from minimum to maximum range (1.6°×21+0.4°=34°). However, the 20 rev + 1° is required in the

actual result (1.6°×20+1°=33°). It means the actual unit degree is 33°/34° = 0.9706°. Similarly, the β

adjustment also calibrated. In β adjustment range, the adjustment range per rev of warm gear is 2.8°.

The actual unit degree is 44.8°/46° = 0.9739° because the difference is (2.8°×16+1.2°=46°) and

41

(2.8°×16=44.8°). Table 2.4 shows each stage’s scale mark of front indicating α and β orientation angles.

Table 2.4 Scale mark of front indicating α and β orientation angles in Goniometer stage

Stage 1 (=α) Stage 2 (=β)

Orientation angle The scale mark of front Orientation angle The scale mark of front

-17 0.8 -23 1.25

-16 0.170588 -22 2.223913

-15 1.141176 -21 0.397826

-14 0.511765 -20 1.371739

-13 1.482353 -19 2.345652

-12 0.852941 -18 0.519565

-11 0.223529 -17 1.493478

-10 1.194118 -16 2.467391

-9 0.564706 -15 0.641304

-8 1.535294 -14 1.615217

-7 0.905882 -13 2.58913

-6 0.276471 -12 0.763043

-5 1.247059 -11 1.736957

-4 0.617647 -10 2.71087

-3 1.588235 -9 0.884783

-2 0.9588242 -8 1.858696

-1 0.329412 -7 0.032609

0 1.3 -6 1.006522

1 0.670588 -5 1.980435

2 0.041176 -4 0.154348

3 1.011765 -3 1.128261

4 0.382353 -2 2.102174

5 1.352941 -1 0.276087

6 0.723529 0 1.25

7 0.094118 1 2.223913

8 1.064706 2 0.397826

9 0.435294 3 1.371739

10 1.405882 4 2.345652

11 0.776471 5 0.519565

12 0.147059 6 1.493478

42

13 1.117647 7 2.467391

14 0.488235 8 0.641304

15 1.458824 9 1.615217

16 0.829412 10 2.58913

17 0.2 11 0.763043

12 1.736957

13 2.71087

14 0.884783

15 1.858696

16 0.032609

17 1.006522

18 1.980435

19 0.154348

20 1.128261

21 2.102174

22 0.276087

23 1.25

The weight and bias of each axis (a1 and a2) for matching the simulated MFD based on DMP to actual

measurements of sensor were determined by minimizing the error in Eq (2.18). It requires one or more

actual measurements that are not zero value in each axis. The first column of Figure 2.25 shows the

weight and bias of each axis according to the number of actual measurements Ns and the second column

of Figure 2.25 is the calibration error of MFD between simulated MFD and actual measurements.

(a) On the x-axis

43

(b) On the y-axis

(c) On the z-axis

Figure 2.25 Weight and bias of each axis according to the number of actual measurements

The weight and bias of each axis is almost same when Ns is 10 or more. However, if the Ns is less

than 10, the weight and bias have large difference from the value of steady state. Although 10 actual

measurements are enough to find weight and bias, a fewer calibration error can be obtained if 20 or

more actual measurements are applied. Measuring 10 and 20 actual measurements, the weight and bias

were determined and summarized in Table 2.5 with calibration error. Figure 2.26 and 2.27 shows the

MFD calibration of 625 data applying weight and bias with Ns=10 and 20 respectively (625 data is the

total number of data in rc=14 (deg) closed circle).

Table 2.5 Weight, bias and calibration error of each axis

Ns Measuring

axis Weight Bias

Maximum

Error (mT)

Mean

Error (mT)

20

x -1.18268 0.204026 4.059862 1.079803

y 1.178557 0.054248 3.509832 1.044628

z 1.035189 -1.41081 3.215983 1.031387

10

x -1.18131 -0.17279 4.490315 1.130565

y 1.163719 0.309549 4.490286 1.04883

z 1.040254 -1.43487 3.404932 1.064838

44

(a) MFD on the x-axis

(b) MFD on the y-axis

(c) MFD on the z-axis

Figure 2.26 MFD calibration of 625 data applying weight and bias with Ns=10

45

(a) MFD on the x-axis

(b) MFD on the y-axis

(c) MFD on the z-axis

Figure 2.27 MFD calibration of 625 data applying weight and bias with Ns=20

where the red and blue solid line in first column is actual measurements and calibrated MFD by applying

weight and bias with Ns=10 (Figure 2.26) and 20 (Figure 2.27) respectively; second column shows the

calibration error of each axis.

Experiment result

The motion region is given as closed circle with rc = 14 (deg). 625data of MFD were measured using

Goniometer stage and Hall-effect sensor to evaluate measurement performance. The q was computed

46

by putting the MFD of Hall-effect sensor into the trained ANN, which is trained by calibrated MFD,

and compared with a reference by using Goniometer stage.

Figure 2.28 shows experiment result using ANN. The maximum and mean error of and in each

circle, which is approximated closed circle in simulation, were shown and the numerical values are

summarized in Table 2.6. The experiment result is similar to simulation, i.e. the error tends to converge

to 0 when rc decreases and the error in origin is the smallest. It means the measurement is more accurate

in the large gradient as confirmed by simulation.

Figure 2.28. Measurement error according to rc by using ANN in experiment

47

Table 2.6 Numerical value of experiment measurement error (unit: degree)

rc

Mean Max Mean Max

14 0.2130 0.6823 0.2052 0.6678

13 0.2411 0.6640 0.1659 0.5738

12 0.1706 0.5457 0.1580 0.4660

11 0.1692 0.4419 0.1452 0.3989

10 0.1389 0.4174 0.1292 0.4100

9 0.1210 0.3659 0.1108 0.3134

8 0.1599 0.3678 0.0955 0.2604

7 0.1238 0.2844 0.0897 0.2100

6 0.0841 0.2408 0.0652 0.1683

5 0.0758 0.2204 0.0845 0.1926

4 0.0465 0.1457 0.0447 0.1714

3 0.0362 0.1079 0.0618 0.2370

2 0.0476 0.0720 0.0532 0.0840

1 0.0509 0.0610 0.0506 0.0650

q=[0 0]T 0.05368 0.0481

2.3.2 Example 2: Two-DOF Orientation Measurement System with Multi-Sensor

Based on the result of example1 using single sensor, the multi-sensor approach is applied to expand

the motion region and reduce measurement error. It is identified that the orientation can be measured

accurately using multi-sensor in example 2

Magnetic field of PM based on DMP

A cylindrical PM is installed with a constant offset Lp from the origin of global coordinate systems

to the center of PM and rotating by and as same in example 1. The system parameters of the

measurement system is also same as example 1 such as PM specifications, initial sensor position and

DMP parameters. The desired motion region of Euler angles is set as q={[ ]T: -19 ≤ and ≤ 19

degree} and dicretized with 1 degree interval. It is impossible to measure this motion region using single

sensor with the error less than 1 degree. Figure 2.29 shows the coordinate systems of example2. The

four sensors are used in this system and their positions are defined by rotating Pi by s and s

48

respectively. Table 2.7 shows rotation angles of four sensors.

Figure 2.29 Coordinate systems of example 2

Table 2.7 Rotation angles of four sensors (unit : degree)

qs1=[10 -10]T, qs2=[-10 -10]T, qs3=[10 10]T, qs4=[-10 10]T

The magnetic field distribution discretized by DMP of each sensor is same in Figure 2.13 but the

orientation region differs. The center of orientation region in Figure 2.13 was set as qs=[0 0]T because

the magnetic sensor of example 1 is in the initial sensor position. It means the center of orientation

region is the rotating angles of each sensor. Therefore, the center of orientation region for each sensor

is the sensor’s rotating angles.

Multi-sensor approach

The issues related to the orientation measurement system are studied and simulated in the chapter

2.3.1 Example 1. The optimized sensor’ position is Ls=83 maximizing the sensing performance.

Sensor’s orientations are same as global coordinate system because it is easy to install the Hall-effect

sensors in the experiment. Bijective domain is analyzed and B (=[Bx By B]T) is selected as MFD vector.

The measurement accuracy is considered using Jacobian matrix. The gradient in origin is the largest

and becomes small as the distance from the origin increases. The region with boundary condition can

be approximated as closed circle and measurement is more accurate in the large gradient. With this

characteristic of bijective domain, the measurement error of each closed circle could be identified in

Figure 2.28 and Table 2.6. Based on this result, the measurement system with desired accuracy,

49

maximum and mean measurement error, can be designed to expand a sensing region by connecting the

closed circle having desired accuracy.

(a) A closed square in orientation domain (b) Range expansion using a closed square

Figure 2.30 Region expansion using a closed square

The closed circle can be approximated to a closed square with a length ac=rc×cos45°×2 of one side

as shown in Figure 2.30 (a). Closed Square having their motion region can be connected to each other

easily in orientation domain. Thus a closed square with desired accuracy can be used to expand a sensing

region as shown in Figure 2.30 (b). Four closed square are connected each other and the expanded

motion region have a length 2×ac of one side. The expanded range of closed square according to the

number of sensors is organized as Table 2.8.

Table 2.8 Expanded range of closed square according to the number of sensors

The number of sensors Expanded range of closed square with a desired accuracy

1 qc={[c c ]T: -ac/2 ≤ c and c ≤ ac/2 degree}

4 qc={[c c ]T: -2×ac/2 ≤ c and c ≤ 2×ac/2 degree}

9 qc={[c c ]T: -3×ac/2 ≤ c and c ≤ 3×ac/2 degree}

...

m2 qc={[c c ]T: -m×ac/2 ≤ c and c ≤ m×ac/2 degree}

In the example2, the desired square motion region is set as q={[ ]T: -19 ≤ and ≤ 19 degree}.

Based on the result of single sensor in Table 2.6, the number of sensors can be determined to measure

the desired motion region with desired accuracy. The expanded range of square according to the number

of sensors is summarized in Table 2.9.

50

Table 2.9 The number of sensors with desired accuracy (unit: degree)

rc ac/2

The number of sensors

4 9 16 25

Expanded range of square according to the number of sensors

14 9.90 19.80 29.70 39.60 49.50

13 9.19 18.38 27.58 36.77 45.96

12 8.49 16.97 25.46 33.94 42.43

11 7.78 15.56 23.33 31.11 38.89

10 7.07 14.14 21.21 28.28 35.36

9 6.36 12.73 19.09 25.46 31.82

8 5.66 11.31 16.97 22.63 28.28

7 4.95 9.90 14.85 19.80 24.75

6 4.24 8.49 12.73 16.97 21.21

5 3.54 7.07 10.61 14.14 17.68

4 2.83 5.66 8.49 11.31 14.14

3 2.12 4.24 6.36 8.49 10.61

2 1.41 2.83 4.24 5.66 7.07

1 0.71 1.41 2.12 2.83 3.54

where the first column is a radius of each closed circle; the second column is half length of one side of

each closed square; the desired accuracy is in Table 2.6 according to the radius of each closed circle.

The desired motion region can be measured using 4 sensors with the measurement accuracy of the

rc=14 closed circle as shown in Table 2.9. The 9, 16 and 25 sensors can be used to measure the desired

motion region with the measurement accuracy of the rc=9,7 and 6 closed circle respectively. It means

more sensors are used, better accuracy can be obtained. The number of sensors is square of m as shown

in the first column of Table 2.8. When m is odd number, the sensors are connected to each other based

on the sensor where the position is in the initial sensor position. On the contrary, when m is oven number,

the sensors are connected to each other without the sensor where the position is in the initial sensor

position as shown in Figure 2.30 (b).

After the sensor's positions are determined, the switching condition is required to distinguish which

sensor is used. The motion region to measure with MFD of ith sensor (=ΩSi) can be defined using closed

circle. The ΩSi can be defined by using magnitude of MFD in Eq (2.29) because the magnitude of MFD

is also circle shape as shown in Figure 2.13 and the boundary condition can be defined by the magnitude

of MFD.

51

:BSi i i (2.29)

where Bi is the magnitude of MFD on ith sensor; εi is the boundary condition of each circle.

Figure 2.31 shows region expansion using multi-sensor with switching conditions.

Figure 2.31 Region expansion using multi-sensor with switching conditions

were εi is the magnitude of MFD in each closed circle and it means boundary condition of each circle.

When the motion region is connected to each other, the overlapped areas exist and there are many

solutions as mentioned in chapter 2.2.6. As all solutions in the overlapped area are satisfying the

bijective, every solution can be applied to obtain orientation from MFD of each sensor. Therefore, the

sensor having a biggest gradient of B with respect to q is used because it has high accuracy.

Experiment

Figure 2.32 shows an experimental setup consisting of a cylindrical PM and four three-dimensional

Hall-effect sensors. The model of the Hall-effect sensor is TLV49d used in the experiment of example

1 and the Goniometer stage is also same. The Hall-effect sensor located in the initial sensor position is

used to set the sensors in a desired place.

52

Figure 2.32 Experiment setup for example 2

The sensor’s measuring points are calibrated by using XYZ stage implemented before in Figure 2.22.

Table 2.10 summarizes the results.

Table 2.10 Calibration results of multi-sensor’s measuring points in example 2 (unit: mm)

Position of Measuring points (desired + discrepancy)

Index of sensors X axis Y axis Z axis

1 +0.955 +0.245 -0.315

2 +1.03 +0.32 -0.343

3 +1.035 +0.26 -0.303

4 +1.01 +0.225 -0.363

10 or more actual measurements per sensor are required to obtain the weight and bias minimizing the

error in Eq (2.18). The weight and bias of each axis according to the sensor were determined by applying

the 20 actual measurements and summarized in Table 2.11 with calibration error. 81data of MFD were

measured using Goniometer stage and Hall-effect sensor to evaluate measurement performance. Figure

2.33 shows the MFD calibration 81data applying weight and bias of each sensors. The calibration error

is in the Figure 2.34. The ith column of Figure means the MFD calibration and error on the ith axis

(i=1,2,3 represent x,y,z axis) and the jth row of Figure means the MFD calibration and error of each

sensor (j=1,2,…,4 represent index of sensor). The red and blue solid line in Figure 2.33 are the actual

measurements and calibrated MFD by applying weight and bias respectively.

53

Table 2.11 Weight, bias and calibration error of each axis for multi-sensor

Index of

sensor Ns

Measuring

axis Weight Bias

Maximum

Error (mT)

Mean

Error (mT)

1

20

x 1.1221 -0.2113 2.7381 0.4469

y 1.1214 -0.0819 5.0372 0.5338

z 0.9651 0.1057 1.4518 0.3009

2

x 1.2386 -0.1733 4.4158 0.6624

y 1.2510 -0.0658 2.8450 0.3581

z 1.0593 -0.0386 5.3849 0.6494

3

x 1.1377 0.2544 3.4524 0.5071

y 1.1441 -0.1558 3.8422 0.3456

z 0.9995 0.1368 1.8687 0.2565

4

x 1.1199 0.3266 3.6256 0.6328

y 1.1332 0.0740 2.8603 0.4538

z 0.9199 0.1959 6.4288 0.8467

Figure 2.33 MFD calibration of 81 data applying weight and bias of each sensors

54

Figure 2.34 MFD calibration error of 81 data applying weight and bias of each sensors

The ANN of each sensor are trained by using the calibrated MFD of each sensor. After training, the

q was computed by putting the MFD of ith Hall-effect sensor (in the ΩSi) into the trained ith ANN. The

initial desired motion region is set as q={[ ]T: -19 ≤ and ≤ 19 degree}. Due to the adjustment

limitation of Goniometer stage, especially stage, the initial desired motion region cannot be measured.

The modified desired motion region is set as qm={[ ]T: -16 ≤ ≤ 16 and -19 ≤ ≤ 19 degree }.

Figure 2.35 shows experiment result using ANN. Figure 2.35 (a) is the result of the all 81data in the

orientation domain, where blue and red circle are the reference by Goniometer stage and the result of

ANN respectively. The measurement error between reference and result by ANN is shown in Figure

2.35 (b). Numerical maximum and minimum error of and are summarized in Table 2.12.

The predicted measurement accuracy is same as the result of closed circles with rc=14 in Table 2.6

since four closed circles with rc=14 are connected. The measurement accuracy of exmaple 2 using four

sensors is within the range of closed circles with rc=14.

55

(a) Result of all 81data in the orientation domain

(b) Measurement error of 81 data

Figure 2.35 Experiment measurement error of example 2

Table 2.12 Numerical error of experiment in example 2 (unit: degree)

qm=[ ]T

mean max mean max

-16 ≤ ≤ 16

-19 ≤ ≤ 19 0.1653 0.4917 0.1347 0.3334

56

2.4 Conclusion

In this chapter, two-DOF orientation measurement system based on magnetic field and ANN was

presented. Using DMP method, the issues related to this system are studied such as one-to-one mapping

between orientation and magnetic field (=bijective relation), optimized sensor’s position and orientation,

measurement accuracy based on Jacobian matrix, multi-sensor approach and calibration of magnetic

field between simulation and actual measurements. Two examples are introduced to evaluate the

proposed method.

Bijective relation between orientation and magnetic field was determined by using continuous MFD

and orientation domain plot visually and Jacobian matrix. The optimized sensor’s position and

orientation maximizing the sensing performance were determined mathematically. The gradient of

MFD according to orientation and MFD variation model were used to evaluate the accuracy of

measurement for each region in simulation. After the calibration of MFD using weight and bias, the

experiment was implemented. In example 1, it was identified how much the orientation can be measured

in a motion region using single Hall-effect sensor. Based on the result of example 1 using single sensor,

the multi-sensor approach was applied to expand a sensing region and reduce measurement error in

example 2. The number of sensors and switching condition in order to select the sensor in use were

considered. As a result, the number of sensors with desired accuracy, maximum and mean error of

estimated orientation, can be determined in desired motion region.

57

CHAPTER 3

CONCLUSION AND FUTURE WORKS

3.1 Accomplishments and Contributions

This thesis presents to develop an orientation measurement system using magnetic field of PM and

magnetic sensors. The specific contributions are as follow.

A. More effective system design.

B. Analysis of issues related to this system.

A measurement system based on magnetic field in previous research consists of many magnets and

magnetic sensors. Using DMP method, the issues related to this system are studied such as one-to-one

mapping between orientation and magnetic field (=bijective relation), optimized sensor’s position and

orientation, measurement accuracy based on the Jacobian matrix, multi-sensor approach and calibration

of magnetic field between simulation and actual measurements. Considering this issues, effective

orientation measurement system can be presented by determining the sensor’s position that maximizes

the sensing performance and not installing unnecessary sensor.

3.2 Future Works

The future works are summarized as follows:

A. Measurement of three-DOF orientation

B. Magnetic field design

C. Torque estimation and control for spherical motor

The orientation measurement system presented in this thesis can measure two orientation angles.

Some application requires three orientation angle, not only and Euler angle but also spinning motion

γ. Therefore, three-DOF orientation measurement system has to be developed to apply some

applications.

58

The magnetic source is important parts in magnetic tracing system. A cylindrical PM used in this

system can measure only two orientaion angles due to the symmetry of the magnetization of PM along

the z axis. Some researchers suggested to measure position/orienataion of magnetic source using multi-

magnet [24]. Combination of magnets or another shape of PM can overcome this limitation and can

make system more simple.

The dynamic model of spherical motor can be derived using Lagrange formulation [20]. The

orientation can be used not only for orientation/position control of spherical motor but also for torque

estimation because torque is derived by orientation defined as Euler angles (, , γ).

59

REFERENCES

[1] H. Z. Jin, H. Lu, S. K. Cho and J. M. Lee, "Nonlinear compensation of a new noncontact

joystick using the universal joint mechanism," IEEE/ASME Trans on Mechatronics, vol. 12,

pp. 549-556, 2007.

[2] C. Schott, R. Racz, and S. Huber, "CMOS three axis hall sensors and joystick application,”

IEEE Sensors. Conference., vol. 2, pp. 977-980, 2004.

[3] G. S. Chirikjian and D. Stein, "Kinematic design and commutation of a spherical stepper

motor," IEEE/ASME Trans on Mechatronics, vol. 4, pp. 342-353, 1999.

[4] K.-M. Lee and H. Son, “Distributed multipole model for design of permanent-magnet-based

actuators,” IEEE/ASME Trans on Magnetics, vol. 43, pp. 3904-3913, 2007.

[5] L. Zhang and W. Chen, "A robust adaptive iterative learning control for trajectory tracking of

permanent-magnet spherical actuator" IEEE Trans on Industrial Electronics, vol. 63, pp. 291-

301, 2016.

[6] G. Seyfarth, A. Bhatia, O. Sassnick, M. Shomin, M. Kumnagi and R. Hollis, “Initial results

for a ballbot driven with a spherical induction motor,” IEEE International Conference on

Robotics and Automation (ICRA), Stockholm, Sweden, pp. 3771-3776, 2016.

[7] H. Son and K.-M. Lee, "Open-loop controller design and dynamic characteristics of a spherical

wheel motor," IEEE Trans on Industrial Electronics, vol.57, no.10, pp. 3475-3482, 2010.

[8] S. M. Lee and H. Son, "Multi-degree of freedom motion platform based on spherical wheels,"

IEEE/ASME Trans on Mechatronics, vol. 22, pp. 2121-2129, 2017.

[9] H. Arioui, L. Nehaoua, S. Hima, N. S’eguy, and S. Espi’e, "Mechatronics, design, and

modeling of a motorcycle riding simulator," IEEE/ASME Trans on Mechatronics, vol. 15, pp.

805-818, 2010.

[10] K. L. Cappel and N. Marlton, "Motion simulator," U.S patent 32 95 224, Jan.3, 1967.

[11] M. Wapler, V. Urban, T. Weisener, J. Stallkamp, M. Durr, and A. Hiller, "A stewart platform

for precision surgery," Trans. Inst. Meas. Control, vol. 25, pp. 329-334, 2003.

60

[12] K.-M. Lee, R. Roth, and Z. Zhou, "Dynamic modeling and control of a ball joint-like variable-

reluctance spherical motor," Journal of Dynamic Systems Measurement and Control, vol. 118,

pp. 29-40, 1996.

[13] K.-M. Lee and D. Zhou, "A real-time optical sensor for simultaneous measurement of three-

DOF motions," IEEE/ASME Trans on Mechatronics, vol. 9, pp. 499-507, 2004.

[14] H. Garner, M. Klement, and K.-M. Lee, "Design and analysis of an absolute non-contact

orientation sensor for wrist motion control," IEEE/ASME International Conference on

Advanced Intelligent Mechatronics (AIM), pp. 69-74, 2001.

[15] C. Hu, M. Q.-H Meng, T. Mei, J. Pu, c. Hu, X. Wang and Y. Chan, "A linear algorithm for

tracing magnet’s position and orientation by using 3-axis magnetic sensors," IEEE Trans on

Magnetics, vol 43, no. 12, pp. 4096-4101, 2007.

[16] K.-M. Lee and H. Son, “Two-DOF magnetic orientation sensor using distributed multipole

model for spherical wheel motor,” Mechatronics, pp. 156-165, 2011.

[17] G. Jinjun, C. Bak and H. Son, "Design of a sensing system for a spherical motor based on hall

effect sensors and neural networks," IEEE International Conference on Advanced Intelligent

Mechatronics (AIM), Busan, Korea, pp. 1410-1414, 2015.

[18] S. Foong, K.-M. Lee, and K. Bai, "Magnetic field-based sensing method for spherical joint,"

IEEE International Conference on Robotics and Automation (ICRA), Anchorage, AK, pp.

5447-5452, 2010.

[19] S. Foong, K.-M. Lee, and K. Bai, "Harnessing embedded magnetic fields for angular sensing

with nanodegree accuracy," IEEE/ASME Trans on Mechatronics, vol. 17, pp. 687-696, 2012.

[20] K.-M. Lee and K. Bai, “Direct field-feedback control of a ball-joint-like permanent-magnet

spherical motor,” IEEE/ASME Trans on Mechatronics, vol. 19 pp. 975-986, 2014.

[21] E. Paperno, I. Sasada, and E. Leonovich, “A new method for magnetic position and orientation

tracking,” IEEE/ASME Trans on Magnetics, pp. 1938-1940, 2001.

[22] H. Son and W. Fang, “Optimization of measuring magnetic fields for position and orientation

tracking,” IEEE/ASME Trans on Mechatronics, vol. 16 pp. 440-448, 2011.

[23] Product information of Goniometer, [Online]. Available: https://www.owis.eu, 2011.

61

[24] X. Qiu, S. Song, M. Q.-H. Meng, “A novel 6-D pose detection method using opposing-magnet

pair system”, IEEE Sensors Journal, vol. 17, pp. 2642-2643, 2017

62

ACKNOWLEDGEMENT

First of all, I would like to express my sincere gratitude to my advisor Assoc/Prof Hungsun Son for

his academic guidance throughout this research. He is very knowledgeable and always coming up with

new ideas. It inspired me and played an important role in the successful completion of this thesis. He

also gave me a generous help when I started my graduate life and it was very helpful to me. It was an

honor for me to work under Prof. Hungsun Son and I always respect him as my mentor.

I want to thank the research former members Wu Fang and Guo Jinjun. During this research, I was

able to study research background with their previous research.

I also want to thank my companions in the Electromechanical Systems and Control Lab(ESCL) Jiyun

Jeon, Myunggun Kim, Seongmin Lee, Chanbeom Bak, Minho Shin, Hoyoung Kim, Wonmo Chungm,

Sangheon Lee and Soyoon Kim for their helpful support and advices. Especially, I have obtained

numerous suggestions and feedback from group meeting members Jiyun Jeon, Chanbeom Bak and

Wonmo Chung. Their assistances are greatly appreciated.

I would like to thank my family Yeonok Shin, Sugeun Kim and Jisun Kim for their long-time supports

throughout my educational career. Without their support, it would have been impossible for me to

complete this work.