Accurate Tilt Estimation of a Rotating

Platform Using Inertial Sensing

A Thesis

Submitted to the Faculty

of

Drexel University

by

Adam C. Salamon

in partial fulfillment of the

requirements for the degree

of

Master of Science in Mechanical Engineering

June 2014

ii

Acknowledgements

I would like to acknowledge the commitments and support of the many people who

supported my pursuit of this degree and thesis. There are too many people to name

them all individually, but there are a few who must be mentioned.

First, my advisor Dr. B.C. Chang, who encouraged me to pursue my academic goals

and was supportive of me from my undergraduate years through my graduate degree.

His encouragement, support and sage advice were a cornerstone of my academic ca-

reer at Drexel University. Dr. Chang has been a close advisor to me, lending advice

in research and in my career. I would be remiss in not also mentioning the support of

my professors who acknowledged the difficulties in pursuing a graduate degree while

supporting a family and working full time. In particular, Dr. Sorin Siegler, Dr. Harry

Kwatny, and Dr. Antonios Kontsos.

Next, my family provided the backbone of support for me when I would have pre-

ferred to spend time with them instead of my studies. They encouraged me to pursue

academic success to better my family’s life. To my wife, Dr. Katherine Salamon,

who pushed me to complete this degree even when time was at a premium and there

was great distance between us. To my mother, Stacey Hollander, who was a constant

cheerleader for my progress no matter how small it seemed.

Finally, to my friends and colleagues at Lockheed Martin. To, Jerry Franke, who

made it his mission to push and constantly remind me of the benefits of this process.

To my friend, mentor, and boss, Dr. Ned Allen, who provided me with the final push

and motivation to cross the finish line.

iii

I’m thankful for all that each of you have done and for those of you who were not

mentioned, but also showed your support. It is thanks to you that this research was

made possible.

iii

Table of Contents

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 LITERATURE REVIEW . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1 Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.1 Magnetometers . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.2 Gyroscopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.3 Accelerometers . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2.1 Inertial Navigation Systems . . . . . . . . . . . . . . . . . . . 15

2.3 Implementations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3.1 Euler Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3.2 Tait-Bryan Angles . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3.3 Quaternions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.4.1 Sensing on Moving Platforms . . . . . . . . . . . . . . . . . . 22

2.5 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

iv

2.5.1 Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.5.2 Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.5.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3 APPROACH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4 IMPLEMENTATION . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.1 System Functional Description . . . . . . . . . . . . . . . . . . . . . . 37

4.2 Implementation Considerations . . . . . . . . . . . . . . . . . . . . . 46

5 CONCLUSION AND FUTURE WORK . . . . . . . . . . . . . . . 51

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

v

List of Tables

2.1 Gyroscope Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Accelerometer Types . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3 Accelerometer Orientations . . . . . . . . . . . . . . . . . . . . . . . . 14

4.1 List of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.2 System Operational Constants . . . . . . . . . . . . . . . . . . . . . . 40

4.3 Potential Aircraft Orientations . . . . . . . . . . . . . . . . . . . . . . 48

vi

List of Figures

1.1 Magnetic Declination Map as of 2010 [26] . . . . . . . . . . . . . . . . 3

1.2 Gravitational Anomaly Map [24] . . . . . . . . . . . . . . . . . . . . . 4

1.3 Test Platform Design Concept Representation . . . . . . . . . . . . . 6

2.1 Ring Laser Gyroscope [25] . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 Micro Electro-Mechanical Gyroscope [1] . . . . . . . . . . . . . . . . 11

2.3 Euler Angle Definition [6] . . . . . . . . . . . . . . . . . . . . . . . . 18

2.4 Tait-Bryan Angles [16] . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.5 Inclination Sensing of a Moving Platform [30] . . . . . . . . . . . . . 25

3.1 Custom Designed PCB and Sensor Package . . . . . . . . . . . . . . . 33

3.2 Sample Accelerometer Data at 17.5◦ Incline . . . . . . . . . . . . . . 34

4.1 Unbonded Strain Gauge Accelerometer Basic Mechanism . . . . . . . 36

4.2 Suspended Mass Motion Under Acceleration . . . . . . . . . . . . . . 36

4.3 Example Accelerometer Mounting Orientation . . . . . . . . . . . . . 37

4.4 Sample Collinearly Mounted Rotating Accelerometer Data . . . . . . 38

4.5 Rotating Sensor Platform . . . . . . . . . . . . . . . . . . . . . . . . 39

4.6 Single Sensor, Ideal Placement, Purely Rotational Motion, Non-Accelerating 41

4.7 Detailed View of Sensor Offset from Normal . . . . . . . . . . . . . . 42

4.8 Single Sensor Undergoing Acceleration (a) Along, (b) Normal and (c)Vertically Normal to Sensitive Axis . . . . . . . . . . . . . . . . . . . 43

4.9 Single Sensor Undergoing Acceleration (a) Along, (b) Normal and (c)Vertically Normal to Sensitive Axis . . . . . . . . . . . . . . . . . . . 44

4.10 Dual Sensor, Ideal Placement, Purely Rotational Motion, Non-Accelerating 45

vii

4.11 Dual Sensor Undergoing Acceleration (a) Along, (b) Normal and (c)Vertically Normal to Sensitive Axis . . . . . . . . . . . . . . . . . . . 45

4.12 Reference Frame [4] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.13 Described Orientation . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.14 Physical System Description . . . . . . . . . . . . . . . . . . . . . . . 49

viii

AbstractAccurate Tilt Estimation of a Rotating Platform Using Inertial Sensing

Adam C. SalamonAdvisor: BC Chang, Ph.D.

Inertial sensing is a highly desirable methodology of sensing in mobile robotics. In-

ertial sensors provide a fixed reference measurement that can be easily transformed

from the body frame of the robot into a relative world frame. These technologies

provide sound information with high levels of reliability and repeatability. Some in-

ertial sensors are capable of providing rate information pertaining to the motion of a

platform, but pose estimating sensors are incapable of this task. This paper proposes

an approach to the measurement of the roll and pitch of a disc that is rotating at

a speed of at least 5Hz without the need for precise placement of the sensors. This

method of measuring pose can be extended to multiple platforms without the use of

complex and occasionally confusing algorithms.

1

CHAPTER 1: INTRODUCTION

As mobile robotic systems become more complex, the need for increasingly more reli-

able and robust sensing mechanisms grows. “In many applications, the determination

of the pose (orientation in 3D) of an object is often required. This is true both for a

free flying body in space and for machine parts which are connected to a fixed frame

by means of articulated chains” [37]. The vast majority of platforms use inertial mea-

surement units (IMU), which use accelerometers, and gyroscopes to calculate pose [7].

Current technologies rely on steady-state conditions to accurately estimate the pose

of a vehicle, which is rarely a practical possibility. For robots to operate efficiently

and effectively in complex environments, accurate estimates of pose must be obtain-

able while the robot is in motion. Techniques have been devised to achieve this task

for linear motion but research has not largely extended into the domain of rotation

[5]. Rotating platforms introduce new problems that current sensing technologies do

not address [13]. Such platforms are now being used more often to give robots more

degrees of freedom and hence an analysis of rotating platforms is rapidly becoming a

field of research interest.

The most common applications requiring accurate sensing from within a rotating

frame include: munitions and highly-agile unmanned vehicles. The Army Research

Lab (ARL) has been exploring the addition of control and sensing to ”dumb” muni-

tions [14]. In munitions, extreme conditions will be experienced by any sensor system

included within the artillery round, imposing yet another limitation on the system

2

implementation; size, weight and power (SWaP). Highly-agile unmanned vehicles re-

quire designers to minimize SWaP as increases in any of the three components can

have a drastic effect on the others. This paper will focus on unmanned vehicles, but

shares in several design constraints of munition sensing. As such, references to this

area will be used as a corollary to the unmanned vehicle domain.

“A crucial problem in localizing and controlling a UAV is the pose estimation: po-

sition and orientation. A single sensor is not sufficient to measure the orientation

and the position of the vehicle with respect to the inertial frame. In practice, one

has to merge information coming from different sensors in order to provide a correct

state estimation by using relevant properties from each sensor (inertial or absolute

information, accuracy, bandwidth, noise level, etc.)” [23].

An effective sensing platform must use a fixed reference for measurement or it will

be prone to errors, sensor drift, and noise. An example of a fixed reference is the

Earth‘s magnetic field. A magnetometer is a sensor that makes use of this particular

reference. It provides a reading based on the magnetic declination angle incident on

the sensor element, much like a compass. A compass will always point along the lines

of magnetic force (which converge on the magnetic poles). The angle between the

direction of force and the direction of the geographic north pole is called the declina-

tion. Moving across the surface of the globe, lines of constant magnetic declination,

called isogonic lines, continually change. As the Earth’s magnetic field varies over

time, the declination angle gradually changes at the same location.

Magnetometers can be used to detect rotation rate, heading and even pose under

certain conditions [38]. A major shortcoming of magnetometers is their susceptibility

to interference from external magnetic field sources. Additionally, magnetic sensors

3

are rarely aligned along the chosen measurement axis which results in offset errors

and changing zero points [10]. An external reference must be provided to calculate

the magnetic declination angle at a particular location on the Earth’s surface. The

associated look-up table must be updated periodically to account for natural shifts

in the Earth’s magnetic field. Figure 1.1 shows the major declination lines for 2010.

Figure 1.1: Magnetic Declination Map as of 2010 [26]

A second and more consistent fixed reference is the Earth’s gravity. While gravity

tends to vary at different points on the Earth, it is relatively constant across a large

area. Figure 1.2 represents the gravitational map of the Earth as of 2008. This

map does vary, but at a slow rate. Accelerometers measure acceleration, and more

specifically are often used to measure gravitational acceleration. They can accurately

measure the pose of a vehicle in a single dimension (multiple dimensions can be

achieved with multiple sensors). These sensors have a major drawback that requires

them to not be under any acceleration while measuring pose. In a real system, this

is not always possible and usually undesirable. This analysis considers a platform

rotating at a relatively constant speed and accelerometers are a good fit, with some

4

post-processing.

Figure 1.2: Gravitational Anomaly Map [24]

A third sensor type, gyroscopes, are designed to measure the velocities of a rotating

object in a given frame. These sensors meaure changes in momentum and are less

susceptible to outside noise as a result. However, current technologies only allow

these sensors to extend into the range of close to 1000 ◦ per second (165 RPM). For

slowly rotating platforms, this does not pose a serious concern, but for platforms

that can rotate at several hundred RPM, accurate measurements of velocity cannot

be made using gyroscopes. Another problem with using gyroscopes is that they only

provide information about spin rate and cannot provide pose information. Often, this

limitation requires the fusion of multiple sensing modalities.

Combinations of all three sensors are used in devices known as Inertial Measurement

Units (IMU). These devices are designed to trade off the benefit of one sensor to offset

the downside of another. For example, a magnetometer can be used to provide the

rate of rotation of a platform to provide feedback to the accelerometer to isolate and

eliminate spurious measurements related to platform motion.

5

A combination of multiple accelerometers along with a simple algorithm could pro-

vide accurate pose estimation of a vehicle that is rotating at a high rate of speed.

The analysis will show that accurate pose estimation can be obtained using the same

number of sensors as a stationary platform, perhaps less. A key element of perform-

ing these estimates will be in the separation of vehicle-centric forces from external

forces (i.e. gravity). Acceleration of the vehicle due to motion must be identified and

eliminated or the sensing system will only be useful while the vehicle is stationary.

Single sensors are not capable of distinguishing motion of the vehicle from tilt without

an additional reference. This paper will seek to identify a minimum set of sensors

required to perform this differentiation.

Real-world analysis is required to determine the effectiveness of the senor system and

a testing platform was designed to accommodate a subset of these testing require-

ments (Figure 1.3). The testing platform is made of three key elements: a rotating

platform, a motor to drive the rotating platform, and a rotation sensor to provide

rotation rate feedback to the system. This testing platform represents a method with

which the system can be tested under real world conditions. An acceleration rate and

maximum rotation rate can be set using a microcontroller interface which controls

the motor directly. For the purposes of this discussion, the primary analysis will be

performed analytically to verify the validity of the approach proposed.

In this paper, the discussion will focus on the applicability of accelerometers to the

problem as well as the implementation of the sensor and algorithm combination within

the confines of an embedded microcontroller system. A study of the current state of

the art in accelerometers will be presented which describes the operation and function

of an accelerometer. Following the study, a description of the configuration options

for implementing the envisioned sensor platform will be provided. Finally, the ac-

6

Figure 1.3: Test Platform Design Concept Representation

celerometer and algorithm components will be tested analytically to determine if the

approach is sound or if additional design considerations must be undertaken.

7

CHAPTER 2: LITERATURE REVIEW

2.1 Sensors

Inertial sensors are based on the principles of inertia. This type of sensor ranges

from large ring laser gyroscopes to extremely small micro electro-mechanical system

(MEMS) accelerometers, gyroscopes and magnetometers. Inertial sensors all rely on a

fixed reference such as Earth’s gravity or magnetic field. These sensors detect changes

in these references and provide either a digital or analog output which describes the

sensed change. Applications for inertial sensors range from control and stabilization

to navigation to measurement and testing. A review of magnetometers, gyroscopes

and accelerometers is presented.

2.1.1 Magnetometers

A magnetometer is a sensor which measures both the strength and direction of a

magnetic field at a point in space. Typically, these sensors measure the Earth’s mag-

netic field and/or fluctuations in the local magnetic field. Applications range from

measurement of the magnetization of a material to geophysical surveys of magnetic

anomalies to detecting the presence of metals in support of the detection of sub-

merged vehicles such as submarines. Magnetometers are a common component in

many consumer devices such as cell phones, tablet computers, and GPS navigation

systems. The most common application for magnetometers is the detection of the

Earth’s magnetic field as a compass.

8

A magnetic field is a vector quantity which is described by the magnitude and direc-

tion of the field. The field magnitude is measured in units of Teslas. The Earth’s

magnetic field is on the order of 20,000 - 80,000 nanoTesla (nT).

Generally speaking, there are two types of magnetometers: vector and absolute. A

vector magnetometer, as the name implies, measures the vector components of a mag-

netic field. This type of magnetometer typically provides an output which contains

the declination and inclination angles relative to the Earth’s magnetic field. An ab-

solute magnetometer measures the absolute magnitude of the magnetic field. This

sensor is used for measuring variations in magnetic fields. While an absolute magne-

tometer may provide a more accurate assessment of the magnetic field incident on it

and changes to that field, it does not provide information that is generally useful for

navigation like a vector magnetometer.

There are several key specifications for magnetometers which much be analyzed and

traded to determine the sensor most appropriate for an application. A selection of

these key specifications are:

• Sample rate - the number of sensor readings per second (Hz)

• Bandwidth - describes the accuracy of sensor readings during rapid change

(1/Hz)

• Resolution - the smallest change in magnetic field that can be measured

• Drift - the change in the sensor output over time

• Thermal stability - the sensitivity of the sensor accuracy to changes in ambient

temperature

9

2.1.2 Gyroscopes

A gyroscope is comprised of a store of inertia and a sensor for determining the angle

at which the center of inertia is pointing. The store of inertia, known as a rotor, can

be constructed of a light source, a physically spinning disc or wheel, or a vibrating

ring, to name a few. The inertia within the rotor creates a resistance to motion of

the gyroscope frame. This enables the orientation of the rotor to remain fixed even

while the frame moves. Gyroscopes in embedded systems use construction techniques

which rely on the inertial characteristics of vibration rings and light rather than those

of a spinning rotor, as is the case in larger mechanical gyroscopes, to perform their

measurements. There are many classes of gyroscopes available and in use today (Ta-

ble 2.1). The most reliable, and most expensive, embeddable gyroscopes are ring

laser gyroscopes (RLG) and fiber optic gyroscopes (FOG). Micro-Electro Mechanical

System (MEMS) gyroscopes tend to be inexpensive and considerably smaller in size.

The trade-off in cost and size leads to a reduction in accuracy for these sensors, which

often necessitates the addition of a reference sensor.

Table 2.1: Gyroscope Types

Gyroscope Type Sensing Modality ImpactMicro Electro-MechanicalSystem (MEMS)

Vibrating MEMS elementbased on the Foucault pen-dulum

Low cost, average ac-curacy

Fiber Optic Gyroscope(FOG)

Interference of light usingfiber optics

High cost, high accu-racy

Hemispherical ResonatorGyroscope (HRG)

Flexual standing wavesacross a solid-state hemi-spherical shell

High cost, high accu-racy

Vibrating Structure Gyro-scope (VSG)

Metallic alloy resonator Average cost, averageaccuracy

Both the RLG and FOG rely on the effects of inertia on light. Light is either sent

through a set of mirrors or an optical cable in opposing directions (Figure 2.1). These

10

sensors involve physical path lengths in the range of 5km of optical transfer distance.

Rotation of the gyroscope will result in minute, but detectable changes in the arrival

time of light from one direction to the other. These types of sensors provide a level

of accuracy high enough to not require external reference sensors. With accuracy,

comes cost and size in these two cases. An RLG typically measures 50cm on a side

and is priced well outside of the reach of most embedded systems. A typical RLG or

FOG system costs approximately $30,000.00 USD[11].

Figure 2.1: Ring Laser Gyroscope [25]

MEMS gyroscopes address several of the limitations of RLGs and FOGs, but have

their own shortcomings. MEMS sensors in the last two decades have increasingly

become more popular in consumer applications [11]. Due to limitations in range

and accuracy however, MEMS gyroscopes have not become popular in navigation

systems. In recent years, improved error characteristics, resistance to environmental

impacts, increased bandwidth, and the miniaturization of microcontrollers capable of

processing the high rate sensor data and error correction models have all lead to wider

adoption of this sensor class. Precision inertial navigation systems (INS) is a field

that is still largely dominated by RLG and FOG sensors on highly mobile systems.

MEMS based INS for relatively static applications have become more popular, such

11

as antenna array stabilization. Cost of MEMS based INS are significantly less than

RLG and FOG, typically in the $1,000 - 5,000.00 range. INS are discussed in more

detail in Section 2.2.1.

Figure 2.2: Micro Electro-Mechanical Gyroscope [1]

2.1.3 Accelerometers

Acceleration is the time rate of change of velocity. Accelerometers are sensors designed

to detect this change. There are several classes of accelerometers, each sensor class

uses a different sensing modality and is designed for a different purpose [35]. Table

2.2 details some of the types of sensors that are commercially available.

In spite of the multiple types of accelerometers, those listed in Table 2.2 all provide

much the same information as an output. Standard convention for accelerometers

is that the force of gravity is +1g when measured against the sensitive axis. If the

sensor was inverted so as to make the sensitive axis in the same direction as gravity,

the output of the sensor would be -1g. Finally, the sensor will output 0g when the

sensor is normal to the direction of gravity. There are several key terms, defined

below, which describe the function and characteristics of an accelerometer:

• Linearity - the maximum deviation of the calibration curve from a straight line

12

Table 2.2: Accelerometer Types

Accelerometer Type Sensing Modality ImpactCapacitive Micromachined feature produces

a change in capacitance relativeto acceleration

High Cost, high accuracy

Piezoelectric Piezoelectric crystal mounted tomass. The voltage output is con-verted to acceleration

Low cost, low accuracy

Piezoresistive Micromachined feature whose re-sistance changes with accelera-tion

Low cost, low accuracy

Hall Effect Changing magnetic fields aresensed and converted to electricalsignals

Low cost, low accuracy

Magnetoresistive The resistivity of the sensing ma-terial changes in the presence of amagnetic field

Average cost, low accuracy

Heat Transfer A heated mass is tracked by sens-ing temperature fluctuations oneach edge

Low cost, high accuracy

(Volts)

Linearity = Vout,0g −1

2(Vout,+1g + Vout,−1g) (2.1)

• Sensitivity - measure of the degree with which the sensor output changes as the

acceleration being measured changes (Volts/g)

Sensitivity =∆Vout

∆g=Vout,+1g − Vout,−1g

2g(2.2)

• Vcc - the voltage supplied to power the sensor (Volts)

• %Vcc - for analog sensors, sensor readings are represented as a percentage of

the supplied voltage, Vcc

When employing accelerometers in practical systems, it is important to take into

account the fact that the sensor output is ratiometric. As a result, the analog out-

put of the sensor changes as the input voltage (Vcc) changes. Failure to take this

13

into consideration can have dramatic effect, but mitigating the ratiometric change is

easily performed. As an example, a sample sensor has the following characteristics:

Vcc=5.0VDC, Vout,0g=1.800VDC. In terms of %Vcc, the output at 0g is 36%Vcc.

Let us suppose the power supply being employed for the sensor has variability of

+/-0.1VDC. The output voltage then could be as low as 1.764VDC and as high as

1.836VDC for the same 36%Vcc for the 0g sensor output. For applications involving

the sensing and control of autonomous systems, this deviation can mean the differ-

ence between traversing down the center of a hallway and hitting a wall or a human

in that hallway. Mitigation of this problem is as simple as calculating the ratiometric

output utilizing the input and output voltages for each sample taken.

Accelerometers can be used as inclinometers to measure the tilt of the sensor. With

gravity treated as a constant, tilt can be determined by measuring gravity directly.

It is due to this benefit that another significant hindrance is identified; mounting

error. This mounting error can be the source of significant error in certain mounting

orientations. For example, a 1 degree mounting error in tilt for the 0g orientation,

where the sensitive axis is normal to the gravitational acceleration, is equivalent to

a 10 degree tilt in both the +1g and -1g mounting positions where the sensitive axis

is aligned to gravity as shown in Table 2.3 [35]. This sensitivity is also a boon to

certain sensor applications. For applications where fine tilts must be measured, the

0g orientation is ideal. For applications where only gross approximation is required,

the +1g or -1g orientations may serve as more appropriate.

14

Table 2.3: Accelerometer Orientations

+1g Orientation 0g Orientation

gn = g cos(θ);Letθ = 1◦ (2.3)

gn = 0.9998g (2.4)

gn = g sin(θ);Letθ = 1◦ (2.5)

gn = 0.0175g (2.6)

0g orientation senses a 57x larger change in sensed change than the +1g orientation

2.2 Systems

Inertial sensors are designed to sense a particular delta to a measurable force be

it a magnetic field, rotation rate, gravity, etc.. Each sensor has inherent strengths

which enable applications of the sensor to measure changes with a relatively high

degree of sensitivity and accuracy. Conversely, each sensor has specific limitations

that prevent use of the sensor without an external reference to counter the effects of

an environmental variable, sensor instability, and drift or motion, which is outside

of the measurement bounds of that particular sensor. Systems have been developed,

that fuse the outputs of combinations of these sensors to provide a single output

which is stabilized by all three sensor types. These systems are generally referred to

as Inertial Navigation Systems (INS).

15

2.2.1 Inertial Navigation Systems

Inertial navigation systems (INS) are used across a wide array of mobile applications.

This includes robotics, automobiles, aircraft, and surface and subsurface maritime

vessels. Contained within an INS is an inertial measurement unit (IMU). In typical

applications of IMUs, a combination of accelerometers, gyroscopes, and other sensors

are combined [34]. Accelerometers are used to sense linear accelerations and gyro-

scopes are used to sense angular velocity. Additional sensors such as magnetometers

are also often applied to provide heading information. Each of these sensors has inher-

ent limitations and the combination of sensors is chosen to mitigate these limitations.

The IMU data is used in concert with additional sensors, such as GPS and a suite of

algorithms to determine the position and orientation of the platform.

In a typical application, an IMU will contain three accelerometers, three gyroscopes

and optionally three magnetometers; although incarnations of using single sensor

types have been attempted [5][22][34]. The sensors are mounted each orthogonally

to each other so as to measure the roll, pitch and yaw axes independently. Using

this combination of sensors, it is possible to determine the absolute orientation of the

platform relative to the Earth‘s surface. It is important to note that these sensors

do not support the determination of height above the Earth however deviations from

the initial position can be calculated. Through frequent sampling and integration of

sensor data, position changes can be determined through a process known as dead-

reckoning [9]. Dead-reckoning however is subject to large variances as navigation

time increases. There are multiple sources of error, that become compounded with

the sample size including: sensor noise, sensor drift, Abbe errors, and rounding errors.

Sensor noise and sensor drift are characteristics of inertial sensors. Sensor noise

sources include the power supply, environmental influences (magnetic fields, wind

16

gusts, vibration, etc.), and analog-to-digital conversions. Sensor drift is most preva-

lent in gyroscopes; as the temperature changes, the velocities sensed drift. Rounding

errors are a result of the precision of the sensor and the precision of the digital system

processing the data. Rounding errors compound with each iteration of the localization

algorithm. Abbe error, named for Ernst Abbe, also known as sine error, describes the

magnification of angular error over distance [2]. Abbe error is described by Equation

2.7.

ε = h sin(θ) (2.7)

For example, suppose a point is measured 10 meters away at a 45◦ angle, with an

angular error of 1◦. The Abbe error for such a measurement is 17.45cm or 1.745% of

the distance measured. This error is exacerbated by the noise inherent in the sensors

themselves as previously described.

IMUs are made to be more accurate through the use of ”ground truth” sensors or

feedback. Such examples are GPS and other external position systems (i.e. motion

capture cameras). These external references however operate on relatively slow time

intervals which may not support the system needs. GPS typically operates at 1Hz

with newer sensors offering 10Hz. Motion capture cameras offer much higher speeds,

but are limited in the volume within which a system can operate and require rigorous

calibration.

2.3 Implementations

2.3.1 Euler Angles

Introduced by Leonhard Euler to describe the orientation of a rigid body in 1776, [29,

p. 189-207] Euler angles describe the 3-dimensional orientation in Euclidean space of

17

an object relative to a static reference frame. The Euler angles are often referred to

as the ”local” reference frame and are relative to the body. When applied to robotic

applications, Euler angles represent three separate rotations, each about a single axis

of the reference coordinate system. Particular attention must be paid to the order in

which Euler angle operations are applied. The convention chosen initially must be

preserved through all calculations using the Euler angles. Failure to do so will result

in inaccurate rotation calculations. Authors who utilize Euler angles must define the

convention chosen prior to any application of the reference frame clearly.

Any three dimensional, reference frame can be represented by Euler angles based on

the combination of three individual rotations, which originate from a known original

orientation. The fixed frame reference in robotics typically refers to the plane of the

Earth‘s surface. The original frame is typically denoted as x,y,z while the rotated

plane is represented by X,Y, Z. The rotated frame is treated as though it is fixed to

the body being rotated and does not change relative to the body. Applications using

proper Euler angles are usually imagined to have the xyz and XYZ frames aligned

initially prior to rotations being applied. Standard notation for the orientation of the

system following each of the three rotations is represented as:

x-y-z (initial state)

x‘-y‘-z‘ (following first rotation)

x“-y“-z“ (following second rotation)

X-Y-Z (final state)

The angles described by the final state can be represented as three angles α, β, and

γ. This Euler angle definition is simpler and is represented in Figure 2.3.

There are six possibilities of choosing the rotation axes for proper Euler angles. In

all of them, the first and third rotation axis are the same. The six possible sequences

18

Figure 2.3: Euler Angle Definition [6]

are:

z-x‘-z“ (intrinsic rotations) or z-x-z (extrinsic rotations)

x-y‘-x“ (intrinsic rotations) or x-y-x (extrinsic rotations)

y-z‘-y“ (intrinsic rotations) or y-z-y (extrinsic rotations)

z-y‘-z“ (intrinsic rotations) or z-y-z (extrinsic rotations)

x-z‘-x“ (intrinsic rotations) or x-z-x (extrinsic rotations)

y-x‘-y“ (intrinsic rotations) or y-x-y (extrinsic rotations)

When implementing physical systems using Euler angle representations, a singularity

point exists which must be taken into consideration. This is also known as gimbal

lock. If the system being measured pitches up to 90◦, the yaw and roll axes become

parallel and can no longer be differentiated. This results in the loss of a degree of

freedom, which requires intervention to correct. One such approach to this is the

use of a quaternion representation. Quaternions do not suffer from gimbal lock and

allow for a more robust system representation. Quaternions are further examined in

Section 2.3.3.

19

2.3.2 Tait-Bryan Angles

In the simplest of terms, most are familiar with the standard angular orientations

of roll, pitch, and yaw. This representation is referred to as the Tait-Bryan angles

or nautical angles. The use of Tait-Bryan angles are often also referred to as Euler

angles, although for the purpose of this discussion they are will be treated as separate

and distinct [8]. The Tait-Bryan angles can be represented by six separate combi-

nations (roll-pitch-yaw, pitch-yaw-roll, yaw-pitch-roll, roll-yaw-pitch, yaw-pitch-roll,

pitch-roll-yaw). It is in this fact that we find one of the key weaknesses of the

Tait-Bryan representation. Failure to effectively use the proper representation will

result in inaccurate computation and incorrect results; similar to proper Euler angles.

Also due to this fact, it is difficult to read Tait-Bryan angles and interpret them as a

physical representation of a system without prior knowledge of the chosen convention.

Figure 2.4 shows a sample aircraft utilizing the yaw-pitch-roll convention. The fixed,

Earth frame has been shifted back from the aircraft center of gravity for clarity in this

image, by the artist. φ represents roll with a positive roll being a counter-clockwise

rotation, θ represents pitch with a positive pitch being down and ψ represents yaw

with a clockwise rotation being positive.

Figure 2.4: Tait-Bryan Angles [16]

20

2.3.3 Quaternions

Quaternions are a non-commutative extension of the complex numbers. William

Rowan Hamilton first described quaternions in Ireland in 1843 [12]. Quaternions

comprise a four-dimensional vector space with a basis consisting of the real numbers

and the imaginary i, j, k. Quaternions generally follow algebraic rules with the

exception of being noncummative for multiplication (Equation 2.8).

ab 6= ba (2.8)

The basic form of a quaternion is a column vector containing the four quaternion

basis (Equation 2.9).

q =

q0

q1

q2

q3

(2.9)

Where Equation 2.10

|q|2 = q20 + q2

1 + q22 + q2

3 = 1 (2.10)

The quaternion can be associated with a rotation about an axis by Equations 2.11,

2.12, 2.13, and 2.14.

q0 = cos(α

2) (2.11)

q1 = sin(α

2) cos(βx) (2.12)

q2 = sin(α

2) cos(βy) (2.13)

21

q3 = sin(α

2) cos(βz) (2.14)

where α is a simple rotation and βx, βy, βz are direction cosines locating the axis of

rotation.

The orthogonal matrix which corresponds to a rotation by a unit quaternion is given

in the homogeneous form by Equation 2.15.

q =

q2

0 + q21 − q2

2 − q23 2(q1q2 − q0q3) 2(q0q2 + q1q3)

2(q1q2 + q0q3 q20 − q2

1 + q22 − q2

3 2(q2q3 − q0q1)

2(q1q3 − q0q2) 2(q0q1 + q2q3) q20 − q2

1 − q22 + q2

3

(2.15)

The direction cosine matrix is represented by Equation 2.16.

cos(θ) cos(ψ) − cos(φ) sin(ψ) + sin(φ) sin(θ) sin(ψ) sin(φ) sin(ψ) + cos(φ) sin(θ) cos(ψ)

cos(θ) sin(ψ) cos(φ) cos(ψ) + sin(φ) sin(θ) sin(ψ) − sin(φ) cos(ψ) + cos(φ) sin(θ) sin(ψ)

− sin(θ) sin(φ) cos(θ) cos(φ) cos(θ)

(2.16)

Combining the quaternion representation of the Euler rotations we get Equation 2.17.

q =

cos(φ2) cos( θ

2) cos(ψ

2) + sin(φ

2) sin( θ

2) sin(ψ

2)

sin(φ2) cos( θ

2) cos(ψ

2)− cos(φ

2) sin( θ

2) sin(ψ

2)

cos(φ2) sin( θ

2) cos(ψ

2) + sin(φ

2) cos( θ

2) sin(ψ

2)

cos(φ2) cos( θ

2) sin(ψ

2)− sin(φ

2) sin( θ

2) cos(ψ

2)

(2.17)

The resulting matrix allows for the conversion between quaternions and Euler angles

22

by Equation 2.18. φ

θ

ψ

=

arctan(2(q0q1+q2q3)

1−2(q21+q22)

arcsin(2(q0q2 − q3q1)

arctan(2(q0q3+q1q2)

1−2(q22+q23))

(2.18)

Quaternions are often used in robotics as they allow for compact representations of

rotations in 3D space. A quaternion can be represented simply as four numbers

versus an orthogonal representation matrix which uses nine numbers. It is also easy

to read off the axis and angle of a rotation axis whereas this task is much harder

using orthogonal matrices and Euler angles.

2.4 Applications

It is desired to utilize only a single sensor type for the application described by this

paper. For this reason, the applications described will focus on systems which utilize

a single inertial sensor type. The applications typically feature a secondary sensor or

an additional algorithm to address shortcomings of the system design with a single

sensor type.

2.4.1 Sensing on Moving Platforms

There has been extensive research performed in the area of inertial measurement of

mobile platforms. The papers surveyed all cite the same limitation of accelerometers;

raw accelerometer data consists of both the position of the accelerometer relative to

gravity, as well as the linear acceleration component along the sensitive axis. For

stationary measurements this poses little issue, however when attempting to sense

position during motion and under acceleration, the two components must be clearly

isolated. It is in this area that the surveyed literature focuses. Approaches to com-

bine multiple sensor types to form a unified result and the application of high- and

23

low-pass filters appear to be the norm.

In one such paper, Boonstra et al combine single axis accelerometers with a single

gyroscope using a Butterworth filter [5]. Boonstra et al sought to assess the accuracy

of measuring the angle and angular velocity of the upper body and upper leg of a

person rising from a seated position in a chair to a standing position. It can be safely

presumed that the linear accelerations will occur in a higher frequency domain than

the gravitational component in this experiment. Based on this presumption, Boon-

stra et al hypothesized that while the system was undergoing acceleration, a low-pass

filter could be applied to assist in distinguishing the gravitational component from

the motion component.

Their results clearly indicate that using a simple filter improves the quality of the

processed data and generally follows the ground truth of the motions being analyzed.

Simply using two single-axis accelerometers alone proved that reasonable estimates

for angle measurements are possible, but angular velocity measurements had an un-

acceptably high error. Adding in rotation rate information from a gyroscope brought

the angle measurement error down to less than two degrees and angular velocity errors

down to single digits. However, the accuracies were improved through the develop-

ment of filters specialized to the specific motion being tracked. While this approach

shows promise, it is still limited in its applicability to systems that move at higher

rates of change. Additionally, combining multiple sensor modalities, accelerometers,

and gyroscopes in this instance, are undesirable for the application being proposed.

Tan et al sought to assemble an inertial navigation system (INS) using only accelerom-

eters [34]. While an INS typical uses a combination of at least accelerometers and

gyroscopes, Tan et al‘s approach was to utilize six accelerometers arranged about

24

the faces of a cube to develop a complete description of the motion of a rigid body.

Tan et al observed the high cost, high power consumption, slow reaction times, small

dynamic range, and generally large volume of gyroscopes made them unfeasible for

many applications. For ”feasible” configurations of accelerometers, Tan et al hypothe-

sized the angular and linear motions can be separately measured using two decoupled

equations. Feasibility means that the specified number and orientation of sensors is

sufficient to compute the linear and angular motions of a moving body. It is impor-

tant to note, and should be obvious, that not all configurations meet the feasibility

limits, however most do. One such infeasible example is orienting all accelerometers

facing the same direction.

One application space that has been more extensively explored is inclination sensing of

moving ground vehicles. Applications range from g-force measurement to fuel gauge

compensation to GPS navigation enhancement. While inclination sensing requires

only a single sensor, implementing inclination sensing on a moving platform requires

feedback about the motion of the platform itself. For an accelerometer in the 0g

orientation (Figure 2.5(a)), the relationship between the inclination of the platform

and gravity is Equation 2.19; where α is the inclination angle.

ax = g sin(α) (2.19)

Solving Equation 2.19 for α yields Equation 2.20.

α = arcsin(Axg

) (2.20)

In this orientation, acceleration of the platform itself will also be sensed by the ac-

celerometer. In an example scenario, a platform which accelerates from 10MPH to

55MPH in 10 seconds is accelerating at 0.2g. The sensor system would detect an

25

additional inclination of 12◦. Alternatively, the sensor could be mounted in the 1g

orientation (Figure 2.5(b)) to measure only gravity and would not be susceptible to

the acceleration of the platform. The inclination angle in this orientation is repre-

sented by Equation 2.21.

α = arcsin(Ayg

) (2.21)

While this orientation addresses the problem of sensing platform acceleration, this

orientation causes very slight changes in sensed gravity changes. A 10◦ change in in-

clination (α) is equivalent to a 0.985g acceleration. The sensed change would only be

0.015g; a very small signal based on the expected accuracy ranges of most accelerom-

eters. To compensate, MEMSIC Corporation recommends an alternate implementa-

tion utilizing two accelerometers which are mounted orthogonally to each other with

gravity bisecting the intersection of the two sensors (Figure 2.5(c))[22]. This mounting

orientation places the sensitive axes of the sensors in similar, but opposing polarity.

A positive acceleration for one sensor will be sensed as a negative acceleration in the

other. Simply adding the output from each sensor eliminates the acceleration sensed

from the platform motion. Calculating the inclination angle simply takes the form of

Equation 2.22.

α = arccos(0.707[ax + ay]

g) (2.22)

Figure 2.5: Inclination Sensing of a Moving Platform [30]

Harkins et al designed a device for making angular measurements on spinning projec-

26

tiles using magnetic sensors [14]. Harkins et al noted that, “Accurate measurement

of angular motion of spinning bodies with on-board sensors has long been recognized

as a daunting task. Recent advances in magnetic sensor technologies have yielded

devices small enough, rugged enough and sensitive enough to be useful in systems

that make high-speed, high-resolution measurements of attitude relative to magnetic

fields when these sensors are installed on free-flying bodies.” Harkins et al designed a

system which measures the angular orientation of a spinning projectile while in flight

using only the Earth‘s magnetic field as a reference.

This system features two magnetic sensors which are mounted normal to the rota-

tion plane and 30◦ out of phase with the rotation plane. The system is designed to

measure the point within the rotation that the projectile currently is. The system

does not need a consistent reference frame as it uses zero crossing points and then

performs arithmetic permutations to adjust for fluctuations in the readings. This

creates a fairly robust system, which intrinsically protects itself from sensor drift and

environmental noise.

2.5 Limitations

Each of the inertial sensors, systems and applications presented have limitations to

their broader applicability. Certain inertial sensors are better suited to particular

sensing modalities, certain systems are too large or too expensive and the applications

presented are targeted to specific uses. A discussion of the limitations of each will

describe why they do not meet the needs of the envisioned system and why further

development is required.

27

2.5.1 Sensors

Magnetometers, as applied to vector sensing vice absolute sensing, measure the magni-

tude and direction of the Earth‘s magnetic field incident on the sensor. This provides

a relatively strong signal to perform measurements from over a small, fixed time win-

dow. As described in Section 2.1.1, the Earth‘s magnetic field is constantly changing

and is different at each point on the Earth‘s surface. This changing declination angle

means that any implementation using a magnetometer will need to either be pro-

vided a declination angle for calibration or will need an external reference to enable

a lookup table to determine the appropriate value. Additionally, magnetometers are

susceptible to magnetic fields from the environment. For example, a DC motor oper-

ating in the vicinity of a magnetometer will create a measurable magnetic field. The

strength of the field present at the sensor will vary based on the load experienced by

the motor at a given time. Other sources of magnetic interference, which can all be

expected in the environment are iron deposits, electrical transformers, and buildings.

Gyroscopes are capable sensors, but are susceptible to noise, sensor drift, environ-

mental changes, and are limited in their maximum bandwidth. The most capable

gyroscopes are uniquely robust to most outside interference, but are not only phys-

ically large they are also prohibitively expensive for most embedded applications.

Gyroscopes experience drift as the ambient temperature changes and as the temper-

ature of the sensor itself changes. For well characterized sensors, this drift can be

compensated for however this introduces the need for additional processing and/or

sensors to measure the ambient and sensor temperatures. Some modern gyroscopes

include built-in temperature compensation. The most damaging limitation of gyro-

scopes for the envisioned system is their maximum bandwidth. The envisioned system

rotates well beyond the maximum rates capable of embeddable gyroscopes.

28

Accelerometers are capable of measuring very wide ranges of accelerations and are not

susceptible to interference from environmental changes. Accelerometers are however

incapable of intrinsically separating accelerations due to motion of the sensor from

accelerations due to gravity. This limitation means that a sensor must be mounted in

such an orientation that isolates it from outside motion while still enabling measure-

ment of gravitational acceleration. This is not feasible for most embedded systems,

which can move in multiple dimensions at once. Approaches for isolating acceleration

of the platform and acceleration due to gravity are well studied, but require additional

sensors.

2.5.2 Systems

Inertial navigation systems (INS) are, generally speaking, able to adjust for shortcom-

ings of individual sensors through the use of complex algorithms and multiple sensors

of multiple types. Gyroscopes can be used to detect platform motion to inform the

sensed tilt angle of an accelerometer, for example. These benefits however come at

a high cost, high power, and large physical space. For applications involving large

vehicles, expensive systems or those not requiring battery power, an INS can provide

a robust sensing solution. For the envisioned system, all of these limitations preclude

the use of an INS.

The envisioned system is intended to be on a low-cost platform. Introducing a single

sensor which costs more than the entirety of the remainder of the system combined,

is not feasible or desired. The system utilizes a battery to power the flight controls

surface, motor, control electronics, and sensors. Battery size, and thus weight, are

limited by the lifting capacity of the platform. The combination of large physical size

and high power consumption both move the size, weight and power (SWaP) curves

29

in the wrong direction. For these reasons, an INS is not a feasible sensing solution.

2.5.3 Applications

Boonstra et al’s approach to sensing a person rising from a chair utilizes two single-

axis accelerometers paired with a single-axis gyroscope [5]. Boonstra et al made

several assumptions in their approach that do not hold for all conditions for the en-

visioned system. Of primary concern is the relatively slow speeds assumed for the

motion of the sensors. Boonstra et al leveraged the fact that a person rising from a

chair occurs at a slow rate and can easily be distinguished from gravitational changes

as a result. While this fact still holds true for the envisioned system, it is known that

the system will be operating at high rates of speed and over a wider band that was

seen by Boonstra et al. Additionally, the envisioned system will rely on only a single

inertial sensor type, whereas Boonstra et al rely on two sensor types. Accelerometers

are used for the primary angular measurements, but a gyroscope is used to provide

rotation rate information to a filtering algorithm. For reasons described in Section

2.5.1, gyroscopes are unlikely to be suitable for the envisioned system as it operates at

a higher rate of speed than feasible gyroscopes for this application type can measure.

Additionally, Boonstra et al utilize a Butterworth filter that is tailored to the type of

motion expected when rising from a chair. This is another limitation of the approach.

The envisioned system is not limited to a small, essentially two dimensional motion,

as is the system measured by Boonstra et al. The envisioned system moves in mul-

tiple modes from hovering to vertical motion to lateral motion and combinations of

them. This motion profile does not lend itself to the application of a simple filter.

The approach taken by Tan et al more closely approximates a system that is usable in

30

the envisioned system [34]. The approach taken combines only a single type of inertial

sensor, the accelerometer, which meets the design objectives of the envisioned system.

Tan et al’s approach of finding a ”feasible” set of sensors and orientations also aligns

well with the design approach. However, Tan et al’s system relies on clearly defined

parameters for the alignment, both of the sensor relative to gravity as well as to each

other, of each sensor. This is not desired for the envisioned system. It is desired that

the approach used for the envisioned system would not be sensitive to alignment and

mounting positions of the individual sensors or the sensing system as a whole relative

to the body of the platform.

The MAGSONDE system designed by Harkins et al also utilizes a single type of iner-

tial sensor, a magnetometer [14]. This meets the design objectives of the envisioned

system for simplicity through the use of a single inertial sensor type. One of the

MAGSONDE system’s greatest strengths for its intended application presents one

of its greatest weaknesses for its applicability towards the envisioned system. That

strength is its ability to operate irrespective of the source of the magnetic field it

is operating in. This allows the device to operate even when strong interference is

present, a very desired feature for any real world system. However, the design of the

MAGSONDE system leverages this fact so deeply that it creates a deficiency for use

in other applications.

MAGSONDE is agnostic to the source of the signal provided and only relies on zero

crossings of the signal for reference to rotation. It does not account for tilt or pitch

of, in this case, the projectile being tracked. The resilience to noise is of great benefit

to any system, but the utility of MAGSONDE as it relates to the envisioned system

is minimal.

31

From this review, it is clear that research utilizing limited sets of inertial sensors have

addressed some of the needs for the envisioned system, but not all of them. The

reviewed systems are limited in range, scope or size and do not meet the needs of a

battery-operated, aerial platform moving at high rates of speed. Additional research

is required to create a system which meets all of these needs.

32

CHAPTER 3: APPROACH

The most basic element of the proposed system is that an accelerometer is capable of

sensing tilt while rotating. To validate this capability, a test platform was constructed

to experimentally verify the system will perform as expected. The test platform fea-

tures a digitally controlled stepper motor capable of speeds ranging from 0Hz up to

1kHz. Control of the motor is achieved through a serial interface provided by the

device manufacturer.

Atop the motor assembly is a flywheel with a custom designed sensor package (Figure

3.1). The system is relatively modular and includes a microcontroller to gather the

required sensor readings. The sensor package includes a short-range radio frequency

(RF) serial data link to provide ease of sensor data gathering. The information is

tabulated on the remote serial terminal. Mounted within the flywheel is a series of

lithium-polymer battery cells [32], which provide sufficient power to the system for

approximately 20 minutes of run time. Additionally, the package provides for the

mounting of multiple accelerometers. Each sensor has a direct link to an analog-to-

digital converter (ADC) and can be used as a digital reference when required.

The complete system is small enough to allow for workbench testing and robust

enough to allow for extreme test cases. As a safety precaution, the entire system is

enclosed in a quarter inch thick, clear polycarbonate case. The system is capable of

producing repeatable results and providing a sustained testing environment. The pri-

33

Figure 3.1: Custom Designed PCB and Sensor Package

mary limitation to the system is the life of the batteries powering the sensor package.

Multiple sensor tests are possible on a single set of batteries. The inexpensive cost

of the battery packs allows for multiple sets of batteries being available for extended

testing periods.

Experimental results show that accelerometers are capable of producing promising

results when placed directly at the center of rotation of a moving platform (Figure

3.2). In a sample test, a single-axis accelerometer placed at the center of the test

platform (Figure 1.3) was able to produce a relatively accurate result to within less

than 1◦ of the actual value. The sensor was mounted to a custom designed circuit

board with the sensitive axis along the z-direction.

These results were only obtained after significant post-processing and waveform smooth-

ing. The raw results show large amounts of noise induced by both mechanical vibra-

tion of the test platform as well as limitations of the sensor itself. The post-processing

requires complex algorithms, which are computationally expensive and decrease the

34

amount of usable data and the effective data rate. In the name of both efficiency and

feasibility, such a process does not lend itself to integration with actual systems, but

does validate that tilt can be measured from within a rotating frame.

Figure 3.2: Sample Accelerometer Data at 17.5◦ Incline

Based on these results, the envisioned approach does appear to be supported by tests

utilizing actual hardware in the loop. Further development of the number and orien-

tation of sensors follows.

35

CHAPTER 4: IMPLEMENTATION

The envisioned embodiment, represented by a system of equations that describe the

motion of the vehicle within the world frame and the motion of the sensors within the

body frame of the vehicle, was determined to enable analysis of the system. Variables

used in the analysis have been tabulated in Table 4.1. Outputs of the system will

represent the pose of the vehicle relative to the world frame.

Table 4.1: List of Variables

Variable Type Variable Description Typical Value

Fixed Variablesg Gravitational Acceleration 9.8065 m/s2

π Pi 3.14159265

Independent Variables

θ Angle Radω Angular Velocity rad/secvB Speed m/sect Time sec

Dependent Variablesa acceleration m/s2

α Angular Acceleration rad/s2

V Speed of point m/sec

The model of the sensor system was derived based upon a strain-gauge accelerometer

that uses a precisely calibrated mass for measurement. Figure 4.1 shows an unbonded

strain gauge accelerometer where 1is the mass, 2, 3, 4, and 5 are the springs, 6,

7, 8, and 9 are the strain gauges, 10 is the case and a is the acceleration of the

accelerometer. When the mass deforms the springs due to acceleration, the strain

gauges will form an electrical signal proportional to the acceleration, which can be

measured to determine the rate of acceleration (Figure 4.2). This will allow for

36

the use of standard kinematic equations of motion to describe the motion of the

sensor components as well as the vehicle. To use accelerometers for measurement and

calculations, the initial orientation and angular velocity of the rotating body must

typically be known [19]. This system design will not require this rigorous calibration

procedure. Estimates can be made to represent the mass of the sensor’s core based

on specifications provided by manufacturer datasheets.

Figure 4.1: Unbonded Strain Gauge Accelerometer Basic Mechanism

Figure 4.2: Suspended Mass Motion Under Acceleration

The orientation of the two identical accelerometers is illustrated in Figure 4.3. The

sensitive axis of each accelerometer is mounted in the same plane and normal to the

center of rotation. The designed system does not require precise mounting relative

to the center of rotation. Mounting considerations such as distance from actual

center of rotation and differences in the z-axis mounting location can be accounted

for automatically during the calibration process and is further discussed in Section

4.1.

37

Figure 4.3: Example Accelerometer Mounting Orientation

4.1 System Functional Description

As the sensors begin to rotate, the suspended mass within each accelerometer is

pushed towards the outside of the rotating disc due to centripetal acceleration [15].

This effect is one of the primary physical characteristics that enable this system to

provide accurate sensor readings. Calibration will use this physical characteristic as

well. The weight of each proof mass and the distance from the center of rotation can

be closely estimated for both sensors. This enables an accurate estimate of the proof

mass deflection.

Since the mass of each accelerometer core is equivalent, rotating the sensor package

at any speed on a level surface should produce equal output signals (S1, S2). If the

results are not equivalent, an offset (δ1, δ2) can be applied to one or both sensors to

zero the summation of the sensor outputs (Equation 4.1). This process can be easily

automated, which greatly reduces human induced error. It also provides for the abil-

ity to extend the calibration cycle duration to increase the quality of the calibration.

Calibration can be performed at multiple acceleration rates to further fine-tune the

offsets.

38

∆Sensor1,2 = (S1 + δ1) + (S2 + δ2) = 0 (4.1)

In an example scenario; two accelerometers are attached to the test fixture (Figure

1.3) and are accelerated to full speed, slowed to a stop, reversed, and accelerated at

the same rate in the opposite direction. This process is repeated multiple times until

an acceptable calibration point is reached. By applying Equation 4.1 to the recorded

data, the resulting sum of the two combined sensor streams resolves to zero (Figure

4.4) and eliminates the induced acceleration from the platform’s rotation. This same

principle applies to lateral motion as well as the combination of lateral and rotational

motion cancellation.

Figure 4.4: Sample Collinearly Mounted Rotating Accelerometer Data

The following kinematics equations [36] will be used to describe the system and relate

the world frame to the body frame:

For non-rotational motion of the system,

VB = VW + VB/W (4.2)

39

where VB is the speed of the body, VW is the speed of the world frame, and VB/W

is the speed of the body relative to the world frame. If the world frame is stationary

then we will just need the speed of the moving body and VB = VB/W .

The sensor system is placed on a uniformly shaped, rotating disc Figure 4.5. The disc

is described by its radius, R, the rate of rotation, ω/ θ̇, and O the center of rotation.

The sensor system disc is mounted to the rotating base-plate of the test platform.

This disc provides a consistent mounting of the sensors for each test and allows for

repeatability.

Figure 4.5: Rotating Sensor Platform

The state of the sensors is defined by a series of equations. The first set of equations

specifies

ω =dθ

dt(4.3)

α =dω

dt(4.4)

where ω is the angular velocity of the system, θ is the angle of rotation of the system,

a is the angular acceleration of the system and t is time. If the angle of rotation

is constant then ω = a = 0. In the test scenarios, this information is known and

provides ”ground truth” data used for validation.

40

Simple rotational motion of a point on the body is defined as,

V = rsω (4.5)

at = rsα (4.6)

an = rsw2 (4.7)

a =√a2r + a2

n (4.8)

Where V is the speed of the point, r is the radius from the point to the center of

the body, at is the tangential acceleration of the point on the body, ar is the radial

acceleration of the point on the body, and a is the total acceleration of the point.

Several sensor placements were explored to determine the most effective combination

and orientation of sensors. For the envisioned system, only two degrees of measure-

ment are required; roll and pitch. It is important to consider that the sensors in all

configurations are rotating and thus will sweep through the roll and pitch positions

repeatedly. It is this fact of the system where simplification in the design of the sensor

package can be found. Several constants will be employed for calculations and are

tabulated in Table 4.2.

Table 4.2: System Operational Constants

Constant Valuers 6”ω 50 Hzan Max Error 1%

The first sensor combination is the simplest and involves a single sensor placed on the

test platform, which is exhibiting non-accelerating, simple, rotational motion (Figure

4.6). This sensor is capable of measuring along a single axis. When the system is

41

accelerated, the proof mass of the sensor is pushed towards the outside of the disc

and the acceleration can be measured by the sensor along the sensitive axis; depicted

as an. Assuming the sensor is perfectly aligned with the center of rotation, no tan-

gential forces will be measured and the acceleration can be directly measured. From

the acceleration, information about the rotation rate (ω) can be measured if the

radius from the center of rotation is known. Applying Equation 4.9 and using the

constants provided in Table 4.2 yields an ideal sensed acceleration, which is identical

to the normal acceleration in this case, of 15,000 in/s2.

an = ω2rs (4.9)

Figure 4.6: Single Sensor, Ideal Placement, Purely Rotational Motion, Non-Accelerating

For practical systems, such as the one envisioned, it is very difficult to achieve per-

fect orientation of the sensor itself with the center of rotation; to do so would be a

very involved process and adds undesired costs and difficulty and reduces the overall

utility of the envisioned design. Practical systems will always have some degree of

offset from the center of rotation (Figure 4.7).

The effect of this mounting offset can be readily calculated and bounds to the mount-

ing error can be calculated. Utilizing the ideal sensed value range and solving for φ in

Equation 4.10, yields a maximum mounting error of +/- 8.1◦. This is within a range

that is feasible for a human operator to install and certainly well within the tolerance

42

Figure 4.7: Detailed View of Sensor Offset from Normal

of an automated placement system.

asens = an cos(φ) (4.10)

This sensor design, however, is not capable of performing the desired measurements

under all envisioned operating conditions. A single sensor is not capable of differ-

entiating rotational acceleration from linear acceleration from tilt. External input

would be required to provide rotational acceleration and linear acceleration in or-

der to extract tilt. To demonstrate this limitation, three cases were considered: the

sensitive axis of the accelerometer along the direction of platform acceleration, the

sensitive axis of the accelerometer normal to the direction of the platform accelera-

tion, and the sensitive axis normal to and out of plane with the sensitive axis (vertical

acceleration). In the case where the acceleration is along the sensitive axis, (Figure

4.8(a)), the acceleration of the platform is sensed directly, this poses an immediate

issue for sensing tilt in the direction of the vehicle acceleration. Equation 4.11 rep-

resents this relationship and applying it using a platform acceleration (aplatform) of

183 in/s2 (0.5g) with perfect mounting of the sensor results in a sensed acceleration

of (asens) 14817 in/s2 or a 1.2% change in sensed output. In the normal case (Figure

4.8(b)), the acceleration of the platform is not sensed as only accelerations within the

sensitive axis are and poses no hindrance to the sensor application. Similarly, accel-

eration in the vertical direction (Figure 4.8(c)), which is normal to and out of plane

with the sensitive axis, is not sensed and poses no hindrance to the sensor application.

43

Figure 4.8: Single Sensor Undergoing Acceleration (a) Along, (b) Normal and (c) Vertically Normal to Sensitive Axis

asens = (an cos(φ)− aplatform cos(ψ)) (4.11)

As described by [35], the relationship between sensed acceleration and tilt is rep-

resented by Equation 4.12. Applying this equation to this case and removing the

known acceleration caused by the platforms rotation results in a sensed tilt of 28.7◦.

This is an obviously unacceptable result and this sensor design was ruled out from

consideration. This error is compounded further if the platform is tilted, relative to

gravity, and is accelerating (Figure 4.9). In the worst case, the sensor will measure

not only the acceleration of the platform, it will measure a tilt angle as well. There

is no method using a single sensor which is capable of isolating just the platform tilt

44

from the motion of the platform itself.

θtilt = arcsin(asensg

) (4.12)

Figure 4.9: Single Sensor Undergoing Acceleration (a) Along, (b) Normal and (c) Vertically Normal to Sensitive Axis

In an attempt to compensate for these external forces, a second sensor was added in

line, but mounted in the opposing direction of the first sensor (Figure 4.10 and 4.11).

With the sensors mounted in opposing directions, the effect of acceleration of the

platform can be intrinsically compensated for as well as the rotational acceleration of

the platform itself. A positive reading for one sensor will register as a negative reading

for the other when the platform accelerates parallel to the sensors and is computer

simply by averaging the sensor outputs (Equation 4.13). Similarly, the effect of rota-

tion of the platform itself can be eliminated by subtracting the baseline acceleration

of each sensor when the platform is rotating at a stable speed and is sitting level. In

the case of acceleration normal to the sensors, no effect is seen as in the single sensor

design.

asens =(aS1 + aS2)

2(4.13)

With the dual sensor setup, the maximum mounting error must again be computed

(Equation 4.14). This results in a combined mounting error of +/- 8.1◦, the same as

the single sensor setup. This result is expected, but compounds the mounting accu-

racy problem. Averaging across both sensors, each sensor must be mounted within a

45

Figure 4.10: Dual Sensor, Ideal Placement, Purely Rotational Motion, Non-Accelerating

+/- 4.05◦ accuracy. While still feasible, it increases the likelihood of mounting error

causing undesired results.

Figure 4.11: Dual Sensor Undergoing Acceleration (a) Along, (b) Normal and (c) Vertically Normal to Sensitive Axis

asens =aS1 cos(α) + aS2 cos(β)

2(4.14)

46

An additional source of error is the placement of sensors on the disc radius from the

center of rotation. Offset mounting error is present when the distance from the center

of rotation for each sensor is not identical. This error source was not present for the

single sensor implementation design and was introduced by the inclusion of a second

sensor. This error is compounded when the rate of rotation increases. Equation 4.15

is derived from the normal acceleration of a point on a rotating surface (Equation

4.7).

an = (rideal + rsensor)ω2 (4.15)

It is possible to address this mounting error similarly to the offset angle previously

discussed. If the distance between the two sensors is fixed and known, it is possible to

isolate and remove the error introduced my the mounting offset from center. Equation

4.14 can be modified to incorporate this mounting offset from center and is represented

by Equation 4.16

asense =ω2rS1 cos(α) + ω2rS2 cos(β)

2(4.16)

The analysis of the dual sensor setup indicates feasibility of the design to the desired

implementation. The design was chosen to take forward into the analysis of next

steps and future work; both of which are described in Section 5.

4.2 Implementation Considerations

For simplicity, the initial system design will only be capable of validating the sys-

tem’s ability to measure tilt from within a rotating frame. The system will have

the rotation rate (ω) and yaw position (θ) provided as a known value. The sys-

tem is envisioned as implemented on a simple microcontroller with minimal power

requirements and cost. To that end, it is important that the system can operate ef-

ficiently at the 5Hz designed rotation rate. Inexpensive microcontrollers are capable

of single-digit kHz operation rates which are capable of keeping up with this rotation

47

rate with a sampling rate of below 50Hz. However, memory is often limited to tens

of kilobytes which must contain all software for data retrieval, processing, and output.

Special attention must be paid to the design of the software algorithms to avoid ex-

ceeding the capacity of the microcontroller. Performing a large number of rotations

in sequence using Euler angles is not desirable. Unit quaternions provide a clean set

of coordinates for the surface of a 3-sphere and makes it much easier to reason about

the rotations geometrically. For this reason, it is often more intuitive to incorporate

results using a quaternion when interpolating, as is required for this application due

to the high rate of speed.

For example, an aircraft‘s orientation is represented by three Euler angles: α = 90,

β = 0, γ = 0; where α is the angle between the x-axis and the N-axis or line of nodes.

β is the angle between the z-axis and the Z-axis and γ is the angle between the N-axis

and the X-axis. Simply reading off the angles does not provide clarity as to the actual

orientation of the craft as the assignment of axes to the fixed frames has not been

defined.

The reference frame used for this example is Figure 4.12.

Figure 4.12: Reference Frame [4]

After applying the first rotation, α, the orientation of the craft could be any one of

the possibilities illustrated in Table 4.3.

By changing the angle descriptor to Tait-Bryan angles, it is easily communicated that

the orientation described by the angles given above is a roll of 90 degrees (Figure 4.13).

48

Table 4.3: Potential Aircraft Orientations

Tait-Bryan angles, as described in Section 2.3.1, are an application of Euler angles.

Several limitations of Euler angles were described and there is an additional consider-

ation, which must be analyzed for implementation in the system described; memory.

Matrices representing quaternions use less memory than Euler angle matrices, which

allows programming space for additional processing, if needed. Storing a quaternion

requires 16 bytes of memory if using single precision floating point numbers (4 matrix

elements, 4 bytes per element) where storing a rotation matrix requires 36 bytes of

memory (9 matrix elements, 4 bytes per element). Similarly, computation utilizing

quaternions takes fewer operations. As an example, two sequential rotations with a

quaternion requires 28 operations (4 matrix elements each requiring 4 multiplication

and 4 addition operations). In the same example, an Euler angle rotation matrix

approach required 45 operations (9 matrix elements each requiring 3 multiplication

and 2 addition operations).

Figure 4.13: Described Orientation

An embodiment of the envisioned physical system is described in Figure 4.14. The

system is described by three key component angles; θ the nutation angle, φ the yaw

angle, and ψ the projected angle on the XY plane. It is from these angles that

49

the system description in quaternions is derived using the tranfer derivation between

quaternions and Euler angles resulting in Equations 2.18 and 2.17. To transform the

system from the Euler representation, it must first be described analytically. This

analysis was performed in Section 4.1 and is represented by Equation 4.12 and 4.16

and further derived here.

Figure 4.14: Physical System Description

As was previously stated, the rotation angle and rate is assumed to be a known value.

Utilizing this knowledge, the values for pitch and roll can be stated by Equations 4.17

and 4.18, respectively. The value of asense is defined by Equation 4.16.

φ = arcsin(asenseg

),where ψ = 90◦ or 180◦ (4.17)

θ = arcsin(asenseg

),where ψ = 0◦ or 270◦ (4.18)

50

ψ = (provided value) (4.19)

Utlizing Equations 4.17, 4.18, and 4.19 with Equation 2.17 (the conversion between

Euler and Quaternion representation) results in the quaternion representation of the

system based on sensor values of Equation 4.20.

q =

cos(arcsin

(atsense2

2) cos(

arcsin(apsense

2

2) cos(ψ

2) + sin(

arcsin(atsense

2

2) sin(

arcsin(apsense

2

2) sin(ψ

2)

sin(arcsin

(atsense2

2) cos(

arcsin(apsense

2

2) cos(ψ

2)− cos(

arcsin(atsense

2

2) sin(

arcsin(apsense

2

2) sin(ψ

2)

cos(arcsin

(atsense2

2) sin(

arcsin(apsense

2

2) cos(ψ

2) + sin(

arcsin(atsense

2

2) cos(

arcsin(apsense

2

2) sin(ψ

2)

cos(arcsin

(atsense2

2) cos(

arcsin(apsense

2

2) sin(ψ

2)− sin(

arcsin(atsense

2

2) sin(

arcsin(apsense

2

2) cos(ψ

2)

(4.20)

This representation will be important for future work, described in Section 5. In

particular, the quaternion representation will be used for simulation and eventual

hardware implementation of the envisioned sensor suite.

51

CHAPTER 5: CONCLUSION AND FUTURE WORK

This paper has presented a design for a sensor system which can accurately measure

the tilt of a rotating platform, using only inertial sensors, from within the rotating

frame. The system employs two accelerometers mounted in opposing directions, which

are connected to a low-cost, digital microcontroller driven system that can provide

tilt measurements of a system rotating at 5Hz. The system design addresses errors

induced from mounting errors due to placement of sensors relative to the center of

rotation both in radius and angle.

A review of the current state of the art in inertial sensors systems was presented

which showed the possibilities present as well as key limitations of each. Ranging

from limited sensing bandwidth to poor applicability to the envisioned system, each

sensor was limited as a standalone component. Combinations of sensors were ex-

plored which addressed the shortcomings of individual sensors. These applications,

while improvements to the sensors themselves, all failed to meet the requirements

set forth in this project. Sensor systems were explored as well, but were ultimately

discounted due to their size, weight, and power needs, as well as cost.

Finally, a sensor design concept which employs single-axis accelerometers was de-

scribed. This system uses the rotation of the vehicle to its advantage and explores a

potentially minimal set of sensors, which meet the goal of sensing tilt of the platform

from within the rotating frame. The system addresses the issues of signal isolation

52

under acceleration and provides a sensing modality that is largely robust against en-

vironmental factors.

Special considerations were paid to the planned implementation in hardware of the

system. Specifically, issues surrounding memory and operational steps were explored.

Approaches for representing the system state digitally were explored. Euler angles

were shown to be the most common format, but exhibited limitations in terms of

complexity, propensity for error, and memory. Quaternions were shown to be a ro-

bust solution which, provide for a useful descriptor of the system while minimizing

both memory and computation needs.

Further development is required to parameterize the system to enable the development

of a simulation environment to validate the envisioned system functionality. Simu-

lation can be performed using a standard mathematical evaluation package, such as

Matlab. The primary simulation components, which must be developed, include a

physical model of the accelerometer, a physical model of a rotating disc and a method

for perturbing the rotation rate and tilt angle of the simulated disc. The first val-

idation steps are to prove that the system produces the expected output when the

two raw accelerometer values are summed as described previously. Upon validation,

sensor mounting error both in placement and orientation will be added and detection

of this error will be tested and validated. Finally, perturbations of the disc tilt will

be added. This simulation will enable arbitrary angle assignments to be tested and

validated.

This paper has shown that it is possible to develop an inertial sensor system, which

enables the measurement of an agile system from within the motion frame of the

system. It employs tried and true sensors and mathematical techniques in a novel

53

way, which minimize the cost and risk associated with the system development. The

author looks forward to continued work developing and refining the system as well as

developing an algorithm suite that is portable to additional applications.

54

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