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Institutionen för systemteknik Department of Electrical Engineering Examensarbete
Institutionen för systemteknik
Department of Electrical Engineering
Examensarbete
Chest Observer for Crash Safety Enhancement
Examensarbete utfört i Reglerteknik
vid Tekniska högskolan i Linköping
av
Christian Blåberg
LITH-ISY-EX--08/4049--SE
Linköping 2008
Department of Electrical Engineering
Linköpings universitet
SE-581 83 Linköping, Sweden
Linköpings tekniska högskola
Linköpings universitet
581 83 Linköping
Chest Observer for Crash Safety Enhancement
Examensarbete utfört i Reglerteknik
vid Tekniska högskolan i Linköping
av
Christian Blåberg
LITH-ISY-EX--08/4049--SE
Handledare:
Christian Lundquist
isy, Linköpings universitet
Tohid Ardeshiri
Autoliv Sverige AB
Examinator:
Thomas Schön
isy, Linköpings universitet
Linköping, 13 June, 2008
Avdelning, Institution
Division, Department
Datum
Date
Division of Automatic Control
Department of Electrical Engineering
Linköpings universitet
SE-581 83 Linköping, Sweden
Språk
Language
Rapporttyp
Report category
ISBN
Svenska/Swedish
Licentiatavhandling
ISRN
Engelska/English
Examensarbete
C-uppsats
D-uppsats
Övrig rapport
2008-06-13
—
LITH-ISY-EX--08/4049--SE
Serietitel och serienummer ISSN
Title of series, numbering
—
URL för elektronisk version
http://www.control.isy.liu.se
http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-ZZZZ
Titel
Title
Observatör för Bröstkorg
Chest Observer for Crash Safety Enhancement
Författare Christian Blåberg
Author
Sammanfattning
Abstract
Feedback control of Chest Acceleration or Chest Deflection is believed to be a
good way of minimizing the risk of injury. In order to implement such a controller
in a car, an observer estimating these responses is needed. The objective of the
study was to develop a model of the dummy’s chest capable of estimating the
Chest Acceleration and the Chest Deflection during frontal crashes in real time.
The used sensor data come from car accelerometer and spindle rotation sensor
of the belt, the data has been collected from dummies during crash tests. This
study has accomplished the aims using a simple linear model of the chest using
masses, springs and dampers. The parameters of the model have been estimated
through system identification. Two types of black-box models have also been
studied, one ARX model and one state-space model. The models have been tested
and validated against data coming from different crash setups. The results show
that all of the studied models can be used to estimate the dummy responses, the
physical grey-box model and the black-box state-space model in particular.
Nyckelord
Keywords
frontal crash, chest model, dummy, parameter estimation, ARX, N4SID
Abstract
Feedback control of Chest Acceleration or Chest Deflection is believed to be a
good way of minimizing the risk of injury. In order to implement such a controller
in a car, an observer estimating these responses is needed. The objective of the
study was to develop a model of the dummy’s chest capable of estimating the
Chest Acceleration and the Chest Deflection during frontal crashes in real time.
The used sensor data come from car accelerometer and spindle rotation sensor
of the belt, the data has been collected from dummies during crash tests. This
study has accomplished the aims using a simple linear model of the chest using
masses, springs and dampers. The parameters of the model have been estimated
through system identification. Two types of black-box models have also been
studied, one ARX model and one state-space model. The models have been tested
and validated against data coming from different crash setups. The results show
that all of the studied models can be used to estimate the dummy responses, the
physical grey-box model and the black-box state-space model in particular.
Sammanfattning
Genom att använda återkoppling av storheterna bröstacceleration och bröstintryck antas man kunna minska risken för skador vid krockar i personbilar. För
att kunna implementera detta behövs en observatör för dessa storheter. Målet
med denna studie är att ta fram en modell för att kunna skatta accelerationen i
bröstkorgen samt bröstintrycket i realtid i frontala krockar. Sensordata som använts kom från en accelerometer och en givare för att mäta rotationen i bältessnurran. Detta har gjorts genom att modellera bröstkorgen med linjära fjädrar och
dämpare. Dess parametrar har skattats från data från krocktester från krockdockor. Två s.k. black-box-modeller har också tagits fram, en ARX-modell och en på
tillståndsform. Modellerna har testats och validerats mha data från olika sorters
krocktester. Resultaten visar att alla studerade modeller kan användas för att skatta de ovan nämnda storheterna, den fysikaliska modellen och black-box-modellen
på tillståndsform fungerade bäst.
v
Acknowledgments
I would like to thank the people at Autoliv in Vårgårda for giving me a good
introduction to the area of crash safety, especially my supervisor Tohid Ardeshiri
and Mika Himiläinen. Many thanks also go to my supervisor at LiTH Christian
Lundquist for answering a lot of questions on system identification. I would like
to express my gratitude to my wife Hanna for her love and support.
vii
Contents
1 Introduction
1.1 Background . . . . . . . . . . . . . . .
1.1.1 The Modern Restraint System
1.1.2 Developing a Restraint System
1.1.3 Evaluating a Restraint System
1.1.4 Adaptive Restraint Systems . .
1.2 Aim of Thesis . . . . . . . . . . . . . .
1.3 Limitations . . . . . . . . . . . . . . .
1.4 Method . . . . . . . . . . . . . . . . .
1.5 Tools . . . . . . . . . . . . . . . . . . .
1.6 Autoliv . . . . . . . . . . . . . . . . .
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2 Crash Tests
2.1 Anatomy of Dummy . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Sensors and Data . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3 Theory & Method
3.1 Physical Model of Dummy’s Chest . . . . . . . . . . . .
3.1.1 Calculating Belt Displacement . . . . . . . . . .
3.1.2 Free-Body Diagram of Chest . . . . . . . . . . .
3.1.3 Equations of Motion . . . . . . . . . . . . . . . .
3.1.4 State-Space Equations of the Model . . . . . . .
3.2 Parameter Estimation . . . . . . . . . . . . . . . . . . .
3.3 Evaluating Model Performance . . . . . . . . . . . . . .
3.4 Black-Box Modeling . . . . . . . . . . . . . . . . . . . .
3.4.1 ARX Model . . . . . . . . . . . . . . . . . . . . .
3.4.2 Subspace Methods for State-Space Identification
3.5 Validating the Model . . . . . . . . . . . . . . . . . . . .
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4 Results
4.1 R16 setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Sled Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Fullscale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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x
Contents
5 Concluding Remarks
5.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Further Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Bibliography
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A Comparison Plots of Simulated and Measured
A.1 R16 . . . . . . . . . . . . . . . . . . . . . . . .
A.1.1 ARX model Nmax = 6 . . . . . . . . . .
A.1.2 ARX model Nmax = 10 . . . . . . . . .
A.1.3 State-Space N = 16 . . . . . . . . . . .
A.1.4 State-Space N = 7 . . . . . . . . . . . .
A.1.5 Grey-Box . . . . . . . . . . . . . . . . .
A.2 Sled Tests . . . . . . . . . . . . . . . . . . . . .
A.2.1 ARX model Nmax = 6 . . . . . . . . . .
A.2.2 ARX model Nmax = 10 . . . . . . . . .
A.2.3 State-Space N = 16 . . . . . . . . . . .
A.2.4 State-Space N = 7 . . . . . . . . . . . .
A.2.5 Grey-Box . . . . . . . . . . . . . . . . .
A.3 Full-scale . . . . . . . . . . . . . . . . . . . . .
A.3.1 ARX model Nmax = 6 . . . . . . . . . .
A.3.2 ARX model Nmax = 10 . . . . . . . . .
A.3.3 State-Space N = 16 . . . . . . . . . . .
A.3.4 State-Space N = 7 . . . . . . . . . . . .
A.3.5 Grey-Box . . . . . . . . . . . . . . . . .
Outputs
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Chapter 1
Introduction
The purpose of this chapter is to give a background to the problem, which leads
to the aim of the thesis. The chapter ends with a discussion of limitations.
1.1
Background
The topic of vehicle safety has been around ever since the birth of automobile
industry. To reduce the severity of injuries, the seatbelt and the airbag have been
introduced; these are often referred to as the restraint system. The objective of
the first safety belt, patented in 1903 by M. G.-D. Leveau, was to prevent the
occupant from ejecting out of the car during a crash. Later on, the belt’s goal was
to prevent impact between the steering wheel and the occupant.
1.1.1
The Modern Restraint System
During a crash, the velocity of the car decreases rapidly and so does the velocity
of the occupant, see Figure 1.1. Huge forces are applied to the occupant, in order
to decelerate it, and it is the restraint systems job to apply these forces in the
least harmful way possible. Another way of looking at it is by considering the
conservation of energy. The kinetic energy of the occupant has to be transferred
somewhere when its velocity decreases. Worst case would be if all this energy
would be transferred into deformation energy of the body, for example head and
chest. The restraint system needs to absorb this energy in a way that is the least
harmful to the occupant. That is to prevent the occupant from hitting the steering
wheel while exerting as little force against the chest as possible.
The Safety Belt
The three-point safety belt was introduced by Nils Bohlin at Volvo in 1959, and
quickly became standard for most cars. Today’s safety belts consist of webbing,
buckle, and retractor, see Figure 1.2. The webbing is the part of the belt system
that is in contact with the occupant. The retractor pulls the webbing in to tension
1
2
Introduction
(a) 0 ms after impact
(b) 60 ms after impact
(c) 100 ms after impact
Figure 1.1: Model of a dummy in the driver seat during crash, from the rigid-body
simulation tool MADYMO.
it against the occupant’s chest and lap. This is done by the rotation of a spindle
inside the retractor. The buckle is used to lock the belt into the floor pan of the
car. Two important features of the seatbelt are the pre-tensioning and the load
limiting.
(a) Retractor
(b) Buckle
(c) Webbing
Figure 1.2: Three components of the belt system.
The pre-tensioner’s task is to remove slack in the belt by tightening it when
a crash is detected. The reason of doing this is to create as much space between
occupant and steering wheel as possible, and also to give the belt an opportunity
to exert force on the occupant as early in the crash as possible. There are three
kinds of pre-tensioners: buckle pre-tensioner, lap pre-tensioner, and retractor pretensioner. The first two remove the slack by pulling the buckle or anchor plate
downwards, the second one by rotating the spindle. It is common to combine
the lap pre-tensioner with either one of the other two. This is to assure a proper
tightening of the belt over both lap and chest.
The load limiter is located inside the retractor. When the force applied on
the occupant’s chest by the belt become too large, it is appropriate to decrease
the force by paying the belt out. This is done through deformation of certain
1.1 Background
3
mechanical components in the load limiter. Some OEMs use two load limiting
levels; first force is limited to the higher level, then after some time the load
limiter switches down to the lower level.
The Airbag
The airbag system consists of an inflatable cushion (bag) and an inflater. The
inflater generates a gas flow to inflate the bag. The bag has a ventilation hole to
make a smooth impact between head or chest and bag.
1.1.2
Developing a Restraint System
When constructing a safety restraint system for a car, many simulations and crash
tests with crash test dummies are performed. These tests and simulations give us
information about how well the restraint system works in real life. Simulations
are performed as a complement to the real crash tests, since they are more cost
efficient. Both are vital in the development of a restraint system. It is desired that
a restraint system performs satisfactory for as many kinds of load cases (crashes)
and occupants as possible.
1.1.3
Evaluating a Restraint System
Organizations such as Euro NCAP1 perform crash tests and give ratings to new
cars. It is important for OEMs (Original Equipment Manufacturers) that their
restraint systems perform well in both crash tests and in real life. Even though the
correlation between the two is not verfied for the general case, it is often assumed
that one of the two will lead to the other. In crash tests, sensors are installed to
measure the dummy responses, for example chest and head acceleration and chest
deflection. The maximum of these responses during a certain time interval will
then serve as measures of the risk of injury, referred to as injury values. Every
restraint system strives to minimize these injury values in some way.
1.1.4
Adaptive Restraint Systems
State-of-the-art belt and airbag have a limited set of modes of operation [1], chosen
at the start of the crash. For example, a crash sensor can distinguish between a
severe crash and a not so severe crash and the restraint system can therefore
behave in two different ways. These modes are added to reduce the risk of injury
of the occupant. The safety belt and airbag can of course not behave the same way
during a crash with a car velocity of 60 km/h as during a crash with a velocity of
20 km/h. If they would, there is a possibility that they would do more harm than
good. Recent research has shown that occupant characteristics, for example size,
age, and seating position play a large part in the risk of injury [6] [3].
It is desired to increase the number of modes of operation. A way of introducing
an unlimited number of modes is by utilizing feedback control, i.e. measuring the
1 http://www.euroncap.com
4
Introduction
dummy responses and manipulating the restraint system in real time during a
crash.
Figure 1.3 shows a standard feedback setup. This is explained thoroughly in
[1], [2], and [7] among others. In these studies, a rigid-body model of the dummy
in a crash from the simulation tool MADYMO has been used as the plant. The
results are very promising; therefore feedback control is very appealing.
Figure 1.3: Closed loop system of an adaptive restraint system and occupant. ‘Dummy
Response’ consists of one or more of the dummy responses mentioned in Section 1.1.3.
‘Reference’ is the corresponding desired values of the dummy responses over time. ‘Crash
Pulse’ is the deceleration of the car over time.
The next step is to implement the controller in a crash test with dummies,
where sensors will provide the dummy responses instead of the model. However
real life implementation imposes a few problems, some of them are listed below.
• As of today, there is no cost efficient way of measuring the needed control
variables, i.e. it is complicated to mount an accelerometer on the occupant.
In most studies on this subject [8], [1], belt force is used to estimate these
control variables and it is assumed that they can be measured. However there
are currently no suitable sensors available at reasonable price for measuring
it.
• The existing models used in simulation environments, such as the rigid-body
simulation tool MADYMO, or the finite element simulation tool LS-Dyna,
are too complex and currently not possible to run in real time.
• The actuators needed to manipulate the restraint system are not at automotive industry price levels.
This thesis does not cover feedback control and actuators, instead this study
tries to solve the issues with the model and sensors by constructing a simple model
of the chest using inputs from sensor that are either already in cars today or will
be in the near future. Figure 1.4 shows how the model will be connected to the
controller and the restraint system in a real life implementation.
1.2 Aim of Thesis
5
Figure 1.4: Closed loop system with observer. The model provides the controller with
responses that cannot be measured explicitly.
1.2
Aim of Thesis
The aim is to estimate the dummy responses chest acceleration (CR) and chest
deflection (CD) using only the information about acceleration of the car and angle
of the belt spindle. This should be done in real time. To accomplish this, a model
of the dummy during the crash needs to be constructed. The aim of this thesis is to
present a few proposals of a simple model suitable for hardware implementation,
i.e. it should be cost and computationally efficient. Inputs to the model will
be acceleration of the car and belt displacement. Outputs will be CR and CD.
The outputs from the model will be verified against results from crash tests with
dummies. The model is built and tested in Matlab Simulink, and is to be used for
feedback purposes.
1.3
Limitations
The thesis will only cover frontal crashes with crash test dummies. Whether the
derived model of the dummy can be applied to real humans is not covered here.
The study is limited to the restraint systems effect on the chest, other parts of
the dummy are neglected. Although some of the tests looked on include the use
of airbag, the thesis focuses on the belt’s effect on the chest. Modeling the airbag
is more complex because contact with dummy and airbag happens roughly in the
middle of the crash while the belt and dummy has contact throughout the whole
crash. As of now, there is no way of controlling the airbag in the same way the
seatbelt can be controlled. A Hybride III mid-size male crash test dummy 2 has
been used in the crash tests.
2 http://www.edmunds.com/apps/vdpcontainers/do/vdp/articleId=105394/pageNumber=1
6
Introduction
1.4
Method
A total of three model types will be studied, all of them are linear. A physical
model of the dummy chest during crash will be developed. The parameters of
the model will be identified using gray-box modeling (parameter estimation). Two
different types of Black-Box models will also be examined.
ARX is a model using linear regression analysis to estimate its parameters.
N4SID is a subspace method for identification of state-space models.
For both of theses black-box models, two different models with different model
orders will be proposed. One with a lower order and one with a higher. Since the
requirements on the model still are non-precise, it is convenient to provide a few
alternatives.
1.5
Tools
Throughout the thesis, Matlab is used for the calculations. The System Identification Toolbox is the most used toolbox since it is used to create/estimate the
models.
1.6
Autoliv
Autoliv Sverige AB is a company within the Autoliv Inc. group, one of the world
wide leaders in manufacturing airbags, seatbelt and other safety equipment for
passenger cars. Autoliv Inc. has 80 companies, incl. joint ventures, and nearly
42,000 employees in 30 countries globally, 900 of them work at Autoliv Sverige in
Vårgårda. The turnover of Autoliv Inc. is 6 billion USD.
Autoliv is a worldwide leader in automotive safety, a pioneer in both seatbelts
and airbags, and a technology leader with the widest product offering for automotive safety. Most leading automobile manufacturers in the world are customers
of Autoliv’s products. The company consists of 80 subsidiaries and joint ventures
in 28 countries. Autoliv test their products in their 20 different crash test tracks
in 12 countries.
Autoliv develops, markets and manufactures airbags, seatbelts, safety electronics, steering wheels, anti-whiplash systems, seat components and child seats as well
as night vision systems and other active safety systems.
The subsidiary in Vårgårda focuses on developing and manufacturing airbags
and seatbelt components.3
Abbreviations
BRS Bobbin rotation sensor: A component that measures the cumulative angle
of belt spindle, which gives us belt displacement.
3 http://www.autoliv.com
1.6 Autoliv
7
CR Chest Retardation also known as Chest Acceleration: This is the acceleration
of the thoracic spine.
CD Chest Deflection: displacement of sternum relative to the spine.
OEM Original equipment manufacturer: in this thesis this refers to vehicle manufacturers.
Dictionary
Safety Cage Part of the car that is (hopefully) not deformed during crash.
Crash Pulse Acceleration of the safety cage of the car measured from a tunnel
accelerometer mounted underneath the seat.
Injury Value Measure of risk of injury, examples are maximum CD and maximum CR.
Pre-tensioning phase Period of time when the belt is pulled in against the
occupant, either by rotating spindle or pulling down buckle or anchor plate.
Load limiting phase Period of time when belt is paid out to limit the belt force.
Belt Force The force exerted along the belt near the retractor.
MADYMO Software package commonly used in the automotive industry. The
solver included is a general purpose numerical code employing rigid body
dynamics and finite element technology to solve for the solution to the Newtonian equations of motion.
LS-Dyna Advanced general-purpose multi physics simulation software package.
Its core-competency lie in highly nonlinear transient dynamic finite element
analysis using explicit time integration.
Chapter 2
Crash Tests
The available data from crash tests originates from three different crash setups,
which are shown in Figure 2.1.
R16 Setup
The R16 setup is the simplest case, consisting only of a hard seat, seat-belt and
dummy. The seat moves on a track and is accelerated in a way that is similar
to an acceleration from a crash. This thesis focuses on R16 setup tests because
most of the available crash test data were from that setup. It is assumed that this
case will be easier to model since there will be no contact between the dummy
and interior parts of the car, other than the seat. Figure 2.1a shows the dummy
strapped in the seat before an R16 setup test.
Sled Tests
These are similar to R16 setup tests but include more interior parts of a car, for
example steering wheel and most importantly an airbag, see Figure 2.1b.
Full-Scale Tests
This is the closest one can get to a real life crash. Here a full size prototype car
is crashed into a barrier. The resulting acceleration of the car depends on among
others its speed and what type of barrier is used. A Full-scale test is shown in
Figure 2.1c.
2.1
Anatomy of Dummy
The 50th percentile HYBRIDE III male dummy is used in most of the crash test
studied in this thesis. 50th percentile means that 50 percent of the adult population
is smaller in respect to weight and height. The dummy is a mechanical surrogate
for a human and is constructed to have similar dynamics as a human being. The
dummy has numerous sensors installed but only two are of interest in this thesis,
9
10
Crash Tests
(a) R16 setup
(b) Sled test
(c) Full-scale
Figure 2.1: Three different crash test scenarios. Source: Autoliv.
the accelerometer mounted on the thoracic spine, see Figure 2.2a, and the sensor
measuring compression of rib cage, see Figure 2.2b.
The human chest roughly consists of three parts [8]; the thoracic spine, the
ribs, and the sternum. The thoracic spine is connected to the ribs, which are
connected to the sternum.
2.2
Sensors and Data
From each crash test, the measured data signals of interest in this study are Crash
Pulse, Chest Acceleration of the dummy (frontal direction), Chest Deflection of
the dummy, and BRS. An example of what the data from one experiment might
look like is shown in Figure 2.3.
Throughout the thesis, when models are estimated and validated, only the
data from t = 0.01 up until t = 0.15 is used. Data before t = 0.01 is not used
because pre-tensioning has not yet occurred and there is still slack in the belt.
The dynamics of the system will change after pre-tensioning, therefore this thesis
focuses on the time during and after pre-tensioning. At t = 0.15 the crash is more
or less over, most signals are zero or close to zero and there is no need to control
CR and CD at this point.
BRS
The BRS is located inside the retractor. It measures the cumulative angle of the
rotating spindle using Hall sensors. The spindle has a number of equally spaced
2.2 Sensors and Data
(a) Spine
11
(b) Rib cage
Figure 2.2: Pictures showing the human chest. Thoracic spine is where Chest Acceleration is measured from on a dummy. Chest Deflection a measure of the compression of
the rib cage.
magnets, which give a signal to the sensor when a magnet is passed. Because
of this, this sensor will produce data containing non-equal sample times. In this
study the data is re-sampled to a higher sampling rate using zero-order-hold.
In the examined crash data there are two different of BRS versions, BRS 1 and
2. The improvements made in version 2 are the following:
1. The resolution was doubled from 48 samples per revolution to 96 samples.
2. it is now possible to detect movement in the spindle in both directions, BRS
1 could only detect pay-out of the belt.
In Section 1.1.1 it is stated that the belt is pulled in during pre-tensioning by
rotating the spindle, this can only be detected with BRS 2 and not with BRS 1.
However in this study, only BRS data after pre-tensioning is used, so the version
of BRS will only affect the resolution.
Tunnel Accelerometer
This sensor measures the acceleration of the safety cage. It is mounted underneath
the seat.
Chest Acceleration Sensor
This sensor is located on the spine of the dummy and measures the acceleration
in the dummy’s forward direction over time.
Chest Deflection Sensor
Measures the compression of the dummy’s chest through a potentiometer, in other
words the displacement of the sternum relative to the spine.
12
Crash Tests
(a) Crash Pulse
(b) BRS
(c) Chest Acceleration
(d) Chest Deflection
Figure 2.3: Example of the four signals used in the thesis, ‘Crash Pulse’ is measured
from an accelerometer underneath the seat, ‘BRS’ shows the rotation of the belt spindle,
‘Chest Acceleration’ is measured from an acceleromter in the spine of the dummy, and
‘Chest Deflection’ is the relative displacement of the sternum and the spine of the dummy.
Chapter 3
Theory & Method
There are two main approaches to choose from when building a model:
• Modeling from first principle, i.e. using laws of physics.
• System identification, using measurements from the true system.
A simple one-dimensional model is shown in Figure 3.1. In section 2.1 it was
stated that the Chest Acceleration is measured from the thoracic spine of the
dummy, and that the Chest Deflection is the displacement of the sternum relative
to the spine. The mass m1 represents the spine and the mass m2 represents the
sternum in Figure 3.1. The rib cage connecting the spine and the sternum is
modeled as a spring and a damper in parallel. The point ‘Seat Belt’ in the figure
is the point where the belt exerts force on the dummy. The connection between
this point and the sternum is also modeled as a spring and a damper in parallel.
The spring and damper represent clothing, flesh, and also stretching of the belt.
3.1
Physical Model of Dummy’s Chest
The variable w in Figure 3.2 is the displacement of the point on the belt segment
that is exerting force on the dummy’s chest along the longitudinal axis. The belt
is payed out during load limiting to release the pressure on the chest caused by
the belt, this corresponds to w decreasing and thereby decreasing the compression
of the springs in Figure 3.1. This will cause CR and CD to decrease. Increase
in CR and CD will occur by increasing w, i.e. either by pulling the belt in or by
decelerating the car.
3.1.1
Calculating Belt Displacement
For a car standing still, w is directly proportional to belt pay-out or pay-in, i.e.
BRS, with proportional constant R. If the car is accelerating or deaccelerating
13
14
Theory & Method
Figure 3.1: Model of occupant’s chest during crash.
and there is no movement in the belt, w will move as the car does. This gives:
ZZ
w = R × BRS +
d
ẇ = R × BRS +
dt
P ulse(t)dt
(3.1)
Zt
P ulse(τ ) dτ,
(3.2)
0
where P ulse(t) is the crash pulse from Chapter 2 measured in m/s2 . Figure 3.2
shows how the position of the belt is calculated. Displacement of belt relative to
the car is assumed to be directly proportional to angle of belt spindle, measured
from BRS signal. Displacement of car is calculated by taking the time integral
of the acceleration twice. The proportional constant R will be dealt with later in
Section 3.2.
3.1.2
Free-Body Diagram of Chest
Figure 3.3 shows the two masses, the two springs, and two dampers of the system.
In order to determine the position velocity and acceleration of the spine and sternum, their respective free-body diagrams are extracted, see Figures 3.4 and 3.5
respectively.
3.1 Physical Model of Dummy’s Chest
15
Figure 3.2: Displacement of belt w, derived from ‘Pulse’ and spindle rotation ‘BRS’.
3.1.3
Equations of Motion
Applying Newton’s second law of motion on the thoracic spine yields
X
F1 = m1 ẍ
F1,k1 = −k1 (x − s)
F1,c1 = −c1 (ẋ − ṡ),
which gives
m1 ẍ = −k1 (x − s) − c1 (ẋ − ṡ).
Same procedure for the sternum yields
X
F2 = m2 s̈
F2,k1 = −k1 (s − x)
F2,c1 = −c1 (ṡ − ẋ)
F2,k2 = −k2 (s − w)
F2,c2 = −c2 (ṡ − ẇ),
which gives
m2 s̈ = −k1 (s − x) − c1 (ṡ − ẋ) − k2 (s − w) − c2 (ṡ − ẇ).
Since CR is defined as ẍ the expression becomes
CR = ẍ = −
k1
c1
(x − s) −
(ẋ − ṡ).
m1
m1
(3.3)
16
Theory & Method
Figure 3.3: The model of the dummy’s chest.
Figure 3.4: Free body diagram of the
spine.
Figure 3.5: Free body diagram of the
sternum.
Finally, Figure 3.1 show that
CD = x − s.
3.1.4
(3.4)
State-Space Equations of the Model
The equations are written in linear state-space form. A continuous state-space
model is suitable because of the derivatives in the equations of motion.
ẋ(t) = A(θ)x(t) + B(θ)u(t),
(3.5)
y(t) = C(θ)x(t) + D(θ)u(t).
with


ẋ(t)
 x(t) 



x(t) = 
 ṡ(t)  ,
 s(t) 
w(t)

t
R
P
ulse(τ
)
dτ
,
u(t) =  0
d
dt BRS(t)
y(t) =
CR(t)
,
CD(t)
and
θ = k1
c1
k2
c2
m1
m2
R
T
.
(3.6)
3.2 Parameter Estimation
17
Then the matrices A, B, C, and D
 c1
k1
c1
− m1 − m
m1
1
 1
0
0
 c
k1
c1 +c2
1

A =  m2
− m2
m2
 0
0
1
0
0
0
c1
k1
c1
k1
− m1 − m1 m1 m
1
C=
0
1
0 −1
become
k1
m1


0
0
 0
0
0 
 c2

k2 
2

,
B
=
− k1m+k
 m2

m
2
2
 0

0
0
1
0
0
0
0 0
, D=
.
0 0
0

0
0 

c2

m2 R 
0 
R
(3.7)
(3.8)
The parameters in Equation 3.6 are unknown, and there is no straight forward way
of finding them for example by studying the dummy. Therefore the parameters are
estimated with the help of data. This is called paramter estiamtion or Grey-Box
modeling and is discussed in Section 3.2. When the parameters are estimated from
the data, the model in Equation 3.5 will be refered to as the ‘Grey-Box model’ for
the rest of this report.
3.2
Parameter Estimation
For a more detailed version of the following section, see [5]. Estimates to the
parameters in Equation 3.6 are sought, if d equals the number of parameters, then
the parameter vector becomes
 
θ1
 .. 
θ =  . .
(3.9)
θd
The task is now to estimate these parameters with the help of measurements. For
every value of the vector θ, the model provides at time t − 1 a prediction of the
true value of y(t). Denote this prediction
ŷ(t|θ).
(3.10)
This guess is evaluated by calculating the prediction error
(t, θ) = y(t) − ŷ(t|θ).
(3.11)
When data is collected over N sample times the following sum is formed
VN (θ) =
N
1 X 2
(t, θ)
N t=1
(3.12)
as a way of measuring the goodness of our parameter θ. In the case with multiple
outputs, (t, θ) is a vector. The expression then becomes
VN (θ) =
N
1 X T
(t, θ)Q(t, θ)
N t=1
(3.13)
18
Theory & Method
where Q is a quadratic matrix of size equal to the number of outputs. Q is often
also diagonal. It is natural to choose the value for θ that minimizes (3.13), that is
θ̂N = arg minVN (θ).
(3.14)
θ
This optimization problem is often non-linear and can therefore not be solved
analytically, hence a numerical iterative method is used to find the minimum.
3.3
Evaluating Model Performance
In order to determine the performance of the model, one needs to compare the
output given by the model ŷ(t) to the measured outputP
y(t). So the error becomes
t=t
e(t) = y(t) − ŷ(t). A good measure of performance is t=0max e(t)2 . However in
this thesis the models have two outputs with different units and sizes of errors.
Therefore a percentual measure is used. This fit is taken from Matlab’s System
Indetification Toolbox and is defined as
f it = 100(1 −
||y(t) − ŷ(t)||
),
||y(t) − ỹ||
(3.15)
where y(t) is measured signal, ŷ(t) is model output and ỹ is the mean of measured
signal. The norm is the 2-norm. Each model will produce a separate fit for each
experiment and output. Define f itCR,i as the fit for output CR in experiment i,
and the same for CD. Then define
Pn
f itCR,i
(3.16)
f itCR,tot , i
Pn n
f itCD,i
f itCD,tot , i
(3.17)
n
f itCR,tot + f itCD,tot
f itT OT ,
,
(3.18)
2
where n is the number of experiments. The measure f itT OT will be the value used
when comparing two or more models with each other.
3.4
Black-Box Modeling
The term black-box originates from the fact that nothing is known about the system, besides the measured outputs and inputs. The derived model will therefore
solely depend on the measurements of the true system. Model order (or complexity) is chosen by the user. The inputs chosen here are BRS-signal and Crash
Pulse, and the outputs are CR and CD. There are many different forms of blackbox models. In this study two different model types are used, the ARX-model
and a model in state-space form. For each of these two methods, two models are
derived, one with a small order and one with a larger order.
3.4 Black-Box Modeling
3.4.1
19
ARX Model
A more detailed description of ARX models can be found in Chapter 12 in [5].
The ARX model is defined by
A(q)y(t) = B(q)u(t − nk) + e(t)
A(q) = 1 + a1 q
−1
B(q) = b1 + b2 q
(3.19)
+ · · · + ana q
−1
−na
+ · · · + bnb q
(3.20)
−nb+1
,
(3.21)
where e(t) is white noise and q is the shift operator so that q −1 y(t) = y(t − 1).
For multiple-input multiple-output (MIMO) systems like the one in this thesis
where n equals the number of inputs and m equals the number of outputs, then
y(t) ∈ Rn×1 and u(t) ∈ Rm×1 . This means that A(q) and B(q) will be matrices
of sizes n × n and n × m respectively. The input delay is defined by nk ∈ Rn×m .
This matrix is chosen by the user. The remaining parameters to be chosen are na,
a matrix of same size as A(q) and nb, a matrix of same size as B(q).
Choosing Model Order
The process of choosing model orders is an important step in system identification.
In this process, only data from R16 tests have been used since the focus is on
that data set. The System Identification Toolbox in Matlab has functions to
automatically derive the best model order, but unfortunately they only work for
single-output systems. Since the system in this thesis has two inputs and two
outputs, the complete model order N N has 12 elements.
NN = NA
NB
NK =
na11
na21
na12
na22
nb11
nb21
nb12
nb22
nk11
nk21
nk12
nk22
(3.22)
First a good guess of the model orders N N is needed. One way of doing this
is to use the functions arxstruc and struc on each combination of input and
output. Call i output number and j input number. The function struc is called
with three arguments, the range of naij the range of nbij and the range of nkij .
However nkij can be derived by using the matlab function delayest on output i
and input j. This function takes two signals and computes the optimal delay for
best correlation between them (based on ARX computations). The range of naij
and nbij are both chosen as 1 to Nmax . Output of the function struc is a set
of all the combinations of parameters naij ,nbij , and nkij , these are then passed
into the function arxstruc. The output of that function will be the best choice
of naij , nbij , and nkij , called N Ninit . But there is no guarantee that this choice
of N N is the optimal one. Some kind of iteration will be needed.
The iterations are done by adding +/ − 1 to one single element in the N A
and N B matrix. The delay N K will be left untouched because it is assumed that
those values are the best ones. 8 elements to alter in two ways gives a total of 16
new models. The iteration process is described below.
1. Calculate the f itT OT for the model with order N Ninit
20
Theory & Method
2. Create 16 new models and calculate their respective fits. If any element in
N A or N B becomes larger than Nmax , remove that model.
3. Compare the current fit with the fit of the new ones. If the current model
is best, then stop. Otherwise keep the new model that has the maximum fit
and go back to step 2.
The result of this iteration process can seen in Figure 3.6. The graph shows
that f itT OT increase as the order goes from 2 up to 10, but then the increase
stops. Therefore the more complex ARX model will have Nmax = 10. The other
number that is picked is Nmax = 6. This pick will give a relatively good fit with
a small order.
Figure 3.6: The f itT OT of iterated ARX model for different values of Nmax .
3.4.2
Subspace Methods for State-Space Identification
The theory of subspace methods is quite substantial, therefore just the basics are
briefly discussed here. The methods have their origin in state-space realization
theory as developed in the 1960s. One of the largest differences between subspace
methods and ‘classical’ state-space identification methods, is that the subspace
methods utilize QR factorization and singular-value decomposition as their main
computaional methods, where ‘classical’ methods use iterative optimization algorithms. Subspace methods calculate the Kalman filter states sequence using
projection methods. The matrices A, B, C, and D are then estimated from the
calculated state sequence using least squares calculations. The process of getting
the state sequences falls outside the scope of this thesis. More about it can be
3.5 Validating the Model
21
found in [4]. N4SID is a type of subspace method, it is the one used in this thesis.
The System Identification Toolbox has the function n4sid.
Choosing Model Order
This process is easier compare to the ARX case, since now only one order number
can be altered. Only R16 data is used to determine the order.
Figure 3.7 shows the f itT OT for state-space models of different orders. The
result here is somewhat non satisfactory. There is no obvious trend in the plot,
order 12 even leads to negative fit! Two order numbers are picked nonetheless,
they are 16 and 7. 16 is picked because it gives the maximum fit and 7 because it
gives a good fit for small number. It is important to remember that for another
data set, it may be N = 16 that will give negative fit and the resulting model from
n4sid could perhaps not be trusted.
Figure 3.7: The f itT OT of state-space models for different values of the order N .
3.5
Validating the Model
Since the model is to be used primarily for feedback control and not much is
known about the controller, there are no quantitative requirements on how well
the model should correspond to measurements, i.e. there are no exact requirements
on f itCR and f itCD . A practical way of testing the model would be to use it as an
observer in a feedback control loop, because that is its purpose. If such a controller
manages to control the dummy responses in a sufficiently good manner, the model
has fulfilled its purpose. However such testing is yet to be done. This means that
22
Theory & Method
the best method for validating the model cannot be used. Using other ways of
evaluating whether the model fits its purpose is therefore risky, but an attempt is
made nonetheless.
Frequency analysis is not possible either because impulse- or step-response test
on the true system cannot be performed. The only way of validating the model is
to compare the measured output with simulated output, i.e. analyze the residual.
An obvious goal is to minimize the norm of the residual. Since not much is known
about the future controller, the perfect norm to use is hard to find. For example,
is it important to have a small error during the whole crash or are certain time
periods more important than others? As stated before, the 2-norm is used when
calculating the fit, since the 2-norm is the standard choice.
Sometimes more is needed than just the calculated fit. It is also useful to
compare the measured outputs and simulated outputs by using visual inspection,
and thereby determine whether the simulated output behaves in a similar fashion
as the measured one.
For every crash setup, the available data is split into two halfs: estimation
data and validation data. The validation data consisted of five R16 tests, two sled
tests, and three full-scale tests. They are numbered 1 to 5, 1 to 2, and 1 to 3
respectively. The results of the validation process can be found in Chapter 4.
Chapter 4
Results
When viewing the results of this study, it is important to remember that this study
focuses on finding an optimal model for the R16 setup, see Chapter 2. Testing
how good that model works for the other setups is secondary, but can nontheless
provide valuable information for the future use of similar models in those setups.
This chapter is divided into three parts, one for each crash test setup.
All the fit calculations can be found in this chapter and also some interesting
plots of comparison plots of simulated and measured output. The rest of the
figures can be found in Appendix A.
4.1
R16 setup
Table 4.1 shows that the Grey-Box model have the best performance for both CR
and CD out of the five models, which should indicate that the model is somewhat
accurate. The fact that none of the models come close to 100%, indicate that the
system cannot fully be described with just the two inputs. Figure 4.1 shows an
example of how well the Grey-Box model can perform for R16 data. The plot
shows the performance of smaller ARX model, the larger black-box model, and
the Grey-Box for the same experiment. As can be seen in the figure, all the three
models perform satisfactory for this experiment.
Model
ARX Nmax = 10
ARX Nmax = 6
State-Space N = 16
State-Space N = 7
Grey-Box
f itCR,tot
50.5
44.8
51.4
46.5
55.0
f itCD,tot
49.2
53.1
49.1
63.8
74.3
f itT OT
49.9
48.9
50.2
55.1
64.6
Table 4.1: Percentual fit for R16 setup data for the different models.
23
24
Results
(a) ARX model Nmax = 6
(b) State-Space N = 16
(c) Grey-Box
Figure 4.1: Simulated and measured Chest Deflection for different models, R16 setup
test, Experiment nr 1.
4.2
Sled Tests
Remember from Chapter 2 that sled tests have an airbag and that R16 tests do not.
The airbag introduces an obvious non-linearity in the system. The effect of this can
be seen in the Grey-Box f itCD,tot in Table 4.2, but in the two ARX models even
more so. Apparently the ARX model is least capable of handling the transition
from R16 to sled tests. Another thing that stands out is that the two state-space
models produce the best fit by far. This could be due to the crash pulses being
smoother in sled tests compared to R16 pulses, where sometimes extreme crash
pulses are used. This leads to the output curves also being smoother, it seems as
if the model produced by n4sid take great advantage of that. Figure 4.2 shows
the nice fit of the larger state-space model for sled test data. This is also a good
example of the grey-box model performing a lot better than ARX for sled test
data.
4.3 Fullscale
25
Model
ARX Nmax = 10
ARX Nmax = 6
State-Space N = 16
State-Space N = 7
Grey-Box
f itCR,tot
38.8
25.7
70.7
62.2
63.1
f itCD,tot
45.9
54.7
82.4
77.8
59.1
f itT OT
42.3
40.2
76.5
70.0
61.1
Table 4.2: Percentual fit for sled test data for the different models.
4.3
Fullscale
The black-box state-space models again show the best fit, but this time the smaller
model performs best. An explanation could be what was discussed in Section 3.4.2,
that this model type cannot always be trusted for this data, it could also be the
case that N = 16 overestimates the model for this setup and thereby producing
a worse fit compared to the smaller model. The two ARX model perform similar
to what they perform for sled tests. The grey-box model performs worse now
compared to other data sets. Figure 4.3 shows an example of how badly the greybox perform in full-scale tests. Even though the f itCR,tot is around 53% this is not
a satisfactory result. The peak of 400 sm2 is missed completely. Notice the f itCR,tot
for state-space N = 16, it is just 10 % more than the grey-box, yet the overall
look of the curve is much better. Note also that the small state-space model has
a better fit than the one with a larger model order. But the overall the curve of
the larger model order looks much better.
Model
ARX Nmax = 10
ARX Nmax = 6
State-Space N = 16
State-Space N = 7
Grey-Box
f itCR,tot
49.0
42.6
50.0
59.5
49.1
f itCD,tot
40.0
39.6
59.2
68.8
47.2
f itT OT
44.5
41.1
54.6
64.1
48.2
Table 4.3: Percentual fit for Fullscale data for the different models.
26
Results
(a) ARX model Nmax = 17
(b) State-Space N = 16
(c) Grey-Box
Figure 4.2: Simulated and measured Chest Acceleration for different models, sled tests,
Experiment nr 2.
4.3 Fullscale
27
(a) State-Space N = 7
(b) State-Space N = 16
(c) Grey-Box
Figure 4.3: Simulated and measured Chest Acceleration for different models, Full Scale
test, Experiment nr 2.
Chapter 5
Concluding Remarks
5.1
Conclusions
The general conclusion that can be stated is that all three model types, the ARX
model, model from subspace methods, and the Grey-Box model, can be used to
estimate CR and CD. All of them produce outputs that behave similar to the
measured output. Here is a list of conclusions of the differences between the
different model types
• For R16 data, the grey-box model produced the best f itCR,tot and the best
f itCD,tot . The fact that two linear black-box models fail to produce better
results than the grey-box model, suggests that the physical model is a very
good linear model of the chest during crash.
• The performance of the grey-box model gets worse when an airbag is present.
This is what should be expected. The system of linear springs and dampers
is not capable of handling the non-linear effect of an airbag.
• The ARX models perform much worse with an airbag present. The ARX
model seem only capable of producing good fits for R16 data.
• As of now, the black-box state-space models is the best choice for sled tests
and full-scale data. However, one needs to be careful here, the model type
has shown some dubious results, see Figure 3.7.
The aim of the thesis, to estimate CR and CD from BRS and Pulse, has been
reached. As can be seen from the results, the estimated CR and CD follow the
measured ones fairly well. The question is: Is it good enough? That is hard
to say now, but since the model is to be used for control purposes the demand
for accuracy is not very large. Especially if the controller is a very simple one.
However since little is known about the controller, no quantitative requirements
can be set on the model as of now. The only way to truly test the model is to
implement it in a closed loop with a controller. That is the only way to find out
29
30
Concluding Remarks
if the model fulfills its purpose. The data sets used are not very large; it is very
likely that the results will improve with more available data.
5.2
Further Work
First of all, a proper testing of the model in a feedback loop with a controller has
to be performed. First recommendation is to use the Grey-Box model in the R16
setup. If results show that the model is not giving good enough estimates then
one or more of the following things can be done to increase performance.
Add Belt Force Sensor
Add a sensor for measuring belt force. That is a sensor to measure the force in
the belt segment, i.e. at which force the belt is being pulled. A quick study of
how belt force can be used to estimate the dummy responses, have shown great
potential in the method, especially in estimating CD. This is also shown in [8].
Sensor Fusion
With additional sensors, a sensor fusion approach is very tempting. This will make
it possible to utilize both belt force and BRS.
Increase Model Complexity
Some things are not fully covered by the proposed model, for example change
in geometry and the contact between dummy and airbag. Taking these things
into consideration will make the model non-linear. Since the software for GreyBox models used in this thesis only supports linear models the proposed model is
linear. However there is software that supports non-linear models as well.
New Performance Measure
In this study, the f itT OT is used to determine the accuracy of the models, however
this measure has proven to be insufficient in some cases. Therefore a new measure
that takes the controller dynamics into account is needed.
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[5] Lennart Ljung and Torkel Glad. Modellbygge och Simulering. Studentlitteratur, Lund, 2004.
[6] M. G. McCarthy, B. P. Chinn, and J. Hill. The effect of occupant characteristics
on injury risk and the development of active-adaptive restraint systems. In
Proceedings of the 17 th International Technical Conference on Experimental
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[8] G.M. van der Zalm. Reduction of the chest deflection: A control approach.
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31
Appendix A
Comparison Plots of
Simulated and Measured
Outputs
This chapter contains the plots of the data from all experiments used in the validation process in this thesis. Every plot shows the measured (solid line) and
simulated (dashed line) output of each respective experiment. The percentage fit
can be found in the title of the plots.
33
34
A.1
A.1.1
Comparison Plots of Simulated and Measured Outputs
R16
ARX model Nmax = 6
Chest Acceleration
(a) R16 test, Experiment 1
(b) R16 test, Experiment 2
(c) R16 test, Experiment 3
(d) R16 test, Experiment 4
(e) R16 test, Experiment 5
Figure A.1: Simulated and measured Chest Acceleration for R16 data using the ARX
model with Nmax = 6.
A.1 R16
35
Chest Deflection
(a) R16 test, Experiment 1
(b) R16 test, Experiment 2
(c) R16 test, Experiment 3
(d) R16 test, Experiment 4
(e) R16 test, Experiment 5
Figure A.2: Simulated and measured Chest Deflection for R16 data using the ARX
model with Nmax = 6.
36
A.1.2
Comparison Plots of Simulated and Measured Outputs
ARX model Nmax = 10
Chest Acceleration
(a) R16 test, Experiment 1
(b) R16 test, Experiment 2
(c) R16 test, Experiment 3
(d) R16 test, Experiment 4
(e) R16 test, Experiment 5
Figure A.3: Simulated and measured Chest Acceleration for R16 data using the ARX
model with Nmax = 10.
A.1 R16
37
Chest Deflection
(a) R16 test, Experiment 1
(b) R16 test, Experiment 2
(c) R16 test, Experiment 3
(d) R16 test, Experiment 4
(e) R16 test, Experiment 5
Figure A.4: Simulated and measured Chest Deflection for R16 data using the ARX
model with Nmax = 10.
38
A.1.3
Comparison Plots of Simulated and Measured Outputs
State-Space N = 16
Chest Acceleration
(a) R16 test, Experiment 1
(b) R16 test, Experiment 2
(c) R16 test, Experiment 3
(d) R16 test, Experiment 4
(e) R16 test, Experiment 5
Figure A.5: Simulated and measured Chest Acceleration for R16 data using the StateSpace model with N = 16.
A.1 R16
39
Chest Deflection
(a) R16 test, Experiment 1
(b) R16 test, Experiment 2
(c) R16 test, Experiment 3
(d) R16 test, Experiment 4
(e) R16 test, Experiment 5
Figure A.6: Simulated and measured Chest Deflection for R16 data using the StateSpace model with N = 16.
40
A.1.4
Comparison Plots of Simulated and Measured Outputs
State-Space N = 7
Chest Acceleration
(a) R16 test, Experiment 1
(b) R16 test, Experiment 2
(c) R16 test, Experiment 3
(d) R16 test, Experiment 4
(e) R16 test, Experiment 5
Figure A.7: Simulated and measured Chest Acceleration for R16 data using the StateSpace model with N = 7.
A.1 R16
41
Chest Deflection
(a) R16 test, Experiment 1
(b) R16 test, Experiment 2
(c) R16 test, Experiment 3
(d) R16 test, Experiment 4
(e) R16 test, Experiment 5
Figure A.8: Simulated and measured Chest Deflection for R16 data using the StateSpace model with N = 7.
42
A.1.5
Comparison Plots of Simulated and Measured Outputs
Grey-Box
Chest Acceleration
(a) R16 test, Experiment 1
(b) R16 test, Experiment 2
(c) R16 test, Experiment 3
(d) R16 test, Experiment 4
(e) R16 test, Experiment 5
Figure A.9: Simulated and measured Chest Acceleration for R16 data using the GreyBox model.
A.1 R16
43
Chest Deflection
(a) R16 test, Experiment 1
(b) R16 test, Experiment 2
(c) R16 test, Experiment 3
(d) R16 test, Experiment 4
(e) R16 test, Experiment 5
Figure A.10: Simulated and measured Chest Deflection for R16 data using the GreyBox model.
44
A.2
A.2.1
Comparison Plots of Simulated and Measured Outputs
Sled Tests
ARX model Nmax = 6
Chest Acceleration
(a) Sled test, Experiment 1
(b) Sled test, Experiment 2
Figure A.11: Simulated and measured Chest Acceleration for sled test data using the
ARX model with Nmax = 6.
Chest Deflection
(a) Sled test, Experiment 1
(b) Sled test, Experiment 2
Figure A.12: Simulated and measured Chest Deflection for sled test data using the
ARX model with Nmax = 6.
A.2 Sled Tests
A.2.2
45
ARX model Nmax = 10
Chest Acceleration
(a) Sled test, Experiment 1
(b) Sled test, Experiment 2
Figure A.13: Simulated and measured Chest Acceleration for sled test data using the
ARX model with Nmax = 10.
Chest Deflection
(a) Sled test, Experiment 1
(b) Sled test, Experiment 2
Figure A.14: Simulated and measured Chest Deflection for sled test data using the
ARX model with Nmax = 10.
46
A.2.3
Comparison Plots of Simulated and Measured Outputs
State-Space N = 16
Chest Acceleration
(a) Sled test, Experiment 1
(b) Sled test, Experiment 2
Figure A.15: Simulated and measured Chest Acceleration for sled test data using the
State-Space model with N = 16.
Chest Deflection
(a) Sled test, Experiment 1
(b) Sled test, Experiment 2
Figure A.16: Simulated and measured Chest Deflection for sled test data using the
State-Space model with N = 16.
A.2 Sled Tests
A.2.4
47
State-Space N = 7
Chest Acceleration
(a) Sled test, Experiment 1
(b) Sled test, Experiment 2
Figure A.17: Simulated and measured Chest Acceleration for sled test data using the
State-Space model with N = 7.
Chest Deflection
(a) Sled test, Experiment 1
(b) Sled test, Experiment 2
Figure A.18: Simulated and measured Chest Deflection for sled test data using the
State-Space model with N = 7.
48
A.2.5
Comparison Plots of Simulated and Measured Outputs
Grey-Box
Chest Acceleration
(a) Sled test, Experiment 1
(b) Sled test, Experiment 2
Figure A.19: Simulated and measured Chest Acceleration for sled test data using the
Grey-Box model.
Chest Deflection
(a) Sled test, Experiment 1
(b) Sled test, Experiment 2
Figure A.20: Simulated and measured Chest Deflection for sled test data using the
Grey-Box model.
A.3 Full-scale
A.3
A.3.1
49
Full-scale
ARX model Nmax = 6
Chest Acceleration
(a) Full-scale test, Experiment 1
(b) Full-scale test, Experiment 2
(c) Full-scale test, Experiment 3
Figure A.21: Simulated and measured Chest Acceleration for full-scale test data using
the ARX model with Nmax = 6.
50
Comparison Plots of Simulated and Measured Outputs
Chest Deflection
(a) Full-scale test, Experiment 1
(b) Full-scale test, Experiment 2
(c) Full-scale test, Experiment 3
Figure A.22: Simulated and measured Chest Deflection for full-scale test data using
the ARX model with Nmax = 6.
A.3 Full-scale
A.3.2
51
ARX model Nmax = 10
Chest Acceleration
(a) Full-scale test, Experiment 1
(b) Full-scale test, Experiment 2
(c) Full-scale test, Experiment 3
Figure A.23: Simulated and measured Chest Acceleration for full-scale test data using
the ARX model with Nmax = 10.
52
Comparison Plots of Simulated and Measured Outputs
Chest Deflection
(a) Full-scale test, Experiment 1
(b) Full-scale test, Experiment 2
(c) Full-scale test, Experiment 3
Figure A.24: Simulated and measured Chest Deflection for full-scale test data using
the ARX model with Nmax = 10.
A.3 Full-scale
A.3.3
53
State-Space N = 16
Chest Acceleration
(a) Full-scale test, Experiment 1
(b) Full-scale test, Experiment 2
(c) Full-scale test, Experiment 3
Figure A.25: Simulated and measured Chest Acceleration for full-scale test data using
the State-Space model with N = 16.
54
Comparison Plots of Simulated and Measured Outputs
Chest Deflection
(a) Full-scale test, Experiment 1
(b) Full-scale test, Experiment 2
(c) Full-scale test, Experiment 3
Figure A.26: Simulated and measured Chest Deflection for full-scale test data using
the State-Space model with N = 16.
A.3 Full-scale
A.3.4
55
State-Space N = 7
Chest Acceleration
(a) Full-scale test, Experiment 1
(b) Full-scale test, Experiment 2
(c) Full-scale test, Experiment 3
Figure A.27: Simulated and measured Chest Acceleration for full-scale test data using
the State-Space model with N = 7.
56
Comparison Plots of Simulated and Measured Outputs
Chest Deflection
(a) Full-scale test, Experiment 1
(b) Full-scale test, Experiment 2
(c) Full-scale test, Experiment 3
Figure A.28: Simulated and measured Chest Deflection for full-scale test data using
the State-Space model with N = 7.
A.3 Full-scale
A.3.5
57
Grey-Box
Chest Acceleration
(a) Full-scale test, Experiment 1
(b) Full-scale test, Experiment 2
(c) Full-scale test, Experiment 3
Figure A.29: Simulated and measured Chest Acceleration for full-scale test data using
the Grey-Box model.
58
Comparison Plots of Simulated and Measured Outputs
Chest Deflection
(a) Full-scale test, Experiment 1
(b) Full-scale test, Experiment 2
(c) Full-scale test, Experiment 3
Figure A.30: Simulated and measured Chest Deflection for full-scale test data using
the Grey-Box model.
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