Chest Observer for Crash Safety Enhancement

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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 tekniska högskola

Linköpings universitet Linköpings universitet

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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

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Avdelning, Institution

Division, Department

Division of Automatic Control Department of Electrical Engineering Linköpings universitet

SE-581 83 Linköping, Sweden

Datum Date 2008-06-13 Språk Language Svenska/Swedish Engelska/English Rapporttyp Report category Licentiatavhandling Examensarbete C-uppsats D-uppsats Övrig rapport

URL för elektronisk version

http://www.control.isy.liu.se http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-ZZZZ ISBN — ISRN LITH-ISY-EX--08/4049--SE

Serietitel och serienummer

Title of series, numbering

ISSN

—

Titel

Title

Observatör för Bröstkorg

Chest Observer for Crash Safety Enhancement

Författare

Author

Christian Blåberg

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

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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 bröstin-tryck 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 an-vänts kom från en accelerometer och en givare för att mäta rotationen i bältess-nurran. Detta har gjorts genom att modellera bröstkorgen med linjära fjädrar och dämpare. Dess parametrar har skattats från data från krocktester från krockdock-or. 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 skat-ta de ovan nämnda storheterna, den fysikaliska modellen och black-box-modellen på tillståndsform fungerade bäst.

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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.

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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

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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 tensioners: buckle tensioner, lap tensioner, and retractor pre-tensioner. 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

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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

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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 automo-tive 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.

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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.

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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 Identi-fication 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 auto-motive 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 electron-ics, 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.

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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

man-ufacturers.

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

maxi-mum 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 New-tonian 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.

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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,

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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

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2.2 Sensors and Data 11

(a) Spine (b) Rib cage

Figure 2.2: Pictures showing the human chest. Thoracic spine is where Chest

Acceler-ation 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.

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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.

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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

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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:

w = R × BRS + Z Z P ulse(t)dt (3.1) ˙ w = R × d dtBRS + t Z 0 P ulse(τ ) dτ, (3.2)

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 ster-num, their respective free-body diagrams are extracted, see Figures 3.4 and 3.5 respectively.

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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= m1x¨ F1,k1 = −k1(x − s) F1,c1 = −c1( ˙x − ˙s), which gives m1¨x = −k1(x − s) − c1( ˙x − ˙s). Same procedure for the sternum yields

X F2= m2¨s F2,k1 = −k1(s − x) F2,c1 = −c1( ˙s − ˙x) F2,k2 = −k2(s − w) F2,c2 = −c2( ˙s − ˙w), which gives m2s = −k¨ 1(s − x) − c1( ˙s − ˙x) − k2(s − w) − c2( ˙s − ˙w).

Since CR is defined as ¨x the expression becomes

CR = ¨x = −k1 m1

(x − s) − c1 m1

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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. (3.4)

3.1.4 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.

˙ x(t) = A(θ)x(t) + B(θ)u(t), (3.5) y(t) = C(θ)x(t) + D(θ)u(t). with x(t) =       ˙ x(t) x(t) ˙s(t) s(t) w(t)       , u(t) =   t R 0 P ulse(τ ) dτ d dtBRS(t)  , y(t) =  CR(t) CD(t) , and θ = k1 c1 k2 c2 m1 m2 R T . (3.6)

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3.2 Parameter Estimation 17

Then the matrices A, B, C, and D become

A =       −c1 m1 − k1 m1 c1 m1 k1 m1 0 1 0 0 0 0 c1 m2 k1 m2 − c1+c2 m2 − k1+k2 m2 k2 m2 0 0 1 0 0 0 0 0 0 0       , B =       0 0 0 0 c2 m2 c2 m2R 0 0 1 R       (3.7) C =− c1 m1 − k1 m1 c1 m1 k1 m1 0 0 1 0 −1 0 , D =0 0 0 0 . (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

θ =    θ1 .. . θd   . (3.9)

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

ˆ

y(t|θ). (3.10)

This guess is evaluated by calculating the prediction error

(t, θ) = y(t) − ˆy(t|θ). (3.11)

When data is collected over N sample times the following sum is formed

VN(θ) = 1 N N X t=1 2(t, θ) (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) = 1 N N X t=1 T(t, θ)Q(t, θ) (3.13)

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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

ˆ

θN = arg min θ

VN(θ). (3.14)

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 ˆy(t) to the measured output y(t). So the error becomes e(t) = y(t) − ˆy(t). A good measure of performance isPt=tmax

t=0 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) − ˆy(t)||

||y(t) − ˜y|| ), (3.15)

where y(t) is measured signal, ˆy(t) is model output and ˜y 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

f itCR,tot, Pn i f itCR,i n (3.16) f itCD,tot, Pn i f itCD,i n (3.17) f itT OT , f itCR,tot+ f itCD,tot 2 , (3.18)

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 sys-tem, besides the measured outputs and inputs. The derived model will therefore solely depend on the measurements of the true system. Model order (or com-plexity) 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 black-box 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.

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3.4 Black-Box Modeling 19

3.4.1 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) (3.19)

A(q) = 1 + a1q−1+ · · · + anaq−na (3.20)

B(q) = b1+ b2q−1+ · · · + bnbq−nb+1, (3.21)

where e(t) is white noise and q is the shift operator so that q−1y(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.

N N = N A N B N K =na11 na12 nb11 nb12 nk11 nk12 na21 na22 nb21 nb22 nk21 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.

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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 al-gorithms. 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

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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 itCRand 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

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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.

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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 f itCR,tot f itCD,tot f itT OT

ARX Nmax= 10 50.5 49.2 49.9

ARX Nmax= 6 44.8 53.1 48.9

State-Space N = 16 51.4 49.1 50.2

State-Space N = 7 46.5 63.8 55.1

Grey-Box 55.0 74.3 64.6

Table 4.1: Percentual fit for R16 setup data for the different models.

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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.

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4.3 Fullscale 25

ARX Nmax= 10 38.8 45.9 42.3

ARX Nmax= 6 25.7 54.7 40.2

State-Space N = 16 70.7 82.4 76.5

State-Space N = 7 62.2 77.8 70.0

Grey-Box 63.1 59.1 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 grey-box perform in full-scale tests. Even though the f itCR,totis 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.

ARX Nmax= 10 49.0 40.0 44.5

ARX Nmax= 6 42.6 39.6 41.1

State-Space N = 16 50.0 59.2 54.6

State-Space N = 7 59.5 68.8 64.1

Grey-Box 49.1 47.2 48.2

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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,

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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

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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

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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 Grey-Box 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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Bibliography

[1] R J Hesseling. Active Restraint Systems, Feedback Control of Occupant in Mo-tion. PhD thesis, Technische Universiteit Eindhoven, Eindhoven, The Nether-lands, 2004.

[2] R J Hesseling, M Steinbuch, F E Veldpaus, and T. Klisch. Identification and control of a vehicle restraint system. Journal of Automobile Engineering, 220(4):401–413, 2006.

[3] H. G. Johannessen and M. Mackay. Why intelligent automotive occupant restraint systems? In Proceedings of the 39 th Annual Meeting of the AAAM, Chicago, USA, pages 519–525, 1995.

[4] Lennart Ljung. System Identification - Theory for the User, 2nd edition. PTR Prentice Hall, 1999.

[5] Lennart Ljung and Torkel Glad. Modellbygge och Simulering. Studentlitter-atur, 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 Safety Vehicles, Amsterdam, 2001.

[7] Ewour van der Laan, Frans Veldpaus, and Maarten Steinbuch. Control oriented modeling of vehicular occupants and restraint systems. In IRCOBI Conference, Maastricht, The Netherlands, 2007.

[8] G.M. van der Zalm. Reduction of the chest deflection: A control approach. PhD thesis, Technische Universiteit Eindhoven, Eindhoven, The Netherlands, 2003. DCT 2003.009.

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Appendix A

Comparison Plots of

Simulated and Measured

Outputs

This chapter contains the plots of the data from all experiments used in the val-idation 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.

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34 Comparison Plots of Simulated and Measured Outputs

A.1 R16

A.1.1 ARX model N

max

= 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

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A.1 R16 35 Chest Deflection

Figure A.2: Simulated and measured Chest Deflection for R16 data using the ARX

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A.1.2 ARX model N

max

= 10

Figure A.3: Simulated and measured Chest Acceleration for R16 data using the ARX

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Figure A.4: Simulated and measured Chest Deflection for R16 data using the ARX

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A.1.3 State-Space N = 16

Figure A.5: Simulated and measured Chest Acceleration for R16 data using the

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Figure A.6: Simulated and measured Chest Deflection for R16 data using the

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A.1.4 State-Space N = 7

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A.1.5 Grey-Box

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A.2 Sled Tests

A.2.1 ARX model N

max

= 6

(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

Figure A.12: Simulated and measured Chest Deflection for sled test data using the

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A.2 Sled Tests 45

A.2.2 ARX model N

max

= 10

ARX model with Nmax= 10.

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A.2.3 State-Space N = 16

State-Space model with N = 16.

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A.2 Sled Tests 47

A.2.4 State-Space N = 7

State-Space model with N = 7.

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A.2.5 Grey-Box

Grey-Box model.

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A.3 Full-scale 49

A.3 Full-scale

A.3.1 ARX model N

max

= 6

(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

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Figure A.22: Simulated and measured Chest Deflection for full-scale test data using

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A.3.2 ARX model N

max

= 10

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A.3.3 State-Space N = 16

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A.3.4 State-Space N = 7

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A.3.5 Grey-Box

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Chest Observer for Crash Safety Enhancement

Institutionen för systemteknik

Department of Electrical Engineering

Examensarbete

Chest Observer for Crash Safety Enhancement

Chest Observer for Crash Safety Enhancement

Examensarbete utfört i Reglerteknik

vid Tekniska högskolan i Linköping

av

Abstract

Sammanfattning

Acknowledgments

Contents

Chapter 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

Abbreviations

Dictionary

Chapter 2

Crash Tests

2.1

Anatomy of Dummy

2.2

Sensors and Data

Chapter 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

Chapter 4

Results

4.1

R16 setup

4.2

Sled Tests

4.3

Fullscale

Chapter 5

Concluding Remarks

5.1

Conclusions

5.2