Vehicle Accident Analysis and Reconstruction Methods [3 ed.] 1468604198, 9781468604191

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Vehicle Accident Analysis and Reconstruction Methods Third Edition

R. Matthew Brach | Raymond M. Brach | James J. Mason

Vehicle Accident Analysis and Reconstruction Methods

Vehicle Accident Analysis and Reconstruction Methods THIRD EDITION R. Matthew Brach, Raymond M. Brach and James J. Mason

Warrendale, Pennsylvania, USA

400 Commonwealth Drive Warrendale, PA 15096-0001 USA E-mail: [email protected] Phone: 877-606-7323 (inside USA and Canada) 724-776-4970 (outside USA) Fax: 724-776-0790

Copyright © 2022 SAE International. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of SAE International. For permission and licensing requests, contact SAE Permissions, 400 Commonwealth Drive, Warrendale, PA 15096-0001 USA; e-mail: [email protected]; phone: 724-772-4028

Chief Growth Officer Frank Menchaca

Library of Congress Catalog Number 2021948645 http://dx.doi.org/10.4271/9781468603453

Development Editor Publishers Solutions, LCC Albany, NY

Information contained in this work has been obtained by SAE International from sources believed to be reliable. However, neither SAE International nor its authors guarantee the accuracy or completeness of any information published herein and neither SAE International nor its authors shall be responsible for any errors, omissions, or damages arising out of use of this information. This work is published with the understanding that SAE International and its authors are supplying information but are not attempting to render engineering or other professional services. If such services are required, the assistance of an appropriate professional should be sought. ISBN-Print 978-1-4686-0419-1 ISBN-PDF 978-1-4686-0345-3 ISBN-epub 978-1-4686-0346-0 To purchase bulk quantities, please contact: SAE Customer Service E-mail: [email protected] Phone: 877-606-7323 (inside USA and Canada) 724-776-4970 (outside USA) Fax: 724-776-0790 Visit the SAE International Bookstore at books.sae.org

Publisher Sherry Dickinson Nigam

Director of Content Management Kelli Zilko Production and Manufacturing Associate Erin Mendicino

Dedication

Times with our heads under the hoods of cars, staring at the engines, pointing at parts, naming the components, checking the oil, filling the washer fluid reservoir, getting our hands dirty, and me explaining how engines operated were far too brief. I thank you for eagerly sharing your sense of wonderment. I dearly cherish those moments. —Gramps To Thaddeus Jonas Pemberton, May 13, 2014 - September 2, 2018 My Little Automotive Engineer

Contents

Foreword Preface to the Third Edition Preface to Second Edition Preface to First Edition Acknowledgments

xv xvii xix xxi xxv

CHAPTER 1

Uncertainty and Sensitivity in Measurements and Calculations in Accident Reconstruction

1

Introduction

1

Upper and Lower Bounds Using a Given Model

4

Differential Variations

6

Statistics of Related Variables

9

Linear Functions Arbitrary Functions (Approximate Method)

10 11

Finite Differences

13

Monte Carlo Method

16

Design of Experiments

19

The Bayesian Method

23

Application Issues and Other ­Considerations

31

Other Methods of Evaluating ­Uncertainty

33

CHAPTER 2

Tire Forces

37

Introduction

37

Rolling Resistance

39

Slip, Longitudinal Force, and Lateral Force

40

Longitudinal Slip © 2022 SAE International

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Contents

Comments, the Coefficient of Friction, and the Frictional Drag Coefficient

45

Longitudinal Tire Force

47

Vehicle Event Data Recorders and Longitudinal Slip

49

Lateral Tire Force

50

Friction Circle and Friction Ellipse

52

Idealized Friction Circle and Idealized Friction Ellipse

53

Friction Circle and Friction Ellipse

55

Modeling Combined Steering and ­Braking Tire Forces

58

The Bakker-Nyborg-Pacejka Model for Lateral and Longitudinal Tire Forces

58

Modified Nicolas-Comstock Combined Tire Force Model

60

Application Issues Tire Stiffness Values

66 66

Antilock Braking Systems

69

Light Vehicle (LV) Frictional Drag Coefficients

70

Frictional Drag Coefficients for Heavy Trucks (HT)

72

Hydroplaning

76

Appendix 2A

79

CHAPTER 3

Straight-Line Motion

83

Introduction

83

Uniform Acceleration and Braking Motion

83

Equations of Constant Acceleration

84

Road Grade and Equivalent Drag Coefficients

87

Vehicle Forward-Motion ­Performance Equations

87

Stopping Distance

92

Distance from Speed

93

Speed from Distance

93

Application Issues

95

Stopping Distance

95

Two Objects Decelerating While in Contact

98

Motion Around Curves

101

Vehicle Fall Equations

101

Equations of Motion of a Projectile

101

Equations of Motion of a Vehicle Leading to a Fall Including Rotational Inertia

104

Contents

CHAPTER 4

Critical Speed from Tire Yaw Marks

111

Introduction

111

Estimation of Speed from Yaw Marks

112

Yaw Marks

115

Radius from Yaw Marks

Critical Speed

117

118

CSF on a Flat Surface

119

Roadway with Superelevation

119

Application Issues

123

Tire Marks in Practice

123

Other Curved Tire Marks

123

Frictional Drag Coefficient, f

124

Driver Control Modes

124

Tire Forces in a Severe Yaw

124

The Critical Speed Formula and Edge Drop-Off (Road-Edge Reentry) 126

Uncertainty of Critical Speed ­Calculations Estimation of Uncertainty by Differential Variations

126 126

Accuracy of the Critical Speed Method

127

Statistical Variations

129

Yaw Mark Striations

131

Striation Angles

132

Striation Spacing

135

CHAPTER 5

Reconstruction of Vehicular Rollover Accidents 139 Introduction

139

Rollover Test Methods

141

Documentation of the Accident Site

144

Documentation of the Accident V ­ ehicle

146

Pre-trip Phase

149

Tire Mark Striation

Trip Phase

155

158

Modeling the Trip Phase

159

Complex Vehicle Trip Models

164

Rim Contact

165

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Roll Phase

166

Speed Analysis for the Roll Phase

166

Determining the Roll Motion of the Vehicle

170

Generating a Realistic Roll Velocity Curve

173

Example Rollover Reconstruction

174

Vehicle-to-Ground Impact Model

182

Impulse Ratio (μ)

185

Impact Angle (ϕ)

187

CHAPTER 6

Analysis of Collisions, Impulse-Momentum Theory

189

Introduction

189

Quantitative Concepts

191

Point-Mass Impulse-Momentum C ­ ollision Theory

193

Coefficient of Restitution, Frictionless ­Point-Mass Collisions

199

Collisions Where Sliding Ends before Separation: The Critical Impulse Ratio, μ0

201

Sideswipe Collisions and Common-Velocity Conditions

201

Controlled Collisions Coefficients of Restitution

204 206

Stiffness Equivalent Collision Coefficient of Restitution

206

Mass Equivalent Collision Coefficient of Restitution

208

Summary of the Point-Mass Impact Model

209

Planar Impact Mechanics

210

Overview of Planar Impact Mechanics Model

216

Application Issues: Coefficients, Dimensions, and Angles

219

Coefficient of Restitution and Impulse Ratio Distances, Angles, and Point C Work of Impulses and Energy Loss (Crush Energy)

219 221 223

RICSAC Collisions

225

Summary of Planar Impact ­Mechanics Model

229

Application Issues

230

Crashes with Large Mass Disparity between the Vehicles

230

Underride/Override Crashes

233

Contents

CHAPTER 7

Event Data Recorders and Crash Reconstruction 237 Introduction

237

Light Vehicle EDR Data

242

EDR Reported ΔV

242

Recording Delay

244

Incomplete Recording

245

Clipping

245

Effect of ACM Location

246

EDR Reported Precrash Vehicle Speed

247

Heavy Vehicle EDR Data

251

Summary

252

CHAPTER 8

Reconstruction Applications, Impulse-Momentum Theory

255

Introduction

255

Point-Mass Collision Applications

256

Rigid Body, PIM Applications: Vehicle Collisions with Rotation

263

Collision Reconstruction Using a Solution of the Planar Impact Equations

264

Reconstructions Using a Spreadsheet Solution of the Planar ­Impact Equations

266

Optimization Methods for Collision Reconstruction

Low-Speed In-Line (Central) Collisions

271

282

In-Line Impulse-Momentum Impact Model

283

Bumper Pair Stiffness Characterization Method

289

Airbags, Event Data Recorders, and ΔV

302

Crash Data

303

Precrash Data

304

CHAPTER 9

Collisions of Articulated Vehicles, Impulse-Momentum Theory

311

Introduction

311

Assumptions for Application of ­ Planar Impact Mechanics to ­Articulated Vehicles

313

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Contents

Articulated Vehicle Impact ­Equations

316

Validation of the Articulated Vehicle Impact Equations Using Experimental Data

323

Appendix 9A: Data Sheets for ­Example 9.4

345

CHAPTER 10

Crush Energy and ΔV Introduction The CRASH3 Method

349 349 350

Crush Stiffness Coefficients Based on Average Crush from Rigid Barrier Tests

361

Application Issues

372

Crush Stiffness Coefficients from Vehicle-to-Vehicle Collisions

372

Damage to One Vehicle Unknown

374

Side Crush Stiffness Coefficients, Two-Vehicle, Front-to-Side Crash Tests

374

Nonlinear Models of Crush

374

Arbitrary Number of Crush Measurements

374

CHAPTER 11

Frontal Vehicle-Pedestrian Collisions

377

Introduction

377

General and Supplementary Information

380

Forward Projection (Type I) Model

380

Hybrid Wrap Model

381

Vehicle-Pedestrian Impact (Type II) ­Mechanics Model

382

Pedestrian Motion

383

Vehicle Motion

386

Values of Physical Variables

387

Reconstruction (Hybrid) Model

392

Application to a Motorcycle Rider Thrown after Impact

394

CHAPTER 12

Photogrammetry for Accident Reconstruction

399

Introduction

399

Aerial Photography

401

Contents

Camera Matching

404

Planar Photogrammetry

405

Three-Dimensional (3D) Photogrammetry

413

Fundamental Information Related to Three-Dimensional (3D) Photogrammetry

414

Mathematical Basis of Three-Dimensional (3D) Photogrammetry

415

Projection Equations

415

Collinearity Equations

417

Coplanarity Equations

418

Multiple Image Considerations

418

Considerations of the Use of Three-Dimensional (3D) Photogrammetry in Practice

418

Summary

428

Appendix 12A: Projective Relation for Planar Photogrammetry

428

CHAPTER 13

Railroad Grade Crossing and Road Intersection Conflicts 433 Introduction

433

Clearing a Crossing or Intersection U ­ sing a Sight Triangle

434

Sight Distance for Stopping before a Crossing or Intersection

439

FHWA Grade-Crossing Equations

443

Stopping Distance

444

Stopping Sight Distance

446

Clearing Sight Distance

446

Locomotive Horn Sound Levels at ­Railroad Grade Crossings

448

Calculation of Horn Sound Levels at a Distance from a Point Source 448 Insertion Loss of Light Vehicles

452

CHAPTER 14

Vehicle Dynamic Simulation

457

Introduction

457

Planar Vehicle Dynamic Simulation

458

Tire Side-Force Stiffness Coefficients

461

Light-Vehicle Side-Force Coefficients

461

Heavy-Vehicle Side-Force Coefficients

462

Sensitivity of the Model to Parameters

462

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Contents

Examples

463

Appendix 14A: Differential Equations of Planar Vehicular Motion

484

Notation

485

Appendix A: Units and Numbers

487

Appendix B: Glossary

501

References

529

Bibliography

557

About the Authors

559

Index

561

Foreword

W

hile electronic recordings of the events in vehicle accidents have, in one form or another, been with us for upwards of 40 years, it is in the last ten years, since the publication of the second edition of Ray and Matt Brach’s book, that they have come upon us with a vengeance. It is a rare accident nowadays which does not have data from at least one of these: fixed security and traffic monitoring video cameras; event data recorders (EDRs) associated with occupant protection systems; dedicated high-resolution crash recorders; digital tachographs; “dash cams”; and other vehicle-borne cameras, which also display speeds; personal speed and location logging devices carried by cyclists and joggers; commercial vehicle trackers with speeds; insurance company “black boxes”; enthusiasts’ telematic devices; “infotainment” systems in cars; and a multitude of (often unsuspected) data collection devices in vehicles of all kinds. These extraordinary riches ought to make life a lot simpler for the accident investigator, and of course, they do supply so much information which would simply not be available by classical reconstruction methods, such as what was happening just beforehand. However, they can also be beguiling: to the naïve user, the recording says what it says, if it shows the speed was 100 km/h, then that must have been the speed, and no expert interpretation is needed. But as any discerning engineer will realize, before one can rely on such output, one must understand how it is generated, what exactly is being measured, and in what circumstances it can be misleading. A striking example is the speed and position data in dash cams and other logging devices, which are usually derived from GPS signals. The way the GPS data is processed varies from device to device, but it is usually averaged over a period of perhaps one or more seconds and then displayed as a constant reading during the next averaging period. This will mean that the speed that is shown on a particular video frame—let us say, the frame showing the actual moment of a collision—could very likely be an average speed computed over a period, which ended a second or so beforehand, and not at all the speed at the moment of the impact. Again, the GPS signal will sometimes fail, and the device may or may not try to interpolate potentially misleading figures for speed or location to bridge the gap. Without a good knowledge of how the device at hand works, or at least an intelligent examination of the displayed figures, the investigator cannot use the outputs with any confidence. The particular recording device that has now become part of everyday accident investigation, in North America at least, is the Event Data Recorder, and the new chapter and various examples that use the data in this third edition are very welcome. It is daunting to see how much research there exists into the characteristics of both light-vehicle and heavyvehicle EDRs, and the chapter is a valuable resource to guide investigators through the mass of publications. As matters stand, Europe is woefully behind when it comes to accessing

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and interpreting EDR data (although the digital tachographs that are fitted to the great majority of heavy vehicles have some overlap in their characteristics), but it is to be hoped that EU and ECE regulations will soon bring us into line. Then we shall need the array of knowledge with which this new edition brings us. Richard Lambourn Transport Research Laboratory www.trl.co.uk July 2021

Preface to the Third Edition

S

ince the publication of the second edition of this book in 2011, the transformation of the field of crash reconstruction in North America due to the proliferation of Event Data Recorder (EDR) technology in new vehicles has continued unabated. The effects of this transformation on the methods used in vehicle crash reconstruction are considered in this book. Most of the changes and additions to the book deal in some way with EDR data. A new chapter is included that bridges the early chapters, which largely contain foundational material that is still important and relevant for the understanding of vehicle collisions and vehicle motion, and the later chapters, which deal with applications of the methods and include example reconstructions. The new examples in the later chapters focus largely on the use of EDR data. The breadth of EDR data available to be exploited for reconstruction purposes is vast and continues to grow as EDR technology evolves. Thus, in the interest of a compendious coverage, the treatment herein focuses on the most often used EDR data, the ΔV and precrash vehicle speed. Other sources of physical evidence, such as on-board and off-board video and surveillance video from the environment, add to the availability of electronic data to use in crash reconstructions. The availability of EDR data has reduced the utility of some methods. For example, with precrash speed data from the EDR, the Critical Speed Formula is not used nearly as often as previously. The same is true of methods that compute ΔV from residual crush, such as CRASH3; with the airbag control module directly calculating the ΔV from the acceleration measured during the impact, the need to measure crush and compute the associated energy loss to find the ΔV is now seldom needed. Of course, there are exceptions. Thus, this material remains in the book, although now with a slightly different context. An example of one of these exceptions has been included. Crashes with low closing speeds, so-called low-speed crashes, have traditionally been a difficult class of collisions to reconstruct. The section dealing with that topic has been expanded to consider the effect of EDR data on these types of crashes. In addition to the impulse-momentum method presented previously, a method utilizing stiffness characterization of bumper pairs is presented. This method requires the availability of data from physical tests to be used. These two methods provide the reconstructionist with options to use for these types of collisions depending on the nature of the available physical evidence. This book continues its emphasis as a presentation of methods of accident reconstruction—scientific, engineering, and mathematical methods—with considerable attention paid to uncertainty. The methods presented here are supported by references and/or data to establish their validity. The references provide the readers with additional information regarding the basis for the method as well as resources for additional information when needed. All the examples analyzed using the computer were done with programs written by the authors. Other than the program for the stiffness characterization method for low-speed crashes, the programs are available as a package, VCRware® (https://www.brachengineering. com/about-vcrware). The authors contend that understanding the models presented in this © 2022 SAE International

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book, and their implementation in software, will make a reconstructionist a more effective user of other software programs based on other models. Color has been added and is used throughout the book with the goal that the updated artwork improves the clarity of the concepts and examples for the reader. Typographical errors in the second edition have been corrected (but new ones were likely created). Numerous additions were made to the Glossary in Appendix B to attempt to keep this resource relevant and up to date. The authors intend that this book will serve as a practical resource to new and experienced practitioners working in the field of crash reconstruction. Additionally, the material in the book will assist users of the methods presented herein by placing their reconstructions of vehicle accidents on a firmer, more scientific foundation. As in the previous editions, feedback is welcome, particularly if any errors are found. The authors maintain errata for all three versions of the book. These errata are available at www.brachengineering.com.

Preface to Second Edition

T

his book remains unique as a presentation of methods of accident reconstruction— scientific, engineering, and mathematical methods. In contrast to many other books on this subject, the methods presented here are supported by references and/or data to establish their validity. It is not simply a second edition of an existing book, but is a revised and enhanced version. Typographical errors in the first edition have been corrected. Additional examples have been included in many of the chapters. The Glossary has been updated, particularly to include many new acronyms that have emerged with the use of event data recorders (EDRs). More noteworthy, however, is the improved coverage of tire forces; additional experimental data are presented and the topic of the tire ellipse/circle is taken out of the idealized realm, explained, and placed into a practical, realistic format based on experimental data. Tables and a nomograph for the use of frictional drag coefficients for sliding tires have been added. With appropriate modification, these can be used with applications involving antilock braking systems. Frictional drag coefficients for heavy trucks differ from those for light vehicles; this topic is now covered, with data and references included. Two new chapters have been added. Articulated vehicles are ever-present on our roads and are overly involved in serious crashes. The analysis of collisions of articulated vehicles requires special concepts and methods. The equations of the mechanics of such collisions are covered in a new chapter of this edition. The topic of the conflict of vehicles (road vehicles or rail trains) approaching each other at an intersection of roads or at a railroad grade crossing has received little or no coverage in other books. A new chapter in this edition presents methods for analyzing and reconstructing such events. In addition, this chapter covers the acoustics of train horns to allow estimation of sound levels from locomotives at grade crossings. Overall, the authors hope that this book will assist users of the methods presented here by placing their reconstructions of vehicle accidents on a firmer, more scientific foundation. As before, feedback is welcome, particularly if any errors are found.

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Preface to First Edition

M

ore than 19,000 registered junkyards (junkyards.com) operate in the United States (US). Each can have hundreds, or even thousands, of wrecked vehicles. Each vehicle sent to a junkyard following an accident represents a tragedy of some degree, with accompanying financial loss, human injury, and perhaps loss of life. One of the objectives of the reconstruction of vehicle accidents is to tell a vital part of that story. Automotive accident reconstruction is the process of determining what happened to the vehicles and persons involved in an accident and how it happened, using the information available after the accident occurred. This task must produce results that are reasonably accurate. Reconstruction is a procedure carried out with the specific purpose of estimating in both a qualitative and quantitative manner how an accident occurred using engineering, scientific, and mathematical principles based on evidence obtained through an accident investigation. The collection of facts associated with the circumstances of an accident is referred to as accident investigation. Determining what happened and how it happened is usually referred to as accident reconstruction. A third facet of postaccident analysis attempts to answer the question of why the accident occurred (causation or fault). This is almost always of interest to the various parties involved. Since vehicles are, or should be, controlled by humans, answering the why question more often than not involves motivation and human psychology. Legal issues can also arise in these incidents. These issues include the violation of criminal and traffic laws where the question of fault is placed before a jury. In this book, the topics of accident investigation, human factors, causation, and fault are not explicitly addressed. Often the effects of these topics are interrelated, and this interrelationship must be addressed to some extent. Indeed, it would be rare to be able to carry out a good reconstruction with little or no physical evidence and data. This book concentrates on reconstruction—the determination of how an accident happened. That said, it is sometimes necessary to combine investigation and reconstruction. Often a reconstructionist may recognize a need for information not gathered initially and must obtain it later. An accurate reconstruction cannot be carried out without a good investigation. A number of books already exist on the topic of accident reconstruction. With some notable exceptions, many of them are tomes devoted to how the authors and perhaps a few colleagues used intuition and insight to decide how they thought an accident happened. In a few cases, these books are collections of “war stories” or case histories, usually presentations of one view of the events. In contrast, this book is one of methods. The perspective taken here is that accident reconstruction is a field of applied science, namely, an application of the principles of science, mathematics, and engineering; accident reconstruction is a quantitative endeavor. The same principles of mathematics, physics, and engineering that allow us to safely race vehicles more than 200 mph, build space stations, and navigate the depths of the oceans can be used to reconstruct vehicle accidents. This requires that reconstructions not only be based on the physical evidence and information gathered from an

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accident investigation but also be based on the laws of nature. A concerted goal of this book is to raise the analytical level of accident reconstruction practice such that commonly known scientific, engineering, and mathematical methods increasingly become a more common part of the field. During the preparation of this book, the authors have assumed that the readers and users of this book and accompanying software (available separately from the authors) have a proper educational background and experience to fully comprehend the material. The United States is not the only country with junkyards (repositories of disadvantaged vehicles, to be politically correct). Vehicle accidents, of course, happen all over the world. Fortunately for engineers, and everyone else for that matter, the laws of mathematics and physics are the same everywhere. Consequently, applications of the material contained herein are not limited by geography. Applications using international units do arise. Therefore, dual units, customary US units (based on the units of foot, pound force, and second) and the metric system (based on the meter, Newton, and second), are used throughout the book. Certain items are avoided, such as the confusion between mass and weight. In customary US units, mass sometimes is given the unit of slug. Its use is avoided here. One slug is equivalent to 1 lb-s2/ft; all appearances of the unit pound, abbreviated as lb in this book, refer to the unit of force. Similarly, in the metric system, a kilogram is considered to be a unit of mass and the corresponding unit of force is newton. The practice of stating weight (a force) in kilograms can cause confusion. This is avoided by always using units of newtons for weight and kilograms for mass. The convention followed here is that weight is the measure of the force of gravity, and the metric unit of force is newton, abbreviated as N. A summary of the topical coverage of the book is not given here. The reader can simply look at the table of contents to see the list of topics. Two features of this book are unique, however, and deserve some mention. One is the coverage of the topic of uncertainty, and the other is the use of many examples throughout the book. Many analysis and reconstruction methods can be implemented using spreadsheet technology. This has been done by the authors, and solutions to the examples are in the form of input information and results printed directly from computer output. Tools contained in popular spreadsheets often allow analysis techniques (time forward computations) to be used for reconstructions (time reverse computations). A common omission made by all accident reconstructionists at one time or another is to measure something or make a calculation based on a measured or estimated parameter and come up with the answer, look at it, and, if it seems to make sense, present it as a definite result or finding. For example, we say “that a car was going 25 mph (36.7 ft/s, 11.2 m/s).” But could it have been 26.2 mph or 24.5 mph? How certain are we of the result? How certain can we be of the result? These questions refer to the uncertainty associated with a measurement or calculation based on a measurement, a group of measurements, or a group of calculations. A view is taken here that we actually are estimating values of dynamical variables. Some estimates have high accuracy and some not so high. The uncertainty associated with results based on these estimates of dynamic variables should always be considered. Determining uncertainty is not always easy to do—but difficulty is not a reason for omitting it. The topic of uncertainty is the first technical topic covered in this book. The uncertainty of a reconstruction may be difficult to calculate, but the authors hope that users of this book will appreciate that a reconstructed speed of a vehicle presented with five significant digits of precision has limited accuracy when the only available skid mark length used in the reconstruction calculation was measured by pacing off the distance.

Preface to First Edition

As already mentioned, a distinction is drawn here between accident reconstruction and accident investigation. The latter is considered to be the process of gathering physical and testimonial evidence from an accident scene, vehicles, and eyewitnesses. It is considered a field of its own. Investigation is most often executed by police officers and sometimes by insurance investigators. As with any other human endeavor, it can be done well and can be  done poorly. Several institutes exist across the country, such as the Northwestern University Traffic Institute, Texas A&M Transportation Institute, Institute of Police Technology and Management (University of North Florida), and others, for training investigators to standardize and improve investigation practices. Although there is a need for each to know what the other does and there is an overlap in knowledge and tasks, a trained accident investigator is not the same as an accident reconstructionist, just as an accident reconstructionist is not a trained accident investigator. Different aspects of accident reconstruction frequently are segregated into the categories of human, vehicle, and environment. The study of human performance and behavior as it relates to vehicular accidents belongs to the study of human factors. This topic is not covered here. Another important aspect of accident reconstruction involving humans is that of occupant kinetics, kinematics, and biomechanics—the study of the motion of vehicle occupants and the physical interaction of a body with interior surfaces and restraints. These concepts are not covered. Environmental topics include such things as the design and performance of roadways, poles and barriers, signs, traffic signals, and their interaction with accidents and crashes. These topics are not covered. As in all professions, the work of accident reconstructionists involves communication and reporting of results. Though they can be extremely important, report writing and diagram preparation are not covered. Other topics omitted include those of finite element analysis of vehicle crash deformation and dynamical crush simulation, such as Simulation Model of Automobile Collisions (SMAC), Simulation Model Nonlinear (SIMON), and others. Collectively, the authors of this book have over 45 years of experience in the practice of vehicle accident reconstruction as well as with the research associated with accident reconstruction methods. Based on this experience, the topics covered throughout the 11 chapters and appendices should provide the methods to quantitatively reconstruct the vast majority of vehicular accidents. Not all accidents involve a crash, or collision, of two vehicles, but most do. Planar impact mechanics (Chapters 6 and 7) is used extensively in the reconstruction of crashes, often combined with estimation of crush energy (Chapter 8). Evidence from the motion of vehicles before an impact or following an impact, or both, often supplies vital information to a reconstruction. Vehicle dynamics simulation (Chapter 11) is invaluable in modeling such motion. Simulation of vehicle dynamics requires the knowledge of how tire forces are generated (Chapter 2), a topic that all accident reconstructionists must thoroughly understand. Methods for the analysis of accidents involving a single vehicle, such as rollovers, pedestrians and bicycle riders hit by cars, or simply yaw marks made by a single vehicle during a sudden high-speed turn, are covered individually. Each accident reconstruction is unique as no two accidents are the same. Moreover, the reconstruction of these accidents can also require the use of different methodologies because of variations in physical evidence and investigative information. This leaves plenty of room for ingenuity and insight for the application of the methods presented in this book. The authors hope this book is useful to those who want to find out how accidents occurred.

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Acknowledgments

W

ith the third edition, as with the two prior editions, it has been the many discussions with other crash reconstructionists as well as the many technical papers published by others in the field that have improved our understanding of the ever-expanding breadth of subjects which form the basis of the methods used in the field of vehicle accident reconstruction. The by-product of these encounters, discussions, and papers is woven into the pages of this book. This third edition also benefits from the times Matt and Ray presented material from the first and second editions of this book while teaching the Society of Automotive Engineers (SAE) Accident Reconstruction Methods seminar after the release of the second edition in 2011. Many students who attended the class asked insightful questions and made numerous interesting comments and suggestions regarding the material. These questions and suggestions led the authors to think critically about the material. This process led to many improvements in the class. These improvements have been incorporated into this third edition. The presentation of rollover reconstruction for light vehicles is drastically improved by inviting Nathan Rose to author the chapter on rollover. Nathan updated the material presented in the previous editions and added new material to make the chapter, once again, relevant and useful to the crash reconstruction community. Thank you, Nathan. The inclusion of a Bayesian Statistics approach to analyzing uncertainty in crash reconstructions would not have been possible without the involvement of Keith Thobe and Shawn Capser. Their knowledge of both statistics and engineering, coupled with our discussions about crash reconstruction, will hopefully make the treatment presented in Chapter 1 useful to the accident reconstruction community. Thanks to Kevin Manogue for organizing the original manuscript and preparing examples and to John McManus, Alan Asay, Don Parker, and Jim Sprague for reading some or all of the manuscript for the first edition. They were followed by Matt Londergan and Weimin Yue who each read part of the second edition, who in turn were followed by Rick Mink, Noel Manuel, and Stan Sangdahl, all from Engineering Systems Inc., who read portions of the third edition. Jason Stigge of Engineering Systems Inc. helped identify content used in a low-speed impact example. Particular thanks to Noel Manuel for the many deep and thoughtful discussions about many of the aspects of the material in the book and his insights into vehicle behavior and relating it to vehicle dynamic simulations. Those discussions were, and continue to be, helpful. Thank you  to Matt’s employer, Engineering Systems Inc., and Jim’s employer ARCCA for their support of our efforts. A sincere thank you to the SAE team who assisted in publishing this edition. Sherry Nigam, a special thank you for your guidance but, most of all, your patience and understanding. Thank you to Linda DeMasi and Bruce Sherwin of Publishers Solutions for their pain-staking review of the manuscript.

© 2022 SAE International

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Acknowledgments

Last, but certainly not least, a great deal of support and encouragement was provided by our wives, Paula, Carol, and Maria, while we prepared this edition. We genuinely thank all three of them and our families. R. Matthew Brach, Naperville, Illinois Raymond M. Brach, Notre Dame, Indiana James J. Mason, Oakland, California

1 Uncertainty and Sensitivity in Measurements and Calculations in Accident Reconstruction

Introduction

A

lthough there may be some cases when and/or where an accident can be reconstructed effectively without the use of calculations and without the use of investigative and experimental data, such cases are rare. And they are becoming rarer as the professional level of the field of crash reconstruction advances, as the demands for more professional and accurate reconstructions increase, and as more crash data is available from vehicle sensors. Whenever measurements are made and whenever calculations are based on measurements and/or experimental data, a level of uncertainty exists. It is the purpose of this chapter to provide ways of quantifying uncertainty. This type of uncertainty is parametric uncertainty, and it relates to variations in the values of the input quantities (parameters) used in the reconstruction calculations. Another type of uncertainty that is prevalent in the field of accident reconstruction is modeling uncertainty. This uncertainty arises when two or more different methods (mathematical or engineering models) can be used to calculate the same quantity. For example, Chapter 6 introduces the theory of impact using point-mass concepts. (Point-mass impact theory ignores rotation of the colliding objects.) Point-mass impact mechanics can be, and often is, used to reconstruct vehicle collisions; its primary purpose, however, is to introduce the concepts of impact mechanics that lead to rigid-body impact—that is, planar impact mechanics. (Planar impact mechanics includes the effect of rigid-body rotation of the colliding objects during the impact process.) Planar impact mechanics more rigorously captures the physics of vehicle-to-vehicle collisions, and therefore provides more accurate results. Except in some very special cases, using pointmass impact mechanics and planar impact mechanics to reconstruct the same collision will produce different results. Such differences are an example of modeling uncertainty.

© 2022 SAE International

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Vehicle Accident Analysis and Reconstruction Methods

As an example of parametric uncertainty, consider an example involving measurements. Suppose a car with a conventional (non-ABS) braking system is brought up to a speed, v, and its brakes are applied suddenly to a level where all four wheels are locked. The tires leave visible tire marks, in this case skid marks, on the road as the car skids to a stop. Suppose, further, that the speed of the car at the beginning of the skid marks is measured using a radar gun and the length, d, of the skid marks is measured using a tape measure. A well-known equation from mechanics used to estimate the speed of a car leaving skid marks from an emergency (locked wheel) stop [1.1] is v = 2 fgd (1.1)



where g is the gravitational constant f is the frictional drag coefficient d is the skid distance Because the speed and distance were measured, this equation can be used to compute the corresponding frictional drag coefficient, f. Although this is a common way of measuring f, it is just one way (e.g., [1.2]). Solving Equation (1.1) for f gives f =



v2 (1.2) 2 gd

It is easy to see that if an error exists in the measurement of either v or d, there will be a corresponding error in f. As used here, the term error does not mean a “mistake” (e.g., using the metric value of g when United States (US) Customary units are intended or vice versa). Rather, the error is a difference of the measured f from the “true” value of f because of measurement inaccuracy (e.g., the radar gun was held a few degrees off to the side or not directly behind the vehicle, or perhaps both), or possibly something else. The “something else” could be because the test conditions differed from a straight-line skid, such as if only three wheels are locked and the car yawed before it stopped. Uncertainty can arise in other ways. Suppose each of two independent observers of the same friction experiment reports the result as f1 = 0.45 and f2 = 0.454. One of the differences between f1 and f2 is that the latter has one more significant digit than the first.1 Can they be considered the same? Is one more correct than the other? Is one more accurate than the other? Sometimes these questions are difficult to answer. Suppose the above example is changed so that Equation (1.1) is to be used to reconstruct the speed of a vehicle from the length, d, of a measured skidding distance and a value, f, the frictional drag coefficient. Suppose, further, that the values of f and d are not known exactly but may vary by some amount about a nominal value. For example, suppose the length, d, was measured several hours or more after an accident and the skid mark length may have changed due to the effects of weather and/or traffic. Suppose, further, that a value of f was measured at the accident site using a vehicle different from the subject vehicle.   If the reader is not familiar with the concepts of significant digits and rounding, it is suggested that they read

1

­Appendix A the section on Numbers, Significant Figures, and Rounding.

CHAPTER 1 Uncertainty and Sensitivity in Measurements and Calculations in Accident Reconstruction

Any variations in f and d are “propagated” through Equation (1.1) and result in uncertainty in the reconstructed speed. In the first example, variations due to measurements of v and d are propagated through Equation (1.2) and result in uncertainty in the calculated value of f. In this case variations in f and d cause uncertainty in the calculated value of v. Both of these examples have the common characteristic that variations in values placed into an equation lead to uncertainty in the result. This notion of variation leading to uncertainty is the primary subject of this chapter. In addition to uncertainty, this chapter also considers the topic of sensitivity. These two topics are closely related, but are separate and distinct. Whereas uncertainty analysis is concerned with the uncertainty of the output of an engineering (reconstruction) model due to the variations in the inputs to the models, sensitivity analysis is concerned with the apportionment, or contribution, of each input to the total uncertainty of the analysis. In a reconstruction setting, uncertainty calculations are typically a component of the reconstruction project resulting directly from recognized uncertainties (variations) in the inputs. (For clarity, uncertainties in the inputs are referred to in this chapter as “variations,” and these variations lead to “uncertainties” in the output.) As will be shown in the examples in this chapter, uncertainty in the frictional drag coefficient, f, for a given roadway introduces variation in the input for this parameter in the vehicle skid-to-stop model and, thereby, produces uncertainty in the output of the model, the vehicle speed, v. On the other hand, sensitivity calculations are not typically done in a reconstruction project. Sensitivity calculations can provide the analyst with insight into the behavior of the model as it pertains to the overall uncertainty in the output of the model from each of multiple input parameters. The corresponding change in the uncertainty of a model result to variation in one input parameter, for example, a 5% change, may be larger (or smaller) than the uncertainty due to a 5% change in another model parameter. Knowledge of the sensitivity of a model can be useful to the reconstructionist in the approach to calculating uncertainty. An example presented in this chapter illustrates this sensitivity of the skid-to-stop model. The distinction between the uncertainty of a result and the sensitivity of a model will be maintained in this chapter. The terminology and notation used in this book are similar to other references, but do have some specific differences. Although the word “error” is commonly used, particularly with respect to measurements (see [1.3, 1.4, 1.5]), its use here is avoided. This approach prevents any connotation that nonexact results are caused by some sort of mistake or blunder, and because error is defined differently by different authors. In general, the quantity to be calculated, such as v or f on the left-hand sides of Equations (1.1) and (1.2), is given the symbol y. The quantity, y, is expressed as a sum of a reference value, Y, and a variation, δy. So the result of an operation (measurement or calculation) of a quantity, y, results in y = Y ± δy. The closeness of the reference value to some “true” value often is referred to as the accuracy of y. In some circumstances, the difference y − Y is called a bias or systematic error. The variation, δy, about the reference value often is referred to as the precision of y or as the imprecision of y [1.6]. After Mandel, the reference value, Y, can be placed into any of three categories based on how it is defined: a. A “true” value that, generally, is unknown. Any value used for Y is an estimate, and values are chosen using a method appropriate for the application. b. An “assigned” value that is agreed upon among experts in a field. An example of this is the value of the acceleration of gravity, g, at sea level, which is assigned the value g = 9.80665 m/s2 [1.7].

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Vehicle Accident Analysis and Reconstruction Methods

c. The mean of a randomly distributed population. With such a statistical definition, Y never is truly known but is estimated by the sample mean of a set of experiments. To generalize the above examples, the problem approached in this chapter is to determine Y and δy when they depend on variations of other variables, say, x1, x 2,…, xn. That is, suppose

y

f x1 ,x2 , ,xn (1.3)

Note that Equation (1.3) can represent a complex sequence of more than one calculation carried out by a computer. In such cases, the variables x1, x 2,…, xn could be functions of other variables, that is, x1 = x1(u1, u2, …, un), x2 = x 2(u1, u2, …, un), xn = xn(u1, u2, …, un), etc., where u1, u2, …, un are different independent variables. Seven different common approaches for calculating uncertainty and sensitivity are covered in this chapter. The first approach is to determine upper and lower bounds on y. Another is to use the analytical form of Equation (1.3) and use calculus to relate variations in the independent variables x1, x2,…, xn to Y and δy. A statistical approach is taken where the independent variables have known statistical properties. This is followed by a method that uses an approach to calculate model sensitivity based on finite differences. Then the Monte Carlo method, based on a statistical approach but implemented via a well-established computer algorithm, is presented. Next, a method using Design of Experiments (DOE) in which reconstruction methods (equations, models, etc.) are treated as processes and a means to calculate the sensitivity of these processes to the factors (variables, parameters, etc.) is presented. Lastly, a method is presented that utilizes Bayesian statistics to calculate uncertainty. The meaning and interpretation of each of the components, Y and δy, differ in the first three approaches and will be discussed in more detail. A common example is used for six of the methods for direct comparison of the results. The overall goal of this chapter is to present the reader with tools to calculate the uncertainty in the results of their analyses (crash reconstructions) and to calculate the sensitivity of models. The nature of the reconstruction work, and the supporting data, frequently involves uncertainty, which manifests itself in the variation of parameters. This variation can come from uncertainty in measurements of physical evidence (length of tire marks, etc.), through estimated values of various parameters (frictional drag, coefficient of restitution, etc.) as well as other sources. Six different methods are covered with details sufficient for the reader to understand the nature of each method and the basis for the use. Examples are provided to understand applications.

 pper and Lower Bounds Using a U Given Model One of the simplest ways of quantifying uncertainty is to establish upper and lower bounds on the dependent variable, y, caused by variations in all independent variables that possess significant variations. First, those quantities in the equation that possess a significant degree of variation are identified as variables of interest. (This identification task is based on knowledge of the sensitivity of the model and may come from the experience of the reconstructionist and/or through sensitivity calculations.) Then an appropriate range of the variation of each variable is determined. Finally, the lowest and highest values of the dependent variable are calculated for all possible combinations of the high and low values

CHAPTER 1 Uncertainty and Sensitivity in Measurements and Calculations in Accident Reconstruction

of the ranges of the independent variables. Values of all specific combinations of the independent variables are used in Equation (1.3) in a way that produces the maximum and minimum values of y. From this δy and Y are assigned the values

δy =

1 ( y max − y min ) 2

and Y =

1 ( y max + y min ) (1.4) 2

Consider again the example given by Equation (1.1). In this example, y is the speed, v, and so Y = V and δy = δv are sought. Under typical circumstances, f and d are known with less than perfect certainty. These are the independent variables. It is assumed that the acceleration of gravity, g, is known with sufficient accuracy and with negligible uncertainty and is a known constant. Suppose that d and f are unknown but the variation of each can be bounded such that dmin ≤ d ≤ dmax and that fmin ≤ f ≤ fmax. The corresponding upper and lower bounds on the estimate of the initial speed, v, are

2f min gd min ≤ v ≤ 2f max gd max (1.5)

The uncertainty δv is taken to be  half of the difference between the upper and lower bounds,



δv =

1 2

(

)

2 f max gdmax − 2 f min gdmin (1.6)

and the reference value, v, is the average of the upper and lower values of v, that is,



V=

1 2

(

)

2 f max gdmax + 2 f min gdmin (1.7)

The result is that v = V ± δv. For example, if fmin = 0.6,  fmax = 0.8,  dmin = 32.0 m, and dmax = 34.0  m, then 19.4 m/s ≤ v ≤ 23.1 m/s. Then δf = 0.1,  δd = 1 m, and v = 23.1 ± 1.8 m/s. This example illustrates what may be considered the simplest and most versatile method for determining uncertainty. It applies to any formula, or system of equations, no matter how complex, and is easy to execute. It is even possible to use this approach with computer simulations using multiple runs of a computer-based model with different inputs. Care must be used when the formula for Y involves differences and division. For example, the lower limit of y = (x1 − x2)/x3 is obtained by using the lower limit of x1 and the upper limits of x2 and x3. Care must also be used when handling negative numbers, particularly when identifying the lower and upper limits of the parameters that produce the corresponding upper and lower limits of the dependent variable as this will depend on the characteristics of the model. A drawback of this (and the next) method is that any likelihood of y to tend to be near the center of, or near either limit of, the range of Y ± δy cannot be assessed and is therefore unknown. Attributing the upper and lower bounds to a specific percentage of a population should not be done; statistical conclusions should follow the use of statistical methods and always be based on statistical data.

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Vehicle Accident Analysis and Reconstruction Methods

Example 1.1 Suppose a vehicle leaves skid marks of length, d, from an emergency stop with locked wheels over a road surface with a frictional drag coefficient, f. The distance d is not known exactly, but is known to be greater than 32 m and less than 34 m. The frictional drag coefficient is known to be somewhere between f = 0.6 and f = 0.8. Determine the initial speed of the vehicle and associated bounds on the uncertainty. Solution The maximum stopping distance occurs when d = dmax = 34 m and f = fmax = 0.8. From Equation (1.1), vmax = 23.1 m/s (51.7 mph). For d = dmin = 32 m and f = fmin = 0.6, Equation (1.1) gives vmin = 19.4 m/s (43.4 mph). Equations (1.6) and (1.7) give V = 21.3 m/s (47.6 mph) and δv = 1.8 m/s (4.1 mph). The end result is that the reconstructed speed of the vehicle at the beginning of the skid marks is v = 21.3 ± 1.8 m/s (47.6 ± 4.1 mph).

Differential Variations Another common method of estimating uncertainty, often referred to as propagation of error, is covered in many laboratory courses taken in science and engineering (e.g., [1.4] and [1.3]). As above, an equation or formula is being used as a reconstruction model to calculate a physical quantity, y, representing a speed, time, distance, speed change, etc. The method uses differential calculus to relate y to the dependent variables x1, x 2, …, xn. As before, the variables x1, x 2 , …, x n could be  functions of other variables, that is, x1 = x1(u1, u2, …, un), x 2 = x 2(u1, u2, …, un), xn = xn(u1, u2, …, un), and so on. Using calculus, y = y(x) can be expressed in a Taylor series near x = a as



y x

y a

y x

x a a

1 2y 2! x 2

x a

2

 (1.8)

a

If (x − a) is small, where a is a reference value of x, then δx = (x − a) is small and (x − a)2 and like terms of higher powers, are considered small and can be  neglected. Let δy = y(x) − y(a), where y(a) = Y is the reference value of y. For a function of several independent variables, the nominal or reference values of x1, x 2, …, xn are given by X1, X 2, …, Xn. Under these conditions, a general formula for uncertainty can be found by replacing the variable differentials with variations, such that



y

y x1 x1

y x2  x2

y xn (1.9) xn

The partial derivatives in each of the terms in Equation (1.9) often are referred to as sensitivity coefficients because their signs and magnitudes indicate how each of the variations, δxi, influences the uncertainty, δy. In applications, the absolute values of the sensitivity

CHAPTER 1 Uncertainty and Sensitivity in Measurements and Calculations in Accident Reconstruction

coefficients sometimes are used to prevent cancellation of the terms when using Equation (1.9) to estimate uncertainty. Equation (1.9) is an approximation that amounts to a linearization of the function y(x1, x 2, …, xn) around its reference value Y(X1, X 2, …, Xn). Note that the derivatives are evaluated at the reference or nominal values. The relative uncertainty often is used and found by dividing Equation (1.9) by Y, giving X ∂y δ xn δ y X1 ∂y δ x1 X 2 ∂y δ x2 = (1.10) + + + n Y Y ∂x1 X1 Y ∂x2 X 2 Y ∂xn X n



Note that y = Y + δy, and for relative uncertainty, y δy =1+ (1.11) Y Y



The independent variables in the function y(x1, x 2, …, xn) given by Equation (1.3) often are thought to be representable in an n-dimensional vector space. In such circumstances, it is common to define a norm2 of δy, where

2

2

2

 ∂y   ∂y   ∂y  2 2 2 δy =   δ xn (1.12)  δ x1 +   δ x2 +  +  ∂ x x x ∂ ∂  1  2  n



Equations (1.9) and (1.12) are different expressions for the same quantity, and it is natural to ask which is correct or, at least, which is better? If the variations δx1, δx2, …, δxn are viewed as orthogonal components of an n-dimensional vector, then it can be shown that the value from Equation (1.12) will always be less than or equal to the value from Equation (1.9), and so Equation (1.9) gives a larger, or more conservative, value. However, the value from Equation (1.12) generally provides a realistic estimate of uncertainty, and so it is commonly used. Another important difference is that each of the terms in Equation (1.9) can have signs that depend on the form of the function y and its derivatives ∂y/∂xi. This means that positive and negative variations can cancel each other. Although cancellations can occur, even to the extent where δy could be zero, this is not something that can be expected. Consequently, absolute value signs sometimes are used with each term of Equation (1.9). This is not done here, but caution must be used when applying Equation (1.9). The use of Equation (1.12) avoids such problems due to each term being squared, and the use of this equation is recommended.   In mathematics, a norm is a measure of the size or length of a vector.

2

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Vehicle Accident Analysis and Reconstruction Methods

In Example 1.1, y = v and there are two variables, x1 = f and x2 = d. After using Equation (1.10), the relative uncertainty of v is given by v V



1 2

f F

d (1.13) D

where the quantities D and F are the reference values of the independent variables. Using the values from the example, it is clear that the variation in friction often has a considerably greater effect than the variation in the distance measurements. This is because, for values typically encountered in practice, δf/F is larger than δd/D. This is particularly true when the friction coefficient is small, such as under icy conditions when F may be small. This use of the relative uncertainty using differential variations, rather than yielding uncertainty of the result of the model, reveals the sensitivity of the result of the model to the selected variations of the input parameters. This use of relative uncertainty will be included in an example. Recall that the derivation of Equation (1.9) involves the replacement of infinitesimal differentials by finite variations. Consequently, there is some degree of approximation involved when using differential variations. As an approximation, higher-order terms were dropped after the expansion of y in a Taylor series about the reference values X1, X 2, …, Xn. It is difficult to assess the error of the approximation because it depends on the functional form of the expression for y, as well as the size of the variations in the dependent variables. The following example demonstrates that the agreement can be reasonably close.

Example 1.2 A vehicle skids to rest over a distance d = 33 m (108.3 ft) in a straight line with its wheels locked. The value of the frictional drag coefficient deemed appropriate for the conditions of the skid is f = 0.70. Variations of the measured values of d and f are established as δd = ±1 m and δf = ±0.1. Determine the initial speed and its uncertainty by computing (a) upper and lower bounds and (b) using differential variations. Solution The reference, or nominal, value of the reconstructed speed can be obtained from Equation (1.1) using D = 33 m and F = 0.70. This gives V = 21.3 m/s (69.8 ft/s, 47.6 mph). Using the same equation, computation of lower and upper bounds for (dmin, fmin) = (32,0.60) and (dmax, fmax) = (34, 0.80) gives lower and upper bounds (vmin, vmax) = (19.4,23.1) m/s [(63.7, 75.8) ft/s, (43.4, 51.7) mph], respectively. Using the method of differential variations, the uncertainty in the speed, δv, can be found using Equation (1.13) after multiplying by the reference value of the speed. This gives δv = 1.8 m/s (6.0 ft/s). The reference value, V, is the same as before, so the reconstructed speed using differential variations is v = 21.3 ± 1.8 m/s (69.7 ± 6.0 ft/s, 47.5 ± 4.1 mph). If Equation (1.12) is used instead of Equation (1.13), the uncertainty is δv = 1.6 m/s (5.1 ft/s) and the reconstructed speed is δv = 1.6 m/s (5.1 ft/s). As mentioned above, the use of Equation (1.13) gives more conservative results than Equation (1.12). To summarize, upper/ lower bounding gives the broadest uncertainty, δv = ±1.8 m/s; the differential variation method gives δv = ±1.6 m/s [or δv = ±1.8 m/s from Equation (1.13)]. The relative uncertainty [Equation (1.13)] can be examined, with δf = 0.1 and F = 0.7 and δd = 1.0 m and D = 33 m. These values give δf/F = 0.14 and δd/D = 0.03. The ratio of these two values is 4.7. This shows that, for these variations in f and d, the result of the model is nearly five times more sensitive to the changes in f than d. Thus, in this case, the relative uncertainty is actually a sensitivity calculation.

CHAPTER 1 Uncertainty and Sensitivity in Measurements and Calculations in Accident Reconstruction

Statistics of Related Variables Sometimes the statistical distribution of a quantity or variable is known. For example, it is known that the height, h, of males in the United States is normally distributed with a certain mean, μh, and variance, 2h, or that the distribution of skid numbers (see Appendix B, Glossary) measured at intervals along roads is, at least approximately, normally distributed (see Figure 1.1). Furthermore, it often happens that the variable, say, x, whose statistical properties are known, is related through a mathematical equation to another variable, say, y. Let

y

f x (1.14)

© SAE International.

 FIGURE 1.1   Histogram distribution of a sample of skid numbers for 230 test sections of a two-lane country road with a wide range of average daily traffic [1.8]. The average is 0.392 and the standard deviation is 0.077.

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Vehicle Accident Analysis and Reconstruction Methods

be the mathematical equation where the function f(x) is given and the statistical distribution of x is known. In this context, x and y are referred to as random variables. The question often arises: what is the statistical distribution of y? This is not an easy question to answer with mathematical rigor for an arbitrary function f(x), but there are some important cases where useful results can be found. One case is when the function is linear; another is when an approximate method of answering the question is satisfactory, or it is the only means available. Each of the variables x and y has a certain statistical distribution (such as a normal distribution, a uniform distribution, a chi-square distribution, etc.). Each distribution has parameters (mean, median, variance, etc.). When random variables are related, such as by Equation (1.14), it is common to find that knowing the statistical distribution of x doesn’t mean the distribution of y is known or can be found. In fact, relating distributions is a very difficult problem. The following section concentrates on finding the parameters μy and σy from the parameters μx and σx.

Linear Functions When the function in Equation (1.14) is linear, that is,

y

f x

ax b (1.15)

where a and b are constants, the statistical properties of y are well known (e.g., see [1.9] and [1.10]) and relatively simple. If x has any statistical distribution with mean, μx, and variance, 2 x , then the mean of y is

a

y

b (1.16)

x

and the variance of y is

2 y

a2

2 x

(1.17)

For linear functions, such as Equation (1.15), if x is normally distributed, then y will be normally distributed. Knowing this is helpful in relating uncertainty to variations through equations using statistics.

Example 1.3 Suppose that under certain circumstances, the perception-decision-reaction time, tpdr, for drivers is known to be normally distributed with a mean, μt, of 1.75 s and a standard deviation of σt = 0.2 s. What are the mean and standard deviation of the distance, d, traveled by a vehicle with a uniform speed of 20 m/s (65.6 mph) under such circumstances?

CHAPTER 1 Uncertainty and Sensitivity in Measurements and Calculations in Accident Reconstruction

Solution From mechanics, it is known that distance is the product of speed, v, and time, τ, so



d v

Here, v is a constant, and the relationship between d and τ is linear. If the time τ = τpdr, then d has a normal distribution, where the mean, μd, is v

d

20.0 1.75

t

35.0 m 98.4 ft

and the variance of d is



2 d

v2

2 t

202 0.2

2

16.0 m2 172.2 ft2

The standard deviation of d is σd = 4.0 m (13.1 ft). With this information and based on the properties of the normal distribution, it can be stated that under the given conditions, drivers will fully react at a distance of 35.0 ± 2(4.0) = (27.0,43.0) m [(88.6, 141.1) ft], 95% of the time. Because of the bell shape of the normal probability curve, values near the center are more likely than those near the tails.

Arbitrary Functions (Approximate Method) For an arbitrary function, f(x), a way of relating the statistical parameters of x and y is to expand the function in a Taylor series about the mean of x, neglect higher-order terms (i.e., linearize the expansion), and then use Equations (1.16) and (1.17). The Taylor series of y about the point x = μx is



y x

f

f x

x

x

1 2f 2 x2

x

x

x

2 x

 (1.18)

x

If terms with a power of 2 and higher are neglected, then this reduces to



y x

f

x

f x

x x

x

(1.19)

11

12

Vehicle Accident Analysis and Reconstruction Methods

Evaluating this at the mean of x, μx gives the mean of y

y

f

x

x

y

(1.20)

Substituting this into Equation (1.19) and rearranging gives

y x

f x

y

x

x

(1.21)

x

Viewing Equation (1.21) as a linear relationship, such as Equation (1.15), and using Equation (1.17) gives the variance of y as



2

df dx

2 y

2 x

(1.22)

x

This process can be generalized to a function of many variables, y = f(x1, x 2, …, xn), which gives



2 y

n i 1

2

df dx i

n

2 xi

n

2 i 1 j 1

xi

f xi

df dx j

xi

xi x j

(1.23)

xj

where the variables in all of the terms in Equation (1.23) are evaluated at their mean values. The quantity, σxixj, is the covariance of xi and xj, and is zero if variables xi and xj are statistically independent. When all the variables xi and xj are statistically independent, then

y

2 x1

2 x2



2 xn

(1.24)

This shows that the standard deviation, σy, of the result of a calculation involving many variables (n  ≥  2) grows as the root-mean-square (rms) of the variance of the input ­quantities, xi. Some words of caution: These expressions are approximate because the Taylor series expansion was linearized. In addition, although expressions for the mean and variance of y have been found, the statistical distribution of y generally is unknown, even when the distribution of x is known. Whereas the distribution of y is the same as the distribution of x when x is normally distributed and when x and y are related linearly, the same is not true when the function f(x) is nonlinear. Additionally, these equations are valid when xi and xj are statistically independent. This may not always be the case in reconstruction applications.

CHAPTER 1 Uncertainty and Sensitivity in Measurements and Calculations in Accident Reconstruction

Example 1.4 The well-known formula from mechanics, Equation (1.1),



v

2fgd

gives the initial velocity, v, for a vehicle that slows to rest uniformly with deceleration fg over a distance, d, where f is a constant frictional drag coefficient. Suppose that f is normally distributed with a mean, μf = 0.4, and a standard deviation, σf = 0.08. What are the mean and standard deviation of the initial speed if the stopping distance is known to be exactly d = 20 m? Solution From Equation (1.20), the mean of the distribution of the initial speed is



2 fg

v

d

2 0.4 9.80665 20

12.53m / s 41.1ft / s

and from Equation (1.22) the standard deviation of v is



sv

gd 2fgd

sf

15.66 0.08

1.25 m / s 4.1 ft / s

f mf

Finite Differences This method provides a procedure for evaluating the uncertainty of a model using a base, or nominal, case and calculating the “deviations” of the model response to prescribed changes in a number of the input parameters. The method has been referred to in the literature as utilizing finite differences [1.11]. But the use of that phrase, at least in the traditional sense from mathematics where finite difference methods are used to solve differential equations, seems to be a misnomer. The presentation here continues with the use of that phrase for consistency with previous literature. The method as presented is based on prior literature, primarily [1.11] but also [1.12]. The literature provides the steps to implement the scheme numerically, thus allowing the approach to be applied directly to simple models and complex models. This method has additional utility as it can be used to calculate the sensitivity of the model by ranking the contributions to the uncertainty to its various parameters. This use of the method can be used to establish the few that are significant for simplification of future uncertainty calculations. The application shown here is rather simple, involving two parameters, but illustrates that the method can be extended to include more parameters. This example uses the same problem from Example 1.2 so that a direct comparison between the results of the various methods can be made.

13

14

Vehicle Accident Analysis and Reconstruction Methods

The premise of the finite difference method is to assign an appropriate range via high and low values for any/all of the input parameters of a model and calculate the model response at each of these value pairs in turn, while keeping all the other parameters at the nominal value. The deviation of the model response from the nominal value of the response for each of these conditions, for both the high and low values of each parameter, is then evaluated. Each set of the deviations for each parameter (high and low) is squared and averaged. The square root of the averaged sums of these squares is the range of uncertainty relative to the nominal response. Moreover, the (high and low) deviations for each parameter represent the uncertainty of the result for that parameter for the given variation in the parameter, which is essentially an upper/lower bound calculation for this parameter. These variances can be used to evaluate the relative uncertainty of each parameter. The authors recommend that the range of variation of the input variables of the model be of the same confidence level. For example, if one variable is selected to represent a 95% confidence band (referred to as the “two-sigma” confidence range), then the ranges of the other variables need to be defined with a similar confidence range. This is particularly important when using the method to rank the sensitivity of the uncertainty of the response to the model variables (inputs). Selecting a high-low range for one of the variables which is larger than the rest risks producing a larger corresponding range of the response, producing larger deviations. A larger range of the response may mistakenly lead to the conclusion that the model is more sensitive to the parameter with the larger range. Note that the selection of an appropriate range, and a range consistent with the other variables, in this method and other methods, is not a simple task. The reconstructionist is encouraged to give critical thought to the selection of these ranges and seek references that provide data for each parameter for guidance. In the reference [1.11], the authors describe the implementation of the method in six steps.

•• Step 1: Determine the nominal (mean) value for each independent variable in the equation or equations.

•• Step 2: Determine the high and low values for each independent variable at the selected confidence level, or the range about the mean values one wishes to consider. Values are commonly reported as the nominal value plus-or-minus one standard deviation. This defines a range of values that will occur 68.3% of the time. Any level of confidence can be selected as long as the same probability level is selected for all values. Selection of high and low bounding values two standard deviations from the mean will include 95.5% of all cases.

•• Step 3: Calculate the nominal result with all the parameters of interest assigned their nominal values.

•• Step 4: With all other parameters at their nominal values, set each parameter in turn at its highest value, find the result for the selected dependent variable, and find the departure of this result from the nominal result.

•• Step 5: Repeat Step 4 for the lowest values of each parameter. •• Step 6: Average the squares of the results of Steps 4 and 5. Take the square root of the sum of these averaged squares. This value is the uncertainty range around the nominal result to the selected confidence level.

The output of Step 6 gives the uncertainty range of the result (output variable) in the vicinity of the nominal result for the assigned range of variation of each of the parameters considered in the uncertainty analysis. The authors state that if the confidence level of the variations for each of the input parameters is the same, then the calculated uncertainty can be stated to the same confidence.

CHAPTER 1 Uncertainty and Sensitivity in Measurements and Calculations in Accident Reconstruction

Example 1.5 Consistent with the conditions listed in Example 1.1, calculate the uncertainty in the pre-skid speed of a vehicle using 32 m ≤ d ≤ 34 m and 0.6 ≤ f ≤ 0.8 using the finite difference method. Solution For this example, the model given by Equation (1.1) is repeated here:



v = 2fgd 

(1.1)

Following the six steps outlined above, the reference (nominal, mean) value for the speed is calculated as 21.3 m/s (69.9 ft/s, 47.6 mph). In Step 4, the “departure” of the response due to the selected variations in the parameters is calculated. The values of the departures are given in Table 1.1. Squaring and averaging the values for each of the high/low pairs and then taking the square root of the sums of the squares of these values gives the uncertainty about the mean of the method. This leads to the following range for the response: v = 21.3 ± 1.6 m/s. Examining the relative uncertainty using the results of the finite difference method, these ranges show that, for the variations of the inputs selected, the magnitudes of the departures for the variation in friction (δf = ±0.1) are significantly greater than for the variation in the length of the skid (δd = ± 1.0 m). This is consistent with the analytical relative uncertainty [see Equation (1.13)]. In the analysis using relative uncertainty, it was shown that the variation in friction has a considerably greater effect than the variation in the distance measurements because, for values typically encountered in practice, δf/F is larger than δd/D [see Equation (1.13) and Example 1.2]. For the values of this example, the calculation shows that the model is 4.7 times more sensitive to the stated variations of f as it is to the stated variations of d. Using the finite difference method, the values of the departures from the mean are 1.52 for f and 0.32 for d (see Table 1.1). The ratio of these departures is 1.52/0.32 = 4.8. We see that the two methods give approximately the same sensitivity of the model to the two parameters for the stated ranges of variation. TABLE 1.1  Deviations about the base value for Example 1.5. High d Deviation about the 0.320 mean

Low d

High f

Low f

−0.325

1.470

−1.579 © SAE International.

15

16

Vehicle Accident Analysis and Reconstruction Methods

Monte Carlo Method The Monte Carlo method, as it applies to crash reconstructions, is a computational algorithm used to evaluate a reconstruction model when the variation of the inputs to the model can be defined using statistical distributions. The evaluation is accomplished with the Monte Carlo algorithm randomly sampling each of the statistical distributions defining the inputs and then calculating the result of the model with the sampled set of inputs. This sampling process is executed numerous (tens of thousands) of times whereby the response of the model is thereby evaluated for nearly all combinations (high values, low values, and all manner of combinations) of the values of the inputs as dictated by the distributions. (The larger the number of trials the larger the number of combinations.) The algorithm saves all of the thousands of values of the model response selected by the analyst, and the distribution of the response is plotted as a histogram. The histogram shows the distribution of the response variable. Analysis of the stored values provides statistics of the response, such as the mean and standard deviation, as well as the percentile distribution of the result. This method is extremely powerful in crash reconstruction applications for several reasons: (1) the calculations can be done for virtually any model, regardless of complexity; (2) relatively little analytical work, such as partial derivatives, is needed; and (3) the output of the method allows the reconstructionist to illustrate the likelihood of the results, which is not possible with the other methods. Other methods such as upper and lower bounds, finite differences, and differential variations do not allow the reconstructionist to address the likelihood of the results. The most important limitation of the application of this method is that the reconstructionist needs to assign a statistical distribution for each of the input values. These distributions are not always specifically known. In some of these circumstances, the Monte Carlo analysis can be facilitated by using a classical distribution such as a uniform distribution (all values equally likely over an appropriate range) for input parameters for which sufficient measurements are not available, but a suitable range can be defined. The example for this section illustrates the use of a uniform distribution. This limitation, and perhaps the burden of having to describe the nature and theory of the method to non-technical people (lawyers, etc.), indicates that its use does not appear to be as prevalent in typical project work by reconstructionists as some of the other methods. However, because of its relative simplicity in application and broad coverage of results, this method is particularly useful as a research tool in illustrating, analyzing, and understanding a wide variety of reconstruction models. A number of papers have been published over the last three decades that present the concepts of, and illustrate the application of, the Monte Carlo method in crash reconstruction work. Two early papers on this topic [1.13, 1.14] provided the foundation of the method and example applications. These were followed by two papers [1.15, 1.16] that provided more background for both the topic and the application of the method. The first of these two papers discussed the nature and selection of the distributions, provided more details regarding the application to a crash reconstruction, and considered the use of filtering the results to limit the solution space. The second of these papers looked specifically at facilitating Monte Carlo analysis using spreadsheet functions without the need for specialty programs or statistical packages. Two more papers were presented [1.17, 1.18] that further considered the method and applications before a couple of papers by Daily [1.19, 1.20] broadened the discussion of the use of Monte Carlo methods by considering the sensitivity

CHAPTER 1 Uncertainty and Sensitivity in Measurements and Calculations in Accident Reconstruction

of the input parameters to the output distribution and the notion that the variables in the Monte Carlo analysis can be correlated. More recent literature includes [1.21, 1.22, 1.23]. An example is presented here using the skid-to-stop model with the same information provided in Example 1.1.

Example 1.6 From Example 1.1, the inputs of the frictional drag, f, and the distance, d, are treated as values that have variations defined using statistical distributions. These definitions facilitate the use of Monte Carlo analysis. The frictional drag, f, is defined using a normal distribution (see Figure 1.1), and the skid distance, d, is defined using a uniform distribution. For this example, these values are defined as 1. Frictional drag coefficient: f = Normal(μf, σf ) = Normal(0.7, 0.03) 2. Skid distance: d = Uniform(dlow, dhigh) = Uniform(32, 34) The example’s analysis was carried out using a commercially available Monte Carlo analysis program [1.24]. A total of 30,000 trials were run. For the purposes of this example, the input variables are considered statistically uncorrelated3. Some of the specific results of the analysis are listed in Table 1.2. The mean of the speed is 21.3 m/s and the standard deviation is 0.49 m/s. The distribution of the speed of the vehicle is shown in Figure 1.2. Visual examination of the resulting distribution shows pronounced symmetry about the mean, i.e., little skewness. The skewness value calculated by the Monte Carlo analysis program is −0.04, a very small value. The lack of skewness (i.e., a symmetrical distribution) allows standard statistical inference procedures typically applied to normal distributions, such as confidence intervals, to be validly applied. Thus, statements about the confidence of the results can be made: the 10th and 90th percentile values show that, with 80% confidence, the reconstructed speed of the vehicle lies between 20.7 m/s and 21.9 m/s (74.5 kph and 78.8 kph). The statistics show that for 95% confidence, the reconstructed speed lies between 20.3 m/s and 22.2 m/s (73.1 kph and 79.9 kph). TABLE 1.2  Results of the Monte Carlo analysis.

© SAE International.

Minimum value

18.9 m/s (62.0 ft/s)

Maximum value

23.2 m/s (76.1 ft/s)

Mean value

21.3 m/s (69.9 ft/s)

Standard deviation

0.49 m/s (1.6 ft/s)

90th percentile

20.7/21.9 m/s (67.9/71.9 ft/s)

10th percentile

20.5/22.1 m/s (67.3/72.5 ft/s) continues

  This assumption was checked using the Monte Carlo data to calculate the correlation coefficient. The

3

correlation coefficient was 5.6 × 10−4, very near zero. Thus the assumption that the input parameters are uncorrelated is supported. Checking correlation is another capability of the Monte Carlo method.

17

Vehicle Accident Analysis and Reconstruction Methods

 FIGURE 1.2   Distribution of the initial speed in m/s (30,000 trials).

© SAE International.

18

Equation (1.1) has long been used in the field of crash reconstruction to calculate the speed of a vehicle based on the value of deceleration and the skid distance [1.53, 1.1]. The value of g is a known constant and has no uncertainty. Therefore, the question of the reliability of the Monte Carlo method in this example resides with the interpretation of the statistical distributions assigned to the frictional drag and skid distance. In the case of the distribution of the frictional drag as normal, data supports this selection. Figure 1.1 shows experimental data supporting the notion that frictional drag data is normally distributed. The selection of a uniform distribution for the skid distance is an admission that the actual length of the skid distance is uncertain and cannot be defined by a single value. There may be variation based on the visibility/darkness of the mark, particularly at the start of the skid. Or there may be variation that would result from repeated measures of the same mark by different investigators [1.26]. Therefore, variation is introduced in this parameter. Some limited data suggest that, depending on the method of measurement, the distribution of measurements for longer distances (~11 m) may appear normal [1.26]. Ultimately, the selection of the distribution (any distribution used in a Monte Carlo simulation) is based on the judgment of the reconstructionist using whatever data is available pertaining to the parameter in question. The selection of a uniform distribution in this example was based as much on illustration purposes (see comments below) as on statistical rigor. An interesting aspect of the results of this two-parameter Monte Carlo analysis is that, while one of the distributions in the analysis is normal (f), the other (d) is not, but the

CHAPTER 1 Uncertainty and Sensitivity in Measurements and Calculations in Accident Reconstruction

distribution of the final result, the vehicle speed, also appears to be normal. It is intuitive that as both of the distributions are symmetric, that the symmetry is preserved in the analysis. It is not intuitive that the resulting distribution for the speed would necessarily appear normal given that one of the input parameters is modeled as a uniform distribution. Additionally, had both the distributions been defined using a uniform distribution, the distribution of the result would also have characteristics of a normal distribution such as lower frequency values near the extremes of the range of the result. (Interested readers are encouraged to try this.) This phenomenon is typically attributed to the Central Limit Theorem [1.14], but the Law of Large Numbers may also play a role.

Design of Experiments The formulation of the concepts of DOE originates in the beginning of the twentieth century. Numerous detailed treatments of the topic have been written [1.27, 1.28]. DOE is an analytical method in which changes in the system/process response due to changes in the input values (the factors) are systematically and numerically determined and analyzed in a convenient, algorithmic manner. The results of the analysis provide insights primarily into the sensitivity of the system/process to changes in the inputs. Generally, the DOE method is used with experimental measurements of a physical system or process. However, the method can be applied to computer-based “experiments,” i.e., computer simulations. The application of the use of DOE applied to computer simulations has been a more recent development. Early work was done in the late 1970s and early 1980s [1.29, 1.30] with the widespread availability of digital computers and the development of simulation models. The topic now is given its own treatment in various books [1.27, 1.31, 1.32] and has been included in literature dealing with forensic engineering applications [1.33]. The treatment of the topic presented here closely follows a portion of the treatment in a prior publication [1.25]. The presentation introduces the basic concepts of DOE and is followed by an example using the simple skid-to-stop example that has been used throughout the chapter. The terminology and the basic approach shown here are consistent with other more in-depth treatments. Consider a process with an output (response), y, that is dependent on multiple input parameters (factors) x1, x 2,…, xk as shown in Figure 1.3. The process is implemented or run repeatedly for different, preselected values (levels) of the factors, where the j-th run, the j-th combination of factors, produces the j-th response value, yj. The primary function of DOE is to determine a quantitative measure of the contribution (effect) of each of the factor changes on the response. The response must be quantitative or measurable. The factors can be quantitative but also can be attribute variables (such as fast or slow, large or small, etc.). In the coverage here, each factor will be allowed to take on only two values: a low level (−) and a high level (+). The low and high levels of the factors are selected by the experimenter or analyst to represent a practical and relevant range of values (viewed from the application and context of the process) large enough to have an influence, yet small enough to determine the local behavior of the process. It is not uncommon to first carry out an exploratory DOE to establish ranges of variables, or a reduced set of variables, followed by another DOE, based on the results of the first, for a more refined analysis. In the case of attribute variables, the choice of low and high levels is arbitrary, but generally

19

Vehicle Accident Analysis and Reconstruction Methods

 FIGURE 1.3   Schematic diagram of a process treatable by DOE.

© SAE International.

20

defined by the nature of the process or the factor. In the case of quantitative variables, the choice is usually intuitive, but may be defined by available data, research in the literature, the experience, and knowledge of the analyst about the process, etc. From the point of view of the analysis, these ranges are arbitrary, but care and judgment by the analyst is needed as the results depend directly upon the choices. The use of an exploratory DOE can be useful in evaluating the appropriateness of the ranges of the quantitative factors or whether a factor has little effect on the result and can be excluded from the analysis. Consider a simple case with two factors, k = 2: x1, with low and high values x1 and x1 , and x 2, with low and high values x2 and x2 . There are n = 2k = 4 runs (possible combinations of response values, yj) for all the combinations of the low and high values of the factors, as shown in Table 1.3. Corresponding estimates of the main effect, ME, of changes in the response are calculated using the differences in the average response at each of the different levels. For factor x1,

ME x1

1 2

y2

y1

y4

y 3 (1.25)

ME x2

1 2

y3

y1

y4

y 2 (1.26)

For factor x 2,

One of the distinct benefits of the DOE method is that it can provide a measure for the effect of the interaction between two or more factors. For the two-factor example, this is done by taking the difference in the diagonal values in Table 1.3

1 2

ME x1x 2

y1

y4

y2

y 3 (1.27)

TABLE 1.3  Experimental layout for DOE with two factors each with two levels. Factors

x1

x1

x2

y1

y2

x2

y3

y4 © SAE International.

CHAPTER 1 Uncertainty and Sensitivity in Measurements and Calculations in Accident Reconstruction

© SAE International.

TABLE 1.4  Standard experimental layout for a DOE with two factors each with two levels. Run

x1

x2

x1x2

1

−

−

+

y y1

2

+

−

−

y2

3

−

+

−

y3

4

+

+

+

y4

The overall average process response is

MEavg

1 y1 4

y2

y3

y 4 (1.28)

This analysis can be placed into a convenient, more standard, format as shown in

Table 1.4. The main effect of each factor and interaction becomes the inner product of the

sign in each factor column with the corresponding value in the response (y) column, each divided by 2k − 1. For example, it can be seen that for k = 2,



ME x1

1 2k 1

y1

y2

y3

y 4 (1.29)

produces the same result as Equation (1.25).

Example 1.7 This example uses the same information provided in Example 1.1 with the process being the skid-to-stop formula given by Equation (1.1). The response of the process is the speed of the vehicle, v. The factors (inputs) to the process are the frictional drag, f, and the distance, d. The ranges used here for these factors, the high and low values, are the ranges typically consistent with available information: the range of the distance d is known to be greater than 32 m and less than 34 m and the range of the frictional drag coefficient is known to be between 0.6 f 0.8. For the analysis here, the frictional drag, f, will be the first factor, x1, and the distance, d, will be the second factor, x2. Thus x1 = f− = 0.6, x1 = f+ = 0.8 and x2 = d− = 32 m, x2 = d+ = 34 m. Table 1.5 shows the experimental layout with the response values computed using Equation (1.1).

© SAE International.

TABLE 1.5  Experimental layout for Example 1.7 with response values. Run

x1

x2

x1x2

y=v

1

−

−

+

y1 = 19.41 m/s

2

+

−

−

y2 = 22.41 m/s

3

−

+

−

y3 = 20.00 m/s

4

+

+

+

y4 = 23.10 m/s continues

21

22

Vehicle Accident Analysis and Reconstruction Methods

The main effects can be calculated using Equations (1.25) through (1.28), or the inner products with Equation (1.29). These are



ME x 1

1/2

19.41 22.41 20.00 23.10

3.05m / s



ME x 2

1/2

19.41 22.41 20.00 23.10

0.64m / s



ME x 1 x 2

1/2

19.41 22.41 20.00 23.10

0.05m / s



ME avg

1/ 4

19.41 22.41 20.00 23.10

21.23m / s

The main effect values are the uncertainties/sensitivities of the (skid-to-stop) process to the changes (variations) of the frictional drag coefficient values and the skid distance values, that is, the calculations show that the initial speed is 3.05/0.64 = 4.8 times more sensitive to the ±0.10 variation in f (x1) than to the ±2.0 m variation in d (x2). The value of the interaction effect ME x1x2 = 0.05 m/s is small relative to the main effects of the factors, so the interaction between the two factors in this model can be considered insignificant. Because of the square root sign [Equation (1.1)], the skid-to-stop model is nonlinear; thus, the process is nonlinear. In general, the complexity of the process has little effect on the ease or difficulty of calculating sensitivities using DOE. Furthermore, the results from this method are virtually identical to using the method of differential variations (relative uncertainty) and finite difference to estimate the sensitivity of the calculation of initial speed. This is seen by comparing the results of Example 1.5 with the example here. In Example 1.5, it was shown that using relative uncertainty by way of differential variations [Equation (1.13)], the sensitivity of the model is 4.7 times higher to f than it is to d. Using differential variations and examining the ratio of the departures, the effect of f to d is calculated to be 4.8. Using the main effects calculated using DOE, we see that the effect of f to d on the speed is 4.8 times as larger. The three methods predict essentially the same sensitivity between the two factors for this model. This is because the DOE method, the method of differential variations, and the method of finite differences all linearize the process response. It is important to point out that as part of the analysis, the DOE method calculates the effects of interactions of the factors on the response to assess the sensitivity to combinations of factors. None of the other methods provides this information. For this example, the analysis shows that the interaction between the two input variables is insignificant, but that may not always be the case, and will likely be difficult to predict. Situations certainly exist where interactions between factors will be significant. The DOE method is the only method of the six presented here that can identify whether/which of the interactions are significant.

CHAPTER 1 Uncertainty and Sensitivity in Measurements and Calculations in Accident Reconstruction

Another advantage to the DOE method is its ability to handle large numbers of factors. It does this by various strategies not covered in this treatment that allow for the significant reduction in the number of trials to be run using a process referred to as a fractional factorial design. These strategies are covered in detail elsewhere [1.28, 1.27]. This fractional factorial approach permits the number of trials to be reduced with little loss in effectiveness with the trade-off being that some information is lost regarding interactions. Fortunately, this trade-off is reasonable as experience shows that, in general, the effect of interactions on the response is negligible. Thus, the sensitivity of crash reconstruction models that have numerous inputs, such as an impact model or vehicle dynamics simulation model, can be analyzed accommodating a large number of factors and interactions. Three papers present this approach [1.25] analyzing an impact model and [1.34, 1.54] analyzing a vehicle dynamics simulation models. Research into the application of DOE methods to crash reconstruction continues with the publication of a variation of the method presented here [1.35].

The Bayesian Method Another method for calculating the uncertainty of a result in an accident reconstruction, based on Bayesian statistics [1.36], has been proposed over the last couple of decades [1.37, 1.38]. These two papers referenced earlier works [1.39, 1.40] that used the Monte Carlo method (see prior topic on Monte Carlo Method) coupled with the notion of restrictions (additional information about the incident not formally part of the Monte Carlo analysis) to eliminate some of the results generated by the analysis. While these papers that use Bayesian statistics generated some interest in the crash reconstruction community, the method has not been widely adopted in the reconstruction of crashes or as a topic of further research in the field. This may be due to the level of complexity as alternative statistical concepts that are not typically taught in the probability and statistics class required in collegiate engineering programs are the foundation of the method. A Bayesian approach to the estimation of accident reconstruction model parameters and the calculations related to their uncertainty is presented here to offer the reader an introduction to the topic. Before presenting the application of Bayesian statistics to crash reconstruction, a brief treatment of the topic is presented. As presented here and elsewhere [1.37, 1.38], the method applies to reconstruction models that are algebraic in their formulation. No consideration is given to the application of the method to reconstruction models that require time integration such as vehicle dynamics simulation programs. In contrast to the other methods presented in this chapter for the calculation of uncertainty, the Bayesian approach is not intended to specifically assess the uncertainty in reconstruction calculations. As will be shown later in this section, the method is intended to obtain the point estimates for expected values of a response, i.e., to use knowledge about the behavior of a system (referred to as a “prior”), with an appropriate model to predict the probability distribution, including the statistical intervals, of the response of the model. The uncertainty of the response that is obtained via the (credible) intervals is actually a secondary output of the analysis. In this way, the Bayesian approach, while providing the uncertainty of the result, is fundamentally different in the way that it is applied to reconstruction problems than the other methods described in this chapter. The subjects of probability and statistics are related in that they both deal with the frequency of events or outcomes. Probability is predictive in nature; it deals with the likelihood of future events given information about the process, whereas statistics deals with the analysis of the frequency of events that have been accumulated from trials that are often

23

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Vehicle Accident Analysis and Reconstruction Methods

used to determine the nature of the process that produced the outcomes. All the methods for the calculation for point estimation and uncertainty presented in the earlier sections of this chapter (and the accompanying discussion) use this standard approach within the field of probability and statistics, which is known as the frequentist approach. This descriptive title is used since the output of the analysis, the probabilities of the outputs given by the statistical model, represent the frequency of the expected results. For example, rolling a single six-sided (fair) die hundreds of times and tallying the outcomes is a classic example of the probabilities of the outcomes seen as frequencies; any single number on the die is expected to show up approximately ⅙ of time, with that number approaching exactly ⅙ as the number of trials nears infinity. However, situations in real life are often such that only one outcome to an event (or series of events) is possible (and hence no frequency information is available or can be accumulated). In these situations, what is required is the probability of the inputs that created this outcome. To solve this problem, an alternative school of thought has arisen over the last several decades that uses a simple formula to reverse probabilities, known as Bayes’ Theorem. The theorem is named after the Reverend Thomas Bayes who first published it (posthumously) in 1764 [1.41]. While the formula is straightforward and has been known for over two centuries, the application of the formula in real-world situations can be quite complex and requires modern computing power, using Markov Chain Monte Carlo (MCMC) methods, to become tractable (e.g., WinBUGS, a free software package that performs the analysis and that can also visually show the Bayesian model [WinBUGS]). Starting first in the insurance industry during the mid-twentieth century [1.41], the method has now evolved to become, among other uses, the tool of choice for finding such one-time events as missing aircraft [1.42, 1.43]. One advantage of the Bayesian method is that it can, at least in some situations, be constantly updated based on new information, adjusting the inputs to the original statistical model to provide new predictions. Whereas traditional frequentist statistics often use their own language (such as the “confidence” in the confidence interval of a model response, which is not, despite common misconceptions, a probability) to accurately express the required concepts when dealing with frequencies, Bayesian statistics simply presents its results in regular probabilities. A primary difference between the frequentist approach to the Bayesian approach applies to the view of the probability distributions of the model parameters. The frequentist view is that the probability distribution for a parameter is selected (uniform, normal, etc.) and the values that define the distribution (mean, variance, etc.) are unknown (desired) but fixed. In the Bayesian approach, the distribution of the observed data is fixed but the underlying values that define the distribution (mean, variance, etc.) are allowed to vary under the notion of the “prior.” These “prior” values are refined using the Bayesian method into “posterior” values and are evaluated. The downside of Bayesian statistics, and the aspect of the method on which critics focus most of their attention, is that the application requires the user to provide a “prior,” for prior distribution, for one or more of the model parameters. The criticism is that this prior distribution can be a subjective assessment on the part of the analyst, particularly when data for the model parameter are unavailable. While subjectivity can be an issue with any statistical model, even frequentist models, the Bayesian approach can give very different answers depending on the choice of a prior. And since this prior is often subjectively derived at the start of the analysis, simpler distributions, such as the Uniform Distribution, are chosen initially (like what was done in the section on Monte Carlo Method), the effect can be viewed as not much better than guessing.

CHAPTER 1 Uncertainty and Sensitivity in Measurements and Calculations in Accident Reconstruction

Before going further, the actual Bayes’ theorem should be reviewed in more detail. Bayes’ theorem is in generic terms as follows



P ( A|B ) =

P ( B|A ) P ( A ) P (B)

(1.30)

In plain language this formula states that the probability (P) of getting A, given that B has occurred, is equal to the quantity of the probability of B happening given that A occurred (i.e., the exact opposite of what we’re trying to find) multiplied by the probability that A occurs in general, all divided by the probability that B occurs in general. Certain terms are used to describe portions of the formula, and these are, following the same order and structure as given in Equation (1.31),



Posterior distribution =

Sampling distribution ∗ Prior distribution (1.31) Normalizing distribution

The prior distribution is based on prior knowledge of the situation, and the posterior distribution represents how that prior distribution has been modified based on the data. Note that the normalizing distribution is not generally needed for software (e.g., WinBUGS) to perform the calculations. The Bayesian formula is true regardless of application. It is simply a result stemming from the logic of chance, allowing conversion of conditional probabilities. It is a general tool that can be potentially used in a wide variety of situations. One common use of Bayes’ theorem is for parameter estimation. In the frequentist school of thought, a variable X follows a known probability distribution with an unknown parameter (or parameters). Data is then used to estimate this parameter. But in no way is the parameter considered random; instead, it is a fixed quantity, just currently unknown. In stark contrast, the Bayesian approach instead sees even the parameters themselves as random (resulting in an almost hierarchical view of statistics, distributions nested within other distributions). In this view, Bayes’ theorem becomes



f (θ |x ) =

f (θ ) ⋅ L ( x|θ )

∫ f (θ ) ⋅ L ( x|θ ) dθ

(1.32)

Θ

where x is a vector of observed data f(θ) is a prior density supporting the belief for the probability distribution around the parameter θ for x, e.g., the probability parameter (π) associated with a binomial distribution L(x| θ) is the “likelihood” of observing x GIVEN the value set for θ4 f(θ| x) is the posterior density for θ GIVEN the observed data x 4

  It should be noted that θ can also be a vector of parameters, e.g., the mean (μ) and standard deviation (σ) of the normal distribution, where θ = {μ, σ}.

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Vehicle Accident Analysis and Reconstruction Methods

Or, in more plain terms,

f parameter|data

L data|parameter f parameter P data



The sampling distribution [see Equation (1.31)] here is what would typically be used in the frequentist approach, where the parameter is a fixed but unknown value. The addition of the other two terms converts this sampling distribution to a posterior distribution for the parameter, given the data. It is here where one can see how Bayes allows input of prior knowledge to lead to potentially more informative answers. As noted above, what makes the Bayesian school somewhat contentious is the prior distribution, which here is the probability of getting the parameter regardless of anything else. Not only is this not even an option in the frequentist school, but this is also the major question of what this would mean or what it should be exactly. Mathematically, there are certain distributions that make the equation solvable (i.e., no computer program is required, only good math skills), so those can be good initial guesses, but in all other cases, the calculations are beyond directly solving and therefore require numerical simulation. Kelly and Smith [1.44] describe a technical approach to Bayesian Inference as follows: 1. Specify the prior density that represents the knowledge, as well as the uncertainty, in the parameter of interest, e.g., the mean. 2. Specify the stochastic model that represents the process of observing the random events, i.e., the likelihood. 3. Collect data. 4. Calculate the posterior density by updating the prior based on the observations. 5. Validate the model, e.g., using Monte Carlo simulation. Computationally, the integral in the denominator of Equation (1.32) can become extremely complex, if not impossible, to solve. To overcome this challenge, simulation techniques support the estimation of f(θ| x) have been developed. Many of these methods are rooted in MCMC algorithms that actually remove the integral of Equation (1.3). More popular approaches to MCMC are Gibbs Sampling [1.45] and Metropolis-Hastings algorithm [1.45]. Such algorithms use a ratio of Equation (1.3) that allows the denominator to cancel out of the calculation and eventually the simulation for the density f(θ| x). Software programs such as WinBUGS [1.46] and RStan [1.47] provide efficient means for estimating the density of f(θ| x) that only require an introductory-level experience in programming. With the notions and concepts concerning the Bayesian method identified, the application to accident reconstruction is now considered. But first note that all the earlier examples in this chapter were about error propagation (or its relative, sensitivity analysis), with the calculations being what amounts to a one-way conversion from one variable (or group of variables) to another, and that probabilities, if known, were regarding inputs and only through the conversion were the probabilities of the response determined. The Bayesian analysis here is, instead, a two-way analysis involving probabilities of both input(s) (f(θ)) and observation (L(x| θ)) at the same time, specifically with the intent of estimating the parameter of a distribution. Consider the previous examples based on Equation (1.1) used throughout this chapter. For this example, it is assumed that the frictional drag coefficient, f, is known and the skid distance, d, has been measured. The goal of the analysis is to determine the original vehicle speed, v, prior to the skid. In the following two examples, WinBUGS scripts are provided to aid readers who wish to implement this method for their use. Comments in the scripts follow the # symbol.

CHAPTER 1 Uncertainty and Sensitivity in Measurements and Calculations in Accident Reconstruction

Example 1.8 For this example, the frictional drag coefficient f will be considered normally distributed as in Example 1.6, with a mean of 0.7 and standard deviation of 0.03, and the measured skid distance is d = 33 m and follows a Normal (μ, σ) distribution, where the parameter μ follows a prior distribution of Uniform (32, 34) and the parameter σ is a constant at 0.033 m. This example will follow the Bayesian approach of specifying uncertainty for the statistical parameters of the variables being analyzed. Or, in Bayesian formula terms, f

|distance

L distance|

f

P distance

MCMC simulations can be run using the WinBUGS software to arrive at the final, posterior distributions for μ, as well as the distribution for future predictions of distance, and therefore speed. The results, using the following script below, are in Figure 1.4 and Table 1.6.

 FIGURE 1.4   Distributions from WinBUGS model for Example 1.8. dist.pred sample: 50,000

frict sample: 50,000

10.0 7.5 5.0 2.5 0.0

15.0 10.0 5.0 0.0 32.8

33.0

33.2

0.5

mu sample: 50,000

0.7

0.8

speed.pred sample: 50,000

15.0

© SAE International.

0.6

1.0 0.75 0.5 0.25 0.0

10.0 5.0 0.0 32.8

32.9

33.0

33.1

19.0

20.0

21.0

22.0

23.0

© SAE International.

TABLE 1.6  Statistics from WinBUGS model for Example 1.8 (50,000 trials). Node

Mean

St. dev.

2.5%

Median

97.5%

dist.pred

33.0

0.04643

32.91

33.0

33.09

frict

0.7

0.02998

0.6415

0.7001

0.7592

mu

33.0

0.03275

32.94

33.0

33.06

speed.pred

21.28

0.4563

20.38

21.29

22.17 continues

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Vehicle Accident Analysis and Reconstruction Methods

model{ g