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SAE TECHNICALPAPER SERIES 2003-01-0489

Evaluating Uncertainty in AccidentReconstruction with Finite Differences

Wade BartlettMechanical Forensics Engineering Services

Albert FondaFonda Engineering Associates

Reprinted From: Accident Reconstruction 2003(SP-1773/SP-1773CD)

2003 SAE World CongressDetroit, Michigan

March 3-6, 2003

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2003-01-0489

Evaluating Uncertainty in Accident Reconstruction with Finite Differences

Wade Bartlett Mechanical Forensics Engineering Services

Albert Fonda Fonda Engineering Associates

Copyright © 2003 SAE International

ABSTRACT

The most effective allocation of accident investigation resources requires knowledge of the overall uncertainty in a set of calculations based on the uncertainty of each variable in real-world accident analyses. Many of the methods currently available are simplistic, mathematically intractable, or highly computation-intensive. This paper presents the Finite Difference method, a numeric approach to partial differentiation with error analysis that requires no high-level mathematical ability to apply, uses very little computation time, provides good results, and can be used with analysis packages of any complexity.

The Finite Difference method inherently incorporates an error treatment which provides investigators a basis to qualitatively rank from dominant to trivial the effects of uncertainty and errors in measured and estimated values. In this way, greater efforts in an accident investigation can be directed to the most influential of the measurements, while less effort need be expended on the values which have trivial effect on the analysis results. Three examples of uncertainty evaluation using the Finite Difference method in accident analyses are presented.

INTRODUCTION

In accident investigation, as in any scientific endeavor, no measurement is ever exact; uncertainty and measurement errors in some variables in an analysis may result in a large reduction in accuracy. There are several methods available for assessing the overall uncertainty in the result of a set of calculations based on the uncertainty in or probable error of each input variable. Classic treatment of possible error combi- nations uses a Root Sum of Squares method in cases of simple combinations, as described in Lindgren and McElrath (1959) and Taylor and Kuyatt (1994). For more complicated systems, Brach (1994) described three methods (upper and lower bounding, and algebraic partial differentiation to evaluate a Taylor series, first

without and then with use of a statistical method). The brute force “get a bigger hammer” approach of Monte Carlo Analysis was demonstrated by Kost and Werner (1994), Wood and O'Riordan (1994), and Rose, Fenton, and Hughes (2001). Application of the algebraic partial differentiation method to the momentum equation was demonstrated by Slakov and McInnis (1991) and Tubergen (1995), but to date the application to more complex systems has not been discussed.

This paper describes the Finite Difference method and presents its mathematical basis. This means of evaluating the overall uncertainty in an analysis is essentially a numeric approach to partial differentiation of the equation under consideration. This method obviates the requirement for algebraic partial differentiation in terms of each input variable, and requires significantly less computation than the Monte Carlo method. It provides meaningful statistical information, and can be automated with relative ease given today’s computer capabilities. It can be applied manually to simple closed-form analyses in a spread sheet environment or more complicated reconstruction algorithms, or can be embedded directly in such programs. In the late 1980's, Fonda embedded the method in an automated fashion in the CRASHEX program (Fonda 1995). However, absent a valid array of probable measurement uncertainties none of these approaches could be exercised to their full potential. The Finite Difference method inherently provides a choice of a structured sensitivity analysis or a Pareto-type evaluation of the input affects to identify the “vital few among the trivial many” as described by Juran (1951, 1974).1 This isolates the most influential

1 For further reading on the history of Pareto charts, see J.M. Juran’s article “The Non-Pareto Principle; Mea Culpa” at www.juran.com/research/articles/SP7518.html

uncertainties or measurement errors on the basis of objective and qualitative criteria rather than on the usual subjective basis. Such an evaluation can assist the

investigator in determining where best to spend resources to improve the result of an analysis.

The steps to apply the method in practice are presented. A comparison of the results as calculated by the Finite Difference method with those calculated by the Monte Carlo Analysis method, partial differentiation method, and Taylor series expansion method for one non-linear analysis commonly undertaken in accident reconstruction is presented. The Finite Difference method results for two more complex cases are also shown and discussed.

MATHEMATICAL BASIS

Given an equation describing a system:

)...,( 21 jXXXfr = EQ.1

Coleman and Steele (1998) noted that the uncertainty of the result can be expressed as:

and that the finite-difference approximation to solve EQ.2 can be written as:

This assumes that the equation under consideration is nearly linear near the nominal values and the independent variables are not related to one another statistically. The ranges selected for each variable must have the same confidence-level (1-sigma or 2-sigma, for instance).

Shigley and Miscke (1986) published an equivalent form of EQ.2 using standard deviations in place of the uncertainty terms above, specifically noting that the standard deviation of the combined result can be evaluated based on the standard deviations of each variable.

DATA SOURCES

One of the limitations to the use of this and other uncertainty analysis methods has been the dearth of useful data defining actual uncertainty ranges in many of the measurement tasks common to accident reconstruction. Other than modest information regarding the measurement of vehicle crush (Smith and Noga 1982; Tumbas and Smith 1988; Brown et al. 1987), the assignment of error ranges was rather speculative. This state persisted until relevant experimental studies performed at the WREX-2000 Conference2 were analyzed and reported by Bartlett, et al. (2002) Some of

the pertinent data are summarized in Appendix A. Only with the availability of the new data could any of the pre-existing methods become efficacious.

IMPLEMENTATION

The series of steps necessary to conduct a Finite Difference analysis is fairly short, and can be used with both spreadsheet analyses and more complicated accident reconstruction analysis packages:

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. One can use any level of confidence one prefers, 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 for the dependent variable of interest using all the nominal values of the independent variables using either a spreadsheet or a more complex algorithm.

Step 4: With all other independent variables at their nominal values, set each independent variable 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 variable.

Step 6: For each input variable, average the squares of the results of steps 4 and 5. This represents the mean of the output variances for positive and negative input deviations, where the variance is the square of the deviation.

Step 7: Take the square root of the sum of these averaged squares, on the basis that the variance of the sum is the sum of the variances. This gives the

2 World Reconstruction Exposition 2000 (WREX-2000), September 24-29, 2000. College Station, Texas, hosted by the Texas Association of Accident Reconstruction Specialists (TAARS) and 21 other accident investigation and accident reconstruction organizations

uncertainty range around the nominal result to the selected confidence level.

If in Step 2 the range about the mean values is 1.0, that is, if one chooses to consider variation of one unit of the input variable whatever that may be, then the result of

Steps 4 and/or 5 is a sensitivity analysis, giving the influence coefficients or sensitivities as X units of each dependent variable per unit of increase and/or decrease of the independent variable. Alternatively, if one selects a range of a particular likelihood (1-sigma or 2-sigma for instance) for each input variable, the result of Steps 4 and/or 5 reveals which input parameters have the greatest influence on the uncertainty of the final answer.

This process can be automated. In the late 1980's, as a part of the development of an algorithm for accident reconstruction, Fonda adopted the Finite Difference method, as he reported in 1995, as "a programmed procedure for evaluating the bands of uncertainty about a developed base case, due to the estimated errors of observation for that case. This 'Deviation Analysis' procedure begins [as the algorithm is exercised] with a return from the output screen to a data-less input screen, where likely perturbations or deviations on all the base inputs can be entered. The differential effects of each input deviation then are found in one run per input, [repeated] for all inputs of interest. These results can be used as-is for sensitivity studies and imposition of constraints. Furthermore, the root of the sum of the squares of all the deviations of each output can be found." (Fonda 1995) On the assumption that the inputs were all independent and equally probable at any given confidence level, the latter option gives the equally probable combined output deviations for the input polarities selected. Both polarities may be investigated, but as averaging of the resulting variances is rarely critical, in order that sensitivity analysis may also be performed variance averaging is not provided in Fonda’s automated procedure.

EXAMPLE 1: SIMPLE CASE

A common non-linear accident reconstruction calculation is finding the distance required for a vehicle to stop given an average deceleration value once braking is initiated, including pre-braking perception time during which the vehicle travels at a constant-speed. The equation of interest can be written as:

Where d = travel distance to stop, feet v = initial velocity, feet/sec t = time to perceive and react, seconds a= average acceleration, feet/sec2

The means and standard deviations of each of the four variables xa can be written as µa and σa respectively.

Using the following input values: v = 45 ± 4 ft/sec t = 1.4 ± 0.3 seconds a = -24 ± 2 ft/s/s (-0.745 ± 0.062 g’s)

the nominal value is found to be 105.2 feet. Combining the 1-sigma bounds so as to generate absolute high and low bounding values yields a range of 77.4 to 137.9 feet. The nonlinearity in the system is hinted at here in that the low bound is 27.8 feet lower than the nominal value, but the high bound is 32.7 feet higher.

Shigley and Mischke (1989) reported that the mean result of EQ.4 could be found by evaluating the equation using the mean input values, and the standard deviation for the distance traveled could be estimated using the Taylor-series expansion shown in EQ.5:

The Shigley and Mischke estimate for mean value was the same nominal value as calculated above (105.2 feet), with an estimated standard deviation of 19.8 feet. This result is approximately 4% higher than most of the other methods shown here. The truncation of higher-order terms in the Taylor series may be the reason for this difference.

To evaluate the uncertainty in the fashion of Coleman and Steele, EQ.2 would be rewritten as follows:

After partially differentiating EQ.4 for each variable, the uncertainty of the result can be expressed as:

Substituting the nominal and 1-sigma uncertainty values for each variable yields the following:

or an overall 1-sigma uncertainty of 19.14 feet.

Using the Finite Difference method yields the nominal value of 105.2 feet and a standard deviation of 19.15 feet. This method required fewer than 100 calculations in approximately 40 cells, as shown in Figure 1. A Monte-Carlo type analysis was conducted using a typical spreadsheet program using techniques discussed by Bartlett (2003). With 25,000 trials using normally distributed input variables, the average distance was found to be 105.7 feet and the standard deviation was 19.1 feet. A histogram of these results is shown in Figure 2. The nonlinearities noted above reveal themselves in the histogram in the form of a slight

Figure 1: Spreadsheet evaluation of uncertainty using Finite Difference method, for Example 1.

skewing from the ideal normal curve based on the sample mean and standard deviation (shown as the single line). The high-value tail is much longer than the low-value tail, weighting the overall average somewhat higher than the median value of 104.9 feet, and pushing the apparent peak slightly lower than predicted. A second run of 25,000 repetitions confirmed this result, with a mean of 106.00 ± 19.29.

Each 25,000-repetition Monte-Carlo analysis required approximately 700,000 calculations in 225,000 spreadsheet cells. Simplified analysis algorithms could dramatically reduce the number of cells used but would be unlikely to significantly reduce the number of calculations required.

The benefits of conducting a sensitivity analysis of accident reconstruction analyses have long been recognized (Metz and Metz 1998; Limpert and Andrews 1991; Jones 1974). An auxiliary benefit of the Finite Difference method is that a sensitivity analysis is integral

(originally an input) is unconstrained, as if it were an output. This amounts to solving the original set of equations for a different set of parameters (equal to the number of equations), which is always permissible, and which as an algorithm is a quicker means to this particular end. It is this logic which renders differing solutions of the same set of equations equivalent, even if not equally efficacious; whatever can be accomplished with one arrangement can be accomplished with the other, but the values of all parameters involved must be examined and confirmed to have credible values.

When the Finite Difference method is used not for sensitivity analysis but for uncertainty analysis, it gives the user the ability to assess which input uncertainties have the greatest effect on the outcome of the calculation. In the preceding example, for instance, increasing the acceleration rate by 1-sigma changed the final result by less than 4%, but changing either the reaction time or the initial speed by the 1-sigma levels defined altered the result by over 12%. Clearly, a tighter range on either of the latter two variables will yield the greatest benefits in terms of narrowing the result.

0

500

1000

1500

2000

2500

40-4

448

-52

56-6

064

-68

72-7

680

-84

88-9

296

-100

104-

108

112-

116

120-

124

128-

132

136-

140

144-

148

152-

156

Distance, feet

Freq

uenc

y

Figure 2: Normal curve based on results average and standard deviation superimposed on histogram results

of Monte-Carlo analysis for Example 1.

to the method. From a sensitivity analysis, one may rule out parts of the range of uncertainty of some inputs on the basis of excluding results which are not credible. For instance, the preliminary uncertainty analysis might permit too much rebound between the vehicles; but if it is noted that the rebound is quite sensitive to an input which has a high uncertainty, the extreme values of the input can be reduced, to reduce the rebound to a more credible level. In the end this amounts to a change of constraints; the rebound (originally an output) is constrained as if it were an input, and the other variable

High-Low range 107.7* ft n/a

Partial Differentiation 105.19 19.14

Monte Carlo 105.7 19.10

Finite Difference 105.19 19.15

Taylor Series (Shigley & Mischke)

105.19 19.80

Table 1: Summary of uncertainty results for Example 1, as calculated by several methods. *High-Low Range method generated a high of 137.9 and a low of 77.4.

Method Mean 1-Sigma

EXAMPLE 2: FINITE DIFFERENCES WITH A/R SOFTWARE

The Finite Difference method can be used with any closed-form accident analysis software package (including CRASH3-type programs) with some hand calculation. For purposes of accident reconstruction this method is not conveniently performed with simulation (SMAC-type) packages because many of the dependent variables in simulation are independent variables in accident reconstruction; and vice versa. This concept was explored, though, by Spek (2000) and Moser, et al. (2003). The momentum analysis module in a commonly used accident analysis program, RECTEC, was used to evaluate a 90-degree intersection collision, with the nominal values shown in Figure 3: These inputs resulted in a calculated vehicle 2 (V2) impact speed of 50.2 mph.

By altering each input value in turn first by 2-sigma high then by 2-sigma low, a table of results was created in a spreadsheet, shown in Figure 4. For instance, with all variables at their nominal value, the approach angle for Vehicle 1 was changed to +3 degrees and then -3 degrees, resulting in a calculated V2 impact speed of 47.62 mph and then 52.74 mph. The change from the nominal value was calculated for each result (Steps 4 and 5). In the first line, the nominal result was subtracted

confident that the true result is 50.18 ± 4.92 mph (between 45.3 mph and 55.1 mph).

From Figure 4, it can be seen that the input variables which had the most effect on the V2-impact speed result were the approach and departure angles for V1.

Figure 3: Screen shot of RECTEC momentum analysis module input screen used for Example 2.

Finally, the square root of the sum of these was taken (Step 7), to find the combined 2-sigma range of 4.92 mph. This indicates that given the 2-sigma inputs selected and the nominal values used, one can be 95%

from the new result to get (47.62 - 50.18) = -2.56 and (52.74 - 50.18) = +2.56 mph. These and all other such differences were then squared and averaged (Step 6).

Figure 4: Spreadsheet used to calculate overall uncertainty in Vehicle 2 impact speed for Example 2.

EXAMPLE 3: INTERSECTION IMPACT

The uncertainty in the results of another right-angle intersection collision were evaluated with the CRASH3 solution and the Finite Difference (“Deviation”) Analysis module contained in CRASHEX using the experimental measurement errors summarized in Appendix A. The CRASH3 option (zero impact duration) was chosen to exemplify a simple vector solution as used in RECTEC.

The case selected is a right-angle intersection impact with both vehicles traveling at 40 mph, Vehicle 2 having been Eastbound when impacted on the right side by the Northbound, smaller Vehicle 1, giving the trajectories shown in Figure 5. This collision configuration was used by Severy in staged impacts in the 1950's, McHenry in the 1970's in the development of both SMAC and CRASH, Fonda in the 1980's in the development of CRASHEX, and was the most-severe case within the family of EDSMAC simulations provided as an accident reconstruction baseline by Kinney and Woolley (1994). An EDSMAC simulation of RICSAC 10 with both vehicles at exactly 40 mph (and with the struck vehicle reversed) established the scene and vehicle data to be reconstructed. The location and path of Vehicle 1 were considered to have established the reference axes, which are otherwise arbitrary.

Figure 6 shows the input data for the CRASH3 analysis that formed the basis for Figure 5. These inputs, which along with crush-data comprised a total of 60 variables, produced a 7 by 10 array of output parameters, not shown (see Fonda 1995, 2000) and not all of equal significance to the reconstructionist. By adopting this solution as the base case during a Finite Difference analysis, with unit input deviations one can assess the sensitivity of each of the 70 outputs to variation in each of the 60 inputs, for a potential of 60x70=4,200 sensitivities in this (typical) instance. Or, by inputting instead the uncertainties and likely errors listed in the Appendix, one can determine the individual uncertainties in each output and thence the 4200 individual and 70 combined (root-sum-square) uncertainties. The task then is to locate the vital few variables which most significantly affect the outcome.

Identification of “the vital few” begins with selection of outputs of interest. For the present study, there are only two: the impact speeds based on momentum. This means that for present purposes the crush-data inputs will be ignored.

For the inputs of Figure 6 the CRASH3 impact speeds for Vehicle 1 and Vehicle 2 respectively were found to be 38.2 and 37.7 mph. Thus the errors of treatment were -1.8 and -2.3 mph using CRASH3. Taking that solution as a base case all variable uncertainties were set to twice the 1SD values outlined in the table in the Appendix for Low, Medium, and High uncertainties. Then in each instance the three largest resulting uncertainties in the two impact speeds, which are for the present the vital few we seek, are found to have the values shown (in order of magnitude) in Table 2.

Table 2 shows that for all levels of error the calculation was found to be most sensitive to errors of measurement of the tire-road friction or vehicle drag (which were varied jointly), with the exception of a greater sensitivity to a High (6 degree) error in the relative directions of approach (and heading) of the vehicles. Otherwise there was secondary or tertiary (and quite minor) sensitivity to path curvature errors, and to error in the weight of the striking vehicle if all errors were Low or Medium.

The High levels of error (11.3 and 5.5 mph) might correspond to a quite tentative reconstruction; yet such results could be quite adequate when a gross approximation would settle the issue at hand. Otherwise all of the data should be collected with greater care,

especially (if nothing else) with regard to the angle of convergence between the vehicles.

The Medium levels of error (3.0 and 2.8 mph) require at least the better friction look-up values of Ebert (1989) (see Appendix A), an allowance for likely occupant and cargo loading of the generic vehicles, and careful treatment of well-documented tire tracks; yet if initially

Figure 6: Site and vehicle input values for Example 3.

Figure 5: Intersection impact for complex example

well investigated by others a further site visit may not be necessary. At this level the results could be quite acceptable when approach speeds were not crucial to the issue at hand. Significant improvements in the end result could be made, however, by on-site friction measurements (see Appendix A), on-site studies to refine the vehicle paths, and accompanying adoption of a method of treatment not presuming instantaneous impact (as do CRASH3 and RECTEC). As noted in Fonda 2000, "If all relevant inputs have been perturbed and each relevant root-sum is at least twice the likely error of treatment as indicated by the present study or others, the treatment is as good as it needs to be, and there need be no specific allowance for errors of treatment." In an intersection impact such as described here, with Medium errors this would apply if CRASHEX were used, but not if CRASH3 or the like were used.

The Low levels of error (both 0.6 mph) presume the use of laser instruments applied to precisely defined targets along tire marks still visible or recreated at the site, careful estimation of specific occupant and cargo loading and tare weights of replicate or (preferably) the involved vehicles, and onsite friction measurements. While such use of computer-aided electronic instrumentation always may be justifiable for its speed, convenience, credibility, and avoidance of inadvertently gross error, its lavish accuracy is effective only in a similarly superior method of treatment, one not degraded by the presumption of instantaneous impact.

Once the measurement influences have been identified in this manner, the more critical data may be collected initially or eventually with better equipment and more care. Conversely as to all the quantities not explicitly noted in Table 2; the end results would hardly be affected even by reasonably large (but admissible) changes in value thereof.

Typically a preliminary Finite Difference study for a given impact configuration will show which of the possible observational uncertainties are the trivial many, and do not significantly affect the issue at hand, leaving the vital few to be more carefully investigated. Thus,

Table 2: Ranked uncertainties of Example 3 reconstruction due to measurement uncertainties.

expenditures available for investigation, always limited, can be more judiciously allocated through the application of the Finite Difference Method.

CONCLUSIONS

With the advent of quantitative, juried data as to likely errors of measurement in accident investigation, a long-known but little-used tool, Finite Difference Analysis, has become useable in accident reconstruction. This procedure has been shown to be useful in identifying the vital few measurements among the trivial many by quantifying the effects of input uncertainty on calculated results in accident reconstruction analyses. The technique has been described and demonstrated.

In evaluating a non-linear equation with three variables of normal probability, the Finite Difference method was shown to compare favorably with a traditional analysis as conducted with the Monte Carlo method. Both methods can narrow the range of results as compared to simple high-low bounding. Histogram representation of

Monte Carlo results provides more graphic information regarding the non-linearities in the result, but requires three orders of magnitude more computation to generate.

The Finite Difference method provides an immediate uncertainty evaluation both when implemented by means of hand calculations and with more complicated accident reconstruction algorithms. When combined with an existing algorithm for accident reconstruction it preserves the sophistication of that specialized procedure while adding the benefits of a sensitivity analysis and statistical information on the accuracy of the outputs of the reconstruction.

Without even any further application of the analytical procedure, the numerical results in the present paper provide the accident investigator an initial guide for the allocation of resources for the field investigation of a typical intersection impact. In future Finite Difference studies, regardless of the method of reconstruction similar results can be developed for other dependent variables given the same impact configuration and for a variety of other impact configurations as well.

ACKNOWLEDGMENTS

The authors would like to thank Ray Brach for his kind assistance.

CONTACT

Wade Bartlett can be contacted through the website http://mfes.com or by email at wade@mfes.com

Albert Fonda can be contacted through the website www.crashex.com or by email at afonda@crashex.com.

REFERENCES

Bartlett, W.D., Wright, W., Masory, O., Brach, R., Baxter, A., Schmidt, B., Navin, F., Stanard, T., Quantifying The Uncertainty in Various Measurement Tasks Common to Accident Reconstruction, SAE Paper 2002-01-0546

Bartlett, W.D., Monte Carlo Analysis Using Spreadsheet Programs, SAE Paper 2003-01-0487

Brach, R., Uncertainty in Accident Reconstruction Calculations, SAE Paper 940722

Brown, D.R., Wiechel, J.F., Stansifer, R.L., Guenther, D.A., Practical Application of Vehicle Speed Determination from Crush Measurements, SAE Paper 870498

Coleman, H., Steele, W.G. Jr, Experimentation and Uncertainty Analysis for Engineers, 2nd Edition, John Wiley & Sons, 1998, ISBN: 0471121460 (pp 78-79)

CRASHEX Operations Manual, Fonda Engineering Associates, 649 S Henderson Rd Suite C307, King of Prussia PA 19406

Ebert, N., Tire Braking Traction Survey Comparison of Public Highways and Test Surfaces, SAE Paper 890638

Eubanks, J.J., Haight, W.R., Malmsbury, R.N., Casteel, D.A., A Comparison of Devices Used to Measure Vehicle Braking Deceleration, SAE Paper 930665

Fonda, A.G., Nonconservation of Momentum During Impact, SAE Paper 950355

Fonda, A.G., Partially-Braked Impact and Trajectory Benchmarks, and Their Application to CRASH3 and CRASHEX, SAE Paper 2000-01-1315

Garrott, W. R., Measured Vehicle Inertial Parameters—NHTSA's Data Through September 1992, SAE 930897, as treated in Fonda, A G, Report of the Yaw Inertia Evaluation Forum to the Accident Investigation Practices Committee of the SAE, March 5, 1993

Jones, I.S., Results of Selected Applications to Actual Highway Accidents of SMAC Reconstruction Program, SAE Paper 741179

Jones, I.S., Baum, A.S., Research Input for Computer Simulation of Automobile Collisions, Volume IV. Staged Collision Reconstructions, December 1978. DOT HS 805 040.

Juran, J.M., Editor, Quality Control Handbook, First Edition, McGraw-Hill Book Company, New York, 1951; third edition, 1974

Kost, G., Werner, S., Use of Monte Carlo Simulation Techniques in Accident Reconstruction, SAE Paper 940719

Limpert, R., Andrews, D.F., Linear and Rotational Momentum for Computing Impact Speeds in Two-car Collisions (LARM), SAE Paper 910103

Lindgren, B.W., McElrath, G.W., Introduction to Probability and Statistics, Macmillan Company, 1959

Metz, L.D., Metz, L.G., Sensitivity of Accident Reconstruction Calculations, SAE Paper 980375

Maryland Association of Traffic Accident Investigators, Joint Conference in Waldorf Maryland, September 1998

Moser A., Steffan H., Spek A., Makkinga W., Application of the Monte Carlo Methods for Stability Analysis within the Accident Reconstruction Software PC-CRASH, submitted to SAE for publication 2003

Rose, N., Fenton, S., Hughes, C., Integrating Monte Carlo Simulation, Momentum-Based Impact Modeling, and Restitution Data to Analyze Crash Severity, SAE Paper 2001-01-3347

Shigley, J.E., Mischke, C.R., Standard Handbook of Machine Design, McGraw Hill, 1986, ISPB: 0-07-056892-8 (pp 4.15)

Shigley, J.E., Mischke, C.R., Mechanical Engineering Design, McGraw-Hill, NY, 5th Ed, 1989

Slakov, G.A., and MacInnis, D.D., The Uncertainty of Pre-Impact Speeds Calculated using Conservation of Linear Momentum, CMRSC7-242, June 17-19, 1991.

Smith, R.A., and Noga, J.T., Accuracy and Sensitivity of CRASH, DOT HS 806-152, March 1982. (The SAE version is 821169, Accuracy and Sensitivity of CRASH, but omits the analytic derivation and the juried crush measurement results.)

Spek, A., Implementation of Monte Carlo Technique in a Time-Forward Vehicle Accident Reconstruction Program, presented at the seventh IES Conference, Krakow, Poland, 2000

Taylor, B., and Kuyatt, C., Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results, National Institute of Standards and Technology, Technical Note 1297, 1994

Tubergen, R.G., The Technique of Uncertainty Analysis as Applied to the Momentum Equation for Accident Reconstruction, SAE Paper 950135

Tumbas, N.S., Smith, R.A., Measurement Protocol for Quantifying Vehicle Damage from an Energy Basis Point of View, SAE Paper 880072

Wood, D., O'Riordain, S., Monte Carlo Simulation Methods Applied to Accident Reconstruction and Avoidance Analysis, SAE Paper 940720

Woolley, R.L., Kinney, J.R., Reference Cases for Comparison of Collision Algorithms Used in Accident Reconstruction, SAE Paper 940567

APPENDIX A

The choice of Finite Difference inputs will be aided by the 2002 SAE paper “Quantifying The Uncertainty in Various Measurement Tasks Common to Accident Reconstruction” by Bartlett, et al., and Smith and Tumbas’ 1982 SAE paper “Accuracy and Sensitivity of CRASH.” Some representative findings from both papers are summarized below (see the papers for clarification).

Distance by Total Station. In the WREX-2000 studies, variations using laser instruments applied to precisely defined targets were vanishingly small for accident reconstruction purposes (±0.02 feet or less, depending on equipment).

Distance. The standard deviation of errors in measurement of distances exceeding about 30 feet or 10 meters is typically not over ±0.07% using a tape (taken as Medium uncertainty in the present study), or ±0.21% using a wheel (taken here as High uncertainty).

Arc. Assuming that the chord is measured precisely, the standard deviation of the uncertainty in measurement of the middle-ordinate of an arc fitted to tire marks may be ±0.042 ft (0.012 m) if the arc is scribed (taken here as Low uncertainty), or ±0.14 feet (0.040 m), when an actual yaw-mark is measured (taken here as a Medium uncertainty). In cases where the arc cannot be clearly and positively identified, the High uncertainty value, taken here as 0.7 feet, should be assessed at the site given scene evidence.

Angle. The WREX-2000 participants, when asked to measure an angle by striking and measuring two sides of a right triangle, reported a mean of 36 degrees with SD = 0.56 degrees. However, the participants expressed aversion to the task, perhaps because the locations of points rather than the magnitudes of angles usually are measured. But if so this implies that, lacking approach tire marks, the directions of approach to impact are often taken to be exactly along the known right-of-way; no feature is actually measured (and none may be detectable). To allow for error in this regard this SD for angles is here taken as Medium, while as a “High” error a significantly larger SD of 3 degrees (a 2 SD error of 6 degrees) allows for unobserved, occasionally significant angulation during an approach. It would be unduly adverse, however, to apply this to every angular input; any one input direction should be exempt, being taken to be the exact reference direction. Likewise any one input X, Y location should be exempt.

Right Angle. If Cartesian coordinates are used (setting aside as beyond the scope of this paper the method of triangulation, measurement to each target from two fixed points), such measurements would be made normal to possibly inexact reference axes. It can be shown that, using a tape a right angle may be established with useful accuracy, such that for Medium accuracy we may estimate the 1SD errors in the location of a point (X, Y)

as Y=0.003X and X=0.003Y. The High level of error is about 10 times that amount, based on the WREX-2000 data.

Weight. Smith (1982) reported a 1SD range for passenger vehicle weight as 65 pounds, taken here as Medium uncertainty. In cases where the number of passengers or their weights are unknown, or contents of the vehicle may have exited the vehicle during collision, the High uncertainty level, taken here as 400 pounds, and will have to assessed on a case-by-case basis.

Mass Dispersion. Yaw moment of inertia in inertial units can hardly be estimated by eye, is difficult to actually measure, and even if measured and published is easy to misquote. To avoid these difficulties, in CRASHEX it is entered in the form of a dimensionless ratio, k²/ab. The mass of any vehicle tends to be distributed as if it were all at the axles, a condition for which k²/ab = 1 where k is the radius of gyration, a is the distance from the front axle to the center of mass, and b is the distance from the rear axle to the center of mass. More exactly, this ratio is about 1.1 for most vehicles; or still more exactly k is just under 30% of the vehicle’s overall length (Garrott, 1993). Errors in mass dispersion tend to be modest because most vehicles are morphologically similar. Errors, taken here to be 0, 0.02, or at worst 0.20, affect the relative effects of impact forces (during impact) and tire forces (after impact) on linear as opposed to angular speed changes.

Wheelbase. In data included in the summary WREX-2000 report but not included in the SAE paper (Bartlett et al. 2002), 16 participants measured the left and right wheelbase of a damaged passenger vehicle using steel tape measures. The standard deviation of their results from the average of their results was approximately 1.0 inch (1% of the measured value), taken here as Medium uncertainty; and 3 inches was taken to be High uncertainty.

Tire-Road Friction, in situ. WREX-2000 follow-up testing showed that given a common instruction set, many people could generate very tight data if given the same drag sled and measurement tool. In the limited number of tests where a variety of drag sleds have been employed by their individual owners and the results compared to vehicle testing, the results have generally been reasonably close. For instance, ten participants at WREX-2000 using their own drag sleds measured µ on a concrete surface to be 0.81 with SD = 0.019, while an ASTM drag-testing trailer measured µ=0.846, 0.802, and 0.821 at 22, 31, and 41.5 mph respectively (35, 50, and 66 kph), averaging 0.823.

At an earlier conference (Maryland, 1998), 20 participants, using their own drag sleds, measuring µ on an asphalt surface on which a Maryland DOT trailer measured µ=0.829, reported a mean µ of 0.81 with SD = 0.028. This value should be representative of the scatter from results of testing the accident vehicle or a

replicate at the scene with accurate measurement equipment (in essence using the vehicle as a drag sled). This result is taken here as Low uncertainty.

Tire-Road Friction, generic. Ebert (1989) reported the peak and slide friction values as measured in 1986 during 102 tests on an array of US-highways by six tire manufacturers using cars from two manufacturers, two test speeds, and two tire loads. Though this data may be a little dated, it is newer than the data now found in many of the commonly used “reference tables.” For the present study the Medium and High uncertainties were taken to be 0.07 (as below) and 0.12, respectively. Note that the latter uncertainty allows 5% of the values to fall outside the range of (say) 0.66 to 0.94, which should indeed encompass nearly all the values likely with but little knowledge of the particular tire and surface involved.

Average SD Low High

Dry Peak 0.90 0.069 0.75 1.08

Dry Slide 0.69 0.075 0.45 0.87

Wet Peak 0.65 0.072 0.47 0.81

Wet Slide 0.43 0.065 0.28 0.58

Lateral Friction, generic. Multiple tests of the lateral dry tire-road friction on Road and Track’s dry skid pad found the maximum possible lateral acceleration to be 0.80±0.056g at 1SD for an array of commonly available passenger cars. However, for lack of a driver in control, after impact steady limit cornering is rarely if ever achieved, and these values were not used in the present study.

Post-Impact Unbraked Drag. In collisions commonly reconstructed, panic braking to rest is the exception. More typically there is only parasitic (engine) drag, perhaps assisted by one or more wheels locked by damage. The same relative accuracies of measurement were assumed as for the lateral limits of tire-road friction.

Crush Depth. The errors of crush measurement by digital means should be trivially small, other than misidentification of what counts as crush. At WREX-2000, 16 participants measured the crush on a vehicle after a partial-overlap collision. The average crush depth reported was 19.9 inches, with 1SD of 5.2 inches (26% of the average). The crush measurements individually reported at each of 6 locations had average uncertainties of slightly over 40%, reported here as High uncertainty. Smith (1982) reported crush depth uncertainty at the 1SD level to be ±1.5 inches with an average crush depth of 7.3 inches (20% of the average depth) with a single “trained but less skilled and experienced” investigator, relative to a crack team. These results are reported here as Medium uncertainty.

Crush Width. In measuring damage with an average reported width of 48 inches, Smith found the crush width to have 1SD = 3.0 inches (6% of the total average width). This value is taken here as Medium uncertainty. In measuring damage with an average reported value of 62 inches, Bartlett et al. found 1SD = 9.9 inches (16% of the total average value), taken here as High uncertainty.

Crush Location. Smith reported that for an experienced technician the 1SD value for location of the center of vehicle damage was 1.8 inches.

Crush Direction. The principal direction of force as measured on damaged vehicles was studied by Smith as well as Bartlett, et al. Smith found that 3 of 34 PDOF estimates differed from the others by one clock sector (30 degrees). From this, he concluded that the 1SD level of accuracy was 10 degrees. Though not reported in Bartlett, 15 participants at WREX offered their estimate of the PDOF of the partial-overlap vehicle. The values ranged from 0 to 45. Discarding the one high-outlier (45 degrees) resulted in 14 accepted responses, with an average of 9.6 degrees, and a 1SD of 11.5 degrees.

Tiremark Measurement. Of 24 participants, asked to measure the left front skid-mark length produced when an on-board VC2000 indicated a stopping distance of 67 feet, 14 included the mark made by a rear wheel (wheelbase 11.7 feet), and reported a mean length of 75.2 feet (high by 8.2 feet; or short of 67+11.7=78.7 by 3.5 feet, or 4.4%), with an SD of 1.7 feet (2.2% of 78.7). The remaining 10 participants measured the skidmark as requested, reporting a mean length of 67.0 feet with a SD of 1.7 feet (2.5% of 67).

Thus, as to skid marks prior to impact (not a part of a post-impact reconstruction), or travel after impact if skid marks alone designate point of rest, observers often mistakenly include wheelbase when measuring skid marks. At best, when the single-wheel skid mark is correctly identified, the SD is typically 2.5%.

There is insufficient data at the present time to determine the relationship of the uncertainty to the skid-length under consideration, if one exists. It is recommended that investigators use 1SD=1 ft for Low uncertainty (in cases where marks are absolutely clear and unequivocal in their start/finish points), 1SD=2 ft for Medium uncertainty, and 1SD=5 ft for High uncertainty (in cases of casual estimation).

Values for one standard deviation (1 SD)

Measurement

Low Uncertainty

Medium Uncertainty

High Uncertainty

Units

Range1 0 0.07 0.19 %

X (from Y axis2) 0 0.17 1.4 % of Y

Y (from X axis2) 0 0.17 1.4 % of X

Chord Rise3 0.04 (0.01) 0.14 (0.04) 0.7 (0.2) feet (meters)

Angle4 0 0.6 3 degrees

Weight5 10 (44) 65 (286) 400 (1778) pounds (newtons)

Mass Dispersion 0 0.02 0.20 unitless

Wheelbase 0 1 (2.5) 3 (7.5) inches (cm)

Tire-Road dry-slide µ6 0.03 0.07 0.12 g's

Crush Depth 7 0.5 inches (1.2 cm)

20% 40%

Crush Width 7 0.5 inches (1.2 cm)

6% 16%

Crush Direction8 0 10 20 degrees

Crush Properties9 10 20 40 %

Skidmark Measurement 1 (0.3) 2 (0.6) 5 (1.5) feet (meters)

TABLE A: Typical values for Low/Medium/High uncertainty measurements common to accident reconstruction.

‘Low’ error by laser or digital sensor is trivial for all distances and angles. (1) Range 30 to 90 feet, for points near the major axis of an elongated cluster. (2) Cartesian distance, about equally far from each axis. (3) ‘Low’ for scribed mark, ‘Medium’ for tire mark, ‘High’ for casual estimate. (4) ‘Medium’ by dubious jury, ‘High’ if object uncertain. (5) ‘Low’ established by on-site test, typical scale resolution; ‘Medium’ per Smith (1982), ‘High’ for casual estimate; In cases where the number of passengers or their weights are unknown, or contents of the vehicle may have exited the vehicle during collision, the high-uncertainty level may be much higher, and will have to assessed on a case-by-case basis. (6) ‘Low’ established by on-site test with actual or replicate vehicle and accurate equipment, ‘Medium’ for less accurate on-site tests or Ebert’s generic values, ‘High” for casual estimates. (7) ‘Low’ is based on uncertainty in vehicle dimensional data for very well defined features from WREX-2000, ‘Medium’ per Smith’s NHTSA investigator(s), ‘High’ based on WREX-2000 results. (8) ‘Medium’ per Smith and Bartlett’s results from WREX; ‘High’ denotes a casual estimate. (9) ‘Low’ if from tests of the same or replicate vehicle; ‘Medium’ if based on vehicle size; ‘High’ denotes a casual estimate.