Uncertainty assessment in restraint system optimization for occupants of tactical vehicles.
We have recently obtained experimental data and used them to develop computational models to quantify occupant impact responses and injury risks for military vehicles during frontal crashes. The number of experimental tests and model runs are however, relatively small due to their high cost. While this is true across the auto industry, it is particularly critical for the Army and other government agencies operating under tight budget constraints. In this study we investigate through statistical simulations how the injury risk varies if a large number of experimental tests were conducted. We show that the injury risk distribution is skewed to the right implying that, although most physical tests result in a small injury risk, there are occasional physical tests for which the injury risk is extremely large. We compute the probabilities of such events and use them to identify optimum design conditions to minimize such probabilities. We also show that the results are robust to various assumptions that the statistical simulations use.
CITATION: Drignei, D., Mourelatos, Z., Kosova, E., Hu, J. et al., "Uncertainty Assessment in Restraint System Optimization for Occupants of Tactical Vehicles," SAE Int. J. Mater. Manf. 9(2):2016,
Advanced restraint technologies, such as seatbelt pre-tensioners, load limiters, and airbags, have the potential to provide improved occupant protection in crashes [2,3,6], but they are currently not utilized in military vehicles. Optimally implementing these technologies requires a better understanding of the occupant kinematics and injury risks in crash scenarios with military vehicles. The solutions are not necessarily the same as those used in passenger vehicles because of differences in crash involvement, occupant characteristics, vehicle compartment geometry, and occupant seating posture. Military gear may also affect restraint system interaction and injury risk.
Recently, experimental data and computational models for quantifying occupant impact responses and injury risks in military vehicles have been developed . However, the numbers of both experimental tests and model runs are small due to their high cost. While this is true across the auto industry, it is particularly critical for the Army and other government agencies operating under tight budget constraints. This study supplements our recent findings by investigating through statistical simulation how the injury risk varies if a large number of experimental tests were conducted. Moreover, we conduct studies to assess the robustness of our results to changes in various assumptions of the statistical simulations. Such uncertainty assessment studies are necessary when the available data set is small.
The main result of this paper is that the distribution of the injury risk is skewed to the right. The practical implication of this finding is that, although most physical tests result in a small injury risk, there are occasional experimental tests (as proxies for real situations) for which the injury risk is extremely large, as reflected by the long right tail of its distribution. Consequently, it is important to answer questions such as "how likely is it for each design condition to result in a physical test (or a real situation) in which the injury risk exceeds a tolerable threshold, such as 0.15?" Then one could set the optimization objective to minimizing the occurrence of such events, incorporating both test and model data. Using only test data will result in large uncertainty due to the small sample size. Therefore, it is necessary to use model runs in order to supplement the information provided by the test data and therefore, reduce the uncertainty in the simulation study.
We show that our results are robust to changes in various assumptions related to the statistical distributions. We first considered the scalar injury risk whose components are assumed to have two different distributions: normal and lognormal. Then, we obtained statistical simulations of time-dependent responses with normal and lognormal distributions, which were subsequently summarized in order to obtain the statistical distribution of the injury risk. Regardless of the distribution and/or method used, it appears that the pretensioner and the load limiter are associated with a safer restraint system. In addition, the 5-point restraint seems to be safer than the 3-point system.
TEST, MODEL AND DESIGN DESCRIPTION
In order to achieve the design optimization goal, sled tests and accompanying computational model runs were performed. The sled tests were conducted using a custom-built sled buck which was constructed from 3D scans of a Hummer H1 vehicle (Figure 1a). The buck (Figure 1b) was reconfigurable to represent both the driver and passenger compartments. All tests were performed in a frontal crash configuration with a 30 mph delta-V and a peak acceleration of 25 g (Figure 1c). All anthropomorphic test devices (ATDs) were outfitted with standard issue military combat boots and Advanced Combat Helmet (ACH) for every test. Tests were conducted with three additional military gear configurations (Figure 1d) - Improved Outer Tactical Vest (IOTV) only, IOTV and Squad Automatic Weapon (SAW) Gunner set with a Tactical Assault Panel (TAP), and IOTV and Rifeman set with TAP. ATDs with the SAW Gunner and Rifeman gear sets were tested in the passenger configuration, while ATDs with helmet only and IOTV only were tested in the driver configuration. Two types of seatbelts, 3-point and 5-point, with and without pretensioner(s) and load limiter(s), were also used. Pretensioners were used on the shoulder and lap belts, and were set to fire at 12ms. In tests using load limiters, a 4.9 kN load limiter was used on the shoulder of the 3-point belt, and 2x2.7 kN load limiters were used on the shoulders of the 5-point belt. While the original study was more comprehensive, here we focus on eight specific design conditions (Table 1) corresponding to a mid-size male dummy (i.e. 50th percentile male ATDs).
A set of finite element (FE) models, including the test buck, ATDs, military gear configurations, and different seatbelts were developed and integrated together. The test buck model was developed based on the design CAD data. The geometries of the models for military gears were based on the seated solider study () with simplification and modification. The seatbelt models were developed based on the seatbelt component tests on the webbing, retractor, pre-tensioner, and load limiter.
A sequence of the crash tests were used to validate the FE models. For each model, the ATD model was positioned and postured based on the FaroArm data measured in the tests. The time histories of the ATD head, chest, and pelvis accelerations, chest defection, femur forces, seatbelt forces, as well as the head and hip excursions were used to tune the models, so that occupant kinematics and injury risks can be accurately modeled. The parameters that were calibrated in the models included the seatbelt slack, material properties of the vest and other military gears, seatbelt to vest/gear contact, seatbelt routing, friction, etc.
For each design condition, injury measures, including the head injury criterion (HIC), chest defection (D), and left and right femur values (LF, RF), were calculated based on the time histories of the ATD output, and were used to compute the injury risks of the head, chest, and lower extremities. The HIC is a measure of the likelihood of head injury resulting from an impact, and is defined as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)
where [a.sub.H] (t) is the head acceleration as a function of time, and [t.sub.1] and [t.sub.2] represent a (15-ms) time interval over the acceleration pulse. The chest defection D is obtained as the maximum of the chest defection time-dependent curve [C.sub.D](t), while the femur scalar values LF and RF are the maximal values of femur compression forces (i.e. minimum of the corresponding femur force time history), respectively. The injury risks are defined as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)
where [phi] is the cumulative standard normal distribution,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5) The last two injury risks are combined into
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)
and the overall injury risk is calculated as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)
It is important to emphasize that each crash test and each model run produce an overall injury risk as in (7). Therefore, we distinguish between test injury risk and model injury risk. Equations (1, 2, 3_, 4, 5, 6, 7) were based on those used in [5_], the current US New Car Assessment Program (US-NCAP), although neck injury measures were not considered in this study.
Generally speaking, good agreements between the tests and simulations were achieved. Examples of model kinematic validation are shown in Figure 2. Examples of model injury measure validation are shown in Figures 3 and 4.
UNCERTAINTY QUANTIFICATION METHODOLOGY
Ideally, thousands of physical tests would be conducted for each design condition to generate the appropriate injury risk distribution. Because this is very expensive, we use statistical simulation to generate ensembles of physical test data for each design condition, which will then generate the corresponding distribution for the overall injury risk. While it is not feasible to obtain the actual injury risk distribution in the absence of many physical tests, we can use the information available to infer what this distribution might be. Specifically, in injury risk evaluation one physical test has been conducted for each of the design conditions and the model has been calibrated; i.e., its parameters are estimated so that the model and test data agree statistically. Then the injury risks are computed from the test and model data. We use the difference between the test and model quantities to infer the injury risk distribution. This is a general, well-established statistical method to quantify uncertainty when a model and limited physical data are available (e.g. ).
For each of the n=8 design conditions there are test and model values of the following scalar summaries: HICtest(i), Dtest(i), LFtest(i), RFtest(i) and HICmodel(i), Dmodel(i), LFmodel(i), and RFmodel(i) for i=1,..., n. The goal is to obtain a large number (e.g., R=10000) of simulated test values HICsim (i,r), Dsim (i,r), LFsim(i,r), RFsim(i,r) for r=1,...,R, as proxies for the physical tests. Below we provide details on how to obtain the simulated chest defection Dsim (i,r). The remaining simulated quantities HICsim (i,r), LFsim(i,r), RFsim(i,r) are obtained similarly. We compute the absolute differences | Dtest(i) - Dmodel(i)| for i=1,... ,n and pool the smallest n/2 values in one group while the remaining n/2 are pooled in a separate group. Then, we compute the standard deviations [S.sub.D] and [S.sub.D] of these absolute deviations in each group. Simulated values are finally obtained as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)
for conditions i in the first group, and
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)
for conditions i in the second group, where z(r) are standard normal random numbers. We use two groups because the absolute deviations differ substantially across conditions and a common standard deviation coming from just one large group does not provide a good measure of spread. In order to study the sensitivity of the results to the distributional assumption of normality, we also investigated lognormal simulations by using log(D) values instead of D in the computation of the standard deviations [S.sub.D] and [S.sub.D] . In this case, the final simulations are obtained as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)
The simulated HICsim (i,r), Dsm (i,r), LFsm(i,r), RFsim(i,r) values are used in relations (2, 3, 4, 5, 6, 7) to obtain P ( i, r) for each design condition i=1... n. The R simulated values P (i, r) are used to characterize numerically the distribution of the overall injury risk by drawing histograms and determining the proportion of simulated values above a critical threshold; e.g., 0.15.  is a good introductory monograph to statistical simulation methods.
An alternative to the scalar simulation method presented above is to simulate the original time-dependent responses and use each of them to calculate scalar values of HIC, D, LF, and RF. These, in turn, are used in (2, 3, 4, 5, 6, 7) to determine the simulated injury risk distribution for each design condition. This is part of the investigation into the robustness of our results with respect to different assumed distributions for the component quantities of the overall injury risk. Since the extremes of the time-dependent responses are of interest, we retained values around either the maximum of the head acceleration and the chest defection, or the minimum of the corresponding femur force time history. The time range around the extremes is determined visually, so that the retained time series appear stationary and amenable to Gaussian process modeling. We also translate the model and test responses along the horizontal axis to match their extremes of interest, using a method called curve registration in functional data analysis (e.g., [1, 8]). After the time ranges are matched, we fit time series statistical models to the difference between test and model local time-dependent responses. For example, the retained chest defection curve can be expressed as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)
for t = l,...,T in the vicinity of extreme response, where e(t) represents the error between test and model. Note that for the scalar data above we had to pool data across several design conditions in order to estimate the needed standard deviations for simulations. Here, however, we have a time-dependent sample for each design condition that allows us to carry out the estimation without pooling data across design conditions. We assume that the error vector [e(1),... ,e(T)] follows a multivariate Gaussian distribution with a zero mean vector and a covariance matrix | whose elements are a function of the distance between the time indices. The parameters of this covariance matrix are estimated using maximum likelihood. The statistically simulated local chest defection response is then obtained as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)
for t = 1,...,T and r = 1,...,R, where z is simulated from the estimated multivariate Gaussian distribution. We then compute the simulated chest defection [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], which in turn is used in (3) to obtain [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Similarly, we obtain the other simulated individual injury risks which are used to compute [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (Equation 7) and its distribution. This is done separately for each design condition i to obtain [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] To investigate the robustness of the proposed method to distributional assumptions, we also considered the lognormal distribution. Specifically, we assumed
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)
with the error vector [e(1),... e(T)] following a multivariate Gaussian distribution with zero mean vector and covariance matrix |. After the parameters are estimated using the maximum likelihood method, the local chest defection response is calculated as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)
for t=1,... ,T and r=1,... ,R, where z is simulated from the estimated multivariate Gaussian distribution. Finally, the simulated chest defection is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and the computation of the simulated overall injury risk is done similarly to the normal distribution case.
Note that even if the error vector is assumed multivariate normal (i.e., Gaussian), the distribution of [D.sup.sim] (r) is neither normal nor lognormal (as it was assumed in Section 3.1), but it can be characterized numerically through histograms.
UNCERTAINTY QUANTIFICATION RESULTS
The green histograms in Figures 5 through 8 show the inferred distribution of the overall injury risk (Equation 7) for each design condition, using first scalar data with normal (Figure 5) and lognormal (Figure 6) errors, and then using time-series statistical models for time-dependent data with normal (Figure 7) and lognormal (Figure 8) errors. In addition, each plot shows the test (red dot) and model (blue dot) overall injury risk, as well as the probability that the injury risk exceeds the threshold 0.15 (value in upper right corner of each plot).
The results are relatively insensitive to the type of noise distribution used. For example, the injury risk distribution of the '5pt_ sawgunner_ptll' condition (lower row, left plot) has small support (i.e., less uncertainty), while the probability of injury risk above 0.15 is small and approximately the same (less than 0.05) in all four figures. On the contrary, the condition '3pt_base' (lower row, second plot from the left) is characterized by large uncertainty and the probability of injury risk above 0.15 is very large in all four figures. The results in Figures 5 through 8 show that the smallest injury risk corresponds to condition '5pt_sawgunner_ptll', followed by '5pt_ iotv_ptll' and then by '3pt_iotv_ptll'. Therefore, 'ptll' appears to be an important safety component that lowers the injury risk. Also, '5pt' leads to smaller injury risk than '3pt'. Condition '5pt base' seems to have a low injury risk as well. However, the results from the '5pt base' and '3pt_base' conditions seem to be more sensitive to the choice of error distribution and/or method (i.e., scalar versus time-dependent) than the other conditions of small injury risk. For example, the right tail of the '5pt base' distribution is much longer in Figure 8 than in Figures 5, 6, 7, while the accumulation of probability at 1 for the '3pt_base' condition in Figure 8 is due to simulations from the longer right tail of the lognormal distribution.
SUMMARY AND CONCLUSIONS
In this paper we addressed the issue of uncertainty quantification in the optimization of restraint systems in tactical vehicles. Due to budget constraints, the number of crash tests and model data is very limited. Consequently, the optimization process may be mired in uncertainty. We characterized this uncertainty using statistical simulations from normal and lognormal distributions. We found that the distribution of the injury risk is skewed to the right, indicating that there are occasional tests as proxies for real situations that lead to a very large injury risk. We fully characterized numerically the overall injury risk distributions and computed the probabilities of the injury risk exceeding a critical threshold (e.g., 0.15).
Furthermore, we identified design conditions minimizing the occurrence of high injury risk events. Specifically, we found that pretensioner and load limiter (ptll) lead to very small probabilities of injury risk exceeding the critical threshold. A restraint system including these components is therefore, safer. Also, the 5-point restraint system is in general safer than the 3-point system. As future work, we shall extend our methods to a larger number of design conditions (e.g., 95th percentile dummies) and include other components (e.g., neck) in the computation of injury risk.
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Associate Professor, Math & Statistics Dept Oakland University, Rochester, MI
Zissimos P. Mourelatos
Professor, Mechanical Engineering Dept Oakland University, Rochester, MI
We would like to acknowledge the technical and financial support of the Automotive Research Center (ARC) in accordance with Cooperative Agreement W56HZV-04-2-0001 U.S. Army Tank Automotive Research, Development and Engineering Center (TARDEC), Warren MI. This work is part of Ervisa Kosova's doctoral dissertation at Oakland University.
HIC - head injury criterion
D - chest defection
LF, RF - left and right femur values
[PHI] - cumulative standard normal distribution
n - number of design conditions
[P.sub.head] - head injury risk
[P.sub.chest] - chest injury risk
[P.sub.f] - femur injury risk
[P.sub.join] - overall injury risk
HICtest, Dtest, LFest, RFest - test scalar measures
HICmodel, Dmodel, LFmodel, RFmodel - model scalar measures
HICsim, Dsim, LFsim, RFsim - statistically simulated scalar measures
[S.sup.(k).sub.D] - standard deviations in group k.
[c.sup.test.sub.D] - test chest defection curve
[C.sup.model.sub.D] - model chest defection curve
[C.sup.sim.sub.D] - statistically simulated chest defection curve
z, e - normally distributed errors
DISTRIBUTION STATEMENT A.
Dorin Drignei Zissimos Mourelatos and Ervisa Kosova Oakland Univ. Jingwen Hu and Matthew Reed Univ. of Michigan - Ann Arbor Jonathan Rupp Transportation Research Institute Rebekah Gruber US Army TARDEC Risa Scherer US Army
Table 1. Analyzed test conditions PT: Pre-tensioner, LL: Load limiter Test ID Abbreviation Side IOTV Cear TD1403 5pt_base Driver N N TD1404 5pt_iotv Driver Y N TD1405 3pt_base Driver N N TD1407 5pt_sawgunner Passenger Y SAW Gunner TD1408 3pt_sawgunner Passenger Y SAW Gunner TD1409 5pt_sawgunner_ptll Passenger Y SAW Gunner TD1417 3pt_iotv_ptll Driver Y N TD1418 5pt_iotv_ptll Driver Y N
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|Author:||Drignei, Dorin; Hu, Jingwen; Reed, Matthew; Rupp, Jonathan; Gruber, Rebekah; Scherer, Risa|
|Publication:||SAE International Journal of Materials and Manufacturing|
|Date:||May 1, 2016|
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