- Email: [email protected]
Stochastic vehicle mobility forecasts using the nato reference mobility model
Journal of Terramechanics, Vol. 33, No. 6, pp. 273 280, 1996
Pergamon
Published by ElsevierScienceLtd on behalfof ISTVS Printed in Great Britain 0022~,898/96 $15.00 + 0.00 PII: S0022-4898(97)00010-4
STOCHASTIC VEHICLE MOBILITY FORECASTS USING THE NATO REFERENCE MOBILITY MODEL ALLAN LESSEM,* GEORGE
MASON* and RICHARD
AHLVIN*
Summary--The NATO Reference Mobility Model (NRMM) is a comprehensive means of predicting the speeds of military vehicles in on-road, off-road, and gap-crossing contexts. The model has been in service for many years and helps user communities concerned with vehicle design, wargaming, and strategic planning. Recent developments in computer hardware and software are creating an opportunity for NRMM to serve a tactical role on the battlefield. Adaptation of NRMM to this role requires that its users come to grips with the collection of digital data to describe vehicle, terrain, and scenario data in a real-time environment. This paper discusses the performance of NRMM when selected inputs and algorithms contain random components. A developmental pathway is outlined that leads from current deterministic mobility forecasts to stochastic forecasts capable of suggesting the risks taken when speed predictions must be made in the presence of data and algorithm errors. Concepts that express measures of confidence for wide-area mobility forecasts when errors are known with small-area detail are described. Several numerical examples are given. Published by Elsevier Science Ltd on behalf of ISTVS
INTRODUCTION T h e N A T O Reference M o b i l i t y M o d e l ( N R M M ) is a c o m p u t e r c o d e to predict the perf o r m a n c e o f wheeled a n d t r a c k e d vehicles o p e r a t i n g o n - r o a d a n d o f f - r o a d in v a r i o u s o p e r a t i o n a l settings. Based o n m a n y years o f field a n d l a b o r a t o r y w o r k b y the U S A E W a t e r w a y s E x p e r i m e n t S t a t i o n (WES), the U S A r m y T a n k - A u t o m o t i v e C o m m a n d , a n d the C o l d R e g i o n s R e s e a r c h a n d E n g i n e e r i n g L a b o r a t o r y , a n d o n c o n t r i b u t i o n s f r o m N A T O m e m b e r s , N R M M c o n s i d e r s m a n y terrain, r o a d , a n d tactical g a p a t t r i b u t e s , vehicle geometries, a n d h u m a n factors [1]. Its f u n d a m e n t a l o u t p u t is the vehicle's effective m a x i m u m speed k e y e d to specific areal units o f t e r r a i n a n d to specific lineal p o r t i o n s o f a r o a d n e t w o r k [2]. In a d d i t i o n to its service in user c o m m u n i t i e s c o n c e r n e d with vehicle design, w a r g a m ing, a n d strategic p l a n n i n g , c o n t i n u i n g d e v e l o p m e n t s in c o m p u t e r t e c h n o l o g y are c r e a t i n g a n o p p o r t u n i t y for N R M M to serve a tactical role on the battlefield. T h e battlefield setting requires h i g h - r e s o l u t i o n d a t a a n d e x p e d i e n t d a t a set p r e p a r a t i o n . A d a p t a t i o n o f N R M M to this role requires t h a t its users c o m e to grips with the effects o f errors a n d uncertainties in vehicle a n d t e r r a i n d a t a a n d o f inherent a l g o r i t h m errors.
*Waterways Experiment Station, US Army Corps of Engineers, 3909 Halls Ferry Road, Vicksburg, Mississippi 39180-6199, USA. 273
274
G. Mason et al.
Current use of N R M M yields forecasts of vehicle speeds in relatively small homogeneous units of terrain. Quad sheet size areas require acquisition of data for hundreds to thousands of terrain units. Predictions are currently made without consideration of the quality of terrain, vehicle, and human performance data. Judgments must often be made about appropriate values. In addition, the curve fits in the model lack the scatter bands associated with their experimental origins. As a result, mobility forecasts are incorrectly interpreted as error free. Adaptation of N R M M to a stochastic orientation is imperative if it is to be used in the high-resolution battlefield context. It must deliver measures of quality for speed predictions that reflect the quality of data and algorithms for both perterrain-unit and per-map forecasts. Therefore the main problem addressed in this study is the inevitable error environment that will surround the collection of data describing the physical environment for N R M M in battlefield situations. This problem is viewed as basically unsolvable; so the approach of this work is to live intelligently with the problem, to understand the effects of the errors, and to allow users of N R M M to state its error performance with clarity.
A PROTOTYPE STOCHASTIC MOBILITY F O R E C A S T I N G P R O C E D U R E FOR NRMM Figure 1 shows procedural pathways that can lead to stochastic mobility forecasts. The shaded elements of Fig. 1 use procedures that are similar to those in place for the nonstochastic mobility forecasts now provided by N R M M (in its most current form N R M M "NRMM llr' MODE 0
• ERROR ~
(~ER~RI TORSi . ~ V
ERROR •
~\\~,~.;\\\~
] SENSITIVITY 3-PO,NT I / ] ANALYSIS • ERROR V
~MAGNITUDE
~CENARI~II
r
v FINGER-
"SENSSTAT" & GRAPHICS
[ ~
BASED ON NRMMII (DETERMINISTIC) BASED ON NRMMII (STOCHASTIC)
Fig. 1. Flow chart of stochastic model process.
1
T/U = TERRAIN UNIT
Stochastic vehicle mobility forecasts
275
2.5.7). These are appropriately modified, and other elements are added, to develop the stochastic forecasts. The products delivered as a stochastic mobility forecast consist of four items: a speed display, a "fingerprint", a "mission rating speed", and a range for the mission rating speed. The speed display is a graphical presentation of nominal predicted speeds for one vehicle operating on one specific area (consisting of hundreds to thousands of terrain units), made under the assumption of error free data and an error free model. A conventional speed display of Lauterbach, Germany, is illustrated in Fig. 2 for an M 1 tank operating in wet conditions. The fingerprint, Fig. 3, is a graphical presentation of the error performance of N R M M specific to the one vehicle and the one specific area. Figure 3 illustrates the nominal speeds on the X axis, which are taken from the standard prediction output of N R M M , and the expected range on the Y axis, which indicates the maximum and minimum speeds if the input values are varied. For an analyst concerned about the ability of N R M M to predict immobilization correctly, the areas where data fall on the X axis above the zero nominal predicted speed indicate possible NOGOs. Likewise a solid square is drawn around the speeds that are predicted as possible GOs when N R M M predicts a NOGO. These speeds are associated with specific terrain units and can be mapped back to the speed map along with the reason the GO or N O G O was predicted in the extreme case. This information has potential benefits for field reconnaissance. The mission rating speed is a concept used by N R M M analysts who postulate a mission "profile" expressing on-road and off-road percentages, to arrive at a one number measure of vehicle performance on the specific area. This concept is preserved and extended by
N I
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0
8
Kilometersper Hour IIoIm .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
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123 urban areas
Fig. 2. Nominal NRMM speed predictions for an M 1AI operating in a wet scenario in Lauterbach, Germany.
276
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50
Legend
O tQ.
•^ 4u
Minimum Speed
•
MaximumSpeed
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Nominal Predicted Speed (kilometers per hour) Fig. 3. Fingerprint of predicted versus maximum and minimum speeds due to uncertainties in the terrain, vehicle, scenario, and internal curve-fit error.
expressing its range, thereby indicating in an integrated and quantitative way the quality of the entire NRMM speed display. The pathways to these products begin with data sets assembled for vehicle, terrain, and scenario of interest. Unlike their use with NRMM in its deterministic form, these data are first examined to evaluate the sensitivity of the speed predictions to standardized variations in data set elements taken one at a time. The standard variations are in the range of plus and minus 10% of nominal values. This approach was studied with a version of NRMM modified to repeat speed predictions in a Monte Carlo context with specified parameters updated in accordance with uniform probability distributions. Earlier trials, aimed at performance, considered possible joint variations and suggested it was sufficient to examine individual variation only [3]. Further experimentation and experience suggested that, in spite of NRMM's inherently nonlinear structure, it would suffice in the vast majority of cases simply to associate extreme values (maximum and minimum) of predicted speeds with extreme values of the range of variation assigned to each parameter in the sensitivity test. Thus, three numbers would be required to examine sensitivity: one for the nominal data set value and two for the maximum and minimum limits of variation. In effect, variation is set aside in favor of three fixed points. This element of the procedural pathway has been named "three-point sensitivity analysis" (Fig. 1); its use goes far to remove a computational bottleneck, although the ability to see probability densities is lost. Similar methods such as the "point estimate method" [4] and extensions of this procedure by Grivas and Harr [5] have been used successfully to define the probability of failure for slopes and embankments and structural foundations.
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Determinations of sensitivity to vehicle, terrain, and scenario parameters are augmented by sensitivities to algorithm errors. These errors are inherent to the many regression curve fits worked into the fabric of N R M M . The procedure assumes that all curve-fit algorithms have been identified and that the original field data from which they were derived have been re-examined such that standard errors of estimate can now be assigned to the curve fits (see Fig. 4 for an example). Each standard error of estimate is viewed as the standard deviation of a zero mean Gaussian error probability density assigned to its particular curve fit. Every value taken from that curve fit during the normal running of N R M M is assigned an additive error taken from its associated error probability density. For the purpose of the three-point extremum analysis used in the screening process, the inherent variability is set aside and the three points applicable to each curve fit are the nominal value and values that are plus and minus two standard deviations of the nominal. Quantification of sensitivity is accomplished by formulating a rank indicator specific to a given vehicle, terrain unit, and parameter. A nominal predicted speed is computed using the nominal parameters from the data set. The nominal data set is the input data and curve fit data used before any variation is applied, thus creating a nominal speed. Maximum and minimum speeds are then computed corresponding to values of the parameter that are greater and lesser than the nominal by 10%, or, in the case of curve-fit errors, greater or lesser than the regression value by two standard deviations of the test data. In most cases, the maximum speed will correspond to the maximum value of the parameter and the minimum speed to the minimum value of the parameter. For example, larger values of speed can result from larger values of soil strength. Conversely, larger values of speed can result from smaller values of slope. In any event, extreme values of parameter and speed generally correspond. In reality it is possible for N R M M to deliver maximum or minimum speeds that do not correspond to the range limits of the parameter, but such occurrences have been rare. To define the sensitivity of the parameter (terrain, vehicle, scenario parameter, and/or empirical curve fit), a rank indicator is computed:
(1)
]R ~- ( Vmax -- Vmin)/Vnom
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|
•
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.... L
.
.
1
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Track Slip Fig. 4. Empiricalcurve-fiterror associatedwith NRMM predictions.
1O0
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The rank indicator IR is defined for the specific vehicle, the specific terrain unit, and the specific parameter subject to variation; Vmaxis the maximum speed; Vmin is the minimum speed; Vnomis the nominal speed; and the magnitude of IR is constrained to the interval 0.0 < IR < 2.0. The arithmetic average of the rank indicators over all terrain units for the given vehicle is taken as their rank - - or sensitivity of the parameter. The rank indicator is formulated to express the idea that greater speed differences between maximum and minimum constitute greater sensitivity of NRMM to the parameter of interest in the given vehicle/terrain setting. Division by the nominal speed renders the rank indicator dimensionless and inherently implies that as nominal speeds increase, greater speed differences are appropriate to the same sensitivity. For example, a speed difference of 2 km h-~ at a nominal speed of 10 km h-1 and a speed difference of 12 km h-z at a nominal speed of 60 km h-~ express the same sensitivity of a given parameter. Experimentation with this formula involving many terrains, vehicles, and parameters revealed that most values of the rank indicator fall within the range 0 to 0.8, and that some cases occur as high as 1.8. The formula, of course, has the potential for infinite values when the nominal speed is zero and variation produces a finite speed range. This case has, by definition, been assigned the value of 2. Quantitative comparisons of the ranks of many parameters lead naturally to a screening process. For the given vehicle/terrain setting, variation of parameters in sequence produces a sequence of quantitative ranks; small values indicate low sensitivity and large values indicate high sensitivity. By selecting a certain threshold value, all parameters whose ranks fall below the threshold need be considered no further. Rankings change with the vehicles and terrains. Numerical experiments based on these ideas recommend a threshold of 20% [2] of the maximum rank for consideration. In the numerical experiments, from 32 to 84% of the parameters studied were eliminated from further consideration. Once parameters have been screened and the one(s) that will remain under consideration as part of the stochastic mobility forecast have been identified, the pathway to that forecast requires the formulation of an "error-magnitude scenario". This is simply a list of those parameters (and curve fits) and the actual nature of the variation (distribution and magnitude) to be assigned to each. During the screening process, each parameter was varied plus and minus 10% of the nominal and each curve fit was varied plus or minus 14% of its regression value. These curve-fit and parameter errors were based on observations in the field and are conservative estimates. During the subsequent Monte Carlo simulation, the opportunity is provided to specify the actual distribution type and ranges on an individual basis for the parameters and the actual standard deviations for the curvefit errors. With the determination of an error-magnitude scenario for a given vehicle/terrain combination, the stage is set for the principal event on the pathway to the stochastic mobility forecast. This is the Monte Carlo analysis of predicted speeds wherein the screened parameters and curve-fit algorithms are varied both jointly and independently and probability densities are determined for the speeds predicted for each terrain unit. The per-terrain-unit speed probability densities and data specifying the mission profile are the raw materials from which an analysis of mission rating speed and its range can be made. Other outputs from the Monte Carlo simulation are a listing of nominal speeds by terrain unit from which the speed display is obtained and maximum and minimum speeds by terrain unit from which the fingerprint is made. For conceptual simplicity and to bound
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279
NRMM error performance, initial work with the mission rating speeds was based on maximum and minimum terrain unit speeds rather than on the speed probability densities. The mission rating speed is approached through the "speed profile", a useful concept worked out early in the history of NRMM. A speed profile is specific to a given vehicle/ terrain combination. NRMM is used to form a sequence of records each of which shows the area and the predicted nominal speed for individual terrain units. These records are sorted in descending order by speed thus identifying the terrain units in which vehicle performance is "best" and "worst". The sum of terrain unit areas from the first record (which represents "best") to the Nth record divided by the sum of all areas defines the fraction of total area represented by the first N records. When the sorted speeds are plotted against this fraction, the result is a speed profile based on terrain unit speeds. NRMM calls it a "speed-in-unit" profile. When the area-weighted average of the first N speeds is plotted against the area fraction, an "average speed profile" is produced, indicated by the centerline in Fig. 5. Assuming that tactical usage of the vehicles will stress deployment over the "best" terrain units, the profiles allow qualification of what is meant by "best". Stochastic orientation of NRMM requires the development of stochastic speed profiles based not only on the nominal predicted speeds but also on the minimum and maximum predicted speeds. The very same computational procedures are used to bound the nominal mission rating speed profile. In effect, range limits that bound NRMM error performance are placed on the traditional speed profiles as indicated by the maximum and minimum speed profiles in Fig. 5. Speed profiles form the basis for the calculation of the mission ratings speeds. A mission rating speed is, as mentioned earlier, a one-number measure of vehicle performance that factors in the parameters of a tactical mission defined on a terrain map. Their parameters are (1) percentages of total operation distance spent on-road and offroad and (2) percentages of their best terrain and road units so occupied. There are several ways to computate the mission rating speed, depending on the depth of resolution
80
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Rangeof Percent NOGOsBasedon Uncertainties in Inputs& Curvefits
o
0
20
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80
100
Fig. 5. Stochasticmissionratingspeedprofileof an M1Al operatingoff-roadin Lauterbach,Germany,in a wetslippery scenario.
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designed. For example, are roads to be considered separately as primary, secondary, and so forth? Are predefined "tactical mobility levels" to be considered? Are time penalties for crossing linear features to be considered? See Ref. [5] for insights and typical applications. For the present, this combination is being made according to the following especially simple formula. If P is the off-road operations percentage, and Vc and VR are the corresponding speeds from the off-road and on-road profiles, then the mission rating speed VMR is 100 e ~ 100-e
Vc
(2)
VR
Stochastic mission rating speeds are derived from the stochastic average speed profiles by evaluating the equation using the nominal, minimum, and maximum values of Vc and V R. These values define VMR,NOM , VMR,MIN , and VMR, MAX. The range in the mission rating speeds so computed constitutes a one-number descriptor of NRMM error performance for the given terrain/vehicle combination and the given mission.
CONCLUSION Responding to the need to make NRMM a risk-based analysis for a tactical setting, the procedures discussed above transform its present deterministic orientation into a stochastic one. Based on this study, the following conclusions can be drawn: (a) Stochastic mobility forecasts can be obtained by supplementing current deterministic NRMM mobility forecasts with procedures involving determination of parameter sensitivity, ranking and screening of parameters, Monte Carlo speed simulations, development of fingerprints, and extension of stochastic mobility rating speeds. (b) The stochastic mobility forecasts developed by the prototype procedures studied here define worst-cast error performance of NRMM. (c) The error performance of NRMM will be found to vary widely among terrains and vehicles, underlining the importance of the fingerprint as a means of visually comprehending the clustering of errors. REFERENCES 1. Haley, P. W., Jurkat, M. P. and Brady, P. M., NATO Reference Mobility Model Edition 1, User's Guide. Technical Report 12503, US Army Tank-Automotive Command, Warren, MI, 1979. 2. Ahlvin, R. B. and Haley, P. W., NATO Reference Mobility Model Edition II, NRMM II User's Guide. Technical Report GL-92-19, US Army Engineer Waterways Experiment Station, Vicksburg, MS, 1992. 3. Lessem, A., Ahlvin, R., Mason, G. and Mlakar, P., Stochastic vehicle mobility forecasts using the NATO Reference Mobility Model. Technical Report GL-92-11, Army Engineer Waterways Experiment Station, Vicksburg, MS, 1992. 4. Rosenblueth, E., Design philosophy: structures. Proceedings 2nd International Conference on Application of Statistics and Problems in Soil and Structural Engineering, Aachen, Germany, 1975. 5. Grivas, D. and Harr, M. E., Consolidation a probabilistic approach. A S C E J. Eng. Mech. Div., 1978, 104(EM 3).
Pergamon
Published by ElsevierScienceLtd on behalfof ISTVS Printed in Great Britain 0022~,898/96 $15.00 + 0.00 PII: S0022-4898(97)00010-4
STOCHASTIC VEHICLE MOBILITY FORECASTS USING THE NATO REFERENCE MOBILITY MODEL ALLAN LESSEM,* GEORGE
MASON* and RICHARD
AHLVIN*
Summary--The NATO Reference Mobility Model (NRMM) is a comprehensive means of predicting the speeds of military vehicles in on-road, off-road, and gap-crossing contexts. The model has been in service for many years and helps user communities concerned with vehicle design, wargaming, and strategic planning. Recent developments in computer hardware and software are creating an opportunity for NRMM to serve a tactical role on the battlefield. Adaptation of NRMM to this role requires that its users come to grips with the collection of digital data to describe vehicle, terrain, and scenario data in a real-time environment. This paper discusses the performance of NRMM when selected inputs and algorithms contain random components. A developmental pathway is outlined that leads from current deterministic mobility forecasts to stochastic forecasts capable of suggesting the risks taken when speed predictions must be made in the presence of data and algorithm errors. Concepts that express measures of confidence for wide-area mobility forecasts when errors are known with small-area detail are described. Several numerical examples are given. Published by Elsevier Science Ltd on behalf of ISTVS
INTRODUCTION T h e N A T O Reference M o b i l i t y M o d e l ( N R M M ) is a c o m p u t e r c o d e to predict the perf o r m a n c e o f wheeled a n d t r a c k e d vehicles o p e r a t i n g o n - r o a d a n d o f f - r o a d in v a r i o u s o p e r a t i o n a l settings. Based o n m a n y years o f field a n d l a b o r a t o r y w o r k b y the U S A E W a t e r w a y s E x p e r i m e n t S t a t i o n (WES), the U S A r m y T a n k - A u t o m o t i v e C o m m a n d , a n d the C o l d R e g i o n s R e s e a r c h a n d E n g i n e e r i n g L a b o r a t o r y , a n d o n c o n t r i b u t i o n s f r o m N A T O m e m b e r s , N R M M c o n s i d e r s m a n y terrain, r o a d , a n d tactical g a p a t t r i b u t e s , vehicle geometries, a n d h u m a n factors [1]. Its f u n d a m e n t a l o u t p u t is the vehicle's effective m a x i m u m speed k e y e d to specific areal units o f t e r r a i n a n d to specific lineal p o r t i o n s o f a r o a d n e t w o r k [2]. In a d d i t i o n to its service in user c o m m u n i t i e s c o n c e r n e d with vehicle design, w a r g a m ing, a n d strategic p l a n n i n g , c o n t i n u i n g d e v e l o p m e n t s in c o m p u t e r t e c h n o l o g y are c r e a t i n g a n o p p o r t u n i t y for N R M M to serve a tactical role on the battlefield. T h e battlefield setting requires h i g h - r e s o l u t i o n d a t a a n d e x p e d i e n t d a t a set p r e p a r a t i o n . A d a p t a t i o n o f N R M M to this role requires t h a t its users c o m e to grips with the effects o f errors a n d uncertainties in vehicle a n d t e r r a i n d a t a a n d o f inherent a l g o r i t h m errors.
*Waterways Experiment Station, US Army Corps of Engineers, 3909 Halls Ferry Road, Vicksburg, Mississippi 39180-6199, USA. 273
274
G. Mason et al.
Current use of N R M M yields forecasts of vehicle speeds in relatively small homogeneous units of terrain. Quad sheet size areas require acquisition of data for hundreds to thousands of terrain units. Predictions are currently made without consideration of the quality of terrain, vehicle, and human performance data. Judgments must often be made about appropriate values. In addition, the curve fits in the model lack the scatter bands associated with their experimental origins. As a result, mobility forecasts are incorrectly interpreted as error free. Adaptation of N R M M to a stochastic orientation is imperative if it is to be used in the high-resolution battlefield context. It must deliver measures of quality for speed predictions that reflect the quality of data and algorithms for both perterrain-unit and per-map forecasts. Therefore the main problem addressed in this study is the inevitable error environment that will surround the collection of data describing the physical environment for N R M M in battlefield situations. This problem is viewed as basically unsolvable; so the approach of this work is to live intelligently with the problem, to understand the effects of the errors, and to allow users of N R M M to state its error performance with clarity.
A PROTOTYPE STOCHASTIC MOBILITY F O R E C A S T I N G P R O C E D U R E FOR NRMM Figure 1 shows procedural pathways that can lead to stochastic mobility forecasts. The shaded elements of Fig. 1 use procedures that are similar to those in place for the nonstochastic mobility forecasts now provided by N R M M (in its most current form N R M M "NRMM llr' MODE 0
• ERROR ~
(~ER~RI TORSi . ~ V
ERROR •
~\\~,~.;\\\~
] SENSITIVITY 3-PO,NT I / ] ANALYSIS • ERROR V
~MAGNITUDE
~CENARI~II
r
v FINGER-
"SENSSTAT" & GRAPHICS
[ ~
BASED ON NRMMII (DETERMINISTIC) BASED ON NRMMII (STOCHASTIC)
Fig. 1. Flow chart of stochastic model process.
1
T/U = TERRAIN UNIT
Stochastic vehicle mobility forecasts
275
2.5.7). These are appropriately modified, and other elements are added, to develop the stochastic forecasts. The products delivered as a stochastic mobility forecast consist of four items: a speed display, a "fingerprint", a "mission rating speed", and a range for the mission rating speed. The speed display is a graphical presentation of nominal predicted speeds for one vehicle operating on one specific area (consisting of hundreds to thousands of terrain units), made under the assumption of error free data and an error free model. A conventional speed display of Lauterbach, Germany, is illustrated in Fig. 2 for an M 1 tank operating in wet conditions. The fingerprint, Fig. 3, is a graphical presentation of the error performance of N R M M specific to the one vehicle and the one specific area. Figure 3 illustrates the nominal speeds on the X axis, which are taken from the standard prediction output of N R M M , and the expected range on the Y axis, which indicates the maximum and minimum speeds if the input values are varied. For an analyst concerned about the ability of N R M M to predict immobilization correctly, the areas where data fall on the X axis above the zero nominal predicted speed indicate possible NOGOs. Likewise a solid square is drawn around the speeds that are predicted as possible GOs when N R M M predicts a NOGO. These speeds are associated with specific terrain units and can be mapped back to the speed map along with the reason the GO or N O G O was predicted in the extreme case. This information has potential benefits for field reconnaissance. The mission rating speed is a concept used by N R M M analysts who postulate a mission "profile" expressing on-road and off-road percentages, to arrive at a one number measure of vehicle performance on the specific area. This concept is preserved and extended by
N I
W_~_E Scale (Kilometers)
0
8
Kilometersper Hour IIoIm .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
8 8-40 40+
123 urban areas
Fig. 2. Nominal NRMM speed predictions for an M 1AI operating in a wet scenario in Lauterbach, Germany.
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50
Legend
O tQ.
•^ 4u
Minimum Speed
•
MaximumSpeed
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C A U T I O N
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/
• ~ i ""~"~ q, - ~ e - ~
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~1~ :
,~.~. /I/1/
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0
30
40
Nominal Predicted Speed (kilometers per hour) Fig. 3. Fingerprint of predicted versus maximum and minimum speeds due to uncertainties in the terrain, vehicle, scenario, and internal curve-fit error.
expressing its range, thereby indicating in an integrated and quantitative way the quality of the entire NRMM speed display. The pathways to these products begin with data sets assembled for vehicle, terrain, and scenario of interest. Unlike their use with NRMM in its deterministic form, these data are first examined to evaluate the sensitivity of the speed predictions to standardized variations in data set elements taken one at a time. The standard variations are in the range of plus and minus 10% of nominal values. This approach was studied with a version of NRMM modified to repeat speed predictions in a Monte Carlo context with specified parameters updated in accordance with uniform probability distributions. Earlier trials, aimed at performance, considered possible joint variations and suggested it was sufficient to examine individual variation only [3]. Further experimentation and experience suggested that, in spite of NRMM's inherently nonlinear structure, it would suffice in the vast majority of cases simply to associate extreme values (maximum and minimum) of predicted speeds with extreme values of the range of variation assigned to each parameter in the sensitivity test. Thus, three numbers would be required to examine sensitivity: one for the nominal data set value and two for the maximum and minimum limits of variation. In effect, variation is set aside in favor of three fixed points. This element of the procedural pathway has been named "three-point sensitivity analysis" (Fig. 1); its use goes far to remove a computational bottleneck, although the ability to see probability densities is lost. Similar methods such as the "point estimate method" [4] and extensions of this procedure by Grivas and Harr [5] have been used successfully to define the probability of failure for slopes and embankments and structural foundations.
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Determinations of sensitivity to vehicle, terrain, and scenario parameters are augmented by sensitivities to algorithm errors. These errors are inherent to the many regression curve fits worked into the fabric of N R M M . The procedure assumes that all curve-fit algorithms have been identified and that the original field data from which they were derived have been re-examined such that standard errors of estimate can now be assigned to the curve fits (see Fig. 4 for an example). Each standard error of estimate is viewed as the standard deviation of a zero mean Gaussian error probability density assigned to its particular curve fit. Every value taken from that curve fit during the normal running of N R M M is assigned an additive error taken from its associated error probability density. For the purpose of the three-point extremum analysis used in the screening process, the inherent variability is set aside and the three points applicable to each curve fit are the nominal value and values that are plus and minus two standard deviations of the nominal. Quantification of sensitivity is accomplished by formulating a rank indicator specific to a given vehicle, terrain unit, and parameter. A nominal predicted speed is computed using the nominal parameters from the data set. The nominal data set is the input data and curve fit data used before any variation is applied, thus creating a nominal speed. Maximum and minimum speeds are then computed corresponding to values of the parameter that are greater and lesser than the nominal by 10%, or, in the case of curve-fit errors, greater or lesser than the regression value by two standard deviations of the test data. In most cases, the maximum speed will correspond to the maximum value of the parameter and the minimum speed to the minimum value of the parameter. For example, larger values of speed can result from larger values of soil strength. Conversely, larger values of speed can result from smaller values of slope. In any event, extreme values of parameter and speed generally correspond. In reality it is possible for N R M M to deliver maximum or minimum speeds that do not correspond to the range limits of the parameter, but such occurrences have been rare. To define the sensitivity of the parameter (terrain, vehicle, scenario parameter, and/or empirical curve fit), a rank indicator is computed:
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The rank indicator IR is defined for the specific vehicle, the specific terrain unit, and the specific parameter subject to variation; Vmaxis the maximum speed; Vmin is the minimum speed; Vnomis the nominal speed; and the magnitude of IR is constrained to the interval 0.0 < IR < 2.0. The arithmetic average of the rank indicators over all terrain units for the given vehicle is taken as their rank - - or sensitivity of the parameter. The rank indicator is formulated to express the idea that greater speed differences between maximum and minimum constitute greater sensitivity of NRMM to the parameter of interest in the given vehicle/terrain setting. Division by the nominal speed renders the rank indicator dimensionless and inherently implies that as nominal speeds increase, greater speed differences are appropriate to the same sensitivity. For example, a speed difference of 2 km h-~ at a nominal speed of 10 km h-1 and a speed difference of 12 km h-z at a nominal speed of 60 km h-~ express the same sensitivity of a given parameter. Experimentation with this formula involving many terrains, vehicles, and parameters revealed that most values of the rank indicator fall within the range 0 to 0.8, and that some cases occur as high as 1.8. The formula, of course, has the potential for infinite values when the nominal speed is zero and variation produces a finite speed range. This case has, by definition, been assigned the value of 2. Quantitative comparisons of the ranks of many parameters lead naturally to a screening process. For the given vehicle/terrain setting, variation of parameters in sequence produces a sequence of quantitative ranks; small values indicate low sensitivity and large values indicate high sensitivity. By selecting a certain threshold value, all parameters whose ranks fall below the threshold need be considered no further. Rankings change with the vehicles and terrains. Numerical experiments based on these ideas recommend a threshold of 20% [2] of the maximum rank for consideration. In the numerical experiments, from 32 to 84% of the parameters studied were eliminated from further consideration. Once parameters have been screened and the one(s) that will remain under consideration as part of the stochastic mobility forecast have been identified, the pathway to that forecast requires the formulation of an "error-magnitude scenario". This is simply a list of those parameters (and curve fits) and the actual nature of the variation (distribution and magnitude) to be assigned to each. During the screening process, each parameter was varied plus and minus 10% of the nominal and each curve fit was varied plus or minus 14% of its regression value. These curve-fit and parameter errors were based on observations in the field and are conservative estimates. During the subsequent Monte Carlo simulation, the opportunity is provided to specify the actual distribution type and ranges on an individual basis for the parameters and the actual standard deviations for the curvefit errors. With the determination of an error-magnitude scenario for a given vehicle/terrain combination, the stage is set for the principal event on the pathway to the stochastic mobility forecast. This is the Monte Carlo analysis of predicted speeds wherein the screened parameters and curve-fit algorithms are varied both jointly and independently and probability densities are determined for the speeds predicted for each terrain unit. The per-terrain-unit speed probability densities and data specifying the mission profile are the raw materials from which an analysis of mission rating speed and its range can be made. Other outputs from the Monte Carlo simulation are a listing of nominal speeds by terrain unit from which the speed display is obtained and maximum and minimum speeds by terrain unit from which the fingerprint is made. For conceptual simplicity and to bound
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NRMM error performance, initial work with the mission rating speeds was based on maximum and minimum terrain unit speeds rather than on the speed probability densities. The mission rating speed is approached through the "speed profile", a useful concept worked out early in the history of NRMM. A speed profile is specific to a given vehicle/ terrain combination. NRMM is used to form a sequence of records each of which shows the area and the predicted nominal speed for individual terrain units. These records are sorted in descending order by speed thus identifying the terrain units in which vehicle performance is "best" and "worst". The sum of terrain unit areas from the first record (which represents "best") to the Nth record divided by the sum of all areas defines the fraction of total area represented by the first N records. When the sorted speeds are plotted against this fraction, the result is a speed profile based on terrain unit speeds. NRMM calls it a "speed-in-unit" profile. When the area-weighted average of the first N speeds is plotted against the area fraction, an "average speed profile" is produced, indicated by the centerline in Fig. 5. Assuming that tactical usage of the vehicles will stress deployment over the "best" terrain units, the profiles allow qualification of what is meant by "best". Stochastic orientation of NRMM requires the development of stochastic speed profiles based not only on the nominal predicted speeds but also on the minimum and maximum predicted speeds. The very same computational procedures are used to bound the nominal mission rating speed profile. In effect, range limits that bound NRMM error performance are placed on the traditional speed profiles as indicated by the maximum and minimum speed profiles in Fig. 5. Speed profiles form the basis for the calculation of the mission ratings speeds. A mission rating speed is, as mentioned earlier, a one-number measure of vehicle performance that factors in the parameters of a tactical mission defined on a terrain map. Their parameters are (1) percentages of total operation distance spent on-road and offroad and (2) percentages of their best terrain and road units so occupied. There are several ways to computate the mission rating speed, depending on the depth of resolution
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designed. For example, are roads to be considered separately as primary, secondary, and so forth? Are predefined "tactical mobility levels" to be considered? Are time penalties for crossing linear features to be considered? See Ref. [5] for insights and typical applications. For the present, this combination is being made according to the following especially simple formula. If P is the off-road operations percentage, and Vc and VR are the corresponding speeds from the off-road and on-road profiles, then the mission rating speed VMR is 100 e ~ 100-e
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Stochastic mission rating speeds are derived from the stochastic average speed profiles by evaluating the equation using the nominal, minimum, and maximum values of Vc and V R. These values define VMR,NOM , VMR,MIN , and VMR, MAX. The range in the mission rating speeds so computed constitutes a one-number descriptor of NRMM error performance for the given terrain/vehicle combination and the given mission.
CONCLUSION Responding to the need to make NRMM a risk-based analysis for a tactical setting, the procedures discussed above transform its present deterministic orientation into a stochastic one. Based on this study, the following conclusions can be drawn: (a) Stochastic mobility forecasts can be obtained by supplementing current deterministic NRMM mobility forecasts with procedures involving determination of parameter sensitivity, ranking and screening of parameters, Monte Carlo speed simulations, development of fingerprints, and extension of stochastic mobility rating speeds. (b) The stochastic mobility forecasts developed by the prototype procedures studied here define worst-cast error performance of NRMM. (c) The error performance of NRMM will be found to vary widely among terrains and vehicles, underlining the importance of the fingerprint as a means of visually comprehending the clustering of errors. REFERENCES 1. Haley, P. W., Jurkat, M. P. and Brady, P. M., NATO Reference Mobility Model Edition 1, User's Guide. Technical Report 12503, US Army Tank-Automotive Command, Warren, MI, 1979. 2. Ahlvin, R. B. and Haley, P. W., NATO Reference Mobility Model Edition II, NRMM II User's Guide. Technical Report GL-92-19, US Army Engineer Waterways Experiment Station, Vicksburg, MS, 1992. 3. Lessem, A., Ahlvin, R., Mason, G. and Mlakar, P., Stochastic vehicle mobility forecasts using the NATO Reference Mobility Model. Technical Report GL-92-11, Army Engineer Waterways Experiment Station, Vicksburg, MS, 1992. 4. Rosenblueth, E., Design philosophy: structures. Proceedings 2nd International Conference on Application of Statistics and Problems in Soil and Structural Engineering, Aachen, Germany, 1975. 5. Grivas, D. and Harr, M. E., Consolidation a probabilistic approach. A S C E J. Eng. Mech. Div., 1978, 104(EM 3).