nonparametric approach of vehicles dynamics

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NONPARAMETRIC APPROACH OF VEHICLES DYNAMICS

Univ. Prof. eng. Ion COPAE PhDMilitary Technical Academy, Bucharest, email: [email protected]

ABSTRACTThis paper addresses for the first time in the technical literature nonparametric dynamics of

vehicles. It tackles the issue both theoretically and experimentally. It presents new theoreticalapproach of systems dynamics given the possibility of data acquisition from different transducers thatare already built-in by the vehicle manufacturers. Unlike the traditional approach, which is exclusivelyparametric, the paper eliminates the two classical simplifying assumptions used in the literature. Sothe paper takes into account that the functional parameters do not obey known classic statistic laws

and thus the vehicle dynamic behavior is described by nonparametric mathematical models (withfamilies of parameters or/and with parameters that vary in time). As a consequence, a nonparametricapproach of vehicle dynamics based on experimental data, ensures a correct estimation of distributionlaws, establishing nonparametric mathematical models both analytical and non-analytical as well ascorrect estimation of certain functional parameters [3; 6; 7].

The first aspect refers to estimation of distribution laws. Practice showed us that dynamicexperimental series do not obey any distribution law from classic statistics. Tis is confirmed by thegraphs from figure 1, where we checked if vehicle speed follows the normal distribution; as we cansee from graph 1a for 50 test runs and from graph 1b for 110 test runs. The same conclusion standsno matter the number of experimental data.

Fig.1. Verification of normal distribution law for vehicle speed,Logan Laureate and Daewoo Tacuma vehicles

Added to that, from fig. 1a and 1c, the mentioned aspect is confirmed no matter the testedvehicle. A Logan Laureate vehicle and a Daewoo Tacuma vehicle were subjected to test runs. Thegraphs include both normal theoretical distribution law and the real experimental distribution. The latergraph also presents how that data should have behaved in the case of a Gauss normal distributionlaw. As we can see from fig. 1b in order to check if vehicle speed respects the normal distribution law,The Smirnov Kolmogorov (S-K noted) test is applied; so in this case Ho hypothesis states that`experimental data respect Gauss distribution`. As we can see from the graph, applying S-K test weare lead to an alternative hypothesis H1 (we reject H0 hypothesis with a significance level of =0,05),thus these data do not abide the imposed significance level.

From figure 2 we deduct that the dynamic experimental series for vehicle speed do not abideany other repartition laws from classic statistics. We tried here lognormal distribution, exponential andRayleigh distribution; a similar conclusion is achieved if we try testing other 76 distribution laws used

in classical parametric statistics.

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Fig.2. Data abidance verification for vehicle speed in the case of other repartition laws from classicalparametric statistics; Logan Laureate vehicle

The same conclusion is deducted from figure 3 taking into account various parameters andgraphical representations. Figure 4a presents probability density f(x) for speed (probability p). Figure4b repartition function F(x) of fuel consumption for 100 km. Both cases are taking into account 110

test runs data for Logan vehicle. The graphs also present the mathematical expressions.

Fig.3. Data abidance verification for vehicle speed and 100 km fuel consumption, 110 test runs data;Logan Laureate vehicle

Nonparametric estimation of probability density f(x)=p has its starting point the targeted

parameters histogram. As exemplification, figure 4 and figure 5 present the histogram of four

functional parameters (included in the graphs) (vehicle speed V, vehicle speed variation dv/dt, engine

speed n, throttles position which is a mark the latter marking the engines load), for 50 test runs

data on Logan Laureate. Each of these graphs show the histogram, from which we can achieve the

probability density estimation f(x): the middle of each region is connected for each parameter and

divide it to the maximum number of values. Likewise, each graph presents the values that are most

probable meaning the f(x) maximum; for example, from figure 4a we can see that most values (2037)

are specific for a 0,41 m/s2

acceleration

The graph from figure 5a shows us that the engine functioned with predominance at medium

rev speed and figure 5b indicates that the engine functioned with predominance at medium and high

loads, over 50% from maximum load (the lower values are specific for engine idling). All these graphs

confirm that experimental data do not obey Gauss distribution law (theoretical/ideal repartition), best

for this conclusion is seen in figure 5b.

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Fig.4. Speed and its variation (acceleration deceleration) histogram,50 experimental test runs, Logan Laureate vehicle

Fig.5. Engine speed and throttle position histograms,50 test runs, Logan Laureate vehicle

In the presented examples density probability were given and distribution functions for asingle parameter; to this purpose we called on single-variable distribution. When two functionalparameters are targeted we call on bivariable; for the general case in order to take into considerationseveral functional parameters we call on multiple-variable distributions. For example, figures 6 and 7present bivariable probability density for 50 test runs with Logan vehicle.

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Considering several equi-probable curves (with p=constant), the graphs presented in figure 6and 7 highlight the areas that have a high probability density (probability values p appear in figure 6band figure 7b). For example from figure 6a we can see that most experimental data are found withinCh=5,5-9,5 kg/h value margin specific for the hourly fuel consumption and within V=40-100 km/h valuemargin specific for vehicle speed.

Fig.6. Vehicle speed Fuel consumption (hourly) probability density estimation and experimentaldata, 50 test runs, Logan Laureate vehicle

Fig.7. Vehicle speed Fuel consumption (l/100 km) probability density estimation and experimentaldata, 50 test runs, Logan Laureate vehicle

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The graph form figure 7 also suggests the fact that once the speed in increased, fuelconsumption for 100 covered kilometers is decreasing; in exchange once the vehicle speed isincreasing so does the hourly fuel consumption as we can see from figure 6. So we can say thatvehicle speed is increasing the engines fuel efficiency is decreasing (Ch is increasing), and vehiclefuel efficiency is increasing (C100 is decreasing). If we take into consideration the connection between

these three functional parameters (represents fuel density):

100100h

CC

V (1)

We can deduct that the increase of hourly fuel consumption is less than the vehicles speed increase.As we stated at the beginning of this paper, a second aspect refers to non-parametrical

mathematical models for vehicle dynamics. As we already know, in the classical approach for vehicledynamics, parametrical mathematical models are used. A much known example of this is representedby the well-known differential equation for straight-line vehicle movement. Its terms includeparameters that define the vehicle, the road and so on. The non-parametric approach for vehicledynamic behavior uses mathematical models that are in non-parametrical analytical form ornon-analytical. Analytical mathematical models have families of parameters (several expressions), orits coefficients are time variable thus are not constant as they are in classical parametrical approach;to this purpose we call on neural networks, genetic algorithms, fuzzy states, neuro-fuzzy algorithms,core functions of different shapes (for example spline functions, Radial base function RBF), differenttype of automata etc. Mathematical non-analytical models rely on graphs, trees, topological methodsetc. [2; 3; 4; 5; 6].

For example, figure 8 presents an exemplification on using RBF function in the case of acertain parameter x, function that is expressed containing a Gaussian component and the pparameter:

1 2 22

( , ) e

x p

K x p

(2)

According to this example which contains the experimental data series for fuel consumption at

one of the test runs, we can see from figure 8b the existence of certain variable coefficients that countup to 177 having 175 experimental data.

Fig.8. Non-parametric modeling using RBF function, fuel consumption for 100 covered km,test run LL35, Logan Laureate vehicle

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The fact that these coefficients vary in time, meaning p from expression (2) is not constant,confirms the nonparametric character of the targeted mathematical model.

A third aspect for the non-parametric approach of vehicle dynamics refers to the estimationof functional parameters. To meet this purpose we turn to bootstrap algorithm and to reversedynamics [1; 6; 7]. As the technical literature refers to it, bootstrap techniques are used there whereclassical statistics is insufficient, or where it cannot be applied from various reasons, for example

when the experimental data do not obey no known repartition laws of classical statistics.For example, if the medium value Vm for vehicle speed is imposed as a specific target, than

for test run LL43 that was carried out on Logan vehicle we will achieve the values from figure 9 for a

200 prediction horizon; in other words put, the values from figure 9 are achieved if 200 experimental

test runs would have been carried out, meeting the same test conditions as test run LL43.

As we can see from figure 9b, the medium experimental value for vehicle speed is 80,6 km/h,

and through bootstrap algorithm the medium value is estimated between 79,0 km/h 82,4 km/h.

Fig.9. Bootstrap algorithm estimation of medium values for vehicle speed for 200 predictions,experimental test run LL43, Logan Laureate vehicle

In the case of inverse dynamics, vehicle speed (including its variation) is known fromexperimentation, and in the well-known equation of vehicle dynamics we target to establish those

components from the power balance, and thus those components of energy balance. [6; 7].

BIBLIOGRAFY:

1. Efron B.An Introduction to the Bootstrap. Chapman and Hall, New York, 19932. Gibbons A.Algorithmic Graph Theory, Cambridge Univ. Press, UK, 1985

3. Greblicki W. Nonparametric System Identification, Cambridge University Press, 20084. Hatcher A.Algebraic Topology. Cornell University, 20015. Jantzen J. Neurofuzzy Modelling. Technical University of Denmark, Department of

Automation, Denmark, 19986. Copae I., Lespezeanu I., Cazacu C., Vehicle Dynamics., Ed. Ericom, Bucharest, 20067. Tarantola A. Inverse problem theory and methods for model parameter estimation.

Universit de Paris, 2005

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nonparametric approach of vehicles dynamics

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