HYBRID GENETIC ALGORITHMS / EXTENDED KALMANFILTER APPROACH FOR VEHICLE STATE AND

PARAMETER ESTIMATION

T A Wenzel∗, K J Burnham ∗, R A Williams ∗∗, M V Blundell ∗

∗ Control Theory and Applications Centre, Coventry University∗∗ Jaguar Research Group, Whitley, Coventry

Coventry University, Priory Street, CV1 5FB, CoventryTel: +44 (0) 24 7688 8972 Fax: +44 (0) 24 7688 8052 E-mail: ctac@coventry.ac.uk

Keywords: vehicle dynamics, state and parameter estima-tion, extended Kalman filter, genetic algorithm

Abstract

This paper describes the development of a hybrid approach toestimate the states and parameters of a vehicle. The parameterestimator is based on a genetic algorithm in conjunction witha bank of extended Kalman filters, which are simultaneouslyutilised to estimate the states of the system.This paper shows results of the approach when applied to afour-wheel-vehicle model and the advantages and limitationsare discussed.

1 Introduction

1.1 Background

The last two decades has seen a significant increase in the useof electronic control systems to enhance the safety of automo-tive vehicles. Such systems include antilock braking systemand electronic stability programs, as described by Bauer [1].Common to these systems is a requirement for accurate knowl-edge of the vehicle states, e.g. slip rate of the wheels or bodyslip angle and yaw rate of the vehicle.

Figure 1: Vehicle control systems

Traditionally, component suppliers have produced systems/subsystems having their own built-in state estimators and thesesystems operate as independent modules within the overall sys-tem. Additionally, as the introduction of new subsystems mod-ules increases, there is increased concern over the interoper-ability of these modules and a subsequent need to reconsiderthe structure / architecture has been recognised.In this work, an approach to develop a new total vehicle model-based state estimator is described. This state / parameter es-timator is to be placed hierarchically above the vehicle safetycontrol systems so that this single estimator supplies all elec-tronic control systems simultaneously, as shown in Figure 1.

1.2 Vehicle model

The implemented model is a four-wheel-vehicle model withfour degrees of freedom: the longitudinal directionx, the lat-eral directiony, the yawψ around the vertical axisz and theroll φ around the longitudinal axisx. Table 1 lists the mainvehicle states that are to be regarded.

Symbol Description Unitvx longitudinal velocity [m/s]vy lateral velocity [m/s]ax longitudinal acceleration [m/s2]ay lateral acceleration [m/s2]Γ torque aroundz-axis [Nm]δ steer angle [rad]β body side slip angle [rad]αij wheel slip angles [rad]ψ yaw rate [rad/s]φ body roll angle [rad]sij slip rate of each wheel [1]vij forward wheel speed [m/s]ωij wheel rotational velocity [rad/s]Fxij longitudinal forces on each wheel [N]Fyij lateral forces on each wheel [N]Fzij vertical forces on each wheel [N]ij i = front(f )/rear(r), j = left(l),right(r)

Table 1: Main vehicle states

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Furthermore, there are some parameters, which influence thebehaviour of the vehicle. The most important are listed in Ta-ble 2.

Symbol Description Unitm mass of vehicle [kg]ms sprung mass [kg]Jz moment of inertia around vert. axis [kgm2]tf , tr front and rear track width [m]b, c distances between centre of gravity

and front and rear axles [m]h height of centre of gravity [m]hs sprung mass’s height of centre of gravity [m]` length of wheel base, where` = b + c [m]r tyre radius [m]g gravitational acceleration [m/s2]βφ roll damping constant [Nms]κφ roll stiffness constant [Nm]

Table 2: Main vehicle parameters

The vehicle model used here is based on the formulation of dif-ferential equations for the calculation of the vehicle states. E.g.Wong [12] and Will andZak [11] give differential equations forthe calculation of acceleration, yaw and roll:

vy =1

m(Fyfl cos δ + Fxfl sin δ + Fyfr cos δ

+Fxfr sin δ + Fyrl + Fyrr) − vxψ (1)

ψ =Γ

Jz(2)

φ =1

Jsx(−mshsay + φ(msghs − κφ) − φβφ)) (3)

whereΓ =

tf

2Fxfl −

tf

2Fxfr +

tr

2Fxrl −

tr

2Fxrr

+bFyfl + bFyfr − cFyrl − cFyrr

+Mzfl + Mzfr + Mzrl + Mzrr (4)

ax = vx − vyψ (5)

ay = vy + vxψ (6)

Using the knowledge of these states, further states, such astheslip angles of each of the tyres and the vehicle body, can becalculated:

αf,l/r = δ − arctan(vy + bψ

vx ±1

2tf ψ

) (7)

αr,l/r = arctan(−vy + cψ

vx ±1

2trψ

) (8)

β = arctan(vy

vx) (9)

Furthermore, it is to be noted that the lateral accelerationatthe centre of gravity is not measured; rather it is the lateral ac-celeration acting on the accelerometer, which is measured.Asimplified relationship for this is given by Chee [2]:

ay,sensor = (ay + ψxa) cos φ + g sin φ (10)

wherexa is the longitudinal distance between the accelerome-ter and the centre of gravity.

1.3 Tyre model

The tyre-road interface is a major feature, which has to be re-garded. Since, ignoring aerodynamic effects, the vehicle mo-tion results from forces generated in the four, ‘palm-sized’ con-tact patches between tyre and road, the tyre model is of highimportance in the calculation of vehicle dynamics.

There are different approaches to calculate the forces acting onthe tyres. The work described in this paper makes use of theso-called ‘TMeasy’ tyre model, developed by Hirschberg et al.[4].

Tyre models vary in terms of their complexities, i.e. numbers oftyre model specific coefficients. Such coefficients are usuallyobtained via experimental trials using test rigs. The importantfactors of tyre slip and vertical forces are also required tobeknown. These particular states may be obtained, see Millikenand Milliken [6], as follows:

Fzfl/r = (1

2mg ± m

ayh

tf)c

`− max

h

`(11)

Fzrl/r = (1

2mg ± m

ayh

tr)b

`+ max

h

`(12)

sij =ωijr

vij− 1 (13)

2 Extended Kalman filter

The Kalman filter algorithm was first developed by R. E.Kalman [5] in 1960 for estimating the non-measurable statesoflinear systems. The Kalman filter algorithm was later extendedfor estimating the states of non-linear systems. In this extensionthe state equations are linearised at each working point.

The Kalman filter operates by successive prediction followedby correction, as is illustrated in Figure 2:

Figure 2: Scheme of Kalman filter

Control 2004, University of Bath, UK, September 2004 ID-124

The main steps of the extended Kalman filter (EKF) can thenbe formulated as e.g. described by Haykin [3]. It assumes a for-mulation of the non-linear system with the input statesu, theoutput statesy and the non-measurable, internal statesx. Inthe following formulation,w andv are the process and outputnoise vectors, respectively.

x(t + 1) = f(x(t),u(t),w(t)) (14)

y(t) = h(x(t),v(t)) (15)

Using this non-linear system, the procedure of the Kalman filtercan be described by five main steps:1. Prediction of state estimate

x−(t) = f(x(t − 1),u(t)) (16)

2. Prediction of error covariance

Φ−(t) = J(t)Φ(t − 1)JT(t) + Q (17)

3. Kalman gain matrix

K(t) = Φ−(t)HT[R + HΦ−(t)HT]−1 (18)

4. Correction of state estimate

x(t) = x−(t) + K(t)[y(t) − Hx−(t)] (19)

5. Correction of error covariance

Φ(t) = [I − K(t)H]Φ−(t) (20)

HereQ andR are the user-specified process noise and outputnoise covariance matrices, whileJ andH are the Jacobian ma-trices of the systems functions:

J =

∂f1

∂x1

· · ·∂f1

∂xm

......

∂fm

∂x1

· · ·∂fm

∂xm

(21)

H =

[0 0 1 0 · · · 00 0 0 0 · · · 1

]

(22)

The input vectoru and the output vectory consist of the onlymeasurable vehicle states:

u =

[δ

vcog

]

(23)

y =

[

ψ

ay

]

(24)

The state vector comprises the internal states, required bythedynamic safety control systems.

x = [vx vy ψ ay Γ β ax · · ·

αij Fz,ij sij φ φ ay,sensor]T (25)

Based on the estimation of wheel slip angles, slip rates and ver-tical forces, the TMeasy tyre model is then utilized to calculatethe longitudinal and lateral tyre forces.

3 Genetic Algorithms

3.1 General

Essentially a genetic algorithm is a guided random search tech-nique, which follows a scheme of random selection, evaluationand evolution, and a good summary may be found in Reeves[7].In this work a genetic algorithm is applied to the process ofparameter estimation.Instead of evaluating a single parameter set, a whole group,known as a ‘population’, of parameter sets is evaluated. A bi-nary string, the so-called ‘chromosome’, represents each pa-rameter set.The following scheme shows an example for a chromosome,which comprises genes for the parameters massm, moment ofinertiaJz and position of centre of gravityb.

0110010100︸ ︷︷ ︸

gene form

11101010010︸ ︷︷ ︸

gene forJz

1011001101︸ ︷︷ ︸

gene forb

Each parameter set is assigned with a ‘fitness’, which describesthe ‘quality’ of the parameter set.Once all parameter sets are evaluated, the process of evolutionor ‘breeding’ takes places, i.e. a new population of chromo-somes is generated. There are numerous approaches, howevertypically two chromosomes are chosen by a weighted randomand using crossover technique, a new chromosome is created.As the following scheme demonstrates, a new chromosome isproduced by combining a part of each existing chromosome’sstring.

Current chromosome 1 1 1 0 0 1 0 1 0Crossover point xCurrent chromosome 2 0 0 1 0 1 1 0 0Next chromosome 1 1 0 0 1 1 0 0

The mutation operation, which is effectively realised by ran-dom mutation of one or more binary bits in one or more of thechromosomes in a given population, attempts to prevent the al-gorithm from becoming trapped in local optima.The procedure of a standard genetic algorithm can be sum-marised as follows:

1. Initialise a first population

2. Evaluate fitnesses of all chromosomes

3. Generate new population using selection of fittest

4. Apply mutation to new population

5. If convergence criteria is not fulfilled, continue in step2

Control 2004, University of Bath, UK, September 2004 ID-124

3.2 Combination with extended Kalman filter

There have been already approaches to combine a genetic algo-rithm with other estimators, e.g. Warwick and Kang [10] usedagenetic algorithm in combination with a recursive least squaresalgorithm.The hybrid approach combining genetic algorithms with the ex-tended Kalman filter is implemented by executing the Kalmanfilter operations several times in parallel at each time step.The parameters to be estimated by the genetic algorithms arethe massm, moment of inertiaJz and the position of the centreof gravity b.For each parameter set the actual prediction error∆y fromequation 19 is calculated:

∆y = [y(t) − Hx−(t)] (26)

The chromosome with the best genes has a mean predictionerror. Thus as basis for fitnessFi of the ith chromosome thedifference between its prediction error∆yi and the mean pre-diction error∆y is selected:

Fi =1

(∆yi − ∆y)(27)

Depending on the fitness values a new generation of parametersets is then generated for use at the next time step.Since all evaluated chromosomes differ slightly, it is difficultto choose a best one. Therefore the ranges, in which the genescan vary, are discretized into several subdivisions or bins. Eachtime the fitness of a chromosome is evaluated, then for each pa-rameter the corresponding bin in the histograms, see Figure5,is incremented. Following such a method, the ‘best’ estimatesof the model parameters are obtained.

3.3 Advantages and limitations

The advantage of using genetic algorithms is that a large rangeof parameters can easily be covered, without an exact a-prioriknowledge of the actual parameters.The limitation is the high computational power, which isneeded for the parallel execution of several Kalman filters.Thisis not a problem, while doing the calculation offline and no real-time processing is required. But for an intended online estima-tion, the calculation has to be done in less than real-time, thusrequiring several parallel processors, which would increase thecost of implementation.

4 Results

The state and parameter estimator was implemented using theindustry standard MATLAB [9] environment. For validation,data of a rear wheel drive executive saloon car was available.Data was on hand from both simulation by other software pack-ages as well as actual measured data from a test drive.For the examples shown below, data was used, which was gen-erated by the simulation software-package ve-DYNA [8]. Theadvantage of using simulated data rather than actual measured

data is the availability of non-measurable states as referenceinformation.The performed driving manoeuvre was double cornering on aracetrack with rather large lateral acceleration (4m/s2). Figure3 shows the steer angle and vehicle velocity inputs, as well asthe reference outputs yaw rate and lateral acceleration.For the genetic algorithms a chromosome length of 31 bits anda probability of 99.9% that at each position not only the samealleles, i.e. bits are chosen, is selected. This results in apopula-tion size of 20. For each gene a range of±20% of the actuallyknown value was chosen and split up into 50 subdivisions, bythis limiting the exactness of the final estimations to less than1%.During testing of the estimator it can be noticed, that the qual-ity of the parameter estimation depends not only on the inputdata, but also on the user-specified settings such as populationsize and definition of the fitness function.Figure 4 shows the parameter estimation versus time. It is notedthat the final parameter estimates differ less than 3% from theactual values.The histogram information in Figure 5 shows the occurrence ofaccumulated parameter bin values, here at time step t=12s.

0 2 4 6 8 10 12−2

0

2

δ [ra

d]

0 2 4 6 8 10 12

1214161820

v x [m/s

]

0 2 4 6 8 10 12−0.4−0.2

00.20.4

ψ. [r

ad/s

]

0 2 4 6 8 10 12−5

0

5

a y,se

nsor

[m/s

2 ]

time [s]

Figure 3: Inputs and reference outputs to the extended KalmanFilter

0 2 4 6 8 10 121600

1800

2000

2200

m [k

g]

0 2 4 6 8 10 123200

3400

3600

3800

J z [kg

m2 ]

0 2 4 6 8 10 12

1.3

1.4

1.5

b [m

]

time [s]

referenceestimation

Figure 4: Parameters estimated by genetic algorithm

Control 2004, University of Bath, UK, September 2004 ID-124

1700 1800 1900 2000 2100 22000

200

400

600

m [kg]

3000 3200 3400 3600 3800 4000 42000

200

400

600

800

1000

Jz [kg m2]

accu

mulat

ed pa

rame

ter va

lues

1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5 1.55 1.60

500

1000

b [m]

Figure 5: Histograms showing parameter distrubtion

It is interesting to note (Figure 4) that after approximately 3seconds (or 6000 chromosomes at a sample time of 10ms) themodel parameter values have converged. As such it is clear thatthe model parameters need not be updated as frequently as thestates.Figure 6 shows the estimates for the most important vehiclestates. It should be noted that no reference signals were avail-able for the vertical tyre forces and the tyre slip rates.Figure 6 shows the estimates for the most important vehiclestates. It should be noted that no reference signals were avail-able for the vertical tyre forces and the tyre slip rates.

0 2 4 6 8 10 12−0.5

00.5

v y [m/s

]

0 2 4 6 8 10 12−5

05

a y [m/s

2 ]

0 2 4 6 8 10 12−0.04−0.020

0.020.04

β [ra

d]

referenceestimation

0 2 4 6 8 10 12

−0.050

0.050.1

φ [ra

d]

0 2 4 6 8 10 12−0.05

00.05

α [ra

d]

0 2 4 6 8 10 120

5000

Fz [N

]

0 2 4 6 8 10 12−0.020

0.020.040.060.08

slip

[1]

time [s]

Figure 6: Estimated states

5 Conclusion

This paper has demonstrated a hybrid approach for the reali-sation of a model-based vehicle state and parameter estimator,

using combined genetic algorithm and extended Kalman filter.The system states are estimated at each time step whilst a pop-ulation of ’candidate’ model parameter values are implemented/ evaluated within a genetic algorithm routine.Results to date are encouraging, indicating that system statesmay be updated at each time step whereas model parametervalues can be less frequently updated as new histogram infor-mation becomes available.This is considered to be particularly beneficial given that modelparameters vary much slower than the system states.Further work is currently ongoing to enhance the stability andefficiency of this approach.

References

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[2] W. Chee, “Measuring yaw rate with accelerometers”,Fu-ture Transportation Technology Conference, Costa Mesa,California, (2001).

[3] S. Haykin, Ed.,Kalman Filtering and Neural Networks.New York: John Wiley & Sons, (2001).

[4] W. Hirschberg, G. Rill, and H. Weinfurter, “User-appropriate tyre-modelling for vehicle dynamics in stan-dard and limit situations”,Vehicle System Dynamics, 38(2), pp. 103–125, (2002).

[5] R. E. Kalman, “A new approach to linear filtering and pre-diction problems”,Transactions of the ASME–Journal ofBasic Engineering, 82, Series D, pp. 35–45, (1960).

[6] W. F. Milliken and D. L. Milliken, Race car vehicle dy-namics. Warrendale: SAE Society of Automotive Engi-neers, (1995).

[7] C. R. Reeves, Ed.,Modern Heuristic Techniques for Com-binatorial Problems. Oxford: Blackwell Scientific Pub-lications, (1993).

[8] TESIS Gesellschaft fur Technische Simulation und Soft-ware mbH, “ve-DYNA”, http://www.tesis.de, (2003),[accessed October 2003].

[9] The MathWorks, Inc., “MATLAB”,http://www.mathworks.com, (2003), [accessed April2003].

[10] K. Warwick and Y.-H. Kang, “Self-tuning proportional,integral and derivative controller based on genetic algo-rithm least squares”,Proceedings of the Institution of Me-chanical Engineers: Journal of Systems and Control En-gineering, 212, Part I, pp. 437–448, (1998).

[11] A. B. Will and S. H.Zak, “Modelling and control of an au-tomated vehicle”,Vehicle System Dynamics, 27, pp. 131–155, (1997).

[12] J. Y. Wong,Theory of Ground Vehicles, 3rd ed. NewYork: John Wiley & Sons, Inc., (2001).

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