DETERMINING DRIVING STATE VARIABLES

Abstract

A method (100) of determining driving state variables of a motor vehicle (105) includes scanning an input vector (u) of signals, which influence the driving state of the motor vehicle (105); scanning a first output vector (y) of variables, which describe the driving state of the motor vehicle (105); determining a second output vector () of variables that describe the driving state of the motor vehicle (105) based on the input vector (u), a weighting vector (r) and a state vector ({circumflex over (x)}); and adapting the weighting vector based on the difference between the two output vectors (y, ). In doing so, the observer includes a Kalman filter.

Claims

1-11. (canceled)

12. A method (100) of determining driving state variables of a motor vehicle (105) using an observer (110), the method comprising: scanning input vectors (u) of variables that determine the driving state of the motor vehicle (105); scanning first output vectors (y) of the variables that describe the driving state of the motor vehicle (105); determining, with the observer (110), second output vectors () of the variables that describe the driving state of the motor vehicle, based on the input vectors (u), weighting vectors (r) and state vectors ({circumflex over (x)}); and adapting (K), with the observer (110), the weighting vectors (r) based on a difference of the first and the second output vectors (y, ); the observer (110) comprising a Kalman filter, which is formed as an Unscented Kalman Filter, and a covariance matrix of measurements (R.sup.n) being adapted by a linear slave Kalman filter.

13. The method (100) according to claim 12, wherein the observer (110) includes a Square Root Kalman Filter.

14. The method (100) according to claim 12, wherein the input vectors (u) comprise a number of revolutions (n) or angular speeds () of wheels (FL, FR, RL, RR) of the motor vehicle (105) and a wheel angle () of the wheels (FL, FR, RL, RR).

15. The method (100) according to claim 12, wherein the first and the second output vectors (y, ) include accelerations (a) of the motor vehicle (105) in longitudinal and transversal directions as well as a yaw rate ({dot over ()}).

16. The method (100) according to claim 12, further comprising determining, on a basis of the observer (110), the driving state variables that include at least a wheel force (F) in a longitudinal, a vertical, or a transversal direction; a wheel slip (S); a slip angle (a); a float angle (); and a vehicle ground speed (V) in either the longitudinal or the transversal direction.

17. The method (100) according to claim 12, further comprising determining the second output vector () based on a physical model (f, h), and determining adhesive coefficients () between tires of the motor vehicle (105) and a roadway on which the motor vehicle is traveling, and adapting the physical model (f, h) based on the coefficients of adhesion ().

18. The method (100) according to claim 12, wherein adapting a covariance matrix of measurement (R.sup.n) as follows: $R_k^n=\frac{1}{m}.Math.\underset{j=1}{\overset{m}{.Math.}}.Math..Math.v_{k-j}.Math.v_{k-j}^T-P_{{\tilde{y}}_k,{\tilde{y}}_{k+1}}.Math.R_{k+1}^n,$ wherein v.sub.kj=y.sub.kj.sub.kj.sup. is fixed and mlIN is arbitrarily chosen.

19. A computer program product using program code means to implement a method (100) of determining driving state variables of a motor vehicle (105) using an observer (110), the method including: scanning input vectors (u) of variables that determine the driving state of the motor vehicle (105); scanning first output vectors (y) of the variables that describe the driving state of the motor vehicle (105); determining, with the observer (110), second output vectors () of the variables that describe the driving state of the motor vehicle, based on the input vectors (u), weighting vectors (r) and state vectors({circumflex over (x)}); and adapting (K), with the observer (110), the weighting vectors (r) based on a difference of the first and the second output vectors (y, ); the observer (110) comprising a Kalman filter, which is formed as an Unscented Kalman Filter, and a covariance matrix of measurements (Rn) being adapted by a linear slave Kalman filter; and the computer program product running on a processing device or is stored on a machine-readable data-storage medium.

20. A device (110) for determining of driving state variable of a motor vehicle (105), the device implements a Kalman filter and being set to execute a method (100) for determining the driving state variable of the motor vehicle including: scanning input vectors (u) of variables that determine the driving state of the motor vehicle (105); scanning first output vectors (y) of the variables that describe the driving state of the motor vehicle (105); determining, with the observer (110), second output vectors () of the variables that describe the driving state of the motor vehicle, based on the input vectors (u), weighting vectors (r) and state vectors ({circumflex over (x)}); and adapting (K), with the observer (110), the weighting vectors (r) based on a difference of the first and the second output vectors (y, ); the observer (110) comprising a Kalman filter, which is formed as an Unscented Kalman Filter, and a covariance matrix of measurements (Rn) is adapted by means of a linear slave Kalman filter.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0017] The invention will now be described in more detail with reference to the attached figures, in which:

[0018] FIG. 1 describes a method; and

[0019] FIG. 2 describes a motor vehicle having different variables

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0020] FIG. 1 shows a schematic representation of a method 100 for determining one or a plurality of driving state variables of a real motor vehicle 105 by means of an observer 110. The observer 110 can be regarded as a method and implemented by means of a programmable microcomputer, for instance. In that sense, the observer 100 can also be viewed as a device for determining driving state variables.

[0021] An input vector u includes measured values of the motor vehicle 105, for example wheel revolutions per minute n.sub.i or, alternatively, angular speeds .sub.i and wheel angles .sub.i of the individual wheels. These measurement values can be sampled using assigned sensors. For instance, an angular speed of a wheel .sub.i can be measured using a magnetic or optical encoder.

[0022] The state of the motor vehicle 105 is described by a state vector x, which can include the vehicle speeds v.sub.x, v.sub.y or a yaw rate {dot over ()}. Typically, not all elements of the state vector x can be observed. A change {dot over (x)} in the state vector x is made based on a current state vector x and the input vector u. This influence can be seen as a function (x,u), which is generally not exactly known. Based on a function h(x), the influence results in an output vector y, which can comprise variables such as vehicle accelerations a.sub.x, a.sub.y or the yaw rate {dot over ()}. These variables variables can again be measured using appropriate sensors. For example, the acceleration can be measured using an inertia sensor and the yaw rate can be measured using a yaw rate sensor. These sensors can be designed micromechanically.

[0023] The mapping of the input vector u by the real motor vehicle 105 should be simulated as accurately as possible by means of an observer 110. In that way, an algorithm for determining driving state variables of the motor vehicle 105, which can be used to determine or predict driving state variables of the motor vehicle 105, is to be generated. Variables that relate to the observer 110 instead of the real motor vehicle 105 are denoted by a caret (e.g. instead of a).

[0024] A physical model of the motor vehicle 115 implements a function ({circumflex over (x)}, u, r), which maps the state vector {circumflex over (x)} of the observer 110 to a change {circumflex over ({dot over (x)})} of the state vector of the observer 110, based on the input vector u and a correction vector r. Based on a function h({circumflex over (x)}), this change results in an output vector of the observer 110. The physical model of the motor vehicle 115 describes the driving characteristics of the motor vehicle 105 based in particular on physical interrelationships.

[0025] The difference between the output vector y and the output vector of the observer 110 is determined and transformed into the vector r by means of a so-called feedback matrix K. In this way, the error of the observer 110 is fed back to minimize it as much as possible.

[0026] The observer 110 has reached its steady state after a few passes through the feedback loop. Then, the output vector approximately equates to the output vector y of the real motor vehicle 105. Therefore, every element of the output vector can be determined quickly and accurately based on all elements elements of the input vector u and the output vector y. In that way, on the one hand, every element can be determined correctly, as potentially many measurement values have to be taken into account, on the other hand, even elements difficult to measure can be determined. For example, a float angle between the direction of movement of the motor vehicle 105 in the center of gravity CoG and the longitudinal axis of the vehicle can be determined without requiring an optical measuring system or a measuring wheel.

[0027] The determined elements typically include state variables of the motor vehicle and, for example, can be used to control the motor vehicle 105. For example, the determined speed of the motor vehicle can be used to control a brake system having anti-skid brakes (ABS), or a speed assistance system for controlling the speed at a predefined value, or be used by an electronic stability program (ESP). Other functionalities to control the movement or a convenience function of the motor vehicle 105 can also be based on the driving state variables determined by the observer 110. Of course driving state variables other than the speed can be utilized as well.

[0028] The method of the observer 110 is now mathematically described in greater detail. FIG. 2 shows related variables of the motor vehicle 105.

Definitions

[0029] In general, the following notations apply:
R rear
F front
FL left front wheel
FR right front wheel
RL left rear wheel
RR right rear wheel
I in longitudinal direction (in the wheel coordinate system)
s in lateral or transversal direction (in the wheel coordinate system)
CoG center of gravity, origin of the motor vehicle/chassis coordinate system
m vehicle mass
Jz gyration moment of inertia
h.sub.COG height of the center of gravity of the vehicle above ground
g gravitational acceleration
b.sub.F track width of the motor vehicle at the front axle (front)
b.sub.R track width of the motor vehicle at the rear axle (rear)
I.sub.F distance of the front axle from the center of gravity along the longitudinal axis
I.sub.R distance of the rear axle from the center of gravity along the longitudinal axis
v speed, in relation to the wheels
V speed, in relation to the center of gravity or the vehicle/chassis coordinate system
a acceleration
{dot over ()} yaw speed={dot over ()}
R.sup.v covariance matrix of the noise due to process noise
R.sup.n covariance matrix of the noise due to measuring noise
B.sub.l; D.sub.l; Ca.sub.l; E.sub.l; B.sub.s; D.sub.s; Ca.sub.s; E.sub.s: parameters of the tire model according to Pacejka
F force
x x direction (longitudinal axis in the vehicle/chassis coordinate system)
y y direction (longitudinal axis in the vehicle/chassis coordinate system)
z z direction (longitudinal axis in the vehicle/chassis coordinate system)
slip angle: angle between the rotation level of a wheel and its direction of movement
0 track correction (in the range of less than 1)
float angle
wheel angle
angular speed of the wheel
Sl slip in longitudinal direction
Ss slip in transversal direction
n number of revolutions of the wheel (alternative to the angular speed of the wheel)
vdiff difference in speed between the peripheral speed of the wheel and the resulting longitudinal speed of the wheel at the wheel contact patch
coefficient of adhesion
factor correction factor for the coefficient of adhesion
kfs correction factor for the lateral forces on the wheel

[0030] Speeds at the Wheel Contact Patch

vx.sub.FL=Vx{dot over ()}.Math.b.sub.F/2

vx.sub.FR=Vx{dot over ()}.Math.b.sub.F/2

vx.sub.RL=Vx{dot over ()}.Math.b.sub.R/2

vx.sub.FR=Vx{dot over ()}.Math.b.sub.R/2

vy.sub.FL=Vy{dot over ()}.Math.l.sub.F

vy.sub.FR=Vy{dot over ()}.Math.l.sub.F

vy.sub.RL=Vy{dot over ()}.Math.l.sub.R

vy.sub.RR=Vy{dot over ()}.Math.l.sub.R

[0031] Computation of the Slip Angle

.sub.FL=.sub.FL+arctan(vy.sub.FL/vx.sub.FL)+0.sub.FL

.sub.FR=.sub.FR+arctan(vy.sub.FR/vx.sub.FR)+0.sub.FR

.sub.RL=arctan(vy.sub.RL/vx.sub.RL)+0.sub.RL

.sub.RR=arctan(vy.sub.RR/vx.sub.RR)+0.sub.RR

[0032] Resulting Longitudinal Speeds of the Wheel at the Wheel Contact Patch

Vl.sub.FL={square root over ((vx.sub.FL).sup.2+(vy.sub.FL).sup.2)}.Math.cos(.sub.FL)

Vl.sub.FR={square root over ((vx.sub.FR).sup.2+(vy.sub.FR).sup.2)}.Math.cos(.sub.FR)

Vl.sub.RL={square root over ((vx.sub.RL).sup.2+(vy.sub.RL).sup.2)}.Math.cos(.sub.RL)

Vl.sub.RR={square root over ((vx.sub.RR).sup.2+(vy.sub.RR).sup.2)}.Math.cos(.sub.RR)

[0033] Switchover Between Drive Slip and Brake Slip

vdiff.sub.FL=R.sub.FL.Math..sub.FLvl.sub.FL

if (vdiff.sub.FL>0):

Sl.sub.FL=1vl.sub.FL/(R.sub.FL.Math..sub.FL);

or, if (vdiff.sub.FL<0):

Sl.sub.FL=(R.sub.FL.Math..sub.FL)/vl.sub.FL1;

or, if (vdiff.sub.FL=0):

Sl.sub.FL=0.

vdiff.sub.FR=R.sub.FR.Math..sub.FRvl.sub.FR

if (vdiff.sub.FR>0):

Sl.sub.FR=1vl.sub.FR/(R.sub.FR.Math..sub.FR);

or, if (vdiff.sub.FL<0):

Sl.sub.FR=(R.sub.FR.Math..sub.FR)/vl.sub.FR1;

or, if (vdiff.sub.FR=0):

Sl.sub.FR=0.

vdiff.sub.RL=R.sub.RL.Math..sub.RLvl.sub.RL

if (vdiff.sub.RL>0):

Sl.sub.RL=1vl.sub.RL/(R.sub.RL.Math..sub.RL);

or, if (vdiff.sub.RL<0):

Sl.sub.RL=(R.sub.RL.Math..sub.RL)/vl.sub.RL1;

or, if (vdiff.sub.RL=0):

Sl.sub.RL=0.

vdiff.sub.RR=R.sub.RR.Math..sub.RRvl.sub.RR

if (vdiff.sub.RR>0):

Sl.sub.RR=1vl.sub.RR/(R.sub.RR.Math..sub.RR);

or, if (vdiff.sub.RR<0):

Sl.sub.RL=(R.sub.RR.Math..sub.RR)/vl.sub.RR1;

or, if (vdiff.sub.RR=0):

Sl.sub.RR=0.

Ss.sub.FL=.sub.FL

Ss.sub.FR=.sub.FR

Ss.sub.RL=.sub.RL

Ss.sub.RR=.sub.RR

[0034] Resulting Slip

Sres.sub.FL={square root over ((Sl.sub.FL).sup.2.Math.(Ss.sub.FL).sup.2)}

Sres.sub.FR={square root over ((Sl.sub.FR).sup.2.Math.(Ss.sub.FR).sup.2)}

Sres.sub.RL={square root over ((Sl.sub.RL).sup.2.Math.(Ss.sub.RL).sup.2)}

Sres.sub.RR={square root over ((Sl.sub.RR).sup.2.Math.(Ss.sub.RR).sup.2)}

[0035] Coefficient of Adhesion Longitudinal According to the Tire Model by Pacejka

l.sub.FL=.sub.factor,l,FL.Math.Dl.Math.sin(Ca.sub.l.Math.arctan(Bl.Math.Sl.sub.FLEl.Math.(Bl.Math.Sl.sub.FLarctan(Bl.Math.Sl.sub.FL))))

l.sub.FR=.sub.factor,l,FR.Math.Dl.Math.sin(Ca.sub.l.Math.arctan(Bl.Math.Sl.sub.FREl.Math.(Bl.Math.Sl.sub.FRarctan(Bl.Math.Sl.sub.FR))))

l.sub.RL=.sub.factor,l,RL.Math.Dl.Math.sin(Ca.sub.l.Math.arctan(Bl.Math.Sl.sub.RLEl.Math.(Bl.Math.Sl.sub.RLarctan(Bl.Math.Sl.sub.RL))))

l.sub.RR=.sub.factor,l,RR.Math.Dl.Math.sin(Ca.sub.l.Math.arctan(Bl.Math.Sl.sub.RREl.Math.(Bl.Math.Sl.sub.RRarctan(Bl.Math.Sl.sub.RR))))

[0036] Adhesion Coefficient Transversal According to the Tire Model by Pacejka

s.sub.FL=.sub.factor,s,FL.Math.Ds.Math.sin(Ca.sub.s.Math.arctan(Bs.Math.Ss.sub.FL.Math.(Bs.Math.Ss.sub.FLarctan(Bs.Math.Ss.sub.FL))))

s.sub.FR=.sub.factor,s,FR.Math.Ds.Math.sin(Ca.sub.s.Math.arctan(Bs.Math.Ss.sub.FR.Math.(Bs.Math.Ss.sub.FRarctan(Bs.Math.Ss.sub.FR))))

s.sub.RL=.sub.factor,s,RL.Math.Ds.Math.sin(Ca.sub.s.Math.arctan(Bs.Math.Ss.sub.RL.Math.(Bs.Math.Ss.sub.RLarctan(Bs.Math.Ss.sub.RL))))

s.sub.RR=.sub.factor,s,RR.Math.Ds.Math.sin(Ca.sub.s.Math.arctan(Bs.Math.Ss.sub.RR.Math.(Bs.Math.Ss.sub.RRarctan(Bs.Math.Ss.sub.RR))))

[0037] Modification of the Adhesion Coefficients

[0038] In another preferred embodiment, the physical model of the motor vehicle described is adapted to the prevalent traction conditions based on the adhesion coefficients described above. It has to be observed that this adaptation can be used with any other non-linear observer algorithm.

[0039] The respective values are determined based on a measured acceleration a and an estimated acceleration and then subtracted from each other. The resulting difference can be filtered before feeding it into a time-discrete integrator

[00002]$\left(\frac{K.Math.\mathrm{Ts}}{z-1}\right).$

The correction factors for every coefficient of adhesion .sub.factor_l_FL, .sub.factor_l_FR, .sub.factor_l_RL and .sub.factor_l_RR, if the respective longitudinal accelerations are used, as well as .sub.factor_s_FL, .sub.factor_s_FR, .sub.factor_s_RL and .sub.factor_l_RR, if the respective transversal accelerations are used, can be determined based on the output of the integrator. The previously determined coefficients of adhesion .sub.s and .sub.l can then be multiplied by a correction factor before further processing is performed.

[00003]${}_{\mathrm{factor},i}=\frac{K.Math.\mathrm{Ts}}{z-1}.Math.\left(.Math.a_x.Math.-.Math.{\hat{a}}_x.Math.\right)$${}_{\mathrm{factor},s}=\frac{K.Math.\mathrm{Ts}}{z-1}.Math.\left(.Math.a_y.Math.-.Math.{\hat{a}}_y.Math.\right)$

[0040] Resulting Coefficients of Adhesion

res.sub.FL={square root over ((l.sub.FL).sup.2+(s.sub.FL).sup.2)}

res.sub.FR={square root over ((l.sub.FR).sup.2+(s.sub.FR).sup.2)}

res.sub.RL={square root over ((l.sub.RL).sup.2+(s.sub.RL).sup.2)}

res.sub.RR={square root over ((l.sub.RR).sup.2+(s.sub.RR).sup.2)}

[0041] Vertical Wheel Forces

[00004]$F_F^{\mathrm{norm}}=m\left(\frac{l_R.Math.g-h_{\mathrm{COG}}.Math.a_x}{l_F+l_R}\right)$$F_R^{\mathrm{norm}}=m\left(\frac{l_F.Math.g+h_{\mathrm{COG}}.Math.a_x}{l_F+l_R}\right)$${\mathrm{Fz}}_{\mathrm{FL}}=\left(\frac{1}{2}-\frac{h_{\mathrm{COG}}}{b_F}.Math.\frac{a_y}{g}\right).Math.F_F^{\mathrm{norm}}$${\mathrm{Fz}}_{\mathrm{FR}}=\left(\frac{1}{2}+\frac{h_{\mathrm{COG}}}{b_F}.Math.\frac{a_y}{g}\right).Math.F_F^{\mathrm{norm}}$${\mathrm{Fz}}_{\mathrm{RL}}=\left(\frac{1}{2}-\frac{h_{\mathrm{COG}}}{b_R}.Math.\frac{a_y}{g}\right).Math.F_R^{\mathrm{norm}}$${\mathrm{Fz}}_{\mathrm{RR}}=\left(\frac{1}{2}+\frac{h_{\mathrm{COG}}}{b_R}.Math.\frac{a_y}{g}\right).Math.F_R^{\mathrm{norm}}$

[0042] Longitudinal Wheel Forces

Fl.sub.FL=Sl.sub.FL/Sres.sub.FL.Math.res.sub.FL.Math.Fz.sub.FL

Fl.sub.FR=Sl.sub.FR/Sres.sub.FR.Math.res.sub.FR.Math.Fz.sub.FR

Fl.sub.RL=Sl.sub.RL/Sres.sub.RL.Math.res.sub.RL.Math.Fz.sub.RL

Fl.sub.RR=Sl.sub.RR/Sres.sub.RR.Math.res.sub.RR.Math.Fz.sub.RR

[0043] Lateral Wheel Forces

Fs.sub.FL=kfs.sub.FL.Math.(Ss.sub.FL/Sres.sub.FL.Math.res.sub.FLFz.sub.FL)

Fs.sub.FR=kfs.sub.FR.Math.(Ss.sub.FR/Sres.sub.FR.Math.res.sub.FRFz.sub.FR)

Fs.sub.RL=kfs.sub.RL.Math.(Ss.sub.RL/Sres.sub.RL.Math.res.sub.RLFz.sub.RL)

Fs.sub.RR=kfs.sub.RR.Math.(Ss.sub.RR/Sres.sub.RR.Math.res.sub.RRFz.sub.RR)

[0044] Forced transformed into the vehicle/chassis coordinate system (specific to the center of gravity CoG)

Fx_CoG.sub.FL=Fl.sub.FL.Math.cos(.sub.FL)Fs.sub.FL.Math.sin(.sub.FL)

Fx_CoG.sub.FR=Fl.sub.FR.Math.cos(.sub.FR)Fs.sub.FR.Math.sin(.sub.FR)

Fx_CoG.sub.RL=Fl.sub.RL

Fx_CoG.sub.RR=Fl.sub.RR

Fy_CoG.sub.FL=Fl.sub.FL.Math.sin(.sub.FL)Fs.sub.FL.Math.cos(.sub.FL)

Fy_CoG.sub.FR=Fl.sub.FR.Math.sin(.sub.FR)Fs.sub.FR.Math.cos(.sub.FR)

Fy_CoG.sub.RL=Fl.sub.RL

Fy_CoG.sub.RR=Fl.sub.RR

[0045] Wind Resistance

Fw=C_AER_X.Math.A_L.Math._AER/2.Math.(Vx).sup.2

ax=1/m.Math.(Fx_CoG.sub.FL+Fx_CoG.sub.FR+Fx_CoG.sub.RL+Fx_Cog.sub.RRFw)+{dot over ()}.Math.VyEquation of motion f1

ay=1/m.Math.(Fy_CoG.sub.FL+Fy_CoG.sub.FR+Fy_CoG.sub.RL+Fy_Cog.sub.RR)+{dot over ()}.Math.VxEquation of motion f2

{umlaut over ()}=1/Jz.Math.(Fx_CoG.sub.FR.Math.b.sub.F/2+Fy_CoG.sub.FR.Math.l.sub.FFx_CoG.sub.FL.Math.b.sub.F/2+Fy_CoG.sub.FL.Math.l.sub.F+Fx_CoG.sub.RR.Math.b.sub.R/2Fy_CoG.sub.RR.Math.l.sub.RFx_CoG.sub.RL.Math.b.sub.R/2Fy_CoG.sub.RL.Math.l.sub.R)Equation of motion f3

[0046] The equations shown above characterize the preferred physical model that forms the basis of observer 110 in FIG. 1. It must be observed that the physical model of the motor vehicle described can be used with any arbitrary non-linear observer algorithm. Conversely, the observer described can work with any other physical model of the motor vehicle.

[0047] The observer 110 can be implemented using different non-linear Kalman filters, but a Standard Unscented Kalman Filter (UKF) is particularly preferred.

[0048] Modification of the Covariance Matrix of Measurements

[0049] Alternative 1:

[0050] When using a standard UKF, the covariance matrix R.sup.n can be modified as follows.

[00005]$R_k^n=\frac{1}{m}.Math.\underset{j=1}{\overset{m}{.Math.}}.Math..Math.v_{k-j}.Math.v_{k-j}^T-P_{{\tilde{y}}_k,{\tilde{y}}_{k+1}}.Math.R_{k+1}^n,$

wherein v.sub.kj=y.sub.kj.sub.kj.sup. is fixed and m1IN can be chosen arbitrarily as needed.

[0051] Also see Covariance matching based adaptive unscented Kalman filter for direct filtering in INS/GNSS integration, Yang Meng (*), Shesheng Gao (*), Yongmin Zhong (**), Gaoge Hu (*), Aleksandar Subic (***). Where: (*) School of Automatics, Northwestern Polytechnical University, Xi'an, China (**) School of Aerospace, Mechanical and Manufacturing Engineering, RMIT University, Australia (***) Swinbume Research and Development, Swinburne University of Technology, Hawthorn, Australia.

[0052] Alternative 2:

[0053] The covariance matrix of measurements R.sup.n of an arbitrary, non-linear Kalman filter can generally be modified by means of a linear slave Kalman filter as described in Adaptive Unscented Kalman Filter and its Applications in Nonlinear Control; Jianda Han, Qi Song and Yuqine He, State Key Laboratory of Robotics, Shenyang Institute of Automation, Chinese Academy of Sciences, P.R. China, chapter 4.

[0054] Kalman Filter

[0055] Below, preferred Kalman filters are described in more detail. The description was taken and adapted from The Square-Root Unscented Kalman Filter for State and Parameter-Estimation; Rudolph van der Merwe, Eric A. Wan; Oregon Graduate Institute of Science and Technology; 20000 NW Walker Road, Beaverton, Oreg. 97006, USA. A person skilled in the art should be familiar with the notations and terms used in the explanations below. Refer to the specified publication for more details.

[0056] In the past years, the Extended Kalman Filter (EKF) has become the preferred algorithm for many non-linear estimates and self-learning algorithms. The EKF applies the method of a linear standard Kalman filter to the linearization of a really non-linear system. This approach is often flawed and results in divergences. Therefore, it is preferred to use a UKF in the application at hand. In that way, a notably improved determination of driving state variables can be achieved. The computational complexity of a standard UKF using (O(L.sup.3)) is comparable to that of an EKF.

[0057] An estimation of the state of a time-discrete, non-linear, dynamic system is to be performed.

x.sub.k+1=F(x.sub.k,u.sub.k)+v.sub.k(eq. 1)

y.sub.k=H(x.sub.k)+n.sub.k,(eq. 2)

where x.sub.k refers to the observed state vector of the system, u.sub.k refers to a known input vector and y.sub.k refers to the observed output vector.

[0058] Initialization:

x.sub.0=[x.sub.0]P.sub.0=[(x.sub.0{circumflex over (x)}.sub.0)(x.sub.0{circumflex over (x)}.sub.0).sup.T](eq. 5)

[0059] for k(1, . . . , )

[0060] Determination of sigma points

.sub.k1=[{circumflex over (x)}.sub.k1{circumflex over (x)}.sub.k1+{square root over (P.sub.k1)}{circumflex over (x)}.sub.k1{square root over (P.sub.k1)}](eq. 6)

[0061] Update:

[00006]$\begin{array}{cc}{}_{kk-1}^{*}=F\left[{}_{k-1},u_{k-1}\right]& \left(\mathrm{eq}..Math.7\right)\\ {\hat{x}}_k^{-}=\underset{i=0}{\overset{2.Math.L}{.Math.}}.Math..Math.W_i^{\left(m\right)}.Math.{}_{i,kk-1}^{*}.Math..Math.P_k^{-}=\underset{i=0}{\overset{2.Math.L}{.Math.}}.Math..Math.{W_i^{\left(c\right)}\left[{}_{i,kk-1}^{*}-{\hat{x}}_k^{-}\right]\left[{}_{i,kk-1}^{*}-{\hat{x}}_k^{-}\right]}^T+R^v& \left(\mathrm{eq}..Math.8\right)\\ {}^5_{kk-1}=\left[{\hat{x}}_k^{-}.Math..Math.{\hat{x}}_k^{-}+.Math.\sqrt{P_k^{-}}.Math..Math.{\hat{x}}_k^{-}-.Math.\sqrt{P_k^{-}}\right].Math..Math.{}_{kk-1}=H\left[{}_{kk-1}\right]& \left(\mathrm{eq}..Math.9\right)\\ {\hat{y}}_k^{-}=\underset{i=0}{\overset{2.Math.L}{.Math.}}.Math..Math.W_i^{\left(m\right)}.Math.{}_{i,kk-1}& \left(\mathrm{eq}..Math.10\right)\end{array}$

[0062] Equations to update the measurements:

[00007]$P_{{\hat{y}}_k,{\tilde{y}}_k}=\underset{i=0}{\overset{2.Math.L}{.Math.}}.Math..Math.{W_i^{\left(c\right)}\left[{}_{i,kk-1}-{\hat{y}}_k^{-}\right]\left[{}_{i,kk-1}-{\hat{y}}_k^{-}\right]}^T+R^n$$\begin{array}{cc}{P}_{x_k.Math.y_k}=\underset{i=0}{\overset{2.Math.L}{.Math.}}.Math..Math.{W_i^{\left(c\right)}\left[{}_{i,kk-1}-{\hat{x}}_k^{-}\right]\left[{}_{i,kk-1}-{\hat{y}}_k^{-}\right]}^T& \left(\mathrm{eq}..Math.11\right)\\ {}_k=P_{x_k.Math.y_k}.Math.P_{{\tilde{y}}_k.Math.{\hat{y}}_k}^{-1}& \left(\mathrm{eq}..Math.12\right)\\ {\hat{x}}_k={\hat{x}}_k^{-}+{}_k\left(y_k-{\hat{y}}_k^{-}\right)& \left(\mathrm{eq}..Math.13\right)\\ P_k=P_k^{-}-{}_k.Math.P_{{\hat{y}}_k.Math.{\tilde{y}}_k}.Math.{}_k^T& \left(\mathrm{eq}..Math.14\right)\end{array}$

[0063] Where R.sup.v represents the covariance matrix of the process noise and R.sup.n represents the covariance matrix of the measurement noise.

[0064] Preferably a Square-Root UKF is used as a refinement of the determination of the state. The following description of this variant of a Kalman filter is also taken from The Square-Root Unscented Kalman Filter for State and Parameter-Estimation.

{circumflex over (x)}.sub.0=[x.sub.n]S.sub.0=chol{(x.sub.0{circumflex over (x)}.sub.0)(x.sub.0{circumflex over (x)}.sub.0).sup.T}(eq. 16)

[0065] for k(1, . . . , ),

[0066] Determining and updating the Sigma point:

[00008]$\begin{array}{cc}{}_{k-1}=\left[{\hat{x}}_{k-1}.Math..Math.{\tilde{x}}_{k-1}+.Math..Math.S_k.Math..Math.{\hat{x}}_{k-1}-.Math..Math.S_k\right]& \left(\mathrm{eq}..Math.17\right)\\ {}_{kk-1}^{*}=F\left[{}_{k-1},u_{k-1}\right]& \left(\mathrm{eq}..Math.18\right)\\ {}_{kk-1}^{*}=\underset{i=0}{\overset{2.Math.L}{.Math.}}.Math..Math.W_i^{\left(m\right)}.Math.{}_{i,kk-1}^{*}& \left(\mathrm{eq}..Math.19\right)\\ S_k^{-}=\mathrm{qr}.Math.\left\{\left[\sqrt{W_1^{\left(c\right)}}.Math.\left({}_{1.Math.\text{:}.Math.2.Math.L,kk-1}^{*}-{\hat{x}}_k^{-}\right).Math..Math.\sqrt{R^v}\right]\right\}& \left(\mathrm{eq}..Math.20\right)\\ S_k^{-}=\mathrm{cholupdate}.Math.\left\{S_k^{-},{}_{0,k}^{*}-{\hat{x}}_k^{-},W_0^{\left(c\right)}\right\}& \left(\mathrm{eq}..Math.21\right)\\ {}^5_{kk-1}=\left[{\hat{x}}_k^{-}.Math..Math.{\hat{x}}_k^{-}+.Math..Math.S_k^{-}.Math..Math.{\hat{x}}_k^{-}-.Math..Math.S_k^{-}\right]& \left(\mathrm{eq}..Math.22\right)\\ {}_{kk-1}=H\left[{}_{kk-1}\right]& \left(\mathrm{eq}..Math.23\right)\\ {\hat{y}}_k^{-}=\underset{i=0}{\overset{2.Math.L}{.Math.}}.Math..Math.W_i^{\left(m\right)}.Math.{}_{i,kk-1}& \left(\mathrm{eq}..Math.24\right)\end{array}$

[0067] Equations to update measurements:

[00009]$\begin{array}{cc}{S}_{{\hat{y}}_k}=\mathrm{qr}.Math.\left\{\left[\sqrt{W_1^{\left(c\right)}}\left[{}_{1.Math.\text{:}.Math.2.Math.L,k}-{\hat{y}}_k\right].Math..Math.\sqrt{R_k^n}\right]\right\}& \left(\mathrm{eq}..Math.25\right)\\ S_{{\hat{y}}_k}=\mathrm{cholupdate}.Math.\left\{S_{{\hat{y}}_k},{}_{0,k}-{\hat{y}}_k,W_0^{\left(c\right)}\right\}& \left(\mathrm{eq}..Math.26\right)\\ P_{x_k.Math.y_k}=\underset{i=0}{\overset{2.Math.L}{.Math.}}.Math..Math.{W_i^{\left(c\right)}\left[{}_{i,kk-1}-{\hat{x}}_k^{-}\right]\left[{}_{i,kk-1}-{\hat{y}}_k^{-}\right]}^T& \left(\mathrm{eq}..Math.27\right)\\ {}_k=\left(P_{x_k.Math.y_k}/S_{{\hat{y}}_k}^T\right)/S_{{\tilde{y}}_k}.Math..Math.{\tilde{x}}_k={\hat{x}}_k^{-}+{}_k\left(y_k-{\hat{y}}_k^{-}\right)& \left(\mathrm{eq}..Math.28\right)\\ U={}_k.Math.S_{{\tilde{y}}_k}& \left(\mathrm{eq}..Math.29\right)\\ S_k=\mathrm{cholupdate}.Math.\left\{S_k^{-},U,-1\right\}& \left(\mathrm{eq}..Math.30\right)\end{array}$

[0068] Where R.sup.v represents the covariance matrix of the process noise and R.sup.n represents the covariance matrix of the measurement noise.

REFERENCE NUMERALS

[0069] 100 Method [0070] 105 Motor vehicle [0071] 110 Observer [0072] 115 Physical model of the motor vehicle