Method for determining a longitudinal vehicle velocity by compensating individual wheel speeds using pitch attitude

[0001] The present invention claims priority to provisional application No. 60/450,143, filed on Feb. 26, 2003, and is related to patent application Ser. No. 10/041,246, filed Jan. 8, 2002, the disclosure of which is incorporated by reference.

[0002] The present invention relates generally to dynamic control systems for automotive vehicles and, more specifically to a system that compensates wheel speed sensor signals to determine a vehicle reference velocity.

[0003] It is a well-known practice to control various operating dynamics of a motor vehicle to achieve active safety. Examples of active safety systems include traction control, yaw stability control and roll stability control systems. A more recent development has been to combine all the available subsystems to achieve better vehicle safety and dynamics performance. The effective operation of the various control systems requires high-accuracy and fast-response-times in the determination of the operating states of the vehicle, regardless of road conditions and driving conditions. Such vehicle operating states include the vehicle longitudinal, lateral and vertical velocities measured along the body-fixed longitudinal, lateral and vertical axes, the attitude of the vehicle body, and the travel course of the vehicle.

[0004] One piece of basic information that forms the aforementioned vehicle state estimation is the linear velocity of the rotating centers of the four wheels. For example, this information can be used to assess the wheel slip used in anti-lock brake controls and traction controls and to estimate the longitudinal velocity of the vehicle. In order to obtain the linear corner velocities, the wheel speed sensors are used. The wheel speed sensors output the products of the wheel rotational speeds and the rolling radii. The wheel rotational speeds are directly measured and the rolling radii are assumed their nominal values. During dynamic maneuvers, the variations of the wheel normal loading will affect the rolling radii. Hence, the nominal rolling radii may not reflect the actual rolling radii and thus cause errors in the calculation of the wheel speeds.

[0005] The determination of the longitudinal velocity is also affected by gravity. That is, if the vehicle is on a pitched road, gravity may affect some of the readings from the acceleration sensors, such as the lateral acceleration signal.

[0006] It would therefore be desirable to provide a more accurate way in which to determine the vehicle speed taking into consideration changes in rolling radii.

[0007] The present invention provides an improved determination of the individual wheel speeds. In the present invention the individual wheel speed calculations may be compensated for by learning the rolling radii of the wheels. Thus, a more accurate determination of the vehicle reference velocity or the longitudinal velocity may be determined.

[0008] In one aspect of the invention, a control system for controlling a safety system of an automotive vehicle includes a plurality of wheel speed sensors generating a plurality of wheel velocity signals, a steering angle sensor generating a steering actuator angle signal, a yaw rate sensor generating a yaw rate signal, a longitudinal acceleration sensor generating a longitudinal acceleration signal and a pitch angle generator generating a pitch angle signal and a controller. The controller generates a longitudinal vehicle velocity in response to the plurality of wheel speed signals, the steering angle signal, the yaw rate signal, the lateral acceleration signal and the pitch rate signal. The controller may determine a slip-related longitudinal velocity and a non-slip longitudinal velocity as intermediate steps.

[0009] In a further aspect of the invention, a method of controlling a safety system for an automotive vehicle having a plurality of wheels includes determining a non-slip longitudinal velocity, determining a slip-related longitudinal velocity, determining a longitudinal velocity of the vehicle in response to the non-slip longitudinal velocity and the slip related longitudinal velocity, and controlling a safety system in response to the longitudinal velocity.

[0010] One advantage of the invention is that the pitch attitude can be taken into consideration resulting in a more accurate longitudinal velocity determination.

[0011] Other features of the present invention will become apparent when viewed in light of the detailed description of the preferred embodiment when taken in conjunction with the attached drawings and appended claims.

[0016] In the following figures the same reference numerals will be used to illustrate the same components.

[0017] Referring now to FIG. 1, various operating parameters and variables used by the present invention are illustrated as they relate to the application of the present invention to a ground based motor vehicle 10 having wheels 12, 14, 16, 18. Those skilled in the art will immediately recognize the basic physics represented by these illustrations, thereby making the adaptation to different types of vehicles easily within their reach. Lateral and longitudinal velocities of the center of gravity are denoted as V_x and V_y, a yaw angular rate is denoted as ω_z, a front wheel steering angle is denoted as δ, lateral acceleration is represented by a_y, and longitudinal acceleration is represented by a_x.

[0018] Using those vehicle motion variables, the velocities of the vehicle at the four corner locations, where the wheels are attached to the vehicle, can be calculated. The velocities are projected along the body fixed longitudinal and lateral directions
V_lfx=V_x−ω_zt_f, V_lfy=V_y+ω_z1_f
V_rfx=V_x−ω_zt_f, V_rfy=V_y+ω_z1_f
V_lrx=V_x−ω_zt_r, V_tfy=V_y+ω_z1_f
V_rrx=V_x−ω_zt_f, V_rry=V_y+ω_z1_f  (1)
where t_f and t_r are the half tracks for the front and rear axles, l_f and l_r are the distances between the center of gravity of the vehicle and the front and rear axles. The variables V_lf, V_rf, V_lr and V_rr are the linear velocities of the four corners along the wheel heading directions (left front, right front, left rear and right rear, respectively), which can be calculated in the following
V_lf=V_ifx cos(δ)+V_lfy sin(δ)
V_rf=V_rfx cos(δ)+V_rfy sin(δ)
V_lr=V_lrx
V_rr=V_rrx  (2)

[0019] Substituting (1) into (2), the following is obtained
V_lf=V_x−ω_zt_f) cos(δ)+(v_y+ω_zl_f) sin (δ)
V_rf=V_x−ω_zt_f) cos(δ)+(v_y+ω_zl_f) sin (δ)
V_lf=V_x−ω_zt_r
V_rr=V_x−ω_zt_r  (3)

[0020] Considering
v_y=v_x tan(β)  (4)

[0021] Therefore, equation (3) can be used to compute both V_x and β. Since there are two unknowns and four constraints in equation (3), there are several ways of computing V_x and β.

[0022] Referring now to FIG. 2, vehicle corner velocity along the wheel longitudinal direction is equal to the sum of the contact patch slip velocity v_cp and the product of the wheel rotational rate ω_whl and its rolling radius r_o.

[0023] Referring now to FIG. 3, stability control system 24 has a controller 26 used for receiving information from a number of sensors which may include a yaw rate sensor 28, speed sensors 30 (at each wheel), a lateral acceleration sensor 32, a roll rate sensor 34, a steering angle (hand wheel position) sensor 35, a longitudinal acceleration sensor 36, a pitch sensor 37, and steering angle position sensor 39. Steering angle position sensor 39 senses the position of the steered road wheels. Lateral acceleration, longitudinal acceleration, yaw rate, roll orientation and speed may also be obtained using a global positioning system (GPS). Based upon inputs from the sensors, controller 26 controls the safety system 40. Depending on the desired sensitivity, the type of safety system and various other factors, not all the sensors 28-39 may be used in a commercial embodiment. Other factors may be obtained from the sensors such as the surface mu and the vehicle side slip angle, β.

[0024] Roll rate sensor 34 and pitch sensor 37 may sense the roll condition to be used with a rollover control system as an extension of the present application. The pitch sensor 37 may be part of a pitch angle generator that generates a pitch angle (longitudinal slope of the vehicle) of the vehicle. In some cases the pitch angle may be measured directly by the pitch sensor if it is a pitch angle sensor. However, for a high volume application such pitch angle sensors are typically cost prohibitive. A pitch rate sensor signal may be used to determine the pitch angle. This: may be performed in a stand alone unit or within controller 26. Of course, other ways of determining pitch angle may be used.

[0025] Safety system 40 may be a number of types of safety systems including a roll stability control system, a yaw control system, antilock brakes, traction control, airbags, or active suspension system. Each of the types of safety systems 40 use the longitudinal velocity for various calculations. Those skilled in the art of safety systems will appreciate the specific use.

[0026] Safety system 40, if implemented as roll control, may control a position of a front right wheel actuator, a front left wheel actuator, a rear left wheel actuator, or a right rear wheel actuator. Although, as described above, two or more of the actuators may be simultaneously controlled as one actuator. Based on the inputs from sensors 28 through 39, controller 26 determines the vehicle dynamic condition and controls the safety system. Controller 26 may also use brake control coupled to front right brakes, front left brakes, rear left brakes, and right rear brakes to dynamically control the vehicle. By using brakes in addition to steering control to prevent rollover some control benefits may be achieved. For example, yaw control and rollover control may be simultaneously accomplished.

[0027] Speed sensor 30 may be one of a variety of speed sensors known to those skilled in the art. For example, a suitable speed sensor may include a sensor at every wheel that is averaged by controller 26. As will be described below, the controller 26 translates the wheel speeds into the speed of the vehicle.

[0028] Referring now to FIG. 4, a method of operating a safety system using a corrected vehicle velocity is determined. In step 60 the wheel speed sensors are read. In one embodiment each wheel has a separate speed sensor. In step 62 the other sensors applicable to the system are read. In step 64 a plurality of non-slip longitudinal velocities are determined. Slip related longitudinal velocity is described further below. As is described below, one of many different calculations may be performed and selected from the non-slip longitudinal velocity determination.

[0029] The average front corner speed as V_f-ave and the rear average corner speed is defined as V_r-ave, i.e. $\begin{array}{cc}{V}_{f-\mathrm{ave}}=\frac{V_{\mathrm{lf}}+V_{\mathrm{rf}}}{2}{V}_{r-\mathrm{ave}}=\frac{V_{\mathrm{lr}}+V_{\mathrm{rr}}}{2}.& \left(5\right)\end{array}$

[0030] Then equation leads to
V_f-ave=V_x[cos(δ)+sin(δ)tan(β)]ω_z1_f sin(δ) V_r-ave=V_x  (6)
which can be used to construct the side slip angle as in the following $\begin{array}{cc}\beta ={\tan }^{-1}\left\{\frac{V_{f-\mathrm{ave}}-V_{r-\mathrm{ave}}\cos \left(\delta \right)-{\omega }_z{l}_f\sin \left(\delta \right)}{V_{r-\mathrm{ave}}\sin \left(\delta \right)}\right\}.& \left(7\right)\end{array}$

[0031] Notice that the vehicle side slip angle β can only be computed from when the vehicle steering angle is non-zero. If the steering angle is around zero, the method proposed in Ford patent application Ser. No. 10/041,246 filed Jan. 8, 2002, the disclosure of which is incorporated by reference, may be used. One method set forth therein is $\begin{array}{cc}{B}_{\mathrm{lat}-\mathrm{yaw}}=\frac{l_r{\omega }_z}{v_x}+\frac{I_z\stackrel{.}{\omega }+M_z+t_{f}M\left[a_y-g\sin \left({\theta }_x\right)\cos \left({\theta }_y\right)\right]}{t_r{c}_r}.& \left(8\right)\end{array}$

[0032] Or in digital environment ${\beta }_{\mathrm{lat}-\mathrm{yaw}}\left(k\right)=\frac{I_r{\omega }_z\left(k\right)}{v_x\left(k\right)}+\frac{\begin{array}{c}-I_z\stackrel{.}{\omega }\left(k\right)+M_z\left(k\right)+\\ t_{f}M\left[a_y\left(k\right)-g\sin \left({\theta \left(k\right)}_x\right)\cos \left({\theta }_y\left(k\right)\right)\right]\end{array}}{t_r{c}_r}$
where M_z is the yaw moment generated from actuators, C_r is the cornering stiffness at the rear axle.

[0033] In this example six ways are set forth for computing the longitudinal velocity from the four corner speeds V_lf, V_rf, V_lr and V_rr. These can be summarized in the following $\begin{array}{cc}{V}_{\mathrm{x1}}=\frac{V_{\mathrm{lr}}+V_{\mathrm{rr}}}{2}{V}_{\mathrm{x2}}=\frac{V_{f-\mathrm{ave}}-{\omega }_z{l}_f\sin \left(\delta \right)}{\cos \left(\delta \right)+\sin \left(\beta \right)\sin \left(\delta \right)}{V}_{\mathrm{x3}}=\frac{V_{\mathrm{lf}}+{\omega }_z\left[t_f\cos \left(\delta \right)-l_f\sin \left(\delta \right)\right]}{\cos \left(\delta \right)+\sin \left(\beta \right)\sin \left(\delta \right)}{V}_{\mathrm{x4}}=\frac{V_{\mathrm{rf}}-{\omega }_z\left[t_f\cos \left(\delta \right)-l_f\sin \left(\delta \right)\right]}{\cos \left(\delta \right)+\sin \left(\beta \right)\sin \left(\delta \right)}{V}_{\mathrm{x2}}=V_{\mathrm{lr}}+{\omega }_z{t}_r{V}_{\mathrm{x2}}=V_{\mathrm{rr}}-{\omega }_z{t}_r& \left(9\right)\end{array}$
and all of those computations should be equal (nearly, in a practical sense) to the actual longitudinal vehicle speed V_x, i.e.
V_x=V_xi, for i=1, 2, . . . , 6  (10)

[0034] Notice that equation (9) is true for all driving and road conditions due to the fact that they are from the kinematic relationships between motion variables.

[0035] From the longitudinal velocity, the four vehicle corner velocities V_lf, V_rf, V_lr and V_rr may be computed. The corner velocities can be measured by mounting four velocity sensors on the four wheels, which sense the longitudinal velocities of the center of the wheels along the heading directions of the wheel (or the wheel longitudinal directions). The velocity sensor-may be further replaced by four acceleration sensors, which sense the linear acceleration of the center of the wheels along the heading directions of the wheels.

[0036] As mainly considered here for a vehicle without the aforementioned corner velocity sensor or corner acceleration sensors, the available sensors are the wheel speed sensors used in anti-lock brake systems. Those ABS wheel speed sensors measure the rotational speed of the wheels. The wheel speed sensor outputs usually are calibrated for providing the linear directional velocities v _sensor=lf, v_sensor-rf, v_sensor-lr and v_sensor-rr by multiplying the wheel rotational angular speed with the nominal rolling radii of the wheels as is shown in FIG. 2. Notice that the wheels experience not only the rotational motion but also the linear sliding motion or longitudinal slip. The longitudinal slip is caused by the relative motion between the wheel and the road at the contact patch.

[0037] If the longitudinal velocities of such relative motions at the contact patches are denoted as v_cp-lf, v_cp-rf, v_cp-lr and v_cp-rr, then the vehicle corner velocities can be expressed as the sums of two speeds as in the following
V_lf=V_cp-lf+V_sensor-lf, V_rf=V_cp-rf+v_sensor-rf
V_lr=V_cp-lr+V_sensor-lr, V_rr=V_cp-rr+V_sensor-rr  (11)

[0038] If there is no slip on the four wheels, v_cp-lf, v_cp-rf, v_cp-lr and v_cp-rr should all be zero, and the wheel speed sensors provide the exact characterization of vehicle corner speeds. Hence the equations in (9) can be used to estimate vehicle side slip angle and the instantaneous longitudinal velocity.

[0039] If V_cp-lf, v_cp-rf, v_cp-lr and v_cp-rr are non-zero, but whose magnitudes are all fractions of the magnitudes of v _sensor=lf, V_sensor-rf, v_sensor-lr and v_sensor-rr, then the minimum of the six calculated variables can be used to characterize the vehicle longitudinal velocity as in the following
{circumflex over (V)}_x=min{{circumflex over (V)}_x1,{circumflex over (V)}_x2,{circumflex over (V)}_x3,{circumflex over (V)}_x4,{circumflex over (V)}_x5,{circumflex over (V)}_x6}  (12)
where {circumflex over (V)}_x1,{circumflex over (V)}_x2,{circumflex over (V)}_x3,{circumflex over (V)}_x4,{circumflex over (V)}_x5,{circumflex over (V)}_x6 are similarly computed as in equation (9) but with the sensor signals replacing the corner velocities as in the following $\begin{array}{cc}{\hat{V}}_{\mathrm{x1}}=\frac{v_{\mathrm{sensor}-\mathrm{lr}}+v_{\mathrm{sensor}-\mathrm{rr}}}{2}{\hat{V}}_{\mathrm{x2}}=\frac{0.5*\left(v_{\mathrm{sensor}-\mathrm{lf}}+v_{\mathrm{sensor}-\mathrm{rf}}\right)-{\omega }_z{l}_f\sin \left(\delta \right)}{\cos \left(\delta \right)+\sin \left(\beta \right)\sin \left(\delta \right)}{\hat{V}}_{\mathrm{x3}}=\frac{v_{\mathrm{sensor}-\mathrm{lf}}+{\omega }_z\left[t_f\cos \left(\delta \right)-l_f\sin \left(\delta \right)\right]}{\cos \left(\delta \right)+\sin \left(\beta \right)\sin \left(\delta \right)}{\hat{V}}_{\mathrm{x4}}=\frac{v_{\mathrm{sensor}-\mathrm{rf}}+{\omega }_z\left[t_z\cos \left(\delta \right)-l_f\sin \left(\delta \right)\right]}{\cos \left(\delta \right)+\sin \left(\beta \right)\sin \left(\delta \right)}{\hat{V}}_{\mathrm{x2}}=v_{\mathrm{sensor}-\mathrm{lr}}+{\omega }_z{t}_r{\hat{V}}_{\mathrm{x2}}=v_{\mathrm{sensor}-\mathrm{rr}}-{\omega }_z{t}_r.& \left(13\right)\end{array}$

[0040] Considering sin(δ)sin(β) is negligible in contrast to cos(δ), hence {circumflex over (V)}_x2,{circumflex over (V)}_x3,{circumflex over (V)}_x4 can be further calculated independent of the vehicle side slip angle $\begin{array}{cc}{\hat{V}}_{\mathrm{x2}}\approx \frac{v_{\mathrm{sensor}-\mathrm{lf}}+v_{\mathrm{sensor}-\mathrm{rf}}}{2\cos \left(\delta \right)}-{\omega }_z{l}_f\tan \left(\delta \right){\hat{V}}_{\mathrm{x3}}\approx v_{\mathrm{sensor}-\mathrm{lf}}\sec \left(\delta \right)+{\omega }_z\left[t_f-l_f\tan \left(\delta \right)\right]{\hat{V}}_{\mathrm{x4}}\approx v_{\mathrm{sensor}-\mathrm{rf}}\sec \left(\delta \right)-{\omega }_z\left[t_z-l_f\tan \left(\delta \right)\right].& \left(14\right)\end{array}$

[0041] In order to confidently use equations (12) and (13) to compute the longitudinal velocity, the magnitude of the contact pitch slip velocities is quantitatively captured in step 66. The variables d_f and d_r are the contact patch velocity differences between left and right wheels in front and rear axles, m_f and m_r as defined as the mean value of the left and right wheels in front and rear axles, i.e., $\begin{array}{cc}\begin{array}{cc}{d}_f=v_{\mathrm{cp}-\mathrm{lf}}-v_{\mathrm{cp}-\mathrm{rf}},& m_f=\frac{v_{\mathrm{cp}-\mathrm{lf}}+v_{\mathrm{cp}-\mathrm{rf}}}{2}\\ d_r=v_{\mathrm{cp}-\mathrm{lr}}-v_{\mathrm{cp}-\mathrm{rr}},& m_r=\frac{v_{\mathrm{cp}-\mathrm{lr}}+v_{\mathrm{cp}-\mathrm{rr}}}{2}.\end{array}& \left(15\right)\end{array}$

[0042] Thus defined d_f, d_r, m_f and m_r can be calculated from the known signals, including the wheel speeds, the yaw rate, the steering angle (at wheel) and the estimated vehicle longitudinal velocity as in the following $\begin{array}{cc}{d}_f=v_{\mathrm{rf}-\mathrm{sensor}}-v_{\mathrm{lf}-\mathrm{sensor}}-2{\omega }_z{t}_f\cos \left(\delta \right)m_f=\frac{v_{\mathrm{rf}-\mathrm{sensor}}{\mathrm{\_v}}_{\mathrm{lf}-\mathrm{sensor}}}{2}-{\hat{V}}_x\left[\cos \left(\delta \right)+\sin \left(\delta \right)\tan \left(\beta \right)\right]-{\omega }_z{l}_f\sin \left(\delta \right)d_r=v_{\mathrm{rr}-\mathrm{sensor}}-v_{\mathrm{lr}-\mathrm{sensor}}-2{\omega }_z{t}_rm_r=\frac{v_{\mathrm{lr}-\mathrm{sensor}}+v_{\mathrm{rr}-\mathrm{sensor}}}{2}-{\hat{V}}_x& \left(16\right)\end{array}$
where m_f can be approximately computed independent of the vehicle side slip angle as in the following $\begin{array}{cc}{m}_f\approx \frac{v_{\mathrm{rf}-\mathrm{sensor}}+v_{\mathrm{lf}-\mathrm{sensor}}}{2}-V_x\cos \left(\delta \right)-{\omega }_z{l}_f\sin \left(\delta \right).& \left(17\right)\end{array}$

[0043] Hence in step 68 if the computed quantities in equation (16) are all smaller than certain thresholds, i.e.,
d_f≦γ₁,
m_f≦γ₂,
d_r≦γ₃,
m_r≦γ₄,
are true for the calibrated parameters Y₁, Y₂, Y₃, Y₄ then the contact patch slip velocities are considered negligible. In this example, the minimum value from equation (13) is chosen. In step 68, if the equations in (18) are not satisfied, further analysis is needed.

[0044] One of such cases, where equation (18) is not satisfied is when the vehicle is driven with steady state steering. One example of this case is where the vehicle is on a corkscrew and the vehicle has very tight turn but with almost constant steering input. Thus, a steady state steering longitudinal velocity may be determined in step 72. In this case, if the rear wheels are negotiating the path of the vehicle, then the following is true for vehicle corner velocities $\begin{array}{cc}\frac{V_{\mathrm{lf}}-V_{\mathrm{rf}}}{V_{\mathrm{lr}}+V_{\mathrm{rr}}}=\frac{V_{\mathrm{lr}}-V_{\mathrm{rr}}}{V_{\mathrm{lf}}+V_{\mathrm{rf}}}.& \left(19\right)\end{array}$

[0045] Based on this condition, the vehicle longitudinal velocity V_x (based on steady state steering) can be estimated as $\begin{array}{cc}{\hat{V}}_{x-\mathrm{ss}-\mathrm{steer}}=\frac{t_f\cos \left(\delta \right)\sin \left(\delta \right)}{t_r-t_f\cos \left(\delta \right)\left[\cos \left(\delta \right)-\sin \left(\delta \right)\tan \left(\beta \right)\right]}{l}_f{\omega }_z.& \left(20\right)\end{array}$

[0046] Considering sin(δ)tan(β) is negligible in contrast to cos(δ), an approximate of equation (20) which is independent of the vehicle side slip angle β can be expressed as $\begin{array}{cc}{\hat{V}}_{x-\mathrm{ss}-\mathrm{steer}}\approx \frac{t_f\cos \left(\delta \right)\sin \left(\delta \right)}{t_r-t_f{\cos }^2\left(\delta \right)}{l}_f{\omega }_z.& \left(21\right)\end{array}$

[0047] If equation (18) is not satisfied and the vehicle is not in steady state steering, then a further correction to recover the errors due to the contact patch slips is needed.

[0048] The instantaneous longitudinal velocity V_x may be written as the sum of two portions: (1) the portion from the spinning of the four wheels as computed in (12), or (21); (2) the longitudinal sliding portion due to the wheel slip. In step 74, the non-slip longitudinal velocity is set as either that of equations (12) or (21) which is denoted as V_x-noslip and is either {circumflex over (v)}_x or v_x-ss-steer from steps 70 or 72.

[0049] The portion contributed by wheel slip is denoted as at the slip related longitudinal velocity V_x-slip, then
V_x=V_x-noslip+v_x-slip  (22)

[0050] The vehicle longitudinal acceleration and vehicle pitch attitude information may be used to perform the correction of the longitudinal velocity of the vehicle and determine the slip related longitudinal velocity in step 76. The longitudinal acceleration sensor signal can be divided into three parts as in the following
a_x-sensor={circumflex over (V)}_x−V_x tan(β)ω_z-sensor−gθ_y  (23)

[0051] Therefore V_x-slip can satisfy the following
{circumflex over (V)}_x-slip−V_x-slip tan(β)ω_z-sensor  (24)
where
f(t)=a_x-sensor−gθ_y−{circumflex over (V)}_x-noslip tan(β)ω_z-sensor  (25)

[0052] The analytic solution for V_x-slip can be obtained as in the following $\begin{array}{cc}{V}_{x-\mathrm{slip}}\left(t\right)=e^{{\int }_0^t\tan \left(\beta \left(\tau \right)\right){\omega }_{z-\mathrm{sensor}}\left(\tau \right)d\tau }{\int }_0^{t}f\left(\tau \right)e^{-{\int }_0^t\left(\beta \left(\tau \right)\right){\omega }_{z-\mathrm{sensor}}\left(\tau \right)d\tau }d\tau .& \left(26\right)\end{array}$

[0053] A digital iterative scheme can be derived to compute V_x-slip from equation (26) as in the following
Γ(k+1)=Γ(k)+tan(β(k+1))ω_z-sensor(k+1)ΔT
Π(k+1)=Π(k)+f(k+1)e^Γ(k+1)Δ
V_x-slip(k+1)=Π(k+1)e^−Γ(k+1)  (27)

[0054] In step 78 the longitudinal velocity of the vehicle may be determined as:
V_x(k)=V_x-nonslip(k)+V_x-slip(k)  (28)

[0055] Once the corrected longitudinal vehicle reference velocity is determined, the safety system 40 may be controlled using this value. The way in which the safety system uses the longitudinal velocity varies depending on the type of safety system.

[0056] While particular embodiments of the invention have been shown and described, numerous variations and alternate embodiments will occur to those skilled in the art. Accordingly, it is intended that the invention be limited only in terms of the appended claims.