Inertial reference for TBA speed limiting

Abstract

A method of controlling a vehicle while the vehicle is backing up with a trailer attached thereto. The vehicle may include a brake system and a power train system. The method includes determining a trailer yaw rate, and estimating a modified trailer curvature. The modified trailer curvature comprises a ratio of the trailer yaw rate to the vehicle speed. The method further includes determining a maximum allowable vehicle speed as a function of modified trailer curvature utilizing predefined criteria that defines a maximum allowable vehicle speed for a given modified trailer curvature. The method further includes limiting the vehicle speed such that the maximum allowable vehicle speed is not exceeded.

Claims

1. A method of controlling a vehicle while the vehicle is backing up with a trailer attached to the vehicle, the method comprising: utilizing a sensor to determine a trailer yaw rate; utilizing a sensor to determine a vehicle speed; determining a ratio of the trailer yaw rate and the vehicle speed; utilizing a controller to determine a maximum allowable vehicle speed as a function of the ratio of the trailer yaw rate and the vehicle speed utilizing predefined criteria that defines a maximum allowable vehicle speed based on the ratio of the trailer yaw rate and the vehicle speed; and utilizing a controller to prevent the vehicle from exceeding the maximum allowable vehicle speed.

2. The method of claim 1, wherein: limiting the vehicle speed includes controlling at least one of a brake system and powertrain system of the vehicle.

3. The method of claim 1, wherein: the predefined criteria for determining maximum allowable vehicle speed comprises a jackknife angle.

4. The method of claim 2, wherein: the vehicle defines a front wheel angle; the jackknife angle comprises the hitch angle at which no front wheel angle will reverse the sign of the modified trailer curvature.

5. The method of claim 2, wherein: the jackknife angle comprises hitch angles at which the vehicle collides with the trailer.

6. The method of claim 5, wherein: a value of the ratio of the trailer yaw rate and the vehicle speed is recalculated at small intervals of time to provide a current ratio of the trailer yaw rate and the vehicle speed.

7. The method of claim 6, wherein: a value of the ratio of the trailer yaw rate and the vehicle speed from a previous time interval is stored if the vehicle speed is below a predefined threshold vehicle speed, and wherein a value of the ratio of the trailer yaw rate and the vehicle speed from the previous time interval is replaced with a value of the current ratio of the trailer yaw rate and the vehicle speed if the vehicle speed is above the predefined threshold vehicle speed.

8. The method of claim 5, including: utilizing a value of the last stored ratio of the trailer yaw rate and the vehicle speed if the vehicle begins to move after stopping.

9. The method of claim 5, including: utilizing a low-pass filter to attenuate current ratio of the trailer yaw rate and the vehicle speed values that are greater than a predefined limit if vehicle speed is above a predefined minimum threshold speed.

10. A method of controlling a vehicle utilizing a trailer backup assist control system comprising: utilizing a controller to calculate a current value of a control parameter by dividing sensor-measured trailer yaw rate by a sensor-measured vehicle speed; utilizing a controller to cause the vehicle to reduce speed if the current value of the control parameter exceeds an allowable limit that is a function of the measured vehicle speed.

11. The method of claim 10, wherein: the vehicle includes at least one wheel defining a wheel angle that can be varied to steer the vehicle; the allowable limit comprises the maximum value the control parameter can take beyond which no wheel angle will reverse the sign of the control parameter.

12. The method of claim 10, wherein: the allowable limit comprises a collision jackknife condition.

13. The method of claim 10, wherein: measured trailer yaw rate is estimated by determining a time derivative of a measured trailer yaw angle or hitch angle.

14. A trailer backup assist control system, comprising: a sensor configured to provide data concerning vehicle speed in a reverse direction; a sensor configured to provide data concerning at least one of a trailer yaw angle and a trailer yaw rate; a controller that is configured to limit vehicle speed in a reverse direction by controlling at least one of a vehicle brake system and a vehicle powertrain system; wherein the control system determines a ratio of trailer yaw rate to vehicle speed as a control parameter, and limits vehicle speed in a reverse direction when the value of the control parameter exceeds the maximum allowable value for the vehicle speed.

15. The trailer backup assist control system of claim 14, wherein: the maximum allowable value of the control parameter comprises a jackknife condition or an impending jackknife condition.

16. The trailer back up assist control system of claim 15, wherein: the jackknife condition comprises a physical collision of a vehicle and a trailer connected to the vehicle.

17. The trailer back up assist control system of claim 15, wherein: the maximum allowable value of the control parameter comprises a control jackknife condition wherein no vehicle wheel angle of a steered wheel will reverse the sign of the control parameter.

18. The trailer back up assist control system of claim 14, including: a steering angle control that controls steering angle based, at least in part, on the control parameter.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) In the drawings:

(2) FIG. 1 is a block diagram showing a vehicle having a trailer backup assist (TBA) system according to one aspect of the present disclosure;

(3) FIG. 2 is a block diagram of the trailer backup assist speed limiting controller of FIG. 1;

(4) FIG. 3 is a partially fragmentary view of a portion of a vehicle interior including a driver input device that may be utilized by a vehicle operator to provide steering input during vehicle backup with a trailer attached to the vehicle;

(5) FIG. 4 is a kinematic model of a vehicle-trailer system;

(6) FIG. 5A shows contours of trailer curvature .sub.2 for L=0 m, W=3.98 m, and .sub.max=32;

(7) FIG. 5B shows contours of modified curvature for L=0 m, W=3.98 m, and .sub.max=32; and

(8) FIG. 6 shows a modified curvature-based speed limit function.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

(9) For purposes of description herein, the terms upper, lower, right, left, rear, front, vertical, horizontal, and derivatives thereof shall relate to the invention as oriented in FIG. 4. However, it is to be understood that the invention may assume various alternative orientations and step sequences, except where expressly specified to the contrary. It is also to be understood that the specific devices and processes illustrated in the attached drawings, and described in the following specification are simply exemplary embodiments of the inventive concepts defined in the appended claims. Hence, specific dimensions and other physical characteristics relating to the embodiments disclosed herein are not to be considered as limiting, unless the claims expressly state otherwise.

(10) Known methods of controlling a vehicle while backup up with a trailer may have various limitations. For example, existing control schemes may require that the hitch angle, trailer length, and hitch offset. Whether the hitch angle or only the trailer yaw rate is measured, reliable estimation of trailer length typically requires a moderate change in hitch angle. Thus the accuracy of an estimate depends on the maneuver executed. Furthermore, when the hitch angle must be estimated on the basis of trailer yaw rate measurements, known methods do not provide either a trailer length or a hitch angle estimate until the maneuver satisfies particular conditions. These factors limit the ability of the controller to meet desired performance criteria during some initial learning period whose duration is maneuver-dependent. During such a learning period, the controller must be conservative with respect to meeting driver-requested curvature, and may fail to prevent jackknife under certain conditions.

(11) The present disclosure addresses this limitation by providing a trailer backup control system and method that uses trailer yaw rate measurements, but does not require knowledge or estimates of instantaneous hitch angle, trailer length, or hitch offset in order to ensure stability and jackknife avoidance. The control system and method is not unduly conservative in the sense that it can control the trailer very close to jackknife. The method is applicable to fifth wheel and conventional trailers, and may utilize modified trailer curvature (the ratio of yaw rate to vehicle speed) as a control input/parameter/variable. When the hitch angle is small, the modified curvature is approximately equal to the true curvature, making it an intuitive quantity for a driver to command. The modified trailer curvature-based control described herein may be utilized during initial backup operations to prevent jackknife or other unacceptable operating conditions during the learning period. Alternatively, the modified curvature control scheme described herein may be used instead of control systems that require knowledge of hitch angle, trailer length, and hitch offset. In general, the method described herein utilizes yaw rate and vehicle speed as control inputs. It will be understood, however, that the present disclosure is not limited to a ratio of yaw rate to vehicle speed, and these inputs may be utilized without taking a ratio of these inputs. In general, trailer yaw rate and vehicle speed can be utilized to control or limit vehicle speed and/or vehicle steering while a vehicle is backing up with a trailer attached thereto.

(12) As discussed in more detail below, the present disclosure includes a method of controlling a vehicle 1 (FIG. 1) having a brake system 4 and a powertrain system 6. The method involves controlling the vehicle while the vehicle is backing up with a trailer 2 attached to the vehicle 1. Trailer 2 defines a hitch angle relative to the vehicle 1. The method includes determining a trailer yaw rate relative to the vehicle 1 (FIG. 4). A control variable or parameter such as a modified trailer curvature () (Equation 3, below) is estimated. The modified trailer curvature comprises a ratio of the trailer yaw rate {dot over ()} (Equation 4) to the vehicle speed V.sub.1 (Equation 5). The method includes determining a maximum allowable vehicle speed as a function of trailer yaw rate and vehicle speed utilizing predefined criteria that defines maximum allowable vehicle speed based on trailer yaw rate and vehicle speed. An example of a predefined criteria is discussed in more detail below in connection with FIG. 6. The method further includes limiting vehicle speed to prevent exceeding the maximum allowable vehicle speed. Vehicle speed may be limited by controlling at least one of the brake system 4 and power train system 6 of vehicle 1. Yaw rate and vehicle sped may also be utilized to control steering of a vehicle while the vehicle is backing up with a trailer attached to the vehicle.

(13) With reference to FIG. 1, vehicle 1 may comprise a motor vehicle having an internal combustion engine, electrical drive system, or a combination thereof (e.g. a hybrid drive system). Vehicle 1 may include a trailer backup assist (TBA) system 10 that is configured to assist a vehicle operator when backing up with a trailer 2 attached to the vehicle 1. It will be understood that vehicle 1 may include a wide variety of engines, drive trains, brake systems, and other such components. In addition to brake system 4, and powertrain system 6, vehicle 1 also includes a steering system 8. Virtually any type of brake system 4, powertrain system 6, and/or steering system 8 may be utilized, and the present invention is not limited to any specific system or configuration.

(14) A brake system control module 4A (FIG. 1) is operably connected to the brake system 4, and a power train system control module 6A is operably connected to power train system 6. The brake system control module 4A and power train system control module 6A are operably connected to a trailer backup assist speed limiting controller 12 of TBA system 10. The TBA speed limiting controller 12 is operably connected to a trailer backup assist steering input apparatus 14, which may comprise a rotatable knob 16 as discussed in more detail below in connection with FIG. 3. A trailer backup assist steering controller 14A is operably connected to the input apparatus 14. The TBA steering controller 14A is operably connected to a power steering assist control module 18 of power steering assist system 20. The power steering assist system 20 is operably connected to steering system 8. The power steering assist system 20 includes a steering angle detection apparatus 22 that is configured to measure a steering angle of steered wheels 24 (FIG. 4) of vehicle 1.

(15) The TBA system 10 also includes a trailer yaw rate detection apparatus 26 (FIG. 1) that is operably connected to a trailer yaw rate detection component 28 of trailer 2. In general, trailer 2 may comprise a fifth wheel/gooseneck trailer, or it may comprise a conventional trailer. As discussed below, the different trailer configurations may have different hitch offsets L (FIG. 4). Yaw rate detection component 28 may comprise a yaw rate sensor (e.g. gyroscope) that is positioned on the trailer 2, and the trailer yaw rate may be directly measured. Alternatively, a hitch angle sensor (not shown) may be utilized to measure the hitch angle , and the trailer yaw rate may be calculated/estimated by taking a time derivative of the hitch angle to determine a hitch angle rate. Specifically, if vehicle 1 includes a yaw rate sensor, the measured (estimated) trailer yaw rate is the sum (when using the sign conventions herein) of the vehicle yaw rate and the hitch angle rate. It will be understood that a trailer yaw angle may also be measured, and a time derivative of the measured trailer yaw angle may be utilized to determine a measured trailer yaw rate.

(16) The speed of vehicle 1 during trailer backup is controlled/limited based on measured trailer yaw rate and vehicle speed. With further reference to FIG. 2, control of vehicle 1 may utilize a modified trailer curvature estimation 30 that is determined utilizing trailer yaw rate and vehicle velocity. The modified trailer curvature 31 and, optionally, other inputs (e.g. driver steering curvature requests) is then utilized to generate or determine a speed limit (maximum allowable vehicle speed) as shown at block 32. The speed limit generation 32 may comprise a jackknife condition or other predefined operating criteria as discussed in more detail below. For example, speed limit generation 32 may comprise determining a predefined maximum speed for the modified trailer curvature (FIG. 6). The speed limit from the speed limit generation 32 is then provided to a power train speed controller 34 and/or a braking speed controller 36. The powertrain and braking speed controllers 34 and 36, respectively, may be configured to generate a powertrain speed limit command and a brake command, respectively. The powertrain and braking speed controllers 34 and 36 may be integrated with the powertrain system control module 6A and brake system control module 4A (FIG. 1), or they may comprise separate components.

(17) With reference to FIG. 3, the steering input 14 utilized by the TBA system 10 may comprise a rotatable knob 16. Knob 16 may be mounted on a center console 38 in a vehicle interior directly adjacent a gear selection lever 40. In use, a user may activate the TBA system 10 by actuating a user input feature such as a button 42. In general, a user may position gear selector 40 in the R position 46, and then actuate input feature/button 42 to activate the TBA system 10. It will be understood that the TBA system 10 may be actuated to assist a user when backing up with a trailer utilizing various inputs, and button 42 is merely an example of one such activation feature. As vehicle 1 is being backed up with a trailer 2 attached thereto, a user may rotate knob 16 to the right or left as shown by the letters R and L. Knob 16 may include indicia 44 that enables an operator to determine what the angle of knob 16 is during backup. The knob 16 may have a maximum rotational angle of +/90 as shown by the arrows designated R (L), R (R).

(18) During vehicle backup with TBA system 10 activated, the power steering assist control module 18 (FIG. 1) controls the angle (FIG. 4) of the steered wheels 24 of vehicle 1 based on user inputs from rotatable knob 16 and other control parameters. Thus, when TBA system 10 is activated and vehicle 1 and trailer 2 are backing up, the user provides steering inputs utilizing rotatable knob 16 rather than the steering wheel (not shown) of vehicle 1. The power steering assist control module 18 may adjust the angle of steered wheels 24 based on various control parameters (e.g. vehicle speed and trailer hitch angle), such that the angle of the steered wheels 24 may not always correspond directly to the steering angle command input provided by rotatable knob 16.

(19) Kinematic Model

(20) The variables utilized in a kinematic model of a vehicle 1 and trailer 2 are shown in FIG. 4. Under standard assumptions, the differential kinematics for A vehicle-trailer model according to the present disclosure may be given by

(21) $\begin{array}{cc}\stackrel{.}{}=\frac{v_1}{D}\sin -\left(1+\frac{L}{D}\cos \right)\frac{v_1}{W}\tan ,& \left(1\right)\end{array}$

(22) where := is the hitch angle, .sub.1 is the velocity of the rear axle of the tow vehicle, is the road wheel angle of the tow vehicle, L is the hitch offset, D is the trailer length, and W is the tow vehicle wheelbase. We assume the road wheel angle is limited to .sub.max.sub.max. We also adopt the convention that .sub.10 when the vehicle is in reverse. These quantities are illustrated in FIG. 4.

(23) For the purpose of control, the term control jackknife is used herein to mean any configuration in which the hitch angle is locally uncontrollable with the vehicle in reverse. This is distinct from the notion of jackknife as a collision between the vehicle and the trailer, which is referred to herein as collision jackknife. In general, the control jackknife angle (as used here) may be greater than or less than the hitch angle at which the trailer and the vehicle collide. Thus, while the controller will attempt to prevent control jackknife, but collision jackknife avoidance may require additional information about the vehicle and trailer geometry.

(24) The control jackknife angle is given by .sub.jk=(.sub.ik, D, L) where .sub.jk=tan(.sub.max)/W is the maximum vehicle curvature and

(25) $\begin{array}{cc}\left({}_1,D,L\right)=\mathrm{sgn}\left(\right){\cos }^{-1}\left(\frac{-\mathrm{DL}{}^2+\sqrt{1-\left(D^2-L^2\right){}^2}}{1+L^2{}^2}\right)& \left(2\right)\end{array}$
Control jackknife is not possible if D.sup.2L.sup.2>1/.sup.2.sub.jk.
Modified Curvature Control

(26) A controller according to the present disclosure may also control steering to prevent control jackknife. A driver input (t)[1, 1] representing a normalized desired trailer curvature and a trailer yaw rate measurement .sub.2(t) may be utilized to determine an appropriate road wheel angle to prevent control jackknife. Furthermore, the controller may be configured to drive the trailer curvature .sub.2(t) approximately to the desired curvature .sub.d(t):=.sub.max (t), where .sub.max=sin(.sub.jk)/(L+D cos .sub.jk). This approximate asymptotic curvature tracking is achieved even if the controller does not know and cannot compute the maximum curvature .sub.max or the control jackknife angle .sub.jk because the controller does not know D or L.

(27) Modified Curvature

(28) A controller according to the present disclosure may utilize a control parameter that takes into account vehicle speed and trailer yaw rate. The control parameter may comprise modified trailer curvature (), which is defined as:

(29) $\begin{array}{cc}=\frac{\sin }{D}-\frac{L\cos }{D}\frac{\tan }{W}& \left(3\right)\end{array}$
the hitch angle dynamics equation (1) can be rewritten in terms of as:

(30) $\begin{array}{cc}\stackrel{.}{}=v_1-v_1\frac{\tan }{W}& \left(4\right)\end{array}$
As {dot over ()}=.sub.2.sub.1 and .sub.1=.sub.1 tan()/W, we have .sub.2=.sub.1, or, when |.sub.1|>0,

(31) $\begin{array}{cc}=\frac{{}_2}{v_1}& \left(5\right)\end{array}$
Thus, if the vehicle is moving, can be determined (computed) from measurements of vehicle velocity and trailer yaw rate without knowing , D, or L. Furthermore, the trailer curvature .sub.2 satisfies

(32) $\begin{array}{cc}{}_2=\frac{{}_2}{v_2}=\frac{1}{\cos +L\sin \frac{\tan }{W}}& \left(6\right)\end{array}$
so =.sub.2 when is small. Just as with the trailer curvature, when taken as an output for the dynamics equation (1), the modified curvature has relative degree one when L=0 and relative degree zero when L0.

(33) It is useful to rewrite the hitch angle dynamics in terms of the distance s.sub.1 traveled by the rear vehicle axle. Because .sub.1=ds.sub.1/dt, the chain law implies that

(34) $\begin{array}{cc}\frac{d}{{\mathrm{ds}}_1}=-\frac{\tan }{W}& \left(7\right)\end{array}$
From the new expression, it is seen that hitch angle equilibrium is equivalent to =tan()/W=.sub.1, where .sub.1 is the vehicle curvature, and control jackknife corresponds to the equilibrium with ||=.sub.max, or ||=j.

(35) The preceding observations justify regarding the driver input (t) as a normalized, desired modified curvature command, which is formalized through the definition.
.sub.d(t):=.sub.jk(t)(8)

(36) FIGS. 5A and 5B compare the trailer curvature .sub.1 (FIG. 5A) with the modified curvature (FIG. 5B) for a particular choice of L, W, and .sub.max. FIGS. 5A and 5B clearly illustrate the relationship between the modified curvature and the jackknife limit (where it exists), which limit is simply the .sub.jk-level curve of the modified curvature . Unlike the true curvature, the modified curvature is finite and continuous even at large hitch angles. Even when L0 the hitch angle at which the trailer curvature .sub.2 is infinite is precisely the hitch angle at which the modified curvature reaches its maximum value for a given D. This can be seen by comparing the denominator in equation (6) with the partial derivative of with respect to .

(37) Control Jackknife Detection and Margin

(38) A measure of the system's proximity to control jackknife can be utilized to implement a speed limiting system/method according to one aspect of the present disclosure. Recalling the previous definition, the system is jackknifed (control jackknife) when the hitch angle is locally uncontrollable, or, in other words, when the hitch angle is such that there is no admissible choice of wheel angle which reverses the sign of d/ds.sub.1 (or, equivalently, of {dot over ()}). The boundary of the control jackknife region corresponds to the equilibrium =j. Based on this definition, one method to detect control jackknife is to detect the sign of {dot over ()} as soon as the vehicle starts moving, then to immediately drive the wheel angle to its limit in the proper direction. If the sign of {dot over ()} changes, then the system is not control jackknifed. The following methods detect control jackknife and determine proximity to control jackknife without saturating the wheel angle.

(39) When it is known that L=0, modified curvature provides a simple solution. In this case, the modified curvature is independent of the wheel angle, so it is sufficient to compare the instantaneous value of to j; if ||j, then the system is control jackknifed. Furthermore, the map .fwdarw.|/j| provides a measure of the proximity to control jackknife, with small values (|/j|1) indicating a large control jackknife margin, and values closer to unity indicating a smaller control jackknife margin. When L=0 and [90, 90], /j=sin /sin j, so this measure is related by a nonlinear transformation to a (parameter-dependent) hitch angle-based measure of proximity to control jackknife.

(40) For general L, the value |/.sub.jk| provides a meaningful measure of the proximity to control jackknife when the system is in equilibrium, that is, when =tan()/W (see (4)). Thus, one heuristic measure of proximity to control jackknife is the absolute value of the function

(41) $\begin{array}{cc}\left(,{}_1\right)=\frac{+c\left(-{}_1\right)}{{}_{\mathrm{jk}}},& \left(9\right)\end{array}$
where 0c<1 is a constant parameter. The term c (.sub.1)/.sub.jk penalizes (by increasing (, .sub.1)|) deviations from equilibrium which will drive the hitch angle closer to control jackknife.
Speed Limiting

(42) It is often more convenient to study the trailer backing problem in a velocity-invariant framework. However, vehicle velocity may be limited to ensure that the TBA system remains active. Furthermore, control jackknife avoidance is improved by reducing the maximum allowed vehicle speed as the hitch angle approaches .sub.jk.

(43) The following disclosure describes a parameterless speed limiting approach based on trailer yaw rate and vehicle speed (e.g. the modified curvature).

(44) Speed limiting can be realized by specifying the desired speed limit .sub.max as a function of the modified curvature . As .sub.max=tan(.sub.max)/W is a function of known vehicle parameters, it is possible to specify a reasonable speed limit (maximum allowable vehicle speed) as a function of modified curvature . One example is illustrated in FIG. 6. When L=0, this method is analogous to known methods that are based on hitch angle. However, when L0, the value (, .sub.1) in place of , and the limits 1 can be used in place of .sub.max.

(45) In FIG. 6, a maximum allowable speed for vehicle 1 during trailer backup with TBA system 10 activated is shown by line 120. Horizontal line segment 110 represents an overall maximum allowable speed for vehicle 1 during backup operations in which the TBA system 10 is activated. This speed may be about, for example, 5 or 10 mph. This speed limit is typically selected to be low enough to ensure that a vehicle operator will have time to react if braking and/or steering inputs are required to avoid an object and to ensure that the operator will be able to control vehicle 1. Referring again to FIG. 6, sloped line segments 112A, 112B extend between horizontal line segment 110 and horizontal outer line segments 114A, 114B. Outer line segments 114A, 114B represent very low vehicle speeds (e.g. 0.5 mph, 1 mph or 2 mph) when vehicle 1 is approaching a jackknife condition. Thus, in the example of FIG. 6, the maximum allowable speed is reduced linearly as the absolute value of the modified curvature increases. This ensures that the speed of vehicle 1 is reduced as a jackknife condition (.sub.max or .sub.max) is approached. The controller may be configured to stop vehicle 1 if a jackknife condition (.sub.max or .sub.max) is reached to prevent any additional motion in a reverse direction, and an alert may be provided to the operator indicating that a jackknife condition has been reached, and that the vehicle must be moved in a forward direction. The alert may comprise an audio alert and/or a visual alert/message.

(46) It will be understood that the speed limit line 120 of FIG. 6 is merely an example of one possible speed limit control arrangement. In general, higher speeds are allowed when modified curvature is zero or near zero, and the allowed vehicle speed is reduced at grater (absolute) values of . The maximum allowable absolute value of may vary depending on the particular vehicle. Typically, the absolute value of .sub.max and .sub.max are about 0.10 or 0.12. Also, it will be understood that the maximum allowable speed may be determined utilizing trailer yaw rate and vehicle speed without calculating modified trailer curvature. For example, control jackknife may be determined using trailer yaw rate taking into account vehicle speed without calculating a ratio of these measured variables.

(47) Estimation of Modified Curvature

(48) In principle, the modified curvature can be determined (calculated) from vehicle velocity measurements and trailer yaw rate measurements using equation (5), provided that the vehicle velocity is nonzero. Furthermore, if the vehicle is in motion and subsequently comes to a stop, we have only to recall the last value of before the velocity reaches zero, as long as it is safe to assume that the hitch angle is unchanged. In practice, however, it is useful to modify this strategy for computing, or, more accurately, estimating, .

(49) The first modification is to specify some positive threshold .sub.1, and to update the estimate of only when |.sub.1|>.sub.1. In addition to avoiding divide-by-zero, this modification may also reduce the effect of velocity sensor nonlinearities.

(50) Also, a low-pass filter may be applied to the estimate of . The low-pass filter attenuates sensor noise (including quantization noise), which could otherwise have a significant effect on the estimate of at low speeds. With a suitable realization, the filter state can be used to hold the value of while updates are disabled. One such realization in discrete-time is given by

(51) $\begin{array}{cc}\hat{}\left[\right]=\{\begin{array}{cc}a\frac{{}_2\left[\right]}{v_1\left[\right]}+\left(1-a\right)\hat{}\left[-1\right],& \mathrm{if}.Math.v_1.Math.>{\mathrm{.Math.}}_{v1}\\ \hat{}\left[-1\right],& \mathrm{otherwise}\end{array},& \left(10\right)\end{array}$
where a (0, 1] is the normalized filter bandwidth.

(52) It is to be understood that variations and modifications can be made on the aforementioned structure without departing from the concepts of the present invention, and further it is to be understood that such concepts are intended to be covered by the following claims unless these claims by their language expressly state otherwise.