FIELD OF THE INVENTION

[0001]
The present invention generally relates to control systems for automotive vehicles, and more particularly relates to an integrated control of an active steering system and a brake system of an automotive vehicle for improving upon a handling, stability, and a maneuverability of the automotive vehicle.
BACKGROUND OF THE INVENTION

[0002]
Some automotive vehicles known in the art utilize an active brake control to enhance a directional stability of the vehicle at or close to a limit of adhesion. Some other automotive vehicles known in the art utilize a limited active control of a rear tire steering angle in order to improve a vehicle handling and maneuverability at low speeds. More recently, automotive vehicles are utilizing a limited active control of a front tire steering angle to introduce a steering correction to a steering angle commanded by a vehicle driver in an effort to improve a vehicle directional stability. The present invention addresses a need for an integrated control of vehicle brakes, and a front tire steering angle and/or a rear tire steering angle.
SUMMARY OF THE INVENTION

[0003]
One form of the present invention is an integrated active steering and braking control method for a vehicle. First, a first corrective yaw moment for the vehicle as a function of a steering angle of an axle of the vehicle is determined, and a second corrective yaw moment for the vehicle as a function of a speed differential between a first tire and a second tire of the vehicle is determined. Second, a corrective steering signal is provided to a steering system of the vehicle whereby the first corrective yaw moment is applied to the vehicle, and a corrective braking signal is provided to a braking system of the vehicle whereby the second corrective yaw moment is applied to the vehicle.

[0004]
A second form of the present invention is also an integrated active steering and braking control method for a vehicle. First, a desired speed differential between the speed of the first tire and the speed of the second tire is determined. Second, a desired steering angle of the axle as a function of said desired speed differential is determined.

[0005]
A third form of the present invention is also an integrated active steering and braking control method for a vehicle. First, a feedforward portion of a corrective front steering angle signal in response to a plurality of operational signals indicative of an operational state of the vehicle is determined. Second, a feedforward portion of a corrective rear steering angle signal in response to said plurality of operational signals.

[0006]
A fourth form of the present invention is also an integrated active steering and braking control system for a vehicle comprising a first controller and a second controller. The first controller is operable to determine a first corrective yaw moment for the vehicle as a function of a steering angle of an axle of the vehicle, and to determine a second corrective yaw moment for the vehicle as a function of a speed differential between a first tire and a second tire of the vehicle. The second controller is operable to provide a corrective steering signal to a steering system of the vehicle whereby the first corrective yaw moment is applied to the vehicle, and to provide a corrective braking signal to a braking system of the vehicle whereby the second corrective yaw moment is applied to the vehicle.

[0007]
A fifth form of the present invention is also an integrated active steering and braking control system for a vehicle. The system comprises a means for determining a feedforward portion of a corrective front steering angle signal in response to a plurality of operational signals indicative of an operational state of the vehicle. The system further comprises a means for determining a feedforward portion of a corrective rear steering angle signal in response to said plurality of operational signals.

[0008]
A sixth form of the present invention is a vehicle comprising an axle, a first tire, a second tire, and an integrated active steering and braking control system. The system is operable to determine a desired speed differential between a speed of the first tire and a speed of the second tire and to determine a desired steering angle of the axle as a function of the desired speed differential.

[0009]
The foregoing forms, and other forms, features and advantages of the present invention will become further apparent from the following detailed description of the presently preferred embodiments, read in conjunction with the accompanying drawings. The detailed description and drawings are merely illustrative of the present invention rather than limiting, the scope of the present invention being defined by the appended claims and equivalents thereof.
BRIEF DESCRIPTION OF THE DRAWINGS

[0010]
[0010]FIG. 1A is a vector diagram illustrating a yaw moment of a vehicle that is generated by a differential braking of a pair of front tires of the vehicle as known in the art;

[0011]
[0011]FIG. 1B is a vector diagram illustrating a yaw moment of a vehicle that is generated by a front tire steering of the vehicle as known in the art;

[0012]
[0012]FIG. 1C is a vector diagram illustrating a yaw moment of a vehicle that is generated by a differential braking of a pair of rear tires of the vehicle as known in the art;

[0013]
[0013]FIG. 1D is a vector diagram illustrating a yaw moment of a vehicle that is generated by a rear tire steering of the vehicle as known in the art;

[0014]
[0014]FIG. 2 is a block diagram of one embodiment of a coordinated control system in accordance with the present invention;

[0015]
[0015]FIG. 3 is a block diagram of one embodiment of a vehicle reference model of FIG. 2 in accordance with the present invention;

[0016]
[0016]FIG. 4 is a graph illustrating three (3) feedforward gain curves for an active rear steer as a function of a vehicle speed in accordance with the present invention;

[0017]
[0017]FIG. 5 is a block diagram of one embodiment of a surface coefficient estimator in accordance with the present invention;

[0018]
[0018]FIG. 6 is a block diagram of one embodiment of a side slip velocity estimator in accordance with the present invention;

[0019]
[0019]FIG. 7 is a block diagram of one embodiment of a vehicle level brake/steer controller in accordance with the present invention; and

[0020]
[0020]FIG. 8 is a graph of a lateral tire force vs. a tire slip angle in accordance with the present invention.
DETAILED DESCRIPTION OF THE PRESENTLY PREFERRED EMBODIMENTS

[0021]
Referring to FIGS. 1A1D, a vehicle 10 including a front axle 11 having a front left tire 12 and a front right tire 13 coupled thereto, and a rear axle 14 having a rear left tire 15 and a rear right tire 16 coupled thereto is shown. As known by those having ordinary skill in the art, a response of vehicle 10 in a yaw plane is primarily dictated by a combination of longitudinal tire forces and lateral tire forces being applied to tires 11, 12, 15, and 16. Good handling of vehicle 10 in the yaw plane requires that a yaw rate (i.e. a rate of rotation of vehicle 10 about a vertical axis 17 passing through the center of gravity of vehicle 10) and a lateral acceleration of vehicle 10 be consistent with driver intentions, subject to a physical limit imposed by a surface coefficient of adhesion. Since the vehicle yaw rate is determined by a yaw moment acting on vehicle 10 (i.e. a moment of forces about vertical axis 17), a main mechanism to control vehicle yaw response is by generating a corrective yaw moment. This can be achieved by applying one or more brakes (not shown) to tires 11, 12, 15, and/or 16; by a change in a steering angle of front axle 11; or by a change in a steering angle of rear axle 14.

[0022]
For example, when vehicle 10 is being driven straight as illustrated in FIG. 1A, a brake force F_{x }can be applied to front right tire 13 to generate corrective yaw moment ΔM_{z1 }in a clockwise direction about vertical axis 17. Corrective yaw moment ΔM_{z1 }can be computed by the following equation (1):

ΔM _{z1} =F _{x}*(t _{w}/2) (1)

[0023]
where t_{w }is a track width. In a linear range of tire operation, brake force F_{x }can be approximated by the following equation (2):

F _{x} =C _{x} *λ=C _{x}*(Δv_{lr1} /v) (2)

[0024]
where C_{x }is a tire longitudinal stiffness; λ is a brake slip; Δv_{lr1 }is a difference in a linear speed of tire 12 and a linear speed of tire 13; and v is a vehicle speed of vehicle 10. Combining equations (1) and (2) yields the following equation (3):

ΔM _{z1} =C _{x}*(t _{w}/2)*Δv_{lr1} /v (3)

[0025]
As illustrated in FIG. 1B, tire 12 and tire 13 can also be controlled to generate corrective yaw moment ΔM_{z2 }as a function of incremental front steering angle Δδ_{f}. Corrective yaw moment ΔM_{z2 }can be computed by the following equation (4):

ΔM _{z2} =F _{y1} *a (4)

[0026]
where a is the distance from axis 17 to front axle 11; and F_{y1 }is the total lateral force on both tire 12 and tire 13, which in the linear range of tire operation can be computed by the following equation (5):

F _{y1}=2*C _{y}*Δδ_{f} (5)

[0027]
where C_{y }is a cornering stiffness coefficient of both tire 12 and tire 13. Thus, corrective yaw moment ΔM_{z2 }can also be computed by the following equation (6):

ΔM _{z2}=2*C _{y} *a*Δδ _{f} (6)

[0028]
Equating yaw moment ΔM_{z2 }to yaw moment ΔM_{z1 }can be accomplished by computing front steering angle Δδ_{f }under the following equation (7) with the assumption that tire longitudinal stiffness coefficient C_{x }and tire lateral stiffness C_{y }are approximately equal:

Δδ_{f}=(C _{x} *t _{w}/(4*C _{y} *a))*(ΔV _{lr1} /v)≈[t _{w}/(4*a)]*(ΔV _{lr1} /v) (7)

[0029]
Also by example, when vehicle 10 is being driven straight as illustrated in FIG. 1C, brake force F_{x }can be applied to rear right tire 16 to generate corrective yaw moment ΔM_{z3 }in a clockwise direction about vertical axis 17. Corrective yaw moment ΔM_{z3 }can be computed by equation (1). In a linear range of tire operation, brake force F_{x }can be approximated by the following equation (8):

F _{x} =C _{x} λ*=C _{x}*(ΔV _{lr2} /v) (8)

[0030]
where C_{x }is a tire longitudinal stiffness; λ is a brake slip; ΔV_{lr2 }is a difference in a linear speed of tire 15 and a linear speed of tire 16; and v is a vehicle speed of vehicle 10. Combining equations (1) and (8) yields the following equation (9):

ΔM _{z3} =C _{x}*(t _{w}/2)*ΔV _{lr2} /v (9)

[0031]
As illustrated in FIG. 1D, tire 15 and tire 16 can also be controlled to generate corrective yaw moment ΔM_{z4 }as a function of incremental rear steering angle Δδ_{r}. Corrective yaw moment ΔM_{z4 }can be computed by the following equation (10):

ΔM _{z4} =F _{y2} *b (10)

[0032]
where b is the distance from axis 17 to rear axle 14; and F_{y2 }is the total lateral force on both tire 15 and tire 16, which in the linear range of tire operation can be computed by the following equation (11):

F _{y2}=−2*C _{y} *Δδr (11)

[0033]
where C_{y1 }is a cornering stiffness coefficient of both tire 15 and tire 16. Thus, corrective yaw moment ΔM_{z4 }can also be computed by the following equation (12):

ΔM _{z4}=−2*C _{y} *a*Δδ _{r} (12)

[0034]
Equating yaw moment ΔM_{z4 }to yaw moment ΔM_{z3 }can be accomplished by computing rear steering angle Δδ_{r }under the following equation (13) with the assumption that tire longitudinal stiffness coefficient C_{x }and tire lateral stiffness C_{y }are approximately equal:

Δδ_{r} =−[C _{x} *t _{w}/(4*C _{y} *b)]*(Δv _{lr2} /v)≈−[t _{w}/(4*b)]*(Δv _{lr2} /v) (13)

[0035]
The present invention is an integrated active steering and braking control method based on equations (7) and (13) that selectively utilizes tire speed differential signal ΔV_{lr1 }to generate corrective yaw moment ΔM_{z1 }and/or to generate corrective yaw moment ΔM_{z2 }when vehicle 10 has an active front steering system, and selectively utilizes tire speed differential signal ΔV_{lr2 }to generate corrective yaw moment ΔM_{Z3 }and/or corrective moment ΔM_{z4 }when vehicle 10 has an active rear steering system.

[0036]
Referring to FIG. 2, an integrated active steering and braking control system 11 for vehicle 10 in accordance with the present invention is shown. System 11 comprises a reference model 20, an estimator 30, a vehicle level brake/steer controller 40, and an actuator controller 50. To implement the principals of the present invention, reference model 20, estimator 30, vehicle level brake/steer controller 40, and an actuator controller 50 may include digital circuitry, analog circuitry, or any combination of digital circuitry and analog circuitry. Also, reference model 20, estimator 30, vehicle level brake/steer controller 40, and an actuator controller 50 may be programmable, a dedicated state machine, or a hybrid combination of programmable and dedicated hardware. Additionally, reference model 20, estimator 30, vehicle level brake/steer controller 40, and an actuator controller 50 may include any control clocks, interfaces, signal conditioners, filters, AnalogtoDigital (A/D) converters, DigitaltoAnalog (D/A) converters, communication ports, or other types of operators as would occur to those having ordinary skill in the art to implement the principals of the present invention.

[0037]
System 11 is incorporated within a processing environment of vehicle 10. However, for the simplicity in describing the present invention, system 11 is illustrated and described as being separate from the processing environment of vehicle 10. Also, for the simplicity in describing the present invention, system 11 will be described herein as if vehicle 10 includes both a front active braking system and a rear active steering system. However, those having ordinary skill in the art will appreciate an applicability of system 11 to a vehicle including only a front active braking system or a rear active steering system.

[0038]
As known by those having ordinary skill in the art, conventional sensors (not shown) provide a plurality of signals indicative of an operational state of vehicle 10 including, but not limited to, a driver steering wheel angle signal δ_{SWA}, a front steering wheel angle signal δ_{f}, a rear steering wheel angle signal δ_{r}, a vehicle yaw rate signal Ω, a lateral acceleration signal a_{y}, a wheel speed signal W_{S1 }(from tire 12), a wheel speed signal W_{S2 }(from tire 13), a wheel speed signal W_{S3 }(from tire 15), a wheel speed signs WS_{S4 }(from tire 16), and an estimated vehicle speed signal V_{x}.

[0039]
Reference model 20 inputs driver steering wheel angle signal δ_{SWA}, lateral acceleration signal a_{y}, and estimated vehicle speed signal v_{x }from vehicle 10. Alternative to lateral acceleration signal a_{y}, reference model 20 can input an estimated surface coefficient of adhesion signal μ_{e }from estimator 30. In response to the inputted signals, reference model 20 provides signals indicative of a feedforward front steering angle correction signal δ_{fdrl}, a feedforward rear steering angle correction signal δ_{rff}, a desired yaw rate signal Ω_{dl}, a desired lateral velocity signal v_{yd}, and a desired slip angle signal β_{d}.

[0040]
Estimator 30 inputs front steering wheel angle signal δ_{f}, rear steering wheel angle signal δ_{r}, vehicle yaw rate signal Ω, lateral acceleration signal a_{y}, and estimated vehicle speed signal v_{x }from vehicle 10. Estimator 30 further inputs desired yaw rate signal Ω_{dl }from reference model 20. In response to the inputted signals, estimator 30 provides an estimated surface coefficient of adhesion signal μ_{e}, an estimated lateral velocity signal V_{ye}, and an estimated slip angle signal β_{e}.

[0041]
Vehicle level brake/steer controller 40 inputs front steering wheel angle signal δ_{f}, rear steering wheel angle signal δ_{r}, vehicle yaw rate signal Ω, lateral acceleration signal a_{y }and estimated vehicle speed signal v_{x }from vehicle 10. Controller 40 further inputs desired yaw rate signal Ω_{d}, desired lateral velocity signal v_{yd}, and desired slip angle signal β_{e }from reference model 20; and estimated surface coefficient of adhesion signal lie, estimated lateral velocity signal v_{ye}, and estimated slip angle signal β_{e }from estimator 30. In response to the inputted signals, controller 40 provides a desired speed differential signal Δv_{lr3t }indicating a desired speed difference between a linear speed of tire 12 and a linear speed of tire 13 (FIGS. 1A1D) or a desired speed difference between a linear speed of tire 15 and a linear speed of tire 16 (FIGS. 1A1D). Controller 40 further provides a desired front steering angle signal δ_{ftd1 }indicative of a desired steering angle of front axle 11 (FIGS. 1A1D), and a desired rear steering angle signal δ_{rtd1 }indicative of a desired steering angle of rear axle 14 (FIGS. 1A1D).

[0042]
Controller 40 only provides desired speed differential signal Δv_{lr3t }and desired front steering angle δ_{ftd1 }for alternative embodiments of vehicle 10 only having a front active steering system.

[0043]
Actuator controller 50 inputs desired speed differential signal Δv_{lr3t}, desired front steering angle signal δ_{ftd1}, and desired rear steering angle signal δ_{rtd1 }from controller 40. Controller 50 further inputs front steering wheel angle signal δ_{f}, rear steering wheel angle signal δ_{r}, wheel speed signal W_{S1}, wheel speed signal W_{S2}, wheel speed signal W_{S3}, and wheel speed signal W_{S4 }from vehicle 10. In response to the inputted signals, actuator controller 50 compares desired tire speed differential signal Δv_{lr3t }to either a speed differential between tire 12 and tire 13 (FIGS. 1A1D) as indicated by wheel speed signs WS_{S1 }and wheel speed signs WS_{S2 }as would occur to those having ordinary skill in the art, or a speed differential between tire 15 and tire 16 (FIGS. 1A1D) as indicated by wheel speed signs WS_{S3 }and wheel speed signs WS_{S4 }as would occur to those having ordinary skill in the art. The result is a corrective braking signal T_{b }that is provided to a braking system (not shown) of vehicle 10. In one embodiment of vehicle 10, a brake actuator of the braking system appropriately adjusts brake pressure to a corresponding brake in response to corrective braking signal T_{b }as would occur to those having ordinary skill in the art.

[0044]
Actuator controller 50 compares desired front steering angle signal δ_{ftd1 }and front steering wheel angle signal δ_{f }as would occur to those with ordinary skill in the art, and compares desired rear steering angle signal δ_{rtd1 }and rear steering wheel angle signal δ_{r }as would occur to those with ordinary skill in the art to thereby provide a corrective front steering signal T_{fs }and a corrective rear steering signal T_{rs }to a steering system (not shown) of vehicle 10. In one embodiment of vehicle 10, a front steering actuator of the steering system adjusts a position of a steering rack for axle 11 (FIGS. 1A1D) in response to corrective front steering signal T_{fs }and a rear steering actuator of the steering system adjusts a position of a steering rack for axle 14 (FIGS. 1A1D) in response to corrective rear steering signal T_{rs}.

[0045]
Referring to FIG. 3, one embodiment of reference model 20 in accordance with the present invention is shown. A block 21 converts steering wheel angle signal δ_{SWA }into a corresponding angle of front tires signal δ_{fdr }as computed by the following equation (14):

δ_{fdr}=δ_{SWA} * K _{f}(v _{x}) (14)

[0046]
where K_{f}(v_{x}) is a ratio between the angle of rotation of a steering wheel of vehicle 10 (FIGS. 1A1D) and front wheels 12 and 13 (FIGS. 1A1D). In the case of active front steer, front ratio K_{f}(v_{x}) may be speed dependent, for example decreasing with speed to promote maneuverability at low speeds and stability at high speeds.

[0047]
A block 22 determines a feedforward part of a steering angle correction by limiting a magnitude of front tire steering angle δ_{fdr }to a reasonable level. A desired value of lateral acceleration is computed from the following equation (15):

a _{yd}=(v _{x} ^{2}*δ_{fdr})/(L+K _{u} * v _{x} ^{2}) (15)

[0048]
where L is a vehicle tirebase and K_{u }is an understeer coefficient. It follows from equation (15) that in order to limit a magnitude of this acceleration to a reasonable level a_{ydmax }(an example value of a_{ydmax }is 12 m/s^{2}), a magnitude of steering angle δ_{fdr }has to be limited in accordance with the following equation (16):

[δ_{fmax}]=[a_{ydmax}]*(L+K _{u} *v _{x} ^{2})/v _{x} ^{2} (16)

[0049]
This limiting can be interpreted as adding a feedforward term to the steering angle δ
_{fff}, as given by the following equation (17):
$\begin{array}{cc}{\delta}_{\mathrm{fff}}=\{\begin{array}{cc}\text{\hspace{1em}}\ue89e0& \mathrm{if}\ue89e\text{\hspace{1em}}\ue89e\uf603{\delta}_{\mathrm{fdr}}\uf604\le {\delta}_{f\ue89e\text{\hspace{1em}}\ue89e\mathrm{max}}\ue8a0\left({v}_{x}\right)\\ \text{\hspace{1em}}& \text{\hspace{1em}}\\ [{\delta}_{f\ue89e\text{\hspace{1em}}\ue89e\mathrm{max}}\ue8a0\left({v}_{x}\right)\uf603{\delta}_{\mathrm{fdr}}\uf604*\mathrm{sign}\ue8a0\left({\delta}_{\mathrm{fdr}}\right)& \mathrm{if}\ue89e\text{\hspace{1em}}\ue89e\uf603{\delta}_{\mathrm{fdr}}\uf604>{\delta}_{f\ue89e\text{\hspace{1em}}\ue89e\mathrm{max}}\ue8a0\left({v}_{x}\right)\end{array}& \left(17\right)\end{array}$

[0050]
After the limitation, front steering angle δ_{fdrl }desired by the driver is computed from the following equation (18):

δ_{fdrl}=δ_{fdr}+δ_{fff} (18)

[0051]
When vehicle 10 is equipped with a traditional steering mechanism, the ratio K_{f }does not depend on speed of vehicle 10 and the limitation of the steering angle cannot be performed, (i.e. δ_{fdrl}=δ_{fdr}).

[0052]
A block 23 determines a feedforward part of the rear tire steering angle δ_{rff }as computed from the following equation (19):

δ_{rff}=δ_{fdr} * K _{rff}(v _{x}) (19)

[0053]
where K_{rff}(V_{x}) is a speed dependant gain that must be selected to achieve an improved maneuverability (to reduce radius of curvature and/or driver steering effort) at low speeds, an improved stability at high speeds and a reduction of vehicle side slip velocity (or side slip angle). One possible choice is requiring that the side slip velocity be equal to zero in a steady state maneuver. Side slip velocity v_{yss }is computed by the following equation (20):

v _{yss}=[(v _{x}*δ_{fdrl})/(L+K _{u} * v _{x} ^{2})]*{b−M*a*v _{x} ^{2}/(C _{r} *L)+K _{rff}(v _{x})*[a+M*b*v _{x} ^{2}/(C _{f} *L)]} (20)

[0054]
where M is mass of vehicle 10, a and b are a distances of vertical axis 17 to front axle 11 and rear axle 14 (FIGS. 1A1D), respectively, and C_{f }and C_{r }are the cornering stiffness coefficients of front tires 12 and 13, and rear tires 15 and 16 (FIGS. 1A1D), respectively. In order to make side slip velocity v_{yss }equal zero, a feedforward gain K_{rff}′(v_{x}) is computed by the following equation (21):

K _{rff}′(v _{x})=−[b−M*a*v _{x} ^{2}/(C _{r} *L)]/[a+M*b*v _{x} ^{2}/(C _{r} *L)] (21)

[0055]
Feedforward gain K_{rff}′(v_{x}) is illustrated in FIG. 4 as curve 1. Gain K_{rff}′(v_{x}) is negative for small speeds and positive for large speeds and it changes sign at a velocity v_{xc }given by the following equation (22):

v _{xc} =[C _{r} *L*b/(M*a)]^{½} (22)

[0056]
Thus, the sign of the rear tire steering angle δ_{rff }is opposite to that of the front steering angle δ_{fdrl }(out of phase steering) at low speeds, which improves maneuverability. At high speeds, rear tires 15 and 16 are steered in phase with the front tires 12 and 13, which improves stability of vehicle 10. In practice, feedforward gain K_{rff}′(v_{x}) given by equation (21) would require too large rear tire steering angle δ_{rff}, which is typically limited to several degrees. Also, yaw rate Ω of vehicle 10 during cornering maneuvers would be very limited at high velocities, thus compromising subjective handling feel. To rectify these problems, feedforward gain K_{rff′(v} _{x}) can be multiplied by a factor η, which is less than 1 in accordance with the following equation (23):

K _{rff}″(v _{x})=−η*[b−M*a*v _{x} ^{2}/(C _{r} *L)]/[a+M*b*v _{x} ^{2}/(C _{f} *L)] (23)

[0057]
with a reasonable value of η=0.4 (the optimal value for a given application depends on the range of steering angle for rear tires 15 and 16). Gain K_{rff}″(v_{x}) given by equation (23) is represented by curve 2 in FIG. 4.

[0058]
According to equation (22), a velocity v_{xc }at which gain K_{rff }changes sign depends on cornering stiffness C_{r }of rear tires 15 and 16. On slippery surfaces, the value of the cornering stiffness C_{r}, and the characteristic velocity v_{xc }(at which gain K_{rff }crosses zero) will be reduced. If the gain determined by equation (23) with the nominal values of cornering stiffness coefficient C_{r }that correspond to a dry surface are used, vehicle 10 will exhibit a tendency to oversteer during driving on slippery surfaces at the velocities just below v_{xc}. This is due to out of phase steering increasing a rate of rotation of vehicle 10. To rectify this problem and make the behavior of vehicle 10 acceptable over the entire range of surfaces, the feedforward gain K_{rff }is chosen to be 0 for speed between approximately 0.4*v_{xc }and v_{xc}, as illustrated by curve 3 in FIG. 4.

[0059]
A block 24 determines a steady state desired values of yaw rate Ω_{dss }and side slip velocity v_{ydss}. These values can be computed from look up tables, which are obtained from vehicle testing performed on dry surface. During tests, the feedforward portion of the rear tire steering angle δ_{rff }must be active and vehicle 10 must be in approximately steady state cornering conditions. Thus, the desired values at a given speed and front steering angle δ_{f }represent the values which vehicle 10 achieves on dry surface in steady state cornering with the feedforward portion of the rear tire steer being active. Another way of obtaining the desired values is by using analytical models. For example, the steady state values of yaw rate Ω_{dss }and side slip velocity v_{ydss }can be computed from the following equations (24) and (25):

Ω_{dss}=[1−K _{rff}(v _{x})]*v _{x}*δ_{fdrl} /[L+K _{u} *v _{x} ^{2}] (24)

v _{ydss}=[(v _{x}*δ_{fdrl})/(L+K _{u} * v _{x} ^{2})]*{b−M*a*v _{x} ^{2}/(C _{r} *L)+K _{rff}(v _{x})*[a+M*b*v _{x} ^{2}/(C _{f} *L)]} (25)

[0060]
In the equations (24) and (25), an understeer coefficient K_{u }depends on the magnitude of lateral acceleration a_{y}. When vehicle 10 is without active rear tire steer, feedforward gain K_{rff}=0. Since yaw rate Ω and side slip velocity v_{y }are overestimated at large steering angles by equations (24) and (25), the desired values obtained from equations (24) and (25) must be limited. A reasonable maximum value for a magnitude of yaw rate Ω can be computed from the following equation (26):

Ω_{dmax} =g/v _{x} (26)

[0061]
where g is acceleration of gravity. The limited value of a desired yaw rate Ω
_{dssl }can be computed from the following equation (27):
$\begin{array}{cc}{\Omega}_{\mathrm{dssl}}=\{\begin{array}{cc}\text{\hspace{1em}}\ue89e{\Omega}_{\mathrm{dss}}& \mathrm{if}\ue89e\text{\hspace{1em}}\ue89e\uf603{\Omega}_{\mathrm{dss}}\uf604\le g/{v}_{x}\\ \text{\hspace{1em}}& \text{\hspace{1em}}\\ \left(g/{v}_{x}\right)*\mathrm{sign}\ue89e\text{\hspace{1em}}\ue89e\left({\Omega}_{\mathrm{dss}}\right)& \mathrm{if}\ue89e\text{\hspace{1em}}\ue89e\uf603{\Omega}_{\mathrm{dss}}\uf604>g/{v}_{x}\end{array}& \left(27\right)\end{array}$

[0062]
The limited value of lateral velocity v_{ydssl }can be computed from the following equation (28):

v _{ydssl}=[Ω_{dss}/(1−K _{rff})]*{b−M*a*v _{x} ^{2}/(C _{r} *L)+K _{rff} *[a+M*b*v _{x} ^{2}/(C_{f} *L)]} (28)

[0063]
A block 25 receives steady state yaw rate Ω_{dssl }and lateral velocity V_{ydss}. Block 25 represents a desired dynamics of vehicle 10 and the delay in the generation of tire lateral forces. In the linear range of handling, the transfer functions between front steering angle δ_{fdrl }and desired yaw rate Ω_{d }and between front steering angle δ_{fdrl }and desired lateral velocity v_{yd }can be computed by the following equations (29) and (30):

G _{Ω}(s)=Ω_{d}(s)/δ_{fdrl}(s)=(C _{f} /M)*[s−z _{Ω}(v _{x})]/[s^{2}+2*ζ(v _{x})*ω_{n}(v _{x})*s+ω_{n} ^{2}(v_{x})] (29)

G _{vy}(s)=v _{yd}(s)/δ_{fdrl}(s)=(a*C _{f} /I _{zz})*[s−z _{vy}(v _{x})]/[s^{2}+2*ζ(v _{x} )*ω _{n}(v _{x})*s+ω _{n} ^{2}(v _{x})] (30)

[0064]
In equations (29) and (30), s is the Laplace operand, I_{zz }is the moment of inertia of vehicle 10 about axis 17, z_{Ω}(v_{x}) and z_{vy}(v_{x}) are zeros of the corresponding transfer functions, ζ(v_{x}) is the damping coefficient, and ω_{n}(v_{x}) is the undamped natural frequency.

[0065]
When vehicle 10 has active rear tire steer, the zeros of the transfer functions depend on feedforward gain K_{rff}. Each one of the above transfer functions can be represented as a product of a steadystate value (corresponding to s=0) and a term representing the dynamics can be computed by the following equations (31) and (32):

G _{Ω}(s)=(Ω_{dss}/δ_{fss})*G _{Ω}′(s) (31)

G _{vy}(s)=(v _{yss}/δ_{fss})*G _{vy}′(s) (32)

[0066]
Where

G _{ω}′(s)=[−ω_{n} ^{2}(v _{x})/z_{ω}(v _{x}) ]*[s−z _{ω}(v _{x})]/[s^{2}+2*ζ(v _{x})*ω_{n}(v _{x})*s+ω _{n} ^{2}(v _{x})] (33)

G _{vy}′(s)=[−ω_{n} ^{2}(v _{x})/z _{vy}(v _{x})]*[s−z _{vy}(v _{x})]/[s ^{2}+2*ζ(v _{x})*ω_{n}(v _{x})*s+ω _{n} ^{2}(v _{x})] (34)

[0067]
Thus, the dynamic values of the desired yaw rate Ω_{d }and lateral velocity v_{yd }can by obtained by passing the steady state values through the differential (or difference) equations (with parameters dependent on speed) representing the dynamics of the transfer functions G_{Ω}′(s) and G_{vy}′(s).

[0068]
In a block 26, the values of desired yaw rate Ω_{d }and side slip velocity v_{yd }are subsequently passed through first order filters representing a delay in generating tire forces due to tire relaxation length. Block 26 can be represented as a transfer function in accordance with the following equation (35):

G _{f}(s)=a _{f}(v _{x})/[s+a _{f}(v _{x})] (35)

[0069]
in which a filter parameter a_{f}(v_{x}) is speed dependent. In the case of vehicle 10 having active rear tire steer, one of the control objectives is to achieve quick response of vehicle 10 to steering inputs. Thus, in this case, the dynamics of vehicle 10 as represented by the transfer functions (31) and (32) can be ignored, since vehicle 10 can respond faster to steering inputs with active rear steer than a conventional vehicle.

[0070]
The desired values of yaw rate Ω
_{d }and lateral velocity v
_{yd }obtained as outputs of block
26 may be subsequently limited in magnitude by a block
27 depending on the surface conditions. A block
27 can utilize either an explicit estimate of surface coefficient of adhesion in lateral direction μ
_{L }or a magnitude of lateral acceleration a
_{y}. In the first case, a limited value of desired yaw rate Ω
_{dl }is computed from the 20 following equation (36):
$\begin{array}{cc}{\Omega}_{\mathrm{dl}}=\{\begin{array}{cc}\text{\hspace{1em}}\ue89e{\Omega}_{d}& \mathrm{if}\ue89e\text{\hspace{1em}}\ue89e\uf603{\Omega}_{d}\uf604\le {\mu}_{L}*g/{v}_{x}\\ \text{\hspace{1em}}& \text{\hspace{1em}}\\ \left({\mu}_{L}*g/{v}_{x}\right)*\mathrm{sign}\ue89e\text{\hspace{1em}}\ue89e\left({\Omega}_{d}\right)& \mathrm{if}\ue89e\text{\hspace{1em}}\ue89e\uf603{\Omega}_{d}\uf604>{\mu}_{L}*g/{v}_{x}\end{array}& \left(36\right)\end{array}$

[0071]
If the magnitude of lateral acceleration a
_{y }is used by block
27, the limited desired yaw rate Ω
_{dl }is computed from the following equation (37):
$\begin{array}{cc}{\Omega}_{\mathrm{dl}}=\{\begin{array}{cc}\text{\hspace{1em}}\ue89e{\Omega}_{d}& \mathrm{if}\ue89e\text{\hspace{1em}}\ue89e\uf603{\Omega}_{d}\uf604\le \left(\uf603{a}_{y}\uf604+\Delta \ue89e\text{\hspace{1em}}\ue89e{a}_{y}\right)/{v}_{x}\\ \text{\hspace{1em}}& \text{\hspace{1em}}\\ \left[\left(\uf603{a}_{y}\uf604+\Delta \ue89e\text{\hspace{1em}}\ue89e{a}_{y}\right)/{v}_{x}\right]*\mathrm{sign}\ue89e\text{\hspace{1em}}\ue89e\left({\Omega}_{d}\right)& \mathrm{if}\ue89e\text{\hspace{1em}}\ue89e\uf603{\Omega}_{d}\uf604>\left(\uf603{a}_{y}\uf604+\Delta \ue89e\text{\hspace{1em}}\ue89e{a}_{y}\right)/{v}_{x}\end{array}& \left(37\right)\end{array}$

[0072]
where Δa_{y }is a constant positive value, for example 2 m/s^{2}. The magnitude of desired lateral velocity V_{yd }is limited by the value obtained from equation (26) with the desired yaw rate at steady state Ω_{dss }replaced by the limited desired yaw rate Ω_{dl}.

[0073]
Block 27 also outputs a desired side slip angle pd that can be computed as an arctangent function of the ratio of desired lateral velocity to longitudinal velocity in accordance with the following equation (38):

β_{d}=Arctan(v _{yd} /v _{x}) (38)

[0074]
Referring to FIG. 5, an embodiment of estimator 30 (FIG. 2) for estimating surface coefficient of adhesion μ_{e }is shown. A block 31 performs preliminary calculations. First, it is recognized that the most robust signal available is yaw rate Ω, and an entry and an exit conditions are dependent mainly on a yaw rate error, i.e. a difference between the desired yaw rate Ω_{dl }and measured yaw rate Ω, and to a lesser extent on measured lateral acceleration a_{y }(entry condition only). Thus, a yaw rate error is calculated and filtered, and lateral acceleration a_{y }is filtered.

[0075]
Second, when vehicle 10 (FIGS. 1A1D) reaches the limit of adhesion in a steady turn, a surface coefficient of adhesion can be determined as a ratio of the magnitude of a filtered lateral acceleration a_{yfilt }to a maximum lateral acceleration a_{ymax }that vehicle 10 can sustain on dry pavement as shown in the following equation (39):

μ_{L}_temp=a _{yfilt‘/a} _{ymax} (39)

[0076]
where μ_{L}_temp is a temporary estimate of surface coefficient of adhesion in the lateral direction, and a_{yfilt }is filtered lateral acceleration, which is also corrected for the effects of measured gravity components resulting from vehicle body roll and bank angle of the road.

[0077]
A block 32 is designed to recognize situations when vehicle 10 operates at or close to the limit of adhesion and estimates a lateral surface coefficient of adhesion μ_{L }from measured lateral acceleration a_{y}. This estimate is calculated by identifying one of the following three conditions.

[0078]
First, entry conditions are tested during a stage S1. Entry conditions are when vehicle 10 is handling at the limit of adhesion and is not in a quick transient maneuver. Under entry conditions, stage S2 sets coefficient of adhesion μ_{L }equal to temporary estimate of surface coefficient of adhesion μ_{L}_temp as calculated by equation (37).

[0079]
Second, exit conditions are tested during a stage S3. Exit conditions indicate vehicle 10 is well below the limit of adhesion (within the linear range of handling behavior). Under exit conditions, a stage S4 resets coefficient of adhesion μ_{L }to a default value of 1.

[0080]
Third, when neither the entry conditions nor the exit conditions are met, a stage S5 holds coefficient of adhesion μ_{L }unchanged from a previous value (i.e. holding conditions). The only exception is when the magnitude of measured lateral acceleration a_{y }exceeds the maximum value predicted using currently held estimate. In this case, stage S5 calculates coefficient of adhesion μ_{L }as if vehicle 10 was in an entry condition.

[0081]
The entry conditions are met during stage S1 when the following three (3) conditions are simultaneously satisfied. The first condition is either (1) the magnitude of the yaw rate error, that is the difference between the desired yaw rate Ω_{d }and the measured yaw rate Ω being greater than a threshold as computed in the following equation (40):

Ω_{d}−Ω_{filt} >Yaw _{—} Threshold1 (40)

[0082]
where the typical value of Yaw_Thershold1 is 0.123 rad/s=7 deg/s); or (2) the magnitude of yaw rate error being greater than a lower threshold Yaw_Threshold2 for some time Te as computed in the following equation (41):

Ω_{d}−Ω_{filt} >Yaw _{—} Threshold2 for Te seconds (41)

[0083]
where Yaw_Threshold2 depends on the magnitude of desired yaw rate Ω_{d }or measured yaw rate Ω. For example, Yaw_Threshold2=4 deg/s +5*Ω_{d}=0.07 rad/s+0.09*1Ω_{d}, where Ω_{d }is the desired yaw rate in [rad/s]. A typical value of the time period Te for which this condition must be satisfied is 0.3 sec. The threshold Yaw_Threshold1 used in equation (40) may also depend on the magnitude of desired yaw rate Ω_{d }or measured yaw rate Ω.

[0084]
The second condition is the signs of the filtered lateral acceleration a_{yfiltl }and the weighted sum of yaw rate Ω and the derivative of yaw rate are the same in accordance with the following mathematical expression (42):

a _{yfilt1}*(Ω+Yaw _{—} Der _{—} Mult*dΩ/dt)>Sign _{—} Comp (42)

[0085]
where Ω is the measured yaw rate and dΩ/dt is its derivative. The magnitude of the filtered lateral acceleration a
_{yfilt }is limited from equation (43):
$\begin{array}{cc}{a}_{\mathrm{yfiltl}}=\{\begin{array}{cc}\text{\hspace{1em}}\ue89e{a}_{\mathrm{yfiltl}}& \mathrm{if}\ue89e\text{\hspace{1em}}\ue89e\uf603{a}_{\mathrm{yfilt}}\uf604\ge {a}_{y\ue89e\text{\hspace{1em}}\ue89e\mathrm{min}}\\ \text{\hspace{1em}}& \text{\hspace{1em}}\\ {a}_{y\ue89e\text{\hspace{1em}}\ue89e\mathrm{min}}*\mathrm{sign}\ue89e\text{\hspace{1em}}\ue89e\left({\Omega}_{d}\right)& \mathrm{if}\ue89e\text{\hspace{1em}}\ue89e\uf603{a}_{\mathrm{yfilt}}\uf604<{a}_{y\ue89e\text{\hspace{1em}}\ue89e\mathrm{min}}\end{array}& \left(43\right)\end{array}$

[0086]
where a_{ymin }is a constant with a typical value of 0.2 M/s^{2}. Thus if a_{yfilt }is very small in magnitude, it is replaced by the a_{ymin }with a sign the same as the desired yaw rate Ω_{d}. This limit is needed to improve estimation on very slick surfaces (e.g. ice) when the magnitude of lateral acceleration a_{y }is comparable to the effect of noise, so that the sign of a_{yfilt }cannot be established.

[0087]
The recommended values in equation (42) for the constant Yaw_Der_Mult is 0.5 and for Sign_Comp is 0.035 (if Ω is in rad/s and dΩ/dt in rad/s^{2}).

[0088]
In order to allow lateral acceleration a
_{y }to fully build up at the beginning of maneuver and after each change in sign, before it can be used for estimation of surface coefficient μ
_{L}, a condition is used that requires both the desired yaw rate Ω
_{dl }and lateral acceleration a
_{y }to have the same signs for a specific time period (necessary for the acceleration to build up). In order to keep track of how long the desired yaw rate Ω
_{d }and lateral acceleration a
_{y }have had the same signs, a timer is introduced. In accordance with an equation (44), the timer becomes zero when the desired yaw rate Ω
_{d }and lateral acceleration a
_{y }have opposite signs and counts the time that elapses from the moment the signs become and remain the same.
$\begin{array}{cc}\mathrm{timer}=\{\begin{array}{c}0\ue89e\text{\hspace{1em}}\ue89e\mathrm{when}\ue89e\text{\hspace{1em}}\ue89e{\Omega}_{d}*{a}_{\mathrm{yfiltl}}<\mathrm{Ay\_sign}\ue89e\mathrm{\_comp}\\ \text{\hspace{1em}}\\ \text{\hspace{1em}}\ue89e\mathrm{timer}+\mathrm{loop\_time}\ue89e\text{\hspace{1em}}\ue89e\mathrm{otherwise}\end{array}& \left(44\right)\end{array}$

[0089]
where Ω_{d }is the desired yaw rate in [rad/s] and Ay_sign_comp is a constant with a typical value of 0.2 m/s^{3}.

[0090]
The third condition is either (1) the signs of the desired yaw rate Ω_{d }and measured lateral acceleration a_{y }are the same and they have been the same for some time in accordance with following equation (45):

timer>hold_time (45)

[0091]
The hold_time in equation (42) can be 0.25 s, or (2) the magnitude of a derivative of lateral acceleration da_{y}/dt is less than a threshold in accordance with the following mathematical equation (46):

da _{y} /dt<Ay _{—} Der _{—} Thresh (46)

[0092]
A recommended value of the threshold, Ay_Der_Thresh=2.5 m/s^{3}. The derivative da_{y}/dt is obtained by passing filtered lateral acceleration a_{yfil }through a high pass filter with a transfer function a_{f}*s/(s+a_{f}) with a typical value of a_{f}=6 rad/s.

[0093]
The exit conditions are met during stage S3 when the following two (2) conditions are simultaneously satisfied. The first condition is the magnitude of yaw rate error filtered is less than or equal to a threshold as illustrated in the following equation (47):

Ω_{d}−Ω_{filt} ≦Yaw _{—} Threshold3 (47)

[0094]
with a typical value of Yaw_Threshold3=0.10 rad/s.

[0095]
The second condition is a lowpass filtered version of the magnitude of the yaw rate error is less than or equal to a threshold as illustrated in the following equation (48):

(Ω_{d}−Ω_{filt})_{filt} <Yaw _{—} Treshold4 (48)

[0096]
where the value of Yaw_Threshold4=0.06 rad/s is recommended and the filter is a first order filter with a cutoff frequency of 1.8 rad/s, e.g. a filter with a transfer function a_{f}/(s+a_{f}) with a_{f}=1.8 rad/s). The thresholds Yaw_Threshold3 and Yaw_Thereshold4 may depend on the magnitude of desired yaw rate Ω_{d }or the measured yaw rate Ω.

[0097]
A block 33 corrects surface estimate μ_{L }for load transfer. Because of the effects of load transfer to the outside tires during cornering, which is smaller on slippery surfaces than on dry roads, lateral acceleration a_{y }is not directly proportional to the surface coefficient of adhesion μ_{L }To account for this effect, the surface estimate μ_{L}_temp computed from equation (37), is corrected using the following equation (49):

μ_{L}=μ_{L}_temp*(c _{1} +C _{2}*μ_{L}_temp) (49)

[0098]
where c_{1}<1 and c_{2}=1−c_{1}, so that on dry surface μ_{L}=μ_{L}_temp=1, while on slippery surfaces μ_{L}<μ_{L}_temp. Example values are c_{1}=0.85 and C_{2}=0.15.

[0099]
A block 34 limits surface estimate μ_{L }from below by a value μ_{Lmin }(a typical value 0.07) and may be limited from above by μ_{Lmax }(a typical value 1.2). Surface estimate μ_{l }can be passed through a slew filter, which limits the rate of change of the estimate to a specified value, for example 2/sec, or a low pass filter.

[0100]
A block
35 estimates total surface coefficient of adhesion μ
_{e }using the following equation (50):
$\begin{array}{cc}{\mu}_{e}=\{\begin{array}{cc}{\mu}_{\mathrm{Lfilt}}& \mathrm{when}\ue89e\text{\hspace{1em}}\ue89e\uf603{a}_{\mathrm{xe}}\uf604\le \mathrm{Ax\_Dz}\\ \text{\hspace{1em}}& \text{\hspace{1em}}\\ {\left\{{\left({\mu}_{\mathrm{Lfilt}}\right)}^{2}+{\left[\left(\uf603{a}_{\mathrm{xe}}\uf604\mathrm{Ax\_DZ}\right)/{a}_{x\ue89e\text{\hspace{1em}}\ue89e\mathrm{max}}\right]}^{2}\right\}}^{1/2}& \mathrm{when}\ue89e\text{\hspace{1em}}\ue89e\uf603{a}_{\mathrm{xe}}\uf604>\mathrm{Ax\_Dz}\end{array}& \left(50\right)\end{array}$

[0101]
where Ax_Dz is the deadzone applied to the estimated longitudinal acceleration (a typical value is 2m/s2) and a_{xmax }is a maximum longitudinal deceleration which vehicle 10 can achieve on dry surface (a typical value is 9 m/s^{2}). The square root function in the above expression can be replaced by a simplified linear equation or by a lookup table. The estimate is finally limited from below by μ_{emin }(typical value is 0.2) and from above by μ_{emax }(1.0).

[0102]
The (unfiltered) estimate of surface coefficient in lateral direction, μ_{L}, was found to be good for estimation of vehicle side slip angle. However, for control purposes, the estimate of the surface coefficient in lateral direction may be too low in some situations (for example during heavy braking on slick surfaces) and may cause unnecessary tight control of slip angle. Therefore, for the purpose of control the estimated surface coefficient is increased when the magnitude of the estimated vehicle longitudinal acceleration exceeds certain value. Note that separate thresholds on yaw rate error for the entry and exit conditions are used, with the thresholds on the exit conditions being a little tighter.

[0103]
Referring to FIG. 6, an embodiment of estimator 30 (FIG. 2) for estimating the actual lateral velocity and slip angle of vehicle 10 (FIGS. 1A1D) as a function of front steering wheel angle signal δ_{f}, rear steering wheel angle signal δ_{r}, yaw rate signal Ω, estimated vehicle speed signal v_{x}, and the estimated lateral surface coefficient of adhesion μ_{L }is shown. The slip angle estimation implements an iterative nonlinear closed loop observer to determine the estimated vehicle lateral velocity v_{ye }and slip angle β_{e}.

[0104]
A block 36 of the observer estimates the side slip angles of front axle 11 and rear axle 14 using the following equations (51a) and (51b):

α_{fe} =[v _{ye}(k−1)+a*Ω]/v _{x}−δ_{f} (51a)

α_{re} =[v _{ye}(k−1)−b*Ω]/v _{x}−δ_{r} (51b)

[0105]
where v_{ye}(k−1) is the estimated lateral velocity from the previous iteration of the observer, and α_{fe }and α_{re }are the estimated front and rear axle side slip angles, respectively. The steering angles δ_{f }and δ_{r }are the actual (measured) steering angles of front tires 12 and 13, and rear tires 15 and 16, respectively, including the corrective terms.

[0106]
A block
37 of the observer estimates lateral forces F
_{yfe }of the front axle
11 according to one of two functions as illustrated in the following equation (52):
$\begin{array}{cc}{F}_{\mathrm{yfe}}=\{\begin{array}{cc}\text{\hspace{1em}}\ue89e{C}_{f}*{\alpha}_{\mathrm{fe}}*\left(1\left({b}_{\mathrm{cf}}*\left(\uf603{\alpha}_{\mathrm{fe}}\uf604/{\mu}_{L}\right)\right)\right),& \mathrm{if}\ue89e\text{\hspace{1em}}\ue89e\uf603{\alpha}_{\mathrm{fe}}\uf604<{\mu}_{L}*{\alpha}_{{f}^{*}}\\ \text{\hspace{1em}}& \text{\hspace{1em}}\\ {N}_{{f}^{*}}*\left(\uf603{\alpha}_{\mathrm{fe}}\uf604/{\alpha}_{\mathrm{fe}}\right)*\left[{\mu}_{L}+{s}_{f}*\left(\uf603{\alpha}_{\mathrm{fe}}\uf604/{\alpha}_{{f}^{*}}{\mu}_{L}\right)\right]& \mathrm{if}\ue89e\text{\hspace{1em}}\ue89e\uf603{\alpha}_{\mathrm{fe}}\uf604\ge {\mu}_{L}*{\alpha}_{{f}^{*}}\end{array}& \left(52\right)\end{array}$

[0107]
where s_{f }is a small nonnegative number (the slope of the F_{yf}−α_{f }curve at the limit of adhesion), e.g., s_{f}=0.05, and where α_{f* }is defined by the following equation (53):

α_{f*}=1/(2*b _{cf},) (53)

[0108]
where b_{cf }is defined by the following equation (54):

b _{cf} =C _{f}/(4*N _{f}), (54)

[0109]
where N_{f }is defined by the following equation (55):

N _{f*} =M*b*(a _{ymax}+Δ_{a})/(a+b) (55)

[0110]
where a_{ymax }is the maximum lateral acceleration that vehicle 10 can sustain on a dry surface in m/s^{2 }and Δ_{a }is a positive constant, e.g., Δ_{a}=0.5 m/s^{2}. M is the nominal value of the total vehicle mass.

[0111]
The observer similarly estimates lateral forces F
_{yre }of the rear axle
14 according to the following equation (56):
$\begin{array}{cc}{F}_{\mathrm{yre}}=\{\begin{array}{cc}\text{\hspace{1em}}\ue89e{C}_{r}*{\alpha}_{\mathrm{re}}*\left(1{b}_{\mathrm{cr}}*\uf603{\alpha}_{\mathrm{re}}\uf604\right),& \mathrm{if}\ue89e\text{\hspace{1em}}\ue89e\uf603{\alpha}_{\mathrm{re}}\uf604<{\mu}_{L}*{\alpha}_{{r}^{*}}\\ \text{\hspace{1em}}& \text{\hspace{1em}}\\ {N}_{{r}^{*}}*\left(\uf603{\alpha}_{\mathrm{re}}\uf604/{\alpha}_{\mathrm{re}}\right)*\left[{\mu}_{L}+{s}_{r}*\left(\uf603{\alpha}_{\mathrm{re}}\uf604/{\alpha}_{{r}^{*}}{\mu}_{L}\right)\right]& \mathrm{if}\ue89e\text{\hspace{1em}}\ue89e\uf603{\alpha}_{\mathrm{re}}\uf604\ge {\mu}_{L}*{\alpha}_{{r}^{*}}\end{array}& \left(56\right)\end{array}$

[0112]
where s_{r }is a small nonnegative number, e.g., s_{r}=0.05 and where α_{r* }is defined by the following equation (57):

α_{r*}=1/(2*b _{cr},) (57)

[0113]
where b_{cr }is defined by the following equation (58):

b _{cr} =C _{r}/(4*N _{r*}), (58)

[0114]
where N_{r* }is defined by the following equation (59):

N _{r*} =M*a*(a _{ymax}+Δ_{a})/(a+b). (59)

[0115]
A block 38 of the observer then estimates a state variable q(k) related to lateral velocity according to the following equation (60):

q(k)=q(k−1)+Δt*{−(1+g _{2})*v _{x}*Ω+((1+g _{3})/M−a*g _{1} /I _{zz}) *F_{yfe}+[(1+g _{3})/M+b*g _{1} /I _{zz} ]*F _{yre}+(g _{2} −g _{3})*a _{y} −g _{4} *ΔA _{yf}} (60)

[0116]
where ΔA_{y }is defined by the following equation (61):

ΔA _{y} =a _{y}−(F _{yfe} +F _{yre})/M, (61)

[0117]
and ΔA_{yf }is ΔA_{y }passed through a first order digital low pass filter, for example, with a cut off frequency of 1 rad/s.

[0118]
A block 39 of the observer uses state variable q(k) to determine estimates of lateral velocity v_{ye }and slip angle β_{e }using equations (62) and (63):

v _{ye}(k)=[q(k)+g _{1}*•_{a}]/(1+g _{2}) (62)

β_{e}=Arctan[v _{ye}(k)/v _{x}]. (63)

[0119]
The gains g_{1}, g_{2}, g_{3 }and g_{4 }are tuning parameters preset by a system designer, typically through experimentation on a test vehicle, and may vary from implementation to implementation. The estimated lateral velocity v_{ye }and the estimated slip angle β_{e }are the main outputs of the observer.

[0120]
Referring to FIG. 7, one embodiment of controller 40 in accordance with the present invention is shown. In controller 40, an overall corrective yaw moment is determined and expressed in terms of a desired speed differential signal Δv_{lr3t }(which is achieved by differential braking) between either front tire 12 and front tire 13 (FIGS. 1A1D), or rear tire 15 and rear tire 16 (FIGS. 1A1D). The corrective yaw moment is also expressed in terms of a summation of front steer angle correction signal Δδ_{f }and front steering angle signal δ_{fdr1 }(FIG. 2) to form the total front steering angle signal δ_{ftd }and in terms of a summation of rear steer angle correction signal Δδ_{r }and rear steering angle signal δ_{rff }(FIG. 2) to form the total desired rear steer angle signal δ_{rtd}. The magnitudes of total desired rear steer angle signal δ_{rtd }and the total desired front steering angle signal δ_{ftd }may be subsequently limited to desired rear steering angle signal δ_{rtd1 }and desired front steering angle signal δ_{ftd1}, respectively.

[0121]
A block
41 calculates desired speed differential signal Δv
_{lr3}, front steer angle correction signal Δδ
_{f }and rear steer angle correction signal Δδ
_{r}. The corrective yaw moment is obtained by a feedback control operating on the yaw rate error and the side slip velocity (or side slip angle) error. The yaw rate error Ω
_{d}−Ω is the difference between the desired yaw rate signal Ω
_{d }and measured yaw rate signal Ω. Similarly, the side slip velocity error is the difference between the desired side slip velocity signal v
_{yd }and the estimated side slip velocity signal v
_{ye}. The control law is essentially a PD (proportional and derivative) feedback control law, in which the control gains depend on vehicle speed signal v
_{x}, estimated surface coefficient of adhesion signal μ
_{e}, and on the magnitude of the estimated vehicle slip angle error. Thus, for the delta velocity signal Δv
_{lr3}, the control law equation (64) may be written as follows:
$\begin{array}{cc}\Delta \ue89e\text{\hspace{1em}}\ue89e{v}_{\mathrm{lr3}}={k}_{\Omega \ue89e\text{\hspace{1em}}\ue89ep}\ue8a0\left({v}_{x},{\mu}_{e}\right)*\left({\Omega}_{d}\Omega \right)+{k}_{\Omega \ue89e\text{\hspace{1em}}\ue89ed}\ue8a0\left({v}_{x},{\mu}_{e}\right)*d\ue8a0\left({\Omega}_{d}\Omega \right)/\mathrm{dt}+{k}_{\mathrm{vyp}}\ue8a0\left({v}_{x},{\mu}_{e},\uf603{\beta}_{d}{\beta}_{e}\uf604\right)*\left({v}_{\mathrm{yd}}{v}_{\mathrm{ye}}\right)+{k}_{\mathrm{vyd}}\ue8a0\left({v}_{x},{\mu}_{e},\uf603{\beta}_{d}{\beta}_{e}\uf604\right)*d\ue8a0\left({v}_{\mathrm{yd}}{v}_{\mathrm{ye}}\right)/\mathrm{dt}& \left(64\right)\end{array}$

[0122]
where k_{Ωp}(v_{x},μ_{e}) and k_{Ωd}(v_{x},μ_{e}) are the proportional and derivative yaw rate gains, while k_{vyp}(v_{x},μ_{e}, β_{d}−β_{e}) and k_{vyd}(V_{x}, μ_{e}, μ_{d}−μ_{e}) are the proportional and derivative lateral velocity gains. The magnitudes of the gains for each velocity and surface coefficient are tuned through vehicle testing and are implemented as look up tables. Typically, the proportional yaw rate gain k_{Ωp}(v_{x},μ_{e}) and derivative yaw rate gain k_{Ωd}(v_{x}, μ_{e}) increase nearly proportionally with vehicle speed v_{x }and decrease as the estimated surface coefficient of adhesion μ_{e }increases. The lateral velocity gains, k_{vyp}(v_{x},μ_{e}, β_{d}−β_{e}) and k_{vyd}(v_{x},μ_{e}, β_{d}−β_{e}), increase with vehicle speed and increase quite rapidly on slippery surfaces. This is done to provide a proper balance between yaw control and side slip control. On dry surfaces, the yaw rate feedback control usually dominates to achieve responsive handling, while on slippery surface the control of side slip increases to achieve better stability. In addition, the slip angle gains may depend on the magnitude of side slip angle error, with the gain generally increasing as the side slip angle error increases. For example, the gain may be zero or close to zero when the magnitude of side slip angle error is below a threshold, and increases as the side slip angle error increases in magnitude.

[0123]
There exist several modifications of the control law, which may be considered the special cases of the control law (64). First, the desired side slip velocity and side slip angle may be set to zero. In this case, the last two terms in equation (64) are proportional and derivative terms with respect to side slip velocity, rather than side slip errors. In this case, the desired side slip velocity does not need to be computed, which simplifies the algorithm. This simplification is justified, because at higher speeds the desired side slip angles are small, especially for active rear steer vehicles. Further simplification may be achieved by deleting the third term in the control law (64), involving the side slip velocity. In this case, the control law includes P (proportional) and D (derivative) yaw rate terms, but only a derivative lateral velocity term. In that manner, the estimation of vehicle side slip velocity is avoided and the algorithm is further simplified. The control gains may depend on whether vehicle is in oversteer or understeer condition.

[0124]
As discussed earlier, differential speed signal Δv_{lr3 }determined for the brake controller can be converted into equivalent steering angle correction signal Δδ_{r }for rear axle 14 and front steering angle correction signal Δδ_{f }for axle 12. Thus the feedback portions of the front or rear steering angles can be computed from equations (65) and (66):

Δδ_{r} =g _{f}(v _{x}, μ_{e})*Δv _{lr3} (65)

Δδ_{f} =g _{r}(v _{x}, μ_{e})*Δv _{lr3} (66)

[0125]
where the gains can vary with speed and the estimated surface coefficient of adhesion.

[0126]
Block 42 determines a vehicle steer flag, which determines whether vehicle 10 is in understeer (flag=1) or oversteer (flag=0). The following is an example of steer flag determination.

[0127]
Vehicle 10 is in understeer if either front steering angle signal δ_{f}, control signal Δv_{lr3 }and lateral acceleration signal a_{y }are all in the same direction or when vehicle 10 is plowing on a slippery surface. Vehicle 10 is in oversteer if either front steering angle signal δ_{f }is in different direction from control signal Δv_{lr3}; or front steering angle signal δ_{f }and control signal Δv_{lr3 }are in the same direction, but lateral acceleration signal a_{y }is in opposite direction. If neither oversteer nor understeer conditions are satisfied, previous steer definition is held. It is theoretically possible that vehicle 10 is plowing (understeer) and front steering angle signal δ_{f }and control signal Δv_{lr3 }have opposite signs (oversteer). In this case vehicle state is considered oversteer (i.e. oversteer overrides understeer if both are true).

[0128]
The situation when vehicle 10 is plowing is identified when the magnitude of the desired yaw rate Ω_{d }is significantly larger than the magnitude of measured yaw rate Ω over a predefined period of time, and the measured yaw rate Ω is small. This can happen only on very slippery road surface. In this situation, we do not demand that front steering angle signal δ_{f}, control signal Δv_{lr3 }and lateral acceleration signal a_{y }have the same signs, in order to declare understeer, since lateral acceleration signal a_{y }may be very small in magnitude.

[0129]
The over/understeer flag is used to further influence the control actions. If the brake control system is a four channel system, i.e. it can actively apply brakes to either front tires 12 and 13 (FIGS. 1A1D) or rear tires 15 and 16 (FIGS. 1A1D), then the control command Δv_{lr3 }is applied to tire 12 and/or tire 13 when vehicle 10 is in oversteer and to tire 15 and./or tire 16 when vehicle 10 is in understeer. For a two channel system, the control command Δv_{lr3 }is always applied to tire 12 and/or tire 13. The actual commanded differential speed signal Δv_{lr3 }is corrected for the difference in tire velocities, resulting from kinematics of turn. During cornering maneuvers, free rolling tires have a speed difference equal to the product of vehicle yaw rate Ω, and the track width t_{w}. Thus, the target tire slip difference can be computed from equation (67):

Δv _{lr3t} =Δv _{lr3} +t _{w}*Ω (67)

[0130]
When the driver is not braking, the velocity difference between front tires 12 and 13 is achieved by braking of one or both front tires 12 and 13, and the velocity difference between rear tires 15 and 16 is achieved by braking of one or both front tires 15 and 16. When driver is braking, the braking force may also be reduced on the opposite side, if braking of the desired tire reached a saturation point without achieving the desired speed difference.

[0131]
A block 43 tests entry and exits conditions for applying the brake command Δv_{lr3t }to vehicle 10. The brake command Δv_{lr3t }is applied only if entry conditions for the active brake control are established and only until the exit conditions for active brake control satisfied. First, the estimated vehicle speed signal v_{x }must be above a certain entry speed v_{min}, which is rather low, for example 5 mph. If this condition is satisfied, then the brake system becomes active when the magnitude of yaw rate error exceeds a threshold value, which depends on vehicle speed signal v_{x}, front steering angle signal δ_{f }and over or understeer flag. The yaw rate error consists of a proportional and a derivative terms. Thus the entry condition can be computed from the following equation (68):

Ω_{d} −Ω+k _{e} *d(Ω_{d}−Ω)/dt >Ω _{thresh}(v_{x}, δ_{f} , steer_flag) (68)

[0132]
where k_{e }is a constant and Ω_{thresh}(v_{x}, δ_{f}, steer_flag) is a threshold, which depends on the vehicle speed signal v_{x}, front steering angle signal δ_{f }and steer flag. It is larger in understeer condition than in oversteer. The entry conditions for the brake system are significantly relaxed, or even the system may not be allow to enter, when vehicle 10 is being braked in ABS mode. In this case, the directional control is provided by steering only, until the errors in yaw following are quite large. In the case of braking on split mu surface (a surface with significantly different coefficients of adhesion under left and right tires) the entire correction of the yaw motion is provided by steering alone. This is done in order to avoid compromising the stopping distance.

[0133]
An exit condition is established if the magnitude of the yaw rate error, as defined above, is below a predetermined yaw rate error threshold (which is lower than the entry threshold) for a specified period of time or when vehicle speed drops below a certain value.

[0134]
When entry conditions are not met, the active brake control system is disabled. During this time vehicle dynamic behavior is controlled through active steer control, front or rear, which do not have entry conditions. A block 44 determines total commanded targeted control valves. First, rear steering angle δ_{rtd }is computed as the sum of the feedforward part δ_{rff }and the feedback part Δδ_{r }in accordance with the following equation (69):

δ_{rtd}=δ_{rff}+Δδ_{r} (69)

[0135]
If vehicle
10 is in oversteer, the commanded rear steer angle is limited in order to limit the side slip angle of the rear tires to a maximum value α
_{rmax}(μ
_{e}), which depends on the estimated surface coefficient of adhesion (it decreases when the surface estimate decreases). Typical shapes of the curves relating lateral force to the tire slip angle for two different surfaces are shown in FIG. 8. Increasing slip angle beyond α
_{rmax }leads to decline in the magnitude of lateral force on most surfaces. The purpose is to avoid increasing slip angle beyond that corresponding to the peak lateral force. This yields the following equation (70):
$\begin{array}{cc}{\delta}_{\mathrm{rtdl}}=\{\begin{array}{cc}\left({v}_{\mathrm{ye}}b*\Omega \right)/{v}_{x}{\alpha}_{r\ue89e\text{\hspace{1em}}\ue89e\mathrm{max}}& \mathrm{if}\ue89e\text{\hspace{1em}}\ue89e{\delta}_{\mathrm{rtd}}<\left({v}_{\mathrm{ye}}b*\Omega \right)/{v}_{x}{\alpha}_{r\ue89e\text{\hspace{1em}}\ue89e\mathrm{max}}\\ \left({v}_{\mathrm{ye}}b*\Omega \right)/{v}_{x}+{\alpha}_{r\ue89e\text{\hspace{1em}}\ue89e\mathrm{max}}& \mathrm{if}\ue89e\text{\hspace{1em}}\ue89e{\delta}_{\mathrm{rtd}}>\left({v}_{\mathrm{ye}}b*\Omega \right)/{v}_{x}+{\alpha}_{r\ue89e\text{\hspace{1em}}\ue89e\mathrm{max}}\\ \text{\hspace{1em}}\ue89e{\delta}_{\mathrm{rtd}}& \mathrm{otherwise}\end{array}& \left(70\right)\end{array}$

[0136]
Similarly, the commanded front steer angle correction, Δδ_{ftd }consists of the feedforward part δ_{fff }and the feedback part Δδ_{f }in accordance with following equation (71):

Δδ_{ftd}=δ_{fff}+Δδ_{f} (71)

[0137]
The total desired steering angle δ_{ftd }is the sum of the steering angle correction and the angle commanded by the driver δ_{fdr }as computed from the following equation (72):

δ_{ftd}=δ_{fdr}+Δδ_{ftd} (72)

[0138]
This steering may subsequently be a subject of the following limitation. If vehicle is in an understeer condition, then the total front tire steering angle δ
_{ftd }is limited to by the following equation (73):
$\begin{array}{cc}{\delta}_{\mathrm{ftd1}}=\{\begin{array}{cc}\left({v}_{\mathrm{ye}}+a*\Omega \right)/{v}_{x}{\alpha}_{f\ue89e\text{\hspace{1em}}\ue89e\mathrm{max}}& \mathrm{if}\ue89e\text{\hspace{1em}}\ue89e{\delta}_{\mathrm{ftd}}<\left({v}_{\mathrm{ye}}+a*\Omega \right)/{v}_{x}{\alpha}_{f\ue89e\text{\hspace{1em}}\ue89e\mathrm{max}}\\ \left({v}_{\mathrm{ye}}+a*\Omega \right)/{v}_{x}+{\alpha}_{f\ue89e\text{\hspace{1em}}\ue89e\mathrm{max}}& \mathrm{if}\ue89e\text{\hspace{1em}}\ue89e{\delta}_{\mathrm{ftd}}>\left({v}_{\mathrm{ye}}+a*\Omega \right)/{v}_{x}{\alpha}_{f\ue89e\text{\hspace{1em}}\ue89e\mathrm{max}}\\ {\delta}_{\mathrm{ftd}}& \mathrm{otherwise}\end{array}& \left(73\right)\end{array}$

[0139]
where α_{fmax}(μ_{e}) is a front tires slip angle corresponding to maximum lateral force. It is a function of the estimated surface coefficient of adhesion μ_{e}.

[0140]
Thus, during normal vehicle operation, vehicle 10 is controlled through steering inputs only, which are quite effective in controlling vehicle yaw motion in and close to the linear range of handling behavior. Only if the actual response of vehicle 10 significantly deviates from the desired response, the active brake control is activated in addition to the steering control.

[0141]
While the embodiments of the present invention disclosed herein are presently considered to be preferred, various changes and modifications can be made without departing from the spirit and scope of the invention. The scope of the invention is indicated in the appended claims, and all changes that come within the meaning and range of equivalents are intended to be embraced therein.