Abstract:
A method for controlling slip of a driven wheel on low friction surfaces includes determining the current magnitude of static tire friction potential, current magnitude of maximum dynamic tire friction potential, and magnitude of tire torque demanded by an operator of the vehicle. A lower magnitude of commanded tire force is set less than the estimated current magnitude of static tire friction potential. The magnitude of demanded wheel torque and the estimated current magnitude of maximum dynamic tire friction potential are used to set an upper magnitude of commanded tire force. The magnitude of commanded tire force is changed in a sequence of saw-tooth pulses, each pulse including an increasing ramp of tire force from the lower commanded tire force to the upper commanded tire force, the ramp having a predetermined slope, and a step that reduces the magnitude of commanded tire force between each pulse to the lower commanded tire force. An adaptation mechanism is used to adjust the saw-tooth signal parameters with respect to road condition changes.

Description:
BACKGROUND OF THE INVENTION  
       [0001]     The present invention relates generally to a control method and system for improving traction of a driven wheel of a motor vehicle operating on an icy or snow surface.  
         [0002]     Winter test data show that the maximum value of longitudinal tire force response during a wide-open-throttle tip-in-transient on ice is significantly larger than the force predicted by the tire static curve, indicating that there is significant tire friction potential that is not covered by the tire static curve. Dynamic tire friction potential (DTFP) may be used to design an advanced traction control or antilock brake system (ABS) that provides improved performance when compared with the traditional traction control systems based on the static tire curve.  
         [0003]     It has been hypothesized in that dynamic tire friction potential may be used to design advanced traction control systems with significantly improved performance when compared with the traditional traction control systems based on the static tire curve. Since the maximum dynamic friction potential occurs at high rates of applied force, of approximately 10000 N/s, implementation of the advanced traction control system would require very fast and precise generation of the wheel torque.  
         [0004]     The peak value of the static tire force that can be reached on an ice surface is typically about 700 N. Experiments have shown that the tire force can be increased up to 2000 N, if a high rate of change of the wheel torque is applied for a vehicle starting from standstill. This finding has led to the development of a traction control strategy that is based on generation of consecutive sequences of high-rate linear rise of applied torque. The strategy can be applied in modern electrical or hybrid electrical vehicles, particularly in those equipped with in-wheel electrical motors.  
       SUMMARY OF THE INVENTION  
       [0005]     A method according to this invention for controlling slip of a driven wheel of a motor vehicle includes determining the current magnitude of static tire friction potential, current magnitude of maximum dynamic tire friction potential, and magnitude of wheel torque demanded by an operator of the vehicle. A lower magnitude of commanded wheel force is set slightly below the current magnitude of static wheel friction potential. The magnitude of demanded wheel torque and the current magnitude of maximum dynamic tire friction potential are used to set an upper magnitude of commanded wheel torque. The magnitude of commanded wheel torque is changed such that wheel torque varies in a sequence of saw-tooth pulses, each pulse including an increasing ramp of wheel force from the lower commanded wheel force to the upper commanded wheel force, the ramp having a predetermined slope, and a step that reduces the magnitude of commanded wheel force between each pulse to the lower commanded wheel force.  
         [0006]     The applied wheel torque has a saw-tooth form consisting of the interval of linear torque rise, which is followed by abrupt torque resetting to a value less and close to the maximum static force. This signal has a frequency in the approximate range from 10 to 20 Hz, and is applied in the low-slip tire region (adhesion region). The linear torque rise is reset when the applied wheel force, which is applied wheel torque divided by the effective tire radius, reaches the maximum dynamic friction potential, or if the wheel slip or wheel acceleration exceeds a predetermined threshold. Using the condition based on wheel acceleration means that precise information regarding the wheel slip (i.e. the vehicle speed) is not necessarily required, so that the strategy can be applied to a four-wheel-drive vehicle. If the wheel starts to spin despite the above conditions, a proportional+integral (PI) slip controller is used to abruptly reduce the slip to desired low values. The saw-tooth signal magnitude is adjusted with respect to wheel torque demanded from the driver and current value of maximum dynamic tire friction potential. An adaptation mechanism is used to adapt the saw-tooth signal parameters with respect to road conditions changes. Since the dynamic tire friction potential increases if the vehicle dwells at the road for several seconds, the strategy includes a stop-and-go intervention which helps to increase low-speed traction performance (e.g. climbing over an icy hill).  
         [0007]     As an extension to the above, it is also proposed to use the torsional vibration mode that is inherent to the tire and hub system. This vibration mode is typically in the range between 20-40 Hz. By appropriately timing the torque oscillation pulses from an electromotor or similar device, it is possible to replace the saw tooth signal. One possible advantage of such an approach is that less energy will be needed, since the natural frequency or resonance will be used. Although the invention is applicable to traction control, it also applies to antilock brake systems, and possibly to dynamic stability control where similar approaches will be used for lateral force generation.  
         [0008]     The benefits of the proposed TCS may be significant for very low vehicle speeds only, since the dynamic tire friction potential reduces at higher speeds. Nevertheless, the traction control strategy can be applied to the following important applications. The proposed TCS would significantly increase the traction force when climbing over an icy or snowy hill at low speeds. A steep, icy hill that could not be overcome by a conventional TCS could be overcome by applying the TCS of this invention. Test data show a significant DTFP may exist for snow surfaces as well. Therefore, a TCS according to this invention can be beneficial toward starting a vehicle in deep snow. In addition, it is anticipated that 10-20 percent traction performance gain can be reached for vehicle speeds up to 10 km/h and even more, provided that a higher rate of applied force is used. The 10-20 percent performance gain can have significant practical meaning for critical ABS braking maneuvers and critical vehicle dynamics control maneuvers, when considering lateral force increase through saw-tooth generation of active-steering torque.  
         [0009]     The method is particularly effective for climbing a steep hill, where a stop-and-go vehicle motion is used to increase the dynamic tire friction potential and climb the hill.  
     
    
     DESCRIPTION OF THE DRAWINGS  
       [0010]     These and other advantages of the present invention will become readily apparent to those skilled in the art from the following detailed description of a preferred embodiment when considered in the light of the accompanying drawings in which:  
         [0011]      FIG. 1  is graph of static tire force vs. wheel slip for an ice surface;  
         [0012]      FIG. 2  is graph of the transient response of tire force vs. time during an abrupt tip-in on ice from standstill;  
         [0013]      FIG. 3  is a graph that shows the variation of dynamic tire friction potential with time rate of change of applied wheel force for zero initial vehicle speed;  
         [0014]      FIG. 4  is a graph of a saw-tooth applied wheel force signal vs. time;  
         [0015]      FIG. 5  is a top view of a motor vehicle driveline that includes motor driven wheels and an electronic controller for a traction control system;  
         [0016]      FIG. 6  is a graph illustrating wheel responses to the control signals produced by a traction control system according to this invention;  
         [0017]      FIG. 7  is a flow diagram of control logic for a main control routine;  
         [0018]      FIG. 8  is a flow diagram of hill-ascend control logic;  
         [0019]      FIG. 9  is a flow diagram of control logic of a core traction control strategy based on dynamic tire friction potential; and  
         [0020]      FIG. 10  is a flow diagram of control logic for determining static tire friction potential and suppressing high slips by PI control. 
     
    
     DESCRIPTION OF THE PREFERRED EMBODIMENT  
       [0021]     According to the tire static friction curve for ice surface shown in  FIG. 1 , the maximum static tire force F [N] is about 650 N and occurs when wheel slip (s) is approximately three percent. But the maximum reconstructed tire force F t  [N] for an abrupt throttle tip-in transient from zero vehicle speed reaches the value of 1500 N as shown in  FIG. 2 . These data point to a dynamic tire friction potential, which is 130% higher than the maximum static friction potential. This results in better traction, i.e. in abrupt change of vehicle speed or acceleration during the tip-in interval. As the initial vehicle speed increases, the dynamic friction potential reduces, and is equal to 850-1000 N for initial vehicle speeds larger than 2 km/h. The zero-speed dynamic potential increases by approximately 30%, if the tire dwells at standstill for several seconds before the abrupt transient is executed.  
         [0022]      FIG. 3  shows the dependence of dynamic friction potential on the rate of change of applied force, where the applied force F app  is equal to the applied wheel torque τ divided by the effective tire radius r, and where the initial vehicle speed was zero. The graph of  FIG. 3  is an exponential curve, which comes close to the saturation level at the applied force rate of approximately 10000 N/s.  
         [0023]     The maximum dynamic friction potential is reached if the applied force F app =τ/r is increased approximately linearly on a ramp. As shown in  FIG. 3 , preferably the time rate of change of applied force, {dot over (F)} app , should be in the range from 10000 to 20000 N/s, in order to reach the maximum dynamic friction potential with a minimum actuator energy. When the applied force reaches the maximum dynamic friction potential F max , i.e. immediately before the wheel starts to spin, the applied force is reset to a value between 0 and maximum static force F max,stat , and again increased linearly. This results in the saw-tooth form of the applied force signal, which is shown in  FIG. 4 . As an alternative, the applied force F app  may be kept at the value F low , for a relatively short time to allow a possibility of tire “relaxation” shown as the dashed bold line in  FIG. 4 .  
         [0024]     The critical point of the simple control concept in  FIG. 4  relates to resetting the applied force to the value F low . This should be done when the applied force reaches the predefined upper level F up , which is less than and close to the maximum dynamic friction potential (F up ≈F max ), or when the wheel acceleration or the wheel slip exceeds a threshold value to avoid wheel spinning due to inaccurate or unreliable F max  values. The choice of wheel acceleration signal should generally be preferred, because it does not require the vehicle speed measurement and can be applied to a 4WD vehicle.  
         [0025]     If the wheel starts to spin despite the above conditions on applied force and wheel acceleration, a slip controller can be used to reduce wheel slip to a low magnitude.  
         [0026]     Since the maximum dynamic friction potential does not depend on the initial value of tire force, the force F low  in  FIG. 4  should be chosen close to the maximum static force F max,stat , to provide higher average tire force  F , i.e., better traction performance.  
         [0027]     Since the control concept illustrated in  FIG. 4  is intended to keep the tire in the low-slip (adhesion) region at all times, the tire force response F should be close to the applied force F app , with some relatively small delay and possible overshoot and some oscillations caused by the tire sidewall and tread dynamics.  
         [0028]     In the “idealized” case F=F app , the tire force mean value  F  is:  
               F   _     =         F   low     +     F   up       2             (   1   )             
 
         [0029]     The index of traction performance improvement with respect to idealized traditional traction control system characterized by F=F max,stat  is  
             i   =           (         F   _       F     max   ,   stat         -   1     )     ·   100     ⁢           ⁢   %     =         (           F   low     +     F   up         2   ⁢     F     max   ,   stat           -   1     )     ·   100     ⁢           ⁢   %               (   2   )             
 
         [0030]     The period of the saw-tooth signal in  FIG. 4  is  
               T   0     =         F   up     -     F   low           F   .     app               (   3   )             
 
         [0031]     If F up ≈F max ≈1700 N and F low ≈500 N, then  F =1100 N, and the theoretical traction performance improvement is i≈70% (for F max,stat =650 N,  FIG. 1 ). This percentage will certainly be lower in the real case due to some practical constraints such as tire force drops around F up  and F low  due to tire dynamics, possible need for the relaxation period around F low , lower dynamic friction potential for non-zero speeds, and similar factors.  
         [0032]     Inserting F up ≈F max ≈1700 N and F low ≈500 N in Eq. (3), together with F app =10000 N/s, yields the period of the applied force saw-tooth signal: T 0 =120 ms, i.e. the frequency is 8.3 Hz. When {dot over (F)} app ≈=20000 N/s, this interval is halved to T 0 =60 ms. The saw-tooth signal of  FIG. 4  with the period of 50-100 ms is difficult to realize in a conventional vehicle with an internal combustion engine, but it can be easily implemented in an electrical vehicle, for which the torque control bandwidth is typically higher than 40 Hz.  
         [0033]     Referring now to  FIG. 5 , a powertrain of a motor vehicle to which the present invention can be applied includes front and rear wheels  10 - 13 ; electric motors  14 - 17 , each motor driveably connected to a respective wheel; a source of electric power, such as an electric storage battery; wheel speed sensors  20 - 23  for producing signals (ω 1 -ω 4 ) representing the speed of the respective wheel. An electronic controller includes a central processing unit CPU communicating with the speed sensors, power source, and motors, and electronic memory containing control algorithms stored there in coded form readable by the CPU. An accelerator pedal sensor, which produces a signal representing the desired wheel torque, is controlled by the vehicle operator and communicates with the controller. A brake pedal sensor, controlled by the vehicle operator, communicates with the controller. A wheel braking system produces a signal representing the magnitude of a desired wheel braking torque.  
         [0034]      FIGS. 6A, 6B  and  6 C show typical responses of one of the driven wheels  10 - 13  to traction control according to the present invention, but they do not illustrate high-frequency oscillations of the tire force signal F and wheel speed signal ω.  
         [0035]     Referring now to the flowchart of the main control routine shown in  FIG. 7 , which initializes slip_contrl=0 and sets t est0 =t at step  60 . During each sampling instant t k =kT s , k=0, 1, 2, . . . , the wheel speed signal ω is differentiated at  62  to obtain a wheel acceleration signal {dot over (ω)}. The wheel acceleration is filtered by using a low-pass and/or notch filter to attenuate the oscillatory modes due to tire sidewall and tread compliance. The filtered acceleration signal {dot over (ω)} F  is compared at  64  with a predetermined threshold Δ {dot over (ω)}  to determine if the wheel is spinning, and the slip_cntrl flag is compared with 1 to check if the wheel slip control is already active. If the test at  64  is false, indicating that wheel spinning is not occurring, at  66  the driver&#39;s wheel torque command F* is set equal to the applied force to the wheel (F app =F*), and the traction control system is not used.  
         [0036]     If wheel spinning does occur at  64 , the slip control algorithm shown in  FIG. 10  is activated at  68  in order to suppress the wheel slip excursion, i.e. to bring the wheel speed back to a previous value that approximately corresponds to the vehicle speed divided by the effective tire radius. The slip control algorithm also comprises a tire friction estimation algorithm, which determines the static tire friction potential F max,stat  i.e. the maximum value of tire static curve. Applying an efficient wheel acceleration filtering algorithm makes the spinning detection fast and reliable. In the case when there are non-driven wheels with measurable speed (e.g. 2WD vehicles equipped with ABS), the driven wheel spinning can be effectively detected based on the difference of driven wheel speed and non-driven wheel speed (slip speed larger than a predetermined threshold: ω− 107   nd &gt;Δ 107  ) instead of using the condition based on the wheel acceleration.  
         [0037]     After the slip control routine of  FIG. 10  is completed, slip_cntrl=1 becomes false at step  70  in  FIG. 7 , and the DTFP-based TCS of  FIG. 9  is ready to be initialized at  72 . The lower force F low  of the saw-tooth signal of  FIG. 6  is set to a value close, but lower than the estimated maximum static tire friction potential F max,stat  (κ low &lt;≈1, e.g. κ low =0.85). The upper force F up  of the saw-tooth signal is calculated by equating the commanded force F* and the average dynamic force  F =0.5(F low +F up ), i.e. F up =κ up (2F*−F low ), where κ up &lt;≈1 (e.g. κ up =0.9). The applied force F app  (i.e. an output of controller  18  in  FIG. 5  ) is reset to the lower force F low  and the TCS core algorithm of  FIG. 9  based on saw-tooth applied force signal is ready to be executed beginning at  74 .  
         [0038]     The TCS core algorithm updates the applied force F app  by adding at  76  the value {dot over (F)} app T s  (T s  is the sampling time) to the rate {dot over (F)} app  in order to generate the force ramp. It then reads the wheel speed ω to calculate at  78  its average value ω* during a single period of saw-tooth signal. This average value relates to the vehicle speed divided by the tire radius. Note that the averaging process suppresses the influence of wheel speed oscillations due to tire compliance, and that the effect of vehicle speed increase during the saw-tooth interval is negligible due to the small saw-tooth interval (typically 50 ms). Note also that this averaging step is not needed if the non-driven wheel speed signal ω nd  is available because ω* is simply set to ω nd  at the instant immediately before wheel spinning occurs.  
         [0039]     The remaining part of the TCS core algorithm of  FIG. 9  is illustrated by the response shown in  FIG. 6 . If the saw-tooth counter variable i is equal to zero at  80 , the algorithm proceeds to the identification algorithm, which keeps ramping-up the applied force, shown as the ramp A-B in  FIG. 6 , until wheel spinning is detected, i.e. until the condition {dot over (ω)} F &gt;Δ ω  is satisfied at  82 . The maximum dynamic tire friction potential F max  is estimated as the applied force value at the instant of spinning detection. Alternatively, it may be set to F app  of a few sampling steps before wheel spinning is detected to account for the delay of spinning detection. As explained above with reference to  FIG. 7 , when wheel spinning is detected, the PI controller of  FIG. 10  is activated at  84  to return the wheel slip to low values. This is achieved by controlling the wheel speed ω to its average value before spinning ω*, i.e. the control error ω*−ω is fed at  86  to the PI controller of  FIG. 10 .  
         [0040]     In order to provide a fast reaction of the controller in terms of overcoming the wheel inertia effect, the controller&#39;s integrator is reset at  88  to a value lower than F low  immediately after wheel spinning detection (point C in  FIG. 6A ). The controller is active for a predetermined time T cntrl , which is typically set to 2T e , where T e  is the equivalent time constant of the closed-loop wheel speed control system. This interval is denoted in  FIG. 6A  as the interval C-D, and is implemented in  FIG. 10  at  90  through the condition t−t est0 &lt;T cntrl .  
         [0041]     While executing the PI control of  FIG. 10 , the maximum static friction potential F max,stat  is estimated. The estimation is delayed with respect to PI controller activation for time T dest  (shown in  FIG. 6A  and the condition t−t est0 &lt;T dest  at step  92  in  FIG. 10 ) in order to leave some time needed to transit from the dynamic friction potential in Point B of  FIG. 6A  to the static friction potential that needs to be estimated. The estimated tire force {circumflex over (F)} is calculated by subtracting the filtered acceleration force Ir −1 {dot over (ω)} F  from the applied force F app . Note that the applied force may be filtered by using the same filter as for the wheel acceleration signal if this filter introduces a significant time delay. According to the tire static curve, the estimated tire force typically increases as the slip decreases, and then starts to decrease in the low-slip region. The maximum static friction potential is calculated as the maximum value of the estimated force.  
         [0042]     After the PI control of  FIG. 10  is completed, it may be convenient to leave some more time for the wheel to “stabilize” in the low slip region. This interval is denoted in  FIG. 6A  as the interval D-E, whose length is time T pause , and is implemented in the PI control of  FIG. 10  at step  94  through the condition t−t est0 −T cntrl &gt;T pause . During the D-E interval, the applied force F app  is set to a value close to and lower than the maximum static friction potential (F app =κ low  F max,stat , κ low &lt;≈1), in order to provide good traction without wheel spinning.  
         [0043]     After completing the execution of the PI control and estimation algorithm of  FIG. 10 , the TCS core algorithm of  FIG. 9  uses the estimated force parameters to determine the saw-tooth signal parameters. The lower force value F low  is determined at  98  from the estimated maximum friction potential as F app =κ low F max,stat , κ low  &lt;≈1. The upper force value F app  is set close to and less than the maximum dynamic friction potential F max  (F app =κ up F max ; where κ up &lt;≈1), provided that the driver&#39;s commanded force F* is larger than the maximum average dynamic force 0.5(F low +F max ); otherwise, the upper force value F app  is set in accordance to the commanded force: F up =2F*−F low , i.e., so that the average dynamic force is equal to the commanded force. F max  is obtained as a reached value of Fapp at  81  the moment of spinning detection at  82 .  
         [0044]     At step  100  of the identification algorithm of  FIG. 9 , the saw-tooth counter i is set equal to 1. Therefore, in the next sampling step, the normal TCS operating mode consisting of a sequence of saw-tooth pulses starts to be generated, as represented graphically in  FIG. 6A  as the interval E-F and in accordance with the steps of the left branch of  FIG. 9 . The upper force F up  is continuously being adjusted with respect to the commanded force F* if the commanded force F* is lower than the maximum average dynamic force. The saw-tooth counter i is increment at  102 .  
         [0045]     When a predetermined number of saw teeth i max  is counted out at  104 , the saw tooth counter i is reset at  106  to zero, and consequently the TCS algorithm is again redirected to the identification algorithm, thereby giving the algorithm an adaptive feature. Namely, if a low-mu/high-mu transition occurs during the normal operating mode, i.e., if a transition from a surface of low friction coefficient to a high friction coefficient occurs, the algorithm learns that the static and dynamic friction potentials are increased, and reacts by increasing F low  and F up , and consequently the average dynamic force. For example, if i max =5 is set at step  104 , the adaptation is carried out each quarter of a second for the typical period of the saw-tooth signal, which period is 50 ms for an icy surface.  
         [0046]     If wheel spinning occurs ({dot over (ω)} F &gt;Δ ω  at step  82  in  FIG. 9 , and point H in  FIG. 6A ) during the normal operating mode (i&gt;0), this means that the tire-road friction coefficient has decreased, or that there has been eventually some other cause of dynamic friction potential reduction, e.g. an increase of vehicle speed. If this happens, the algorithm is immediately redirected at step  84  to the PI slip control of  FIG. 10  to reduce the wheel slip. It is then followed by the identification routine (i=0). After the identification routine ends (i=1), the algorithm is again redirected to the normal operating mode.  
         [0047]     The core traction control algorithm of  FIG. 9  is initially called from the superimposed routine of  FIG. 7  with the saw-tooth counter set at step  72  to i=1. This means that the identification algorithm is initially skipped, i.e., the interval A-E in  FIG. 6A  is initially absent, because friction is partially estimated during the slip control algorithm called from the superimposed routine of  FIG. 7 . However, since this algorithm does not provide the information on maximum DTFP F max , the upper force is initially set in the algorithm of  FIG. 7  in accordance to the driver&#39;s command F*. If this command is too large, i.e., if it cannot be realized by the TCS, wheel spinning will occur again, as represented by point H in  FIG. 6A , and the identification algorithm of  FIG. 9  will then be executed with (i=0).  
         [0048]     After each execution of the TCS core algorithm (the DTFP_based_TCS routine of  FIG. 9 ), the main control routine of  FIG. 7  checks at step  104  whether the force commanded by the driver, F*, is lower than the saw-tooth signal lower value F low . If the test at  104  is true, the traction control is suspended, and the applied force is set at  66  to the commanded force (F app =F*). If the test at  104  is false, a check is made at step  110  to determine if vehicle speed is negative (i.e. of opposite sign than the applied force) or if the driver has turned on a hill-ascend button. A negative vehicle speed means that the dynamic traction force has not been high enough to climb the vehicle over the hill. If the test at step  110  is true, control passes at step  112  to the hill ascend control of  FIG. 8 , the vehicle is stopped at  114  by means of active wheel brakes (brake-by-wire) and remains braked on the hill for a period (dwell time T dwell ) at  116 . Dwell time of a few seconds increases the dynamic friction potential on the ice surface by up to 30%. This may be enough to overcome the hill. Thus, after the dwell period expires, the identification routine is executed (i=0) at  118 , and then the normal TCS mode based on the saw-tooth signal is activated (i=1). If the hill-ascend button is turned on, the vehicle stop mode is also activated at vehicle speeds greater than the predetermined speed Δ ν  (typically equal to 1-1.5 km/h) due to the test at  112  of  FIG. 7 . The vehicle stop mode is activated because the dynamic friction potential on ice surface is significantly reduced at speeds greater than 1.5 km/h, and because the driver had requested slow driving during hill climbing. This relates to stop-and-go vehicle operation, which can significantly increase the traction performance on icy hills. Vehicle speed can again be estimated as the average value of driven wheel speed. Alternatively, vehicle speed can be estimated with reference to vehicle acceleration measurements, GPS measurements, or the non-driven wheel speed for 2WD vehicles.  
         [0049]     There may be many variants of the TCS control described above. For example, if the average dynamic force 0.5 (F low +F up ) is just slightly greater than the maximum static friction potential F max,stat ≈F low , the DTFP-based TCS can be turned off to avoid relatively high NVH content related to the saw-tooth applied force variations, and a conventional traction control strategy can be used instead. The driver may turn off the DTFP-based TCS by using a special button, or turn off the traction control. Similarly, if a road condition different than ice or snow is detected, as indicated by F max,stat /F z &gt;0.25, where F z  is the normal force to the wheel, the DTFP-based TCS would again be turned off. Note that the benefits of the strategy are primarily emphasized for low friction coefficient (mu) surfaces. Also, the upper value of the saw-tooth signal, F up , may be made dependent on the vehicle speed, in order to avoid wheel spinning at higher speed due to the reduction of DTFP. The value F up  may also be made dependent on the steering wheel angle, in order to avoid driven wheel spinning for vehicle turning. Similarly, the saw-tooth signal rate {dot over (F)} app  may be made adaptive to reach a good trade-off between the NVH performance and TCS performance for different operating modes and road conditions.  
         [0050]     In the stop-and-go strategy for climbing a hill, it may be convenient to set up the applied force to a constant saturation level, Fapp,sat˜(Fup+Flow)/2, after the first high-rate linear increase of the force is applied, so that a saturated linear rise of the applied force can be used. This will decrease the NVH content without causing significant deterioration of performance.  
         [0051]     The control strategy can easily be implemented in ABS applications. Although the strategy is described with reference to vehicle braking using an electrical motor, it can also be applied using fast hydraulic or electromagnetic actuators for the brakes. In the case of ABS applications, the responses F app  and F in  FIG. 6  would simply have the opposite sign, the control strategy would be adjusted accordingly in terms of changing signs of different variables, and the wheel speed response of  FIG. 6  would have a falling trend and have peaks in the opposite direction, i.e., braking peaks. Furthermore, the strategy could be applied for vehicle dynamics control to increase the lateral force on the vehicle, provided the vehicle is equipped with a steer-by-wire system or an electrically assisted steering system. In this case, the saw-tooth signal would relate to applied steering torque, and would excite the lateral DTFP.  
         [0052]     In accordance with the provisions of the patent statutes, the present invention has been described in what is considered to represent its preferred embodiment. However, it should be noted that the invention can be practiced otherwise than as specifically illustrated and described without departing from its spirit or scope.