DE102008046259B4 - A method and apparatus for determining a vehicle stability during braking and method and apparatus for adjusting a brake pressure for a vehicle - Google Patents

Abstract

A method for determining a driving stability of a vehicle when braking, comprising:
Determining a center of gravity position (11) of the vehicle,
Determining side force coefficients of wheels (12-15) of the vehicle, and
- Determining a value for the driving stability of the vehicle from a speed of the vehicle, the center of gravity (11) of the vehicle, the Seitenkraftbeiwerten the wheels (12-15) of the vehicle, a vehicle mass and a vehicle inertia about a vehicle's vertical axis.

Description

The present invention relates to a method and apparatus for determining a running stability of a vehicle when braking, and an apparatus and a method for adjusting a brake pressure for a vehicle.

The most important criteria when braking vehicles, such as Passenger cars or trucks, are the braking distance, steerability and stability. Steerability and stability are more important because of the increased safety. If a vehicle loses its stability, it is virtually impossible for a normal driver to recapture a rotational movement of the vehicle.

Braking systems of today's vehicles are usually equipped with automatic anti-lock brakes (ABV), so-called anti-lock braking systems (ABS). The ABS counteracts the tendency of the wheels to lock when the vehicle is braked by regulating the brake pressure to ensure lateral support of the wheels and, in most cases, a short braking distance.

The steerability of the vehicle is usually ensured by the blockage avoidance of the steerable wheels, generally the front wheels. The prevention of rear wheel lock is generally a necessary, but not sufficient, condition for driving stability. Especially at higher speeds, the vehicle may make uncontrollable turns, although the rear wheels will not stall. In addition, due to the concept some vehicles, for example vehicles with a higher center of gravity, are difficult to stabilize during certain braking maneuvers, for example from high speeds in a curve or on a one-sided smooth road. The reason for this lies in an unfavorable Radlastverlagerung between the wheels during the braking process. Such a shift affects the lateral guidance of the wheels and leads under certain conditions to the instability of the vehicle.

In this context, the DE 196 23 595 A1 a method for controlling the driving behavior of a vehicle. In the method, at least the vehicle mass and the instantaneous location of the center of mass of the vehicle are determined on the basis of tire sensor signals. As a basic data, a mass distribution model contains the basic mass distribution of the vehicle, ie those masses that are always the same even with different loadings. Furthermore, variable masses such as location and mass of passengers, baggage, etc. are determined via the tire sensor signals and stored in the mass distribution model to a variable mass distribution. Knowing all such data, it is also possible to determine characteristic variables for the driving dynamics of the vehicle so that the yaw angle speed and the vehicle's angle of slip can be calculated by appropriate calculation. Since the tire sensors detect the load on the individual wheels and, in addition, determine the center of gravity, the behavior of the vehicle can easily be reproduced under certain conditions. It can be calculated, for example, which cornering force can apply a particular wheel when driving through a particular curve. The mass distribution model can be incorporated into a method for controlling the driving stability in the sense of a yaw moment control. The controlled system of the control method is formed by the vehicle, which is provided with tire sensors. This tire sensor detects the individual wheel loads as well as lateral and torsional forces which are applied to the individual tires. In addition, the single-wheel speeds are recorded. These can also be determined by suitable tire sensors. In the mass distribution model flow the single wheel speeds and the measured forces such. B. wheel loads, shear forces and longitudinal forces on the individual wheels. On the basis of such a mass distribution model, the yaw angular acceleration and the float angle velocity of the vehicle can be determined. By integration with time, the yaw rate and the slip angle of the vehicle are obtained from the yaw angular acceleration and the slip angle velocity. These flow as current actual states into a known yaw moment control law. The target values for the control are calculated in another way. For this purpose, the single wheel speeds are converted in a speed observer into a vehicle reference speed. A linear single-track model uses this reference speed and the steering angle of the front wheels to calculate target values for the slip angle velocity and the yaw angular acceleration. These values are also integrated into target values for the yaw rate and the slip angle of the vehicle. The slip angles and the yaw rates are compared with each other and converted into an applied target yaw moment. This applied target yaw moment is then converted into a wheel force distributor in applied to the individual wheels wheel forces. These are nominal forces. The actual forces correspond to the measured forces detected by the tire sensors. By subtraction of the desired forces with the actual forces are in a Radkraftregler the individual differential forces, for example, converted into brake pressures. These in turn affect the vehicle.

The DE 10 2004 035 004 A1 relates to a method for increasing the driving stability of a motor vehicle. The method uses a model-based feedforward control to apply a compensating yaw moment to the motor vehicle. Three quantities are used as input signals for a pilot control process: 1. an effective steering angle, which may result from a driver steering angle; 2. a vehicle reference speed, which is provided in all common vehicle control systems; and 3. a value for the longitudinal acceleration, which may preferably be output by a longitudinal acceleration sensor, determined from engine and brake torque values or the time derivative of the vehicle reference speed. Optionally, some of the following variables may optionally also be included in an extended pilot control method: longitudinal torque / longitudinal force distribution between the front and rear axles, which can be calculated in particular from the longitudinal acceleration and / or the longitudinal slip at the front axle and the rear axle; Mean wheel slip on the front axle and rear axle, which can be determined, for example, with the aid of the vehicle reference speed, the four wheel speeds and the yaw rate; Mass, yaw moment of inertia, center of gravity, which can be assumed to be fixed values; average slip angle at front and rear axle by means of suitable sensors; average coefficient of adhesion at the front and rear axle by means of suitable sensors; Medium skew stiffnesses on front and rear axle by means of a suitable observer.

Object of the present invention is therefore to ensure the driving stability of a vehicle when braking to ensure a stable braking operation.

According to the present invention, this object is achieved by a method of determining a running stability of a vehicle when braking according to claim 1, a method of adjusting a brake pressure for a vehicle according to claim 11, a device for determining a driving stability of a vehicle when braking according to claim 16 and a device for adjusting a brake pressure for a vehicle according to claim 18 solved. The dependent claims define preferred and advantageous embodiments of the invention.

In the context of the present invention, a method for determining a driving stability of a vehicle during braking is provided. The method includes determining a speed of the vehicle, for example, by detecting speeds of wheels of the vehicle, determining a center of gravity position of the vehicle, and determining lateral force coefficients of wheels of the vehicle. From the speed of the vehicle, the center of gravity of the vehicle, the Seitenkraftbeiwerten the wheels of the vehicle and a vehicle mass and a vehicle moment of inertia about a vehicle's vertical axis, the driving stability of the vehicle is then determined.

The center of gravity position of the vehicle may include, for example, a distance of a center of gravity of the vehicle from the front axle and a distance of the center of gravity of the vehicle from the rear axle. The lateral force coefficients may be determined for each individual wheel of the vehicle, i. the left front wheel, the right front wheel, the rear left wheel and the rear right wheel are individually determined. The value determined for the running stability of the vehicle may have a value range indicating a stable driving state of the vehicle and a further value range indicating an unstable driving state of the vehicle. For example, a negative value for the driving stability of the vehicle may indicate a stable driving state of the vehicle, and a positive value may indicate an unstable driving state of the vehicle.

According to one embodiment, the lateral force coefficient for a wheel of the vehicle is determined by determining a circumferential force of the wheel, determining a slip of the wheel, for example from the angular velocity of the wheel and the vehicle speed, and determining the lateral force coefficient of the wheel from the circumferential force, the slip and a wheel-specific constant certainly.

According to another embodiment, the circumferential force of the wheel is determined by detecting a brake pressure of a wheel brake cylinder of a brake of the wheel, determining an angular acceleration of the wheel, and determining the circumferential force from the brake pressure, the angular acceleration, and vehicle-specific quantities. The vehicle-specific variables may include a piston area of the wheel brake cylinder, an average friction radius of a brake pad of a brake disk of the brake, a brake characteristic of the brake, a moment of inertia of the wheel and all masses connected thereto, and a dynamic tire radius.

In addition, when determining the circumferential force, a drag torque of the motor acting on the wheel can be additionally detected and taken into account in the determination of the circumferential force.

According to one embodiment, the center of gravity of the vehicle, for example, the distance of the center of gravity of the vehicle from the front axle and the distance of the center of gravity of the vehicle from the rear axle, the peripheral forces of the wheels and the wheelbase of the vehicle can be determined.

With the aid of the method described above, a value for the driving stability of the vehicle can be reliably determined from measured values and vehicle-specific variables. The measured values are already available in many modern vehicles anyway or can be detected in a simple manner with suitable sensors.

According to the present invention, there is further provided a method of adjusting a brake pressure to a vehicle. The method includes determining slip angles of the wheels of the vehicle and determining a value for vehicle stability during braking. The brake pressure at the wheels of the vehicle is then set in dependence on the determined slip angle and the determined driving stability such that a predetermined slip angle is not exceeded and that a predetermined value range for driving stability is maintained. The driving stability can be determined as described above. The predetermined value range for the driving stability in which a vehicle state is considered to be stable may include, for example, a range of -∞ to -0.5.

With the aid of the method according to the invention for adjusting the brake pressure, the driving stability during braking is ensured, thus preventing the vehicle from breaking out.

According to one embodiment of the method, a yaw rate of the vehicle is detected, and determined therefrom and from other variables, a slip angle of a wheel of the vehicle. The further variables may include the speed of the vehicle, the lateral force coefficients of the wheels of the vehicle, the peripheral forces of the wheels of the vehicle, the center of gravity of the vehicle and the vehicle mass and the vehicle inertia about a vehicle's vertical axis.

As described above, the slip angles at the front axle and at the rear axle of the vehicle can be determined by means of easily detectable sizes of the vehicle. The predetermined slip angle, which should not be exceeded when setting the brake pressure of the vehicle, for example, be a maximum slip angle of 6 °. Up to such a slip angle is a stabilization of the vehicle by a steering intervention of the driver in a simple manner possible, so that a stabilization of the vehicle when braking on, for example, greatly different grip surfaces on the right and left wheels of the vehicle is possible.

According to the present invention, there is further provided an apparatus for determining a running stability of a vehicle during braking. The device comprises a processing unit which is coupled to sensors of the vehicle. The sensors provide readings for an angular velocity of each wheel of the vehicle and a brake pressure on each wheel of the vehicle. The processing unit is configured to determine a speed of the vehicle from, for example, the angular velocity of the wheels, a value for the driving stability of the vehicle from the speed of the vehicle, the angular velocities of the wheels of the vehicle, the brake pressures at each wheel of the vehicle, and vehicle-specific information. The value for the driving stability may be determined by the processing unit as previously described in the context of the method for determining a driving stability of a vehicle when braking.

Finally, in accordance with the present invention, an apparatus for adjusting a brake pressure for a vehicle is provided. The device comprises a processing unit which is coupled to sensors and an actuator of the vehicle. The sensors provide readings of an angular velocity of each wheel of the vehicle, a yaw angular velocity of the vehicle, and a brake pressure for each wheel of the vehicle. The actuator is used to set a brake pressure for the vehicle determined by the processing unit. The processing unit is configured such that it determines slip angles of the wheels of the vehicle and a value for driving stability of the vehicle during braking from the measured values of the sensors and further vehicle-specific information. Depending on the particular slip angle and the particular driving stability, the brake pressure at the wheels is then adjusted such that a predetermined slip angle is not exceeded on any of the wheels and that a predetermined range of values for the driving stability is maintained. To determine the driving stability, for example, the device described above for determining a driving stability of a vehicle during braking can be used. The brake pressure adjusting device may determine the slip angles of the wheels and the running stability of the vehicle, for example, as described in connection with the method for adjusting a brake pressure for a vehicle and adjust the brake pressure for the vehicle accordingly.

• 1 zeigt eine schematische Blockdarstellung einer Vorrichtung zum Einstellen eines Bremsdrucks für ein Fahrzeug gemäß einer Ausführungsform der vorliegenden Erfindung.
• 2 zeigt eine schematische Darstellung einer Ansicht eines Fahrzeugs von oben mit auf das Fahrzeug wirkenden Kräften und Momenten.
• 3 zeigt eine schematische Darstellung eines Fahrzeugs mit auf das Fahrzeug wirkenden Kräften zur Bestimmung der Radlasten.
• 4 zeigt ein Diagramm, welches die Radlasten der Räder eines Fahrzeugs über der Zeit beim Bremsen zeigt.
• 5 zeigt ein Diagramm, welches die Seitenkraftbeiwerte der Räder eines Fahrzeugs über der Zeit beim Bremsen zeigt.
• 6 zeigt in vier Diagrammen für jeweils ein Rad des Fahrzeugs jeweils simulierte Seitenkräfte und gemäß einer Ausführungsform bestimmte Seitenkräfte über der Zeit beim einem Bremsvorgang.
• 7 stellt eine für die Fahrstabilität relevante Größe gemäß der vorliegenden Erfindung während eines Bremsvorgangs über der Zeit dar.
• 8 stellt eine charakteristische Größe gemäß der vorliegenden Erfindung für eine Stabilität des Fahrzeugs während eines Bremsvorgangs über der Zeit dar.
• 9 stellt weitere charakteristische Größen gemäß der vorliegenden Erfindung für die Fahrstabilität während eines Bremsvorgangs über der Zeit dar.
• 10 stellt eine schematische Ansicht eines Fahrzeugs von oben mit auf das Fahrzeug wirkenden Kräften dar, wobei sich die rechten Räder des Fahrzeugs auf einem Untergrund befinden, welcher einen anderen Reibungskoeffizienten als der Untergrund, auf dem sich die linken Räder des Fahrzeugs befinden, aufweist.
• 11 zeigt ein Diagramm, welches den Verlauf eines für die Fahrstabilität des Fahrzeugs charakteristischen Wertes gemäß der vorliegenden Erfindung während eines Bremsvorgangs über der Zeit darstellt.
• 12 zeigt eine Darstellung, welche die gemäß der vorliegenden Erfindung geregelten Bremsdrücke an vier Rädern eines Fahrzeugs über der Zeit während eines Bremsvorgangs darstellt.
• 13 zeigt einen Schräglaufwinkel der Hinterräder während eines Bremsvorgangs über der Zeit.
• 14 zeigt einen Verlauf eines Lenkradwinkels während eines Bremsvorgangs über der Zeit.

The present invention will be explained below with reference to the drawings based on preferred embodiments.

• 1 shows a schematic block diagram of a device for adjusting a brake pressure for a vehicle according to an embodiment of the present invention.
• 2 shows a schematic representation of a view of a vehicle from above with forces and moments acting on the vehicle.
• 3 shows a schematic representation of a vehicle with forces acting on the vehicle to determine the wheel loads.
• 4 shows a diagram showing the wheel loads of the wheels of a vehicle over time when braking.
• 5 Figure 11 is a graph showing the lateral force coefficients of the wheels of a vehicle over time during braking.
• 6 shows in four diagrams for each wheel of the vehicle respectively simulated lateral forces and according to one embodiment, certain lateral forces over time in a braking operation.
• 7 represents a relevant for the driving stability size according to the present invention during a braking operation over time.
• 8th represents a characteristic quantity according to the present invention for a stability of the vehicle during a braking operation over time.
• 9 Figure 12 illustrates further characteristic quantities according to the present invention for the driving stability during a braking operation over time.
• 10 FIG. 12 illustrates a schematic top view of a vehicle with forces acting on the vehicle with the right wheels of the vehicle on a ground having a different coefficient of friction than the ground on which the left wheels of the vehicle are located. FIG.
• 11 FIG. 12 is a graph showing the progress of a value characteristic of the running stability of the vehicle according to the present invention during a braking operation over time. FIG.
• 12 FIG. 10 is a diagram illustrating the braking pressures on four wheels of a vehicle over time during a braking operation regulated in accordance with the present invention. FIG.
• 13 shows a slip angle of the rear wheels during a braking operation over time.
• 14 shows a course of a steering wheel angle during a braking operation over time.

1 shows a device 1 for adjusting a brake pressure for a vehicle. The device 1 includes a device 2 for determining a running stability of a vehicle during braking and a processing unit 3 , The device 2 for determining the driving stability comprises a processing unit 5 , which with sensors 4 of the vehicle is coupled. With the help of the sensors 4 become the processing unit 5 Angular velocities ν _VL , ν _VR , ν _HL , ν _HR for a front left wheel, a front right wheel, a rear left wheel, and a rear right wheel of the vehicle and brake pressures p _B, V L _, p _{B, VR} , p _{B, HL} , p _{B, HR} for the front left wheel, the front right wheel, the rear left wheel, and the rear right wheel of the vehicle. The processing unit 5 determined from those of the sensors 4 provided values and other vehicle-specific information, a speed of the vehicle and Seitenkraftbeiwerte c _{α, VL} , c _{α, VR} , c _{α, HL} , c _{α, HR} for the front left wheel, the front right wheel, the rear left wheel and the rear right Rad. Another processing unit 6 the device 2 for determining the driving stability, from the lateral force coefficients, a stability value θ and slip angles α _VL , α _VR , α _HL , α _HR for the front left wheel, the rear left wheel, the front right wheel, and the rear right wheel of the vehicle are determined. A detailed description, as from the measured values of the sensors the values for the driving stability and the skew angles are determined will be described later. The processing units 5 and 6 Of course, it can also be realized as a processing unit.

The slip angles and the value for the driving stability are determined by the device 2 to the processing unit 3 the device 1 transmitted to adjust the brake pressure. In the processing unit 3 the slip angles are compared with a maximum allowable slip angle and a brake pressure 7 that of the processing unit 3 to a braking system 8th of the vehicle is set depending on the determined slip angle. For example, the brake pressure 7 with the help of a regulator 9 the processing unit 3 be reduced as soon as one of the slip angle exceeds a predetermined maximum slip angle, for example, 6 °. In addition, with the help of the regulator 9 and that of the device 2 certain value χ the driving stability of the brake pressure 7 set so that a current value of driving stability χ _is smaller than a predetermined value χ _soll is. The predetermined maximum value for the driving stability χ _should be, for example, -0.5 or another vehicle-specific size.

The to the brake system 8th the vehicle output brake pressure 7 For example, can set a brake pressure in the master cylinder of the vehicle. The brake system 8th The vehicle may further comprise an anti-lock device, a so-called anti-lock braking system (ABS). With the help of the anti-lock brake the brake pressures at the individual wheels are then further reduced upon detection of a wheel lock.

With reference to the 2-9 Below is the operation of the device 2 for determining the driving stability of the vehicle will be described in more detail. After that, referring to the 10-14 the operation of the device 1 for setting the brake pressure for the vehicle will be described in more detail.

In the following, a condition of driving stability of a vehicle during braking and its determination in real driving will be described. To ensure a stable braking process, this stability condition must always be met.

Equations of motion of motor vehicles when braking

First, referring to 2 considered a minimum two-lane overall vehicle model to determine overall vehicle behavior during braking.

With Ψ the yaw angle of the vehicle is compared to a spatially fixed coordinate system x ₀ . y ₀ designated. An angle β between a vehicle longitudinal axis 10 and a direction of the vehicle speed v at the vehicle's center of gravity 11 is called slip angle β. An angle Ψ + β between a parallel to the x ₀ Axis and the driving speed direction is called the course angle. Accordingly, the vector of velocity v rotates at an angular velocity Ψ̇ + β̇̇ in the spatially fixed coordinate system x ₀ . y ₀ , An acceleration of the vehicle's center of gravity 11 in the spatially fixed coordinate system, ie the derivation of the velocity vector of the vehicle's center of gravity 11 in the spatially fixed coordinate system, is thus composed of a so-called centripetal acceleration ν (Ψ̇ + β̇̇) and a relative acceleration v together. These accelerations are in 2 located.

$$\text{tan}\left({\delta }_V-{\alpha }_{VR}\right)=\frac{\nu \text{ sin}\beta +l_V\dot{\psi }}{\nu \text{ cos}\beta +b_V\dot{\psi }},$$

$$\text{tan}\left({\alpha }_{HL}\right)=\frac{-\nu \text{ sin}\beta +l_H\dot{\psi }}{\nu \text{ cos}\beta -b_H\dot{\psi }},$$

$$\text{tan}\left({\alpha }_{HR}\right)=\frac{-\nu \text{ sin}\beta +l_H\dot{\psi }}{\nu \text{ cos}\beta -b_H\dot{\psi }}.$$

Angle α _VL , α _VR , α _HL , α _HR between Radmittelebenen and the respective speed directions of the wheels 12 - 15 are called slip angle. Because of the kinematic constraints they can be expressed by the slip angle β and the yaw angular velocity Ψ̇ by equations (1) to (4): $$\text{tan}\left({\delta }_V-{\alpha }_{VL}\right)=\frac{\nu \text{sin}\beta +l_V\dot{\psi }}{\nu \text{cos}\beta -b_V\dot{\psi }}.$$

$$\text{tan}\left({\delta }_V-{\alpha }_{VR}\right)=\frac{\nu \text{sin}\beta +l_V\dot{\psi }}{\nu \text{cos}\beta +b_V\dot{\psi }}.$$

$$\text{tan}\left({\alpha }_{HL}\right)=\frac{-\nu \text{sin}\beta +l_H\dot{\psi }}{\nu \text{cos}\beta -b_H\dot{\psi }}.$$

$$\text{tan}\left({\alpha }_{HR}\right)=\frac{-\nu \text{sin}\beta +l_H\dot{\psi }}{\nu \text{cos}\beta -b_H\dot{\psi }},$$

It is assumed that the wheels 12 . 13 a steerable axle (in this case, the front axle) with a same steering angle δ _{V are taken in} parallel.

$$\text{tan}\left({\alpha }_H\right)=\frac{\nu \text{ sin}\beta +l_H\dot{\psi }}{\nu \text{cos}\beta }.$$

As a rule, half-track widths b _V and b _{H are} small compared to a course radius ρ = ν / (Ψ̇ + β̇̇), ie b _V << ρ ≈ v / Ψ̇ and b _H << ρ ≈ ν / Ψ̇, ie b _V Ψ̇ << ν, b _H Ψ̇ << ν. The terms b _V Ψ̇ and b _H Ψ̇ in equations (1) to (4) can thus be neglected. The slip angles at the wheels of an axle are practically the same in this case and can thus be determined from equations (5) and (6): $$\text{tan}\left({\delta }_V-{\alpha }_V\right)=\frac{\nu \text{sin}\beta +l_V\dot{\psi }}{\nu \text{cos}\beta }.$$

$$\text{tan}\left({\alpha }_H\right)=\frac{\nu \text{sin}\beta +l_H\dot{\psi }}{\nu \text{cos}\beta },$$

$${\alpha }_H=-\beta +l_H\frac{\dot{\psi }}{\nu }.$$

Since the angles δ _V -α _V , α _H and β are small during normal travel, linearization of equations (5) and (6) is allowed. Thus, equations (7) and (8) result: $${\alpha }_V={\delta }_V-\beta -l_V\frac{\dot{\psi }}{\nu }.$$

$${\alpha }_H=-\beta +l_H\frac{\dot{\psi }}{\nu },$$

noch die Seitenkräfte $F_{VL}^S,F_{VR}^S,F_{HL}^S$

und $F_{HR}^S.$

Diese Kräfte sind ebenfalls in 2 eingezeichnet. Die Luftkräfte werden hier vernachlässigt.During a general braking operation, the vehicle acts in addition to the braking forces (circumferential forces) $F_{VL}^U.F_{VR}^U.F_{HL}^U\text{and}{F}_{HR}^U$

still the lateral forces $F_{VL}^S.F_{VR}^S.F_{HL}^S$

and $F_{HR}^S,$

These forces are also in 2 located. The air forces are neglected here.

With the mass of the vehicle m and a moment of inertia about the vehicle vertical _axis J _Z are set up kinetic equations for the vehicle model:

The momentum balance (force equilibrium) in the vehicle longitudinal direction yields equation (9): $$m\nu \left(\dot{\psi }+\dot{\beta }\right)\text{sin}\dot{\beta }-m\dot{\nu }\text{cos}\beta -\left(F_{HL}^U+F_{HR}^U\right)-\left(F_{VL}^U+F_{VR}^U\right)\text{cos}{\delta }_V-\left(F_{VL}^S+F_{VR}^S\right)\text{sin}{\delta }_V=0$$

From the set of impulses (equilibrium of forces) in the vehicle transverse direction, equation (10) results: $$m\nu \left(\dot{\psi }+\dot{\beta }\right)\text{cos}\beta -m\dot{\nu }\text{sin}\beta -\left(F_{HL}^S+F_{HR}^S\right)-\left(F_{VL}^S+F_{VR}^S\right)\text{cos}{\delta }_V+\left(F_{VL}^U+F_{VR}^U\right)\text{sin}{\delta }_V=0$$

From the spin set (moment equilibrium) around the vehicle vertical axis results equation (11) $$\begin{array}{l}{J}_Z\ddot{\psi }-l_V\left(F_{VL}^S+F_{VR}^S\right)\text{cos}{\delta }_V+l_V\left(F_{VL}^U+F_{VR}^U\right)\text{sin}{\delta }_V+l_H\left(F_{HL}^S+F_{HR}^S\right)\hfill \\ +b_V\left(F_{VR}^U\text{cos}{\delta }_V-F_{VL}^U\text{cos}{\delta }_V+F_{VR}^S\text{sin}{\delta }_V-F_{VL}^S\text{sin}{\delta }_V\right)+b_H\left(F_{HR}^U-F_{HL}^U\right)=\mathrm{0th}\hfill \end{array}$$

$$m\nu \left(\dot{\psi }+\dot{\beta }\right)+m\dot{\nu }\beta -\left(F_{HL}^S+F_{HR}^S\right)-\left(F_{VL}^S+F_{VR}^S\right)+\left(F_{VL}^U+F_{VR}^U\right){\delta }_V=\mathrm{0,}$$

$$\begin{array}{l}{J}_Z\ddot{\psi }-l_V\left(F_{VL}^S+F_{VR}^S\right)+l_V\left(F_{VL}^U+F_{VR}^U\right){\delta }_V+l_H\left(F_{HL}^S+F_{HR}^S\right)\hfill \\ +b_V\left(F_{VR}^U-F_{VL}^U\right)+b_H\left(F_{HR}^U-F_{HL}^U\right)=0.\hfill \end{array}$$

Since the angle quantities are small during a normal braking operation, the equations (9) to (11) are simplified without much deviation from equations (12) to (14): $$\dot{m}\nu +\left(F_{HL}^U+F_{HR}^U\right)+\left(F_{VL}^U+F_{VR}^U\right)=0$$

$$m\nu \left(\dot{\psi }+\dot{\beta }\right)+m\dot{\nu }\beta -\left(F_{HL}^S+F_{HR}^S\right)-\left(F_{VL}^S+F_{VR}^S\right)+\left(F_{VL}^U+F_{VR}^U\right){\delta }_V=0$$

$$\begin{array}{l}{J}_Z\ddot{\psi }-l_V\left(F_{VL}^S+F_{VR}^S\right)+l_V\left(F_{VL}^U+F_{VR}^U\right){\delta }_V+l_H\left(F_{HL}^S+F_{HR}^S\right)\hfill \\ +b_V\left(F_{VR}^U-F_{VL}^U\right)+b_H\left(F_{HR}^U-F_{HL}^U\right)=\mathrm{0th}\hfill \end{array}$$

lassen sich mit den Schräglaufwinkeln α_V, α_H und den Seitenkraftbeiwerten der Räder c_αVL, c_αVR, c_αHL, c_αHR in folgenden linearisierten Formen als Gleichungen (15) bis (18) ausdrücken: $$F_{VL}^S=c_{\alpha VL}{\alpha }_V,$$

$$F_{VR}^S=c_{\alpha VR}{\alpha }_V,$$

$$F_{HL}^S=c_{\alpha HL}{\alpha }_H,$$

$$F_{HR}^S=c_{\alpha HR}{\alpha }_H.$$

Equations (13) and (14) are relevant to driving stability and are considered in more detail. The lateral forces $F_{VL}^S.F_{VR}^S.F_{HL}^S\text{.}{F}_{HR}^S$

can be expressed with the _{slip angles} α _V , α _H and the lateral force _{coefficients of} the wheels c _αVL , c _αVR , c _αHL , c _αHR in the following linearized forms as equations (15) to (18): $$F_{VL}^S=c_{\alpha VL}{\alpha }_V.$$

$$F_{VR}^S=c_{\alpha VR}{\alpha }_V.$$

$$F_{HL}^S=c_{\alpha HL}{\alpha }_H.$$

$$F_{HR}^S=c_{\alpha HR}{\alpha }_H,$$

$$\begin{array}{c}\ddot{\psi }=\frac{l_V{c}_{\alpha V}}{J_Z}\left(-\beta +{\delta }_V-l_V\frac{\dot{\psi }}{\nu }\right)-\frac{l_H{c}_{\alpha H}}{J_Z}\left(-\beta +l_H\frac{\dot{\psi }}{\nu }\right)-\frac{l_{\nu }\left(F_{VL}^U+F_{VR}^U\right)}{J_Z}{\delta }_V\\ +\frac{b_V\left(F_{VL}^U-F_{VR}^U\right)+b_H\left(F_{HL}^U-F_{HR}^U\right)}{J_Z}.\end{array}$$

If these relationships and equations (7) and (8) are used in equations (13) and (14), equations (19) and (20) are obtained: $$\dot{\beta }=\frac{c_{\alpha V}}{m\nu }\left(-\beta +{\delta }_V-l_V\frac{\dot{\psi }}{\nu }\right)+\frac{c_{\alpha H}}{m\nu }\left(-\beta +l_H\frac{\dot{\psi }}{\nu }\right)-\frac{F_{VL}^U+F_{VR}^U}{m\nu }{\delta }_V-\frac{\dot{\nu }}{\nu }\beta -\dot{\psi }.$$

$$\begin{array}{c}\ddot{\psi }=\frac{l_V{c}_{\alpha V}}{J_Z}\left(-\beta +{\delta }_V-l_V\frac{\dot{\psi }}{\nu }\right)-\frac{l_H{c}_{\alpha H}}{J_Z}\left(-\beta +l_H\frac{\dot{\psi }}{\nu }\right)-\frac{l_{\nu }\left(F_{VL}^U+F_{VR}^U\right)}{J_Z}{\delta }_V\\ +\frac{b_V\left(F_{VL}^U-F_{VR}^U\right)+b_H\left(F_{HL}^U-F_{HR}^U\right)}{J_Z},\end{array}$$

Here, c _αV = c _αVL + c _αVR and c _αH = c _αHL , + c _{αHR are} the lateral force _{coefficients of} all wheels of the front and rear axles. Equations (19) and (20) thus essentially correspond to a one-track model with the addition of an additional yaw moment, which will arise with asymmetric braking forces.

$$\begin{array}{c}\ddot{\psi }=\left(-\frac{l_V{c}_{\alpha V}}{J_Z}+\frac{l_H{c}_{\alpha H}}{J_Z}\right)\beta +\left(-\frac{l_V^2{c}_{\alpha V}}{J_Z\nu }-\frac{l_H^2{c}_{\alpha H}}{J_Z\nu }\right)\dot{\psi }+\left(\frac{l_V{c}_{\alpha V}}{J_Z}-\frac{l_V\left(F_{VL}^U+F_{VR}^U\right)}{J_Z}\right){\delta }_V\\ +\frac{b_V\left(F_{VL}^U-F_{VR}^U\right)+b_H\left(F_{HL}^U-F_{HR}^U\right)}{J_Z}.\end{array}$$

oder in Matrixschreibweise die Gleichung (23): $$\dot{x}=Ax+b{\delta }_V+\upsilon$$

mit dem Zustandsvektor $$\mathbf{x}=\left[{}_{\dot{\psi }}^{\beta }\right],$$

der Systemmatrix $$\mathbf{A}=\left[\begin{array}{c}\frac{-c_{\alpha V}-c_{\alpha H}-m\dot{\nu }}{m\nu }\frac{-l_V{c}_{\alpha V}+l_H{c}_{\alpha H}-m{\nu }^2}{m{\nu }^2}\\ \frac{-l_V{c}_{\alpha V}+l_H{c}_{\alpha H}}{J_Z}\frac{-l_V^2{c}_{\alpha V}-l_H^2{c}_{\alpha H}}{J_Z\nu }\end{array}\right],$$

der Eingangsmatrix $$\mathbf{b}=\left[\begin{array}{c}\frac{c_{\alpha V}-F_{VL}^U-F_{VR}^U}{m\nu }\\ \frac{l_V\left(c_{\alpha V}-F_{VL}^U-F_{VR}^U\right)}{J_Z}\end{array}\right]$$

und dem Vektor der Störgrößen $$\upsilon =\left[\frac{b_V\left(F_{VL}^U-F_{VR}^U\right){+}^0{b}_H\left(F_{HL}^U-F_{HR}^U\right)}{J_Z}\right].$$

A corresponding state space representation of equations (19) and (20) yields equations (21) and (22): $$\dot{\beta }=\left(-\frac{c_{\alpha V}}{m\nu }-\frac{c_{\alpha H}}{m\nu }-\frac{\dot{\nu }}{\nu }\right)\beta +\left(-\frac{c_{\alpha V}{l}_V}{m{\nu }^2}+\frac{c_{\alpha H}{l}_H}{m{\nu }^2}-1\right)\dot{\psi }+\left(\frac{c_{\alpha V}}{m\nu }-\frac{F_{VL}^U+F_{VR}^U}{m\nu }\right){\delta }_V.$$

$$\begin{array}{c}\ddot{\psi }=\left(-\frac{l_V{c}_{\alpha V}}{J_Z}+\frac{l_H{c}_{\alpha H}}{J_Z}\right)\beta +\left(-\frac{l_V^2{c}_{\alpha V}}{J_Z\nu }-\frac{l_H^2{c}_{\alpha H}}{J_Z\nu }\right)\dot{\psi }+\left(\frac{l_V{c}_{\alpha V}}{J_Z}-\frac{l_V\left(F_{VL}^U+F_{VR}^U\right)}{J_Z}\right){\delta }_V\\ +\frac{b_V\left(F_{VL}^U-F_{VR}^U\right)+b_H\left(F_{HL}^U-F_{HR}^U\right)}{J_Z},\end{array}$$

or in matrix notation the equation (23): $$\dot{x}=Ax+b{\delta }_V+\upsilon$$

with the state vector $$\mathbf{x}=\left[{}_{\dot{\psi }}^{\beta }\right].$$

the system matrix $$\mathbf{A}=\left[\begin{array}{c}\frac{-c_{\alpha V}-c_{\alpha H}-m\dot{\nu }}{m\nu }\frac{-l_V{c}_{\alpha V}+l_H{c}_{\alpha H}-m{\nu }^2}{m{\nu }^2}\\ \frac{-l_V{c}_{\alpha V}+l_H{c}_{\alpha H}}{J_Z}\frac{-l_V^2{c}_{\alpha V}-l_H^2{c}_{\alpha H}}{J_Z\nu }\end{array}\right].$$

the input matrix $$\mathbf{b}=\left[\begin{array}{c}\frac{c_{\alpha V}-F_{VL}^U-F_{VR}^U}{m\nu }\\ \frac{l_V\left(c_{\alpha V}-F_{VL}^U-F_{VR}^U\right)}{J_Z}\end{array}\right]$$

and the vector of the disturbances $$\upsilon =\left[\frac{b_V\left(F_{VL}^U-F_{VR}^U\right){+}^0{b}_H\left(F_{HL}^U-F_{HR}^U\right)}{J_Z}\right],$$

It can be seen that this is a linear time-varying system because the velocity v in the system and input matrix constantly changes during braking. Accordingly, the characteristics of the system are also variable over the driving speed. In the following investigations, however, the speed is first "frozen" for simplicity. This is also allowed because the longitudinal dynamics of the vehicle according to equation (12) during a normal braking operation can be considered practically decoupled from lateral dynamics. If such quasi-stationary investigations are transferred to other speeds, one obtains an overall picture of the system properties, such as e.g. controllability and stability.

Controllability of motor vehicles when braking

The controllability matrix of the system according to equation (23) is: $$\begin{array}{l}\mathbf{Q}=\left[\mathbf{b}AB\right]=\hfill \\ \left[\begin{array}{c}\frac{c_{\alpha V}-F_{VL}^U-F_{VR}^U}{m\nu }\frac{-c_{\alpha V}-c_{\alpha H}-m\dot{\nu }}{m\nu }\frac{c_{\alpha V}-F_{VL}^U-F_{VR}^U}{m\nu }+\frac{-l_V{c}_{\alpha V}+l_H{c}_{\alpha H}-m{\nu }^2}{m{\nu }^2}\frac{l_V\left(c_{\alpha V}-F_{VL}^U-F_{VR}^U\right)}{J_Z}\\ \frac{l_V\left(c_{\alpha V}-F_{VL}^U-F_{VR}^U\right)}{J_Z}\frac{-l_V{c}_{\alpha V}+l_H{c}_{\alpha H}}{J_Z}\frac{c_{\alpha V}-F_{VL}^U-F_{VR}^U}{m\nu }+\frac{-l_V^2{c}_{\alpha V}-l_H^2{c}_{\alpha H}}{J_Z\nu }\frac{l_V\left(c_{\alpha V}-F_{VL}^U-F_{VR}^U\right)}{J_Z}\end{array}\right]\hfill \end{array},$$

liegt, können die letzteren vernachlässigt werden. Die Steuerbarkeitsmatrix reduziert sich somit auf Gleichung (24): $$\mathbf{Q}=\left[\begin{array}{c}\frac{c_{\alpha V}}{m\nu }\frac{-c_{\alpha V}-c_{\alpha H}}{m\nu }\frac{c_{\alpha V}}{m\nu }+\frac{-l_V{c}_{\alpha V}+l_H{c}_{\alpha H}-m{\nu }^2}{m{\nu }^2}\frac{l_V{c}_{\alpha V}}{J_Z}\\ \frac{l_V{c}_{\alpha V}}{J_Z}\frac{-l_V{c}_{\alpha V}+l_H{c}_{\alpha H}}{J_Z}\frac{c_{\alpha V}}{m\nu }\text{+}\frac{-l_V^2{c}_{\alpha V}-l_H^2{c}_{\alpha H}}{J_Z\nu }\frac{l_V{c}_{\alpha V}}{J_Z}\end{array}\right].$$

Since the lateral force _coefficient c _αV usually much higher than the braking forces $F_{VL}^U+F_{VR}^U$

the latter can be neglected. The controllability matrix is thus reduced to equation (24): $$\mathbf{Q}=\left[\begin{array}{c}\frac{c_{\alpha V}}{m\nu }\frac{-c_{\alpha V}-c_{\alpha H}}{m\nu }\frac{c_{\alpha V}}{m\nu }+\frac{-l_V{c}_{\alpha V}+l_H{c}_{\alpha H}-m{\nu }^2}{m{\nu }^2}\frac{l_V{c}_{\alpha V}}{J_Z}\\ \frac{l_V{c}_{\alpha V}}{J_Z}\frac{-l_V{c}_{\alpha V}+l_H{c}_{\alpha H}}{J_Z}\frac{c_{\alpha V}}{m\nu }\text{+}\frac{-l_V^2{c}_{\alpha V}-l_H^2{c}_{\alpha H}}{J_Z\nu }\frac{l_V{c}_{\alpha V}}{J_Z}\end{array}\right],$$

The system is controllable if and only if the matrix Q is regular, that is, equation (25) is satisfied $$\text{det}\mathbf{Q}=\frac{c_{\alpha V}^2}{m{\nu }^2{\mathrm{<figure-callout\; id="2"\; label="J\; Z"\; filenames="DE102008046259B4\_0100.png,DE102008046259B4\_0103.png"\; state="\{\{state\}\}">J\; </figure-callout>}}_Z^{}{\nu }^2}\left(J_Z\left(l_V+l_H\right)c_{\alpha H}-m{l}_V{l}_H\left(l_V+l_H\right)c_{\alpha H}+m^2{\mathrm{<figure-callout\; id="2"\; label="l\; V"\; filenames="DE102008046259B4\_0100.png,DE102008046259B4\_0103.png"\; state="\{\{state\}\}">l\; </figure-callout>}}_V^{}{\nu }^2\right)\ne \mathrm{0th}$$

in der Regel einen positiven Wert an. Die notwendige Bedingung für die Steuerbarkeit des Fahrzeugs lautet somit: c_αV ≠ 0, d.h. der Seitenkraftbeiwert der Vorderachse muss einen von Null verschiedenen Wert (in der Praxis einen genügend großen Wert wegen der oben getroffenen Annahmen zur Vereinfachung) aufweisen. Aus diesem Grund sollen die lenkenden Vorderräder beim Bremsen nicht blockieren.Since in a car approximately J _Z ≈ ml _V l _H holds, the term in brackets $J_Z\left(l_V+l_H\right)c_{\alpha H}-m{l}_V{l}_H\left(l_V+l_H\right)c_{\alpha H}+m^2{\mathrm{<figure-callout\; id="2"\; label="l\; V"\; filenames="DE102008046259B4\_0100.png,DE102008046259B4\_0103.png"\; state="\{\{state\}\}">l\; </figure-callout>}}_V^{}{\mathrm{<figure-callout\; id="2"\; label="\nu "\; filenames="DE102008046259B4\_0100.png,DE102008046259B4\_0103.png"\; state="\{\{state\}\}">\nu </figure-callout>}}^2$

usually a positive value. The necessary condition for the steerability of the vehicle is thus: c _αV ≠ 0, ie the lateral force _{coefficient of} the front axle must have a non-zero value (in practice a sufficiently large value because of the simplification assumptions made above). For this reason, the steering front wheels should not block when braking.

Stability condition of motor vehicles during braking

The ABS system prevents the wheels from locking. Thus, the controllability of the vehicle is ensured. Theoretically, steering can compensate for the disturbances on the vehicle and even correct the skidding of the stern. The driver acts in this case as a state controller of the form of equation (26). $${\delta }_V=\mathbf{Rx},$$

Where R is a 1 × 2 feedback matrix.

If equation (26) is used in (23), the differential equation of the state control (equation of state of the closed-loop system) according to equation (27) results: $$\dot{x}=\left(A+bR\right)x+\upsilon ,$$

The system matrix, which governs the system properties, is changed by the feedback. That is, one can stabilize an actually unstable vehicle by suitable choice of the feedback matrix. A skilled driver can do that most of the time, a normal driver (and that's the majority), however, often can not.

Studies have shown that a normal skier usually does not steer properly and fast enough during a skid. It must always be expected that the driver can intervene only after a period of time that corresponds approximately to his reaction time, corrective. For these reasons, in the following stability study, the native behavior of the vehicle, ie without Driver interventions, closer look. This is expressed by the homogeneous state differential equation ẋ = Ax according to equation (28). $$\left[{}_{\ddot{\psi }}^{\dot{\beta }}\right]=\left[\begin{array}{c}\frac{c_{\alpha V}-c_{\alpha H}-m\dot{\nu }}{m\nu }\frac{-l_V{c}_{\alpha V}+l_H{c}_{\alpha H}-m{\nu }^2}{m{\nu }^2}\\ \frac{-l_V{c}_{\alpha V}+l_H{c}_{\alpha H}}{J_Z}\frac{-l_V^2{c}_{\alpha V}-l_H^2{c}_{\alpha H}}{J_Z\nu }\end{array}\right]\left[{}_{\dot{\psi }}^{\beta }\right]$$

The solution of this equation (28) for eg the yaw rate is described by equation (29): $$\dot{\psi }={\zeta }_1{e}^{{\lambda }_1\text{'}}+{\zeta }_2{e}^{{\lambda }_2\text{'}},$$

mit $$2\sigma =\frac{m\left(l_V^2{c}_{\alpha V}+l_H^2{c}_{\alpha H}\right)+J_Z\left(c_{\alpha V}+c_{\alpha H}+m\dot{\nu }\right)}{J_{Z}m\nu }$$

und $${\omega }^2=\frac{{\left(l_V+L_H\right)}^2{c}_{\alpha V}{c}_{\alpha H}+m{\nu }^2\left(l_H{c}_{\alpha H}-l_V{c}_{\alpha V}\right)+m\dot{\nu }\left(l_V^2{c}_{\alpha V}+l_H^2{c}_{\alpha H}\right)}{J_{Z}m{\nu }^2}.$$

The coefficients ζ _1, ζ _{2 are} determined by initial conditions of the state variables β (t = 0) and ψ̇ (t = 0). λ ₁ , λ ₂ are the eigenvalues of the system matrix A, ie the solutions of the characteristic equation of the system according to equations (30) to (32): $$\text{det}\left(\lambda \mathbf{I}-\mathbf{A}\right)={\mathrm{<figure-callout\; id="2"\; label="\lambda "\; filenames="DE102008046259B4\_0100.png,DE102008046259B4\_0103.png"\; state="\{\{state\}\}">\lambda </figure-callout>}}^2+2\sigma \lambda +{\mathrm{<figure-callout\; id="2"\; label="\omega "\; filenames="DE102008046259B4\_0100.png,DE102008046259B4\_0103.png"\; state="\{\{state\}\}">\omega </figure-callout>}}^2=0$$

With $$2\sigma =\frac{m\left(l_V^2{c}_{\alpha V}+l_H^2{c}_{\alpha H}\right)+J_Z\left(c_{\alpha V}+c_{\alpha H}+m\dot{\nu }\right)}{J_{Z}m\nu }$$

and $${\omega }^2=\frac{{\left(l_V+L_H\right)}^2{c}_{\alpha V}{c}_{\alpha H}+m{\nu }^2\left(l_H{c}_{\alpha H}-l_V{c}_{\alpha V}\right)+m\dot{\nu }\left(l_V^2{c}_{\alpha V}+l_H^2{c}_{\alpha H}\right)}{J_{Z}m{\nu }^2},$$

It follows immediately equation (33) $${\lambda }_{1.2}=-\sigma \pm \sqrt{{\sigma }^2-{\omega }^2},$$

From the solution of the homogeneous equation (29) it can be seen that the stability requirement is satisfied if the real parts of the eigenvalues λ ₁ and λ _{2 are} negative, ie equation (34) is satisfied: $$\text{re}\left({\lambda }_{1.2}\right)<\mathrm{0th}$$

It is then that the system always strives against a steady state under a fault. Otherwise, no equilibrium position (steady state) is achieved.

The eigenvalues are not only decisive for the stability but also largely influence the time behavior of the system. The more "negative" Re (λ _1,2 ) is, the faster a transition process will sound. As a driver, you immediately notice how easily the vehicle can be controlled.

For real vehicles, σ is almost always a positive quantity. On the other hand, ω ² can take both a positive and a negative value and therefore represents a characteristic variable for the stability.

For the case ω ² > 0, Re (λ _1,2 ) <0 always applies, so the vehicle is stable. If σ ² - ω ² > 0, λ ₁ and λ _{2 are} negatively real. One has in this case a monotonous time process. For σ ² - ω ² <0, λ ₁ and λ _{2 are} conjugate complex. Such a conjugate complex eigenvalue pair includes a decaying vibrational process.

der beim Bremsen zwar einen vergleichsweise kleinen negativen Wert annimmt, führt aber zur weiteren Verschärfung der Situation.For the case ω ² <0, one of the eigenvalues becomes positive. This will change the stability condition ( 34 ) is no longer fulfilled. This situation can occur if the term l _H c _αH - l _V c _{αV falls below} a certain negative value (in this case, the vehicle is overdriven because of I _H c _αH -l _V c _αV <0). The term $m\dot{\nu }\left(l_V^2{c}_{\alpha V}+l_H^2{c}_{\alpha H}\right)$

Although the braking takes a comparatively small negative value, but leads to further aggravation of the situation.

Since the term l _H c _αH -l _V c _{αV is multiplied} by ν ² , instability at higher speeds can already be caused by braking, because of the load on the front axle and the relief of the rear axle in the braking maneuver increases the Seitenkraftbeiwert the front axle and the lateral force coefficient of the rear axle. If a lateral acceleration also acts on the vehicle, the wheel loads will shift between left and right.

Wheel load change of motor vehicles during braking

The following are with reference to 3 Wheel load changes during braking and their influence on the stability of the vehicle with the help of a simple model considered qualitatively. The vehicle of the 3 includes a rear axle 16 , a front axle 17 and wheels 12-15 ,

unter Berücksichtigung von Längs- und Querbeschleunigung in erster Nährung durch Gleichungen (35) bis (38) ausdrücken: $$F_{VL}^Z=\frac{1}{2}mg\frac{l_H}{l}-\frac{1}{2}m{a}_x\frac{h}{l}-\epsilon m{a}_y\frac{h}{2b},$$

$$F_{VR}^Z=\frac{1}{2}mg\frac{l_H}{l}-\frac{1}{2}m{a}_x\frac{h}{l}+\epsilon m{a}_y\frac{h}{2b},$$

$$F_{HL}^Z=\frac{1}{2}mg\frac{l_V}{l}+\frac{1}{2}m{a}_x\frac{h}{l}-\left(1-\epsilon \right)m{a}_y\frac{h}{2b},$$

$$F_{HR}^Z=\frac{1}{2}mg\frac{l_V}{l}+\frac{1}{2}m{a}_x\frac{h}{l}+\left(1-\epsilon \right)m{a}_y\frac{h}{2b}.$$

To 3 let the wheel loads $F_{VL}^Z.F_{VR}^Z.F_{HL}^Z.F_{HR}^Z$

Expressing longitudinal and lateral acceleration in first approximation by equations (35) to (38): $$F_{VL}^Z=\frac{1}{2}mG\frac{l_H}{l}-\frac{1}{2}m{a}_x\frac{H}{l}-\epsilon m{a}_y\frac{H}{2b}.$$

$$F_{VR}^Z=\frac{1}{2}mG\frac{l_H}{l}-\frac{1}{2}m{a}_x\frac{H}{l}+\epsilon m{a}_y\frac{H}{2b}.$$

$$F_{HL}^Z=\frac{1}{2}mG\frac{l_V}{l}+\frac{1}{2}m{a}_x\frac{H}{l}-\left(1-\epsilon \right)m{a}_y\frac{H}{2b}.$$

$$F_{HR}^Z=\frac{1}{2}mG\frac{l_V}{l}+\frac{1}{2}m{a}_x\frac{H}{l}+\left(1-\epsilon \right)m{a}_y\frac{H}{2b},$$

It is assumed simplifying that the gauges of the front and rear axles are identical, namely b _V = b _H = b. With ε the differences of the Aufbaufeder- and the stabilizer hardness between the front and the rear axle are considered. By adjusting the Aufbaufeder- and the stabilizer hardness, the lateral Radlastverlagerung can be distributed due to the lateral acceleration targeted to the front and rear axles.

It can be seen from equations (35) to (38) that the wheel loads are decisively influenced by the center of gravity position l _V / l, l _H / l, h / l and h / 2b. This is primarily due to the vehicle concept and changes after load level. A higher ratio of the center of gravity height to the wheelbase h / l requires a greater wheel load displacement from rear to front wheels during braking. In the presence of a lateral acceleration, a higher ratio of the center of gravity height to the track h / 2b results in a larger wheel load shift between left 12, 15 and right wheels 13 . 14 , On a one-sided slippery road surface (μ-split), such a shift will strongly influence the lateral force coefficients of the axles. Too much relief of the wheels on the grippier side can lead eg to the loss of lateral guidance of the axles.

In addition to the wheel load and the coefficient of friction, the lateral force coefficients still depend on the circumferential slip of the wheels. The tire properties are considered in more detail below.

Tire properties and determination of lateral force coefficients

To describe the tire properties, the HSRI tire model is used, because this based on vehicle-technical basics model provides insight into the relationships and dependencies between the different sizes (forces, slip angle, circumferential slip, etc.).

The calculation is based on the deformations of the tire tread particles and the resulting shear stresses in the laces. Whether the profile particles in the landslide already partially reach the adhesion limit is determined by an auxiliary size s characterized according to equation (39). $$\overline{S}=\frac{\sqrt{{\left(C_s^{0}S\right)}^2+{\left({\mathrm{<figure-callout\; id="0"\; label="c\; \alpha "\; filenames="DE102008046259B4\_0103.png,DE102008046259B4\_0104.png"\; state="\{\{state\}\}">c\; </figure-callout>}}_{\alpha }^{}\text{tan}\alpha \right)}^2}}{\mu F^Z\left(1-S\right)}$$

die Nullpunkttangente der Umfangskraft-Umfangsschlupf-Kurve bei Schräglaufwinkel α = 0. Analog ist $c_{\alpha }^0$

die Nullpunkttangente der Seitenkraft-Schräglaufwinkel-Kurve bei Umfangsschlupf S = 0. Die Größen $c_s^0$

und $c_{\alpha }^0$

hängen in erster Linie von der Radlast nach Gleichungen (40) und (41) ab: $$c_s^0={\kappa }_s{F}^Z$$

$$c_{\alpha }^0=\left({\kappa }_{\alpha 1}+{\kappa }_{\alpha 2}\frac{F^Z}{F_{\text{Nenn}}^Z}\right)F^Z.$$

S denotes the circumferential slip, α the slip angle and μ the friction coefficient of a wheel. Further is $c_s^0$

the zero point tangent of the circumferential force-circumferential slip curve at slip angle α = 0. Analogous $c_{\alpha }^{}$${}_0^{}$

the zero point tangent of the lateral force skew angle curve with circumferential slippage S = 0. The sizes $c_s^{}$${}_0^{}$

and ${\mathrm{<figure-callout\; id="0"\; label="c\; \alpha "\; filenames="DE102008046259B4\_0103.png,DE102008046259B4\_0104.png"\; state="\{\{state\}\}">c\; </figure-callout>}}_{\alpha }^{}$

depend primarily on the wheel load according to equations (40) and (41): $$c_s^0={\kappa }_s{F}^Z$$

$$c_{\alpha }^0=\left({\kappa }_{\alpha 1}+{\kappa }_{\alpha 2}\frac{F^Z}{F_{\text{nominal}}^Z}\right)F^Z,$$

• Eine Umfangskraft wird durch Gleichung (42) beschrieben $$F^U=c_s^{0}S,$$
  

In the case S ≤ 0.5, all profile particles in the laces are still in the adhesion area, so the adhesion limit is not yet reached. The forces result from the integration of the voltages across the Laces and can be expressed in the following way:

• A circumferential force is described by Equation (42) $$F^U=c_s^{0}S.$$
  

A lateral force is described by Equation (43) $$F^S={\mathrm{<figure-callout\; id="0"\; label="c\; \alpha "\; filenames="DE102008046259B4\_0103.png,DE102008046259B4\_0104.png"\; state="\{\{state\}\}">c\; </figure-callout>}}_{\alpha }^{}\text{tan}\alpha$$

• Die Umfangskraft ergibt sich zu Gleichung (44).

$$F^U=\frac{c_s^{0}S}{1-S}\frac{\overline{S}-\mathrm{0,25}}{{\overline{S}}^2},$$

In the case> 0.5, the profile particles slide partially or even completely in the laces. By integrating the stresses in the adhesion and sliding area over the Laces, the forces are as follows:

• The circumferential force is given by equation (44).

$$F^U=\frac{c_s^{0}S}{1-S}\frac{\overline{S}-0.25}{{\overline{S}}^2}.$$

The lateral force results in equation (45). $$F^S=\frac{c_{\alpha }^0\text{tan}\alpha }{1-S}\frac{\overline{S}-0.25}{{\overline{S}}^2},$$

folglich $$c_{\alpha }\approx \frac{F^S}{\text{tan}\alpha }=\frac{c_{\alpha }^0}{c_s^0}\frac{F^U}{S}.$$

It should be noted that equations (42) and (43) or (44) and (45) show a simple relationship between the circumferential and lateral forces according to equations (46) and (47): $$\frac{F^S}{\text{tan}\alpha }=\frac{c_{\alpha }^0}{c_s^0}\frac{F^U}{S}$$

consequently $$c_{\alpha }\approx \frac{F^S}{\text{tan}\alpha }=\frac{c_{\alpha }^0}{c_s^0}\frac{F^U}{S},$$

von der Radlast nach Gleichung (48) ab: $$\frac{c_{\alpha }^0}{c_s^0}=\frac{{\kappa }_{\alpha 1}+{\kappa }_{\alpha 2}\frac{F^Z}{F_{\text{Nenn}}^Z}}{{\kappa }_s}.$$

This relationship states that the lateral force coefficient c _α can be determined during braking by the circumferential force F ^U and the corresponding circumferential slip. According to equations (40) and (41), the coefficient depends ${\mathrm{<figure-callout\; id="0"\; label="c\; \alpha "\; filenames="DE102008046259B4\_0103.png,DE102008046259B4\_0104.png"\; state="\{\{state\}\}">c\; </figure-callout>}}_{\alpha }^{}/{\mathrm{<figure-callout\; id="0"\; label="c\; s"\; filenames="DE102008046259B4\_0103.png,DE102008046259B4\_0104.png"\; state="\{\{state\}\}">c\; </figure-callout>}}_s^{}$

from the wheel load according to equation (48): $$\frac{c_{\alpha }^0}{c_s^0}=\frac{{\kappa }_{\alpha 1}+{\kappa }_{\alpha 2}\frac{F^Z}{F_{\text{nominal}}^Z}}{{\kappa }_s},$$

abweicht, gilt Gleichung (49) $$\frac{c_{\alpha }^0}{c_s^0}=\frac{{\kappa }_{\alpha 1}}{{\kappa }_s}=\kappa =\text{Konstant}$$

und ist in der Regel eine gute Nährung, weil in diesem Fall κ_αl für den Zähler der Gleichung (48) maßgebend ist. Somit wird der Aufwand zur Radlastschätzung erspart. Gleichung (47) zur Berechnung des Seitenkraftbeiwerts kann dann durch Gleichung (50) ausgedrückt werden: $$c_{\alpha }\approx \kappa \frac{F^U}{S}.$$

If the wheel load F ^{Z is} not very strong from its face value $F_{\text{nominal}}^Z$

deviates, equation (49) applies $$\frac{c_{\alpha }^0}{{\mathrm{<figure-callout\; id="0"\; label="c\; s"\; filenames="DE102008046259B4\_0103.png,DE102008046259B4\_0104.png"\; state="\{\{state\}\}">c\; </figure-callout>}}_s^{}}=\frac{{\mathrm{<figure-callout\; id="1"\; label="\kappa \; \alpha "\; filenames="DE102008046259B4\_0103.png,DE102008046259B4\_0104.png"\; state="\{\{state\}\}">\kappa \; </figure-callout>}}_{\alpha }}{{\kappa }_s}=\kappa =\text{Constant}$$

and is usually a good nourishment, because in this case κ _{αl is decisive} for the numerator of equation (48). Thus, the effort for wheel load estimation is spared. Equation (47) for calculating the lateral force coefficient can then be expressed by equation (50): $$c_{\alpha }\approx \kappa \frac{F^U}{S},$$

The coefficient κ must be determined beforehand by measurements.

The circumferential force (braking force in this case) F ^U is not measured directly in the real vehicle and must be calculated via the brake pressure according to equation (51): $$F^U=\frac{p_{B}Ar{C}_B+J_{rad}\dot{\omega }+M_{MOtOr}}{R_{dyn}},$$

Here, p _{B is} the brake pressure, A the piston surface of the wheel brake cylinder, r the mean friction radius of the brake pad on the brake disc and C _B the brake characteristic. With ω̇ the angular acceleration of the wheel is called. Further, J _{Rad is} the moment of inertia of the wheel and all associated rotational masses, R _dyn the dynamic tire radius.

If not disengaged during braking, the required brake pressure of a drive wheel because of the acting thereon motor drag torque M _motor at the same slip is smaller. The higher the vehicle speed and the lower the engaged gear, the stronger the effect of the engine braking torque.

The approach to calculating the lateral force coefficient presented above automatically considers friction coefficient and wheel load changes and also works in combined maneuvers, e.g. with simultaneous occurrence of the circumferential slip and the skew angle. However, it is sensitive to very small circumferential slip values. However, this limitation is not serious for real use, since the braking stability only becomes an issue with larger circumferential slip values.

Determination of the stability condition in real driving

With the approach presented above, lateral force coefficients of the wheels during braking can be determined. Based on this, the quantity ω ² characteristic of the stability or the eigenvalues of the system λ _1,2 can be calculated. The following is a concrete example. This is the straight braking of a vehicle on a one-sided smooth road (ice on the left side and concrete on the right side) with an initial speed of 85 km / h. The vehicle has a high center of gravity and is equipped with an unmatched ABS. A human driver must try to drive the vehicle straight ahead.

The vehicle behaves restless during this braking process and is difficult to control by the driver. This maneuver is reconstructed in the computer with the aid of a multi-body overall vehicle model, so that metrologically complex parameters such as wheel loads, slip angles and lateral forces are made available for further investigations or for the validation of the presented method for determining the driving stability.

In 4 the wheel loads reconstructed by simulation are shown. The graph 18 shows the wheel load on the left front wheel, the graph 19 the wheel load on the right front wheel, the graph 20 the wheel load on the left rear wheel and the graph 21 the wheel load on the right rear wheel.

It can be seen clearly from this figure that the deceleration leads to wheel load shifting from the rear to the front axle, while the lateral acceleration caused by the yawing leads to wheel load shifting between the left and right wheels. The Radlastverlagerungen are pronounced in this case because of the high center of gravity of the vehicle. The rear wheel is particularly relieved on the grippier side of the road when the unequal braking forces turn the vehicle to this side and the driver does not initiate a corresponding correction by steering (eg by the second or the fourth second in 4 ).

In this case, it is also to be expected that the rear axle loses much of its cornering ability. In fact, this can be seen from the lateral force coefficients calculated using the method described above. 5 shows the calculated lateral force coefficients for the individual wheels of the vehicle. graph 22 shows the lateral force coefficient of the left front wheel, graph 23 the lateral force coefficient of the right front wheel, graph 24 the lateral force coefficient of the left rear wheel and graph 25 the lateral force coefficient of the right rear wheel.

To validate the calculated lateral force coefficients, they are multiplied by the wheel slip angles reconstructed by simulation. 6 shows the reconstructed lateral forces for each wheel 26 . 28 . 30 . 32 and the lateral forces calculated according to the above method 27 . 29 . 31 . 33 , 6 shows that the calculated lateral forces are in good agreement with the reconstructed lateral forces.

According to equations (31), (32) and (33), the quantities σ and ω ² and the eigenvalues (poles) λ _{1,2 are} calculated. The results are in 7 . 8th and 9 shown.

While the quantity σ always remains positive in this braking operation, the quantity ω ² becomes smaller by the second second 8th partially negative. Accordingly, the first eigenvalue (see 9 ) in this case a positive real part, which indicates instability of the vehicle. Due to the unexpected drivability change, the driver has difficulty capturing the turning of the vehicle. A common consequence of this is the overriding on the part of the driver, which partially amplifies the problem.

In 5 The constellation of lateral force coefficients did not improve after the fourth second. Consequently, the vehicle remains oversteering, ie, L _H c _{α H} <1 _V c _αV . To 8th Nevertheless, the vehicle has a stable behavior, since the driving speed in this case is not so high and an oversteering vehicle is unstable even at a certain speed. It follows that the braking forces can be increased accordingly with decreasing driving speed in order to achieve the shortest possible braking distance.

As described above, a method for determining a stability condition in braking has been derived. Here, ω ^{2 represents} a characteristic quantity and correlates well with the subjective perception of the driver of the controllability of the vehicle.

In addition, the method is suitable to allow the determination of the stability condition during braking of different types, for example, straight braking on a homogeneous and heterogeneous road, braking in the curve, etc.

nach Gleichungen (35), (36), (37), (38) eine gute Schätzung des Verhältnisses l_V /l_H dar. Da der Radstand l_V + l_H konstant ist, können l_V und l_H berechnet werden.It should be noted that the required sizes l _V . l _H , m, J _Z also dependent on the load condition and therefore are changeable. The variation of these quantities has a negative effect on the accuracy of the determination. With a certain amount of effort, these quantities can also be calculated. For example, the vehicle can be braked straight ahead so easily that the wheel load shift caused by the deceleration remains negligibly small. If the rotational speeds of the front and rear wheels are regulated in the same way, ie if the circumferential slip values of the wheels are the same, the ratio is set $F_V^U/F_H^U$

according to equations (35), (36), (37), (38) a good estimate of the ratio l _V / l _H dar. Since the wheelbase l _V + l _{H is} constant, can l _V and l _H be calculated.

On the other side are the variation ranges of sizes l _V . l _H , m and J _Z usually not significantly big. The resulting inaccuracies or deviations in the specific values of ω ² or Re (λ _1,2 ) will thus practically not prevent the use of the stability condition.

By use is meant here that one not only determines the stability condition, but also uses it selectively in the regulation of brake pressures, as will be described below. The aim is to make the best possible use of the grip of the wheels on the roadway when the stability condition is met.

Method for setting a brake pressure for a vehicle

When braking, the vehicle parameters which are decisive for the driving behavior change. If the parameters are changed too much, unwanted drivability, even extreme instability, can occur. A normal driver is usually surprised and overwhelmed in such situations. To avoid this, certain conditions must be met during braking. The fulfillment of these conditions can be achieved by a targeted regulation of the brake pressures. The aim of the regulation is in addition to the controllability of the vehicle also the shortest possible braking distance. In the following, the conditions to be fulfilled and the corresponding control strategies are presented.

Stability condition of the whole vehicle

The stability of vehicles during braking was previously examined using differential equations (23).

$$2\sigma =-\left(A_{11}+A_{22}\right)-\frac{m\left(l_V^2{c}_{\alpha V}+l_H^2{c}_{\alpha H}\right)+J_Z\left(c_{\alpha V}+c_{\alpha H}+m\dot{v}\right)}{J_{Z}mv}$$

$${\omega }^2=A_{11}{A}_{22}-A_{12}{A}_{21}=\frac{{\left(l_V+l_H\right)}^2{c}_{\alpha V}{c}_{\alpha H}+m{v}^2\left(l_H{c}_{\alpha H}-l_V{c}_{\alpha V}\right)+m\dot{v}\left(l_V^2{c}_{\alpha V}+l_H^2{c}_{\alpha H}\right)}{J_{Z}m{v}^2}$$

The stability of the vehicle is decisively determined by the eigenvalues of the system matrix A from equations (52) to (54): $${\lambda }_{1.2}=-\sigma \pm \sqrt{{\sigma }^2-{\mathrm{<figure-callout\; id="2"\; label="\omega "\; filenames="DE102008046259B4\_0100.png,DE102008046259B4\_0103.png"\; state="\{\{state\}\}">\omega </figure-callout>}}^2}$$

$$2\sigma =-\left({\mathrm{<figure-callout\; id="11"\; label="A"\; filenames="DE102008046259B4\_0101.png"\; state="\{\{state\}\}">A</figure-callout>}}_{11}+A_{22}\right)-\frac{m\left(l_V^2{c}_{\alpha V}+l_H^2{c}_{\alpha H}\right)+J_Z\left(c_{\alpha V}+c_{\alpha H}+m\dot{v}\right)}{J_{Z}mv}$$

$${\mathrm{<figure-callout\; id="2"\; label="\omega "\; filenames="DE102008046259B4\_0100.png,DE102008046259B4\_0103.png"\; state="\{\{state\}\}">\omega </figure-callout>}}^2=A_{11}{A}_{22}-A_{12}{\mathrm{<figure-callout\; id="21"\; label="A"\; filenames="DE102008046259B4\_0103.png"\; state="\{\{state\}\}">A</figure-callout>}}_{21}=\frac{{\left(l_V+l_H\right)}^2{c}_{\alpha V}{c}_{\alpha H}+m{v}^2\left(l_H{c}_{\alpha H}-l_V{c}_{\alpha V}\right)+m\dot{v}\left(l_V^2{c}_{\alpha V}+l_H^2{c}_{\alpha H}\right)}{J_{\mathrm{<figure-callout\; id="2"\; label="Z\; m\; v"\; filenames="DE102008046259B4\_0100.png,DE102008046259B4\_0103.png"\; state="\{\{state\}\}">Z\; </figure-callout>}}m{v}^{}{}^2}$$

oder anders ausgedrückt $${\omega }^2>0.$$

In order to ensure the stability of the vehicle, the real part of the first eigenvalue should assume a negative value (the real part of the second eigenvalue is always negative), namely $$\text{re}\left({\lambda }_1\right)<0$$

or in other words $${\omega }^2>\mathrm{0th}$$

With the method described above for determining the stability of a vehicle, the variables ω ² or λ ₁ can be determined in real driving operation. Therefore, condition ( 55 ) respectively. ( 56 ) included in the regulation of brake pressures. When the brake pressures are controlled so that the condition ( 55 ) respectively. ( 56 ) is always maintained, the vehicle remains theoretically stable. However, one must still make sure that a stable vehicle for the driver is not necessarily easy to control.

Braking (e.g., on a heterogeneous road, cornering or other disturbances) often requires steering correction, otherwise the vehicle will quickly leave the correct lane. An average driver is usually familiar only with the driving behavior in normal driving situations. Too much change in driving behavior during braking might surprise the driver and possibly lead to a wrong driver reaction. Frequency responses can be used to characterize the driving behavior.

Assessment of the driving behavior by means of the yaw frequency response

mit $$K=A_{21}{b}_{11}-A_{11}{b}_{21}=\frac{\left(c_{\alpha H}l+m\dot{v}{l}_v\right)\left(c_{\alpha V}-F_{VL}^U-F_{VL}^U\right)}{J_{Z}mv}$$

und $$T_Z=\frac{b_{21}}{K}-\frac{mv{l}_V}{c_{\alpha H}l+m\dot{v}{l}_v}.$$

The yaw frequency response is a valuable tool for assessing driving behavior. From differential equations ( 23 ) one can use the Laplace transform to use the yaw transfer function ( 57 ) derive: $$G\left(s\right)=\frac{\dot{\psi }\left(s\right)}{{\delta }_V\left(s\right)}=K\frac{T_{Z}s+1}{{\mathrm{<figure-callout\; id="2"\; label="s"\; filenames="DE102008046259B4\_0100.png,DE102008046259B4\_0103.png"\; state="\{\{state\}\}">s</figure-callout>}}^2+2\sigma s+{\mathrm{<figure-callout\; id="2"\; label="\omega "\; filenames="DE102008046259B4\_0100.png,DE102008046259B4\_0103.png"\; state="\{\{state\}\}">\omega </figure-callout>}}^2}=K\frac{T_{Z}s+1}{\left(s-{\lambda }_1\right)\left(s-{\lambda }_2\right)}$$

With $$K=A_{21}{b}_{11}-A_{11}{\mathrm{<figure-callout\; id="21"\; label="b"\; filenames="DE102008046259B4\_0103.png"\; state="\{\{state\}\}">b</figure-callout>}}_{21}=\frac{\left(c_{\alpha H}l+m\dot{v}{l}_v\right)\left(c_{\alpha V}-F_{VL}^U-F_{VL}^U\right)}{J_{Z}mv}$$

and $$T_Z=\frac{b_{21}}{K}-\frac{mv{l}_V}{c_{\alpha H}l+m\dot{v}{l}_v},$$

Where / = 1 _V + 1 _H.

Is used for s in the yaw transfer function ( 57 ) uses the complex frequency j2πf, one obtains the yaw frequency response: $$F\left(f\right)=G\left(j2\pi f\right)=K\frac{j2\pi f{T}_Z+1}{-4{\pi }^2{f}^2+j4\pi f\sigma +{\omega }^2}=K\frac{j2\pi f{T}_Z+1}{\left(j2\pi f-{\lambda }_1\right)\left(j2\pi f-{\lambda }_2\right)},$$

The frequency response thus represents a section of the transfer function, namely the entirety of its functional values on the imaginary axis.

From the yaw frequency response ( 60 ) follows an amplitude response | F (f) | and a phase transition ∠F (f). They describe the amplitude and the phase of the yaw rate after the decay of the transient (which requires stability) against a harmonious excitation on the steering wheel. At a very low frequency, ie f → 0, F (0) = K / ω ² corresponds to the so-called yaw amplification factor, namely the ratio between the yaw rate and the steering wheel angle in a stationary circular drive.

The yaw frequency response should not change too much during the braking process, so that the driver is not overwhelmed.

oder alternativ $$\begin{array}{cc}{\omega }^2>\xi ,& \left(\xi >0\right).\end{array}$$

Both the amplitude response and the phase response of the yaw frequency response fall off very early and very steeply, especially at higher speeds (indicating a large change in drivability) when the quantity Re (λ ₁ ) is negative but not small enough. It therefore makes sense to limit Re (λ ₁ ) as follows: $$\begin{array}{cc}\text{re}\left({\lambda }_1\right)<\tau .& \left(\tau <0\right)\end{array}$$

or alternatively $$\begin{array}{cc}{\omega }^2>\xi .& \left(\xi >0\right),\end{array}$$

The limit values τ and ξ are expediently to be determined by means of applications, so that the inaccuracies in the estimated values and above all the perception of the driving behavior by the driver can be taken into account.

The limits should not be too generous, so that a normal driver can still cope with the change in driving behavior when braking. On the other hand, too conservative values lead to a reduction in brake pressures and consequently a longer braking distance.

The quantity Re (λ ₁ ) decreases rapidly as the vehicle speed decreases. The brake pressures can be increased accordingly without violating the stability condition. However, the braking pressures must be limited because of the tire characteristics, so that the stability of the wheels is ensured both in the circumferential direction (not blocked) and in the lateral direction.

Limiting the slip angle

Similar to the circumferential force over the circumferential slip, the lateral force can only increase up to a certain limit with the skew angle. If this limit is exceeded, the lateral force will not increase or decrease, although the skew angle continues to increase. In this case, the vehicle gets into an unstable area, ie no stationary state is reached. This situation can occur especially when braking on one-sided slippery roads. In 10 is the quasi-stationary (by "quasi" is meant that the driving speed is still changing) state of a straight-braking on such a roadway exemplified. In this case the vehicle drives straight ahead with a slip angle β.

hervorgerufene Störgiermoment muss durch ein entgegen wirkendes Moment kompensiert werden, das durch die Seitenkräfte $F_{VL}^S,F_{VR}^S,F_{HL}^S,F_{HR}^S$

erzeugt wird. Diese Seitenkräfte führen wiederum zu Schräglaufwinkeln α_VL, α_VR, α_HL, α_HR. Es ist leicht einzusehen, dass die Schräglaufwinkel in diesem Fall mit einer steigenden Bremsstärke größer werden. Die Bremsstärke muss deshalb begrenzt werden, damit diese Winkel eine bestimmte Grenze nicht überschreiten. Dabei genügt es im Allgemeinen, die quasi-stationären Schräglaufwinkel heranzuziehen.That by the asymmetrical braking forces $\begin{array}{cc}{F}_{VL}^U<F_{VR}^U.& F_{HL}^U<F_{HR}^U\end{array}$

caused Störgiermoment must be compensated by a counteracting moment, by the side forces $F_{VL}^S.F_{VR}^S.F_{HL}^S.F_{HR}^S$

is produced. These lateral forces in turn lead to slip angles α _VL , α _VR , α _HL , α _HR . It is easy to see that the slip angles increase in this case with increasing braking power. The braking power must therefore be limited so that these angles do not exceed a certain limit. It is generally sufficient to use the quasi-stationary slip angles.

In the quasi-stationary state applies $$\dot{\text{x}}=\left[\begin{array}{c}\dot{\beta }\\ \ddot{\psi }\end{array}\right]=\mathrm{0th}$$

Differential equations ( 23 ) are thus reduced to the following algebraic equations $$0=\text{Ax}+\text{b}{\delta }_V+\upsilon \mathrm{,}$$

$${\delta }_V=\frac{{\upsilon }_{21}{A}_{11}-\left(A_{12}{A}_{21}-A_{11}{A}_{22}\right)\dot{\psi }}{A_{21}{b}_{11}-A_{11}{b}_{21}}.$$

From this, the quasi-stationary slip angle and the required steering angle can be determined at a measured yaw rate: $$\beta =\frac{{\upsilon }_{21}{b}_{11}-\left(A_{12}{b}_{21}-A_{22}{b}_{11}\right)\dot{\psi }}{A_{11}{b}_{21}-A_{21}{\mathrm{<figure-callout\; id="11"\; label="b"\; filenames="DE102008046259B4\_0101.png"\; state="\{\{state\}\}">b</figure-callout>}}_{11}}.$$

$${\delta }_V=\frac{{\upsilon }_{21}{A}_{11}-\left(A_{12}{A}_{21}-A_{11}{A}_{22}\right)\dot{\psi }}{A_{21}{b}_{11}-A_{11}{b}_{21}},$$

$${\alpha }_H=-\beta +l_H\frac{\dot{\psi }}{v}.$$

ergeben sich quasi-stationäre Schräglaufwinkel der Vorder- und Hinterräder.Due to the following relationships $${\alpha }_V={\delta }_V-\beta -l_V\frac{\dot{\psi }}{v}.$$

$${\alpha }_H=-\beta +l_H\frac{\dot{\psi }}{v},$$

result in quasi-stationary slip angle of the front and rear wheels.

control strategy

1. 1. Die Stabilitätsbedingung des Gesamtfahrzeugs nach (61) bzw. (62) muss erfüllt werden;
2. 2. Die Schräglaufwinkel der Räder dürfen eine bestimmte Grenze nicht überschreiten;
3. 3. Die Räder dürfen nicht blockieren.

A stable and controllable braking operation therefore requires three conditions:

1. 1. The stability condition of the whole vehicle according to ( 61 ) respectively. ( 62 ) must be fulfilled;
2. 2. The slip angles of the wheels must not exceed a certain limit;
3. 3. The wheels must not lock.

Under these conditions, the adhesion of the wheels on the road should be optimally utilized to achieve the shortest possible braking distance. This can be achieved by a targeted regulation of the brake pressures or the slip values.

In principle, many control strategies are conceivable. One embodiment is in 1 shown. The locking of the wheels is as usual by a conventional anti-lock 8th prevented. However, the brake pressure in the master cylinder becomes subject to the above conditions 1 and 2 regulated. An advantage of this strategy is that braking forces of the left and right wheels are approximately the same even on one-sided slippery roads, as long as the anti-lock 8th not yet active, eg with a light brake. Although the driver is not informed about the road condition in this case, he also does not need to counter-steer.

This control strategy is now being investigated with the help of a multi-body complete vehicle model. As a controlled variable, Re (λ ₁ ) is used. Alternatively, the size ω ² can be used.

The presented control strategy works in braking of different types, e.g. Straight braking on homogeneous and heterogeneous road surfaces, braking in the curve, etc. As an example, only a few simulation results during straight braking on a one-sided (left in this case) smooth roadway are shown below.

As 11 shows, the default τ = -0.5 was well maintained during the braking process. The condition 1 is thus fulfilled.

Since Re (λ ₁ ) at the beginning of the braking process is significantly smaller than the default, the brake pressure in the master cylinder is built up quickly. The rotational speed of the wheels on the slippery side of the road decreases rapidly. In this case, the anti-lock device responds so that the brake pressure in the respective wheel cylinders does not increase any further and blocking is prevented as in 12 shown. The graph 34 shows the brake pressure on the left front wheel, the graph 35 the brake pressure on the right front wheel, the graph 36 the brake pressure on the left rear wheel and the graph 37 the brake pressure on the right rear wheel.

Since the wheels on the grip side of the road (right) can transmit even more braking forces, the brake pressure under the stability conditions 1 and 3 further increased until the slip angle of a wheel (in this case on the rear axle) reaches the predetermined limit (in this case 6 °). The course of the slip angle is in 13 shown.

The driving behavior in higher speed ranges, ie at the beginning of the braking, changes relatively strongly. In order to improve this, a variable limit τ that is variable with the driving speed is conceivable instead of a constant one.

The required steering wheel angle during braking is in 14 shown. It can be seen that this control strategy allows a smooth steering correction.

LIST OF REFERENCE NUMBERS

1, 2

contraption

3

processing unit

4

sensors

5, 6

processing unit

7

brake pressure

8th

Brake system, anti-lock device

9

regulator

10

vehicle longitudinal axis

11

Center of gravity

12-15

wheel

16

rear axle

17

Front

18-37

graph

Claims (19)

A method for determining a driving stability of a vehicle when braking, comprising: Determining a center of gravity position (11) of the vehicle, Determining side force coefficients of wheels (12-15) of the vehicle, and - Determining a value for the driving stability of the vehicle from a speed of the vehicle, the center of gravity (11) of the vehicle, the Seitenkraftbeiwerten the wheels (12-15) of the vehicle, a vehicle mass and a vehicle inertia about a vehicle's vertical axis. Method according to Claim 1 , characterized in that the value χ for the driving stability from the speed v of the vehicle, a distance l _{V of} a center of gravity (11) of the vehicle from the front axle (17), a distance l _{H of} the center of gravity (11) from the rear axle (16 ), the lateral force _coefficients front left c _αVL , front right c _αVR , rear left c _αHL , rear right c _{αHR of} the wheels (12-15), the vehicle mass m and the vehicle _{moment of inertia} J _Z is determined about a vehicle _{vertical axis} as follows: $$\chi =\text{re}\left(-\sigma +\sqrt{{\sigma }^2+{\omega }^2}\right).$$

in which $$\sigma =\frac{m\left(l_V^2{c}_{\alpha V}+l_H^2{c}_{\alpha H}\right)+J_Z\left(c_{\alpha V}+c_{\alpha H}+m\dot{v}\right)}{2J_{Z}mv}.$$

$${\omega }^2=\frac{{\left(l_V+l_H\right)}^2{c}_{\alpha V}{c}_{\alpha H}+m{v}^2\left(l_H{c}_{\alpha H}-l_V{c}_{\alpha V}\right)+m\dot{v}\left(l_V^2{c}_{\alpha V}+l_H^2{c}_{\alpha H}\right)}{J_{2}m{v}^2}.$$

$$c_{\alpha V}=c_{\alpha VL}+c_{\alpha VR}$$

and $$c_{\alpha H}=c_{\alpha HL}+c_{\alpha HR}$$

applies. Method according to Claim 2 characterized in that determining a lateral force coefficient of the lateral force coefficients for a wheel (12-15) of the vehicle comprises: - determining a circumferential force of the wheel (12-15), - detecting slippage of the wheel (12-15), and - determining the lateral force coefficient of the wheel (12-15) from the circumferential force, the slip and the wheel-specific constant. Method according to Claim 3 , characterized in that the lateral force coefficient c _α for a wheel (12-15) from the circumferential force F ^U , the slip and the wheel-specific constant κ is determined as follows: $$c_{\alpha }=\kappa \frac{F^U}{S},$$

Method according to Claim 3 or 4 characterized in that determining the circumferential force of the wheel (12-15) comprises: - detecting a brake pressure of a wheel brake cylinder of a brake of the wheel (12-15), - detecting an angular velocity of the wheel (12-15), - determining an angular acceleration of the wheel (12-15) from the angular velocity, and - determining the circumferential force from the brake pressure, the angular acceleration and vehicle-specific quantities. Method according to Claim 5 , characterized in that the vehicle-specific variables - a piston surface of the wheel brake cylinder, - a mean friction radius of a brake pad of a brake disc of the brake, - a brake characteristic of the brake, - a moment of inertia of the wheel (12-15) and all masses connected thereto and - a dynamic Include tire radius. Method according to Claim 6 characterized in that determining the circumferential force of the wheel (12-15) further comprises: - detecting an engine drag torque acting on the wheel (12-15) and - additionally determining the circumferential force from the engine drag torque. Method according to Claim 7 , characterized in that the circumferential force F ^{U of} the wheel (12-15) from the brake pressure p _B , the angular acceleration ω̇, the piston surface A of the wheel brake cylinder, the average friction radius r of the brake pad of the brake disc, the brake characteristic C _{B of} the brake, the moment of inertia J _{wheel of} the wheel and all masses associated therewith, the dynamic tire radius R _dyn and the engine drag torque M _motor acting on the wheel (12-15) are determined as follows: $$F^U=\frac{p_{B}Ar{C}_B+J_{Rad}\dot{\omega }+M_{MOtOr}}{R_{dyn}}$$

Method according to one of Claims 5 - 8th , characterized in that the center of gravity (11) of the vehicle from the peripheral forces of the wheels (12-15) and the wheelbase of the vehicle is determined. Method according to Claim 9 , characterized in that the distance l _{V of} the center of gravity (11) of the vehicle from the front axle (17) and the distance l _{H of} the center of gravity (11) from the rear axle (16) from the ratio $F_V^U/F_H^U$

the circumferential force of the front wheels (12, 13) to the peripheral force of the rear wheels (14, 15) and the wheelbase l is determined as follows: $$l_V/l_H=F_V^U/F_H^U\text{and}{l}_V+l_H=l,$$

A method of adjusting a brake pressure for a vehicle, comprising: Determining slip angles of wheels (12-15) of the vehicle, Determining a value for vehicle stability during braking, and Adjusting the brake pressure (7) on the wheels (12-15) of the vehicle depending on the determined slip angle and the determined driving stability such that a predetermined slip angle is not exceeded and that a predetermined range of values for the driving stability is maintained. Method according to Claim 11 , characterized in that the value for the driving stability according to one of Claims 1 to 10 is determined. Method according to Claim 12 , characterized in that said predetermined range of values for driving stability comprises a range of -∞ to -0.5. Method according to one of Claims 11 to 13 characterized in that a yaw angular velocity of the vehicle is further detected, and that the slip angle of a wheel (12-15) of the vehicle is determined from the yaw rate and other magnitudes, the other magnitudes being the speed of the vehicle, lateral force coefficients of the wheels of the vehicle, circumferential forces the wheels of the vehicle, a center of gravity of the vehicle, a vehicle mass and a vehicle inertia about a vehicle vertical axis. Method according to one of Claims 11 to 14 characterized in that the predetermined slip angle is 6 °. Device for determining a driving stability of a vehicle during braking, comprising a processing unit (5, 6) which is coupled to sensors (4) of the vehicle, the sensors (4) providing the following measured values: - An angular velocity of each wheel (12-15) of the vehicle and a brake pressure for each wheel (12-15) of the vehicle, the processing unit (5, 6) being configured, a value for the driving stability of the vehicle from a speed of the vehicle, the angular speeds of each wheel (12-15) of the vehicle, determine the brake pressures for each wheel (12-15) of the vehicle and vehicle-specific information. Device after Claim 16 , characterized in that the device (2) for carrying out the method according to one of Claims 1 to 10 is designed. Device for setting a brake pressure for a vehicle, comprising a processing unit (3) which is coupled to sensors (4) and an actuator (8) of the vehicle, the sensors (4) providing the following measured values: an angular velocity of each wheel (12-15) of the vehicle, a yaw rate and a brake pressure for each wheel (12-15) of the vehicle, wherein the actuator (8) is designed to set a brake pressure (7) for the vehicle determined by the processing unit (3), wherein the processing unit (3) is configured to determine slip angles of the wheels (12-15) of the vehicle and a value for driving stability of the vehicle when braking from the measured values of the sensors (4) and vehicle-specific information, and the brake pressure (7) for the vehicle Adjust vehicle depending on the particular slip angle and the specific driving stability such that a predetermined slip angle is not exceeded and that a predetermined range of values for driving stability is maintained. Device after Claim 18 , characterized in that the device (1) for carrying out the method according to one of Claims 11 to 15 is designed.

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