Attitude sensing system for an automotive vehicle relative to the road

TECHNICAL FIELD

0001The present invention relates generally to a control apparatus for controlling a system of an automotive vehicle in response to sensed dynamic behavior, and more specifically, to a method and apparatus for controlling the system of the vehicle by determining attitude of the vehicle.

BACKGROUND

0002Dynamic control systems for automotive vehicles have recently begun to be offered on various products. Dynamic control systems typically control the yaw of the vehicle by controlling the braking effort at the various wheels of the vehicle. Yaw control systems typically compare the desired direction of the vehicle based upon the steering wheel angle and the direction of travel. By regulating the amount of braking at each corner of the vehicle, the desired direction of travel may be maintained. Typically, the dynamic control systems do not address roll of the vehicle. For high profile vehicles in particular, it would be desirable to control the rollover characteristic of the vehicle to maintain the vehicle position with respect to the road. That is, it is desirable to maintain contact of each of the four tires of the vehicle on the road.

0003In vehicle rollover control, it is desired to alter the vehicle attitude such that its motion along the roll direction is prevented from achieving a predetermined limit (rollover limit) with the aid of the actuation from the available active systems such as controllable brake system, steering system and suspension system. Although the vehicle attitude is well defined, direct measurement is usually impossible.

0004There are two types of vehicle attitudes needed to be distinguished. One is the so-called global attitude, which is sensed by the angular rate sensors. The other is the relative attitude, which measures the relative angular positions of the vehicle with respect to the road surface on which the vehicle is driven. The global attitude of the vehicle is relative to an earth frame (or called the inertia frame), sea level, or a flat road. It can be directly related to the three angular rate gyro sensors. While the relative attitude of the vehicle measures the relative angular positions of the vehicle with respect to the road surface, which are always of various terrains. Unlike the global attitude, there are no gyro-type sensors which can be directly related to the relative attitude. A reasonable estimate is that a successful relative attitude sensing system must utilize both the gyro-type sensors (when the road becomes flat, the relative attitude sensing system recovers the global attitude) and some other sensor signals.

0005One reason to distinguish relative and global attitude is due to the fact that vehicles are usually driven on a 3-dimensional road surface of different terrains, not always on a flat road surface. Driving on a road surface with large road bank does increase the rollover tendency, i.e., a large output from the global attitude sensing system might well imply an uncontrollable rollover event regardless of the flat road driving and the 3-D road driving. However driving on a three-dimensional road with moderate road bank angle, the global attitude may not be able to provide enough fidelity for a rollover event to be distinguished. Vehicular rollover happens when one side of the vehicle is lifted from the road surface with a long duration of time without returning back. If a vehicle is driven on a banked road, the global attitude sensing system will pick up certain attitude information even when the vehicle does not experience any wheel lifting (four wheels are always contacting the road surface). Hence a measure of the relative angular positions of the vehicle with respect to the portion of the road surface on which the vehicle is driven provides more fidelity than global attitude to sense the rollover event.

0006The vehicle rollover sensing system used for deploying safety-related devices has been proposed in U.S. Pat. Nos. 6,002,975, 6,038,495, EP 1002709A2, where a stand-alone sensor module including 5 sensors are used including the roll/pitch angular rate sensors, later/longitudinal/vertical acceleration sensors. These systems sense the global attitude of a vehicle without considering the relative attitude of the vehicle with respect to the road surfaces. Due to the stand-alone nature of the sensing module, it does not share internal information with vehicle dynamics control systems.

0007The rollover control system using brake controls has been proposed in U.S. Pat. No. 6,065,558 (“Anti-Rollover Brake System”), where the claimed sensor setting could be any of the following: (1) a lateral accelerometer; (2) a sensor for measuring the body roll angle; (3) an accelerometer, a gyroscope, a roll rate senor, and sensors measuring the distances between the vehicle and the wheels to measure the roll angle of the vehicle. In the current invention, a different sensor set is used. The used sensors includes those used in the vehicle yaw stability control (lateral/longitudinal accelerometers, yaw angular rate sensor, wheel speeds and steering angle) and an extra roll rate angular sensor. Also, notice that U.S. Pat. No. 6,065,558 does not intend to distinguish between global and relative attitude of a vehicle reflected by the Euler angles.

0008Another vehicle attitude sensing method has been proposed in U.S. Pat. No. 5,408,411 (“System For Predicting Behavior Of Automotive Vehicle And For Controlling Vehicular Behavior Based Thereon”). Where a sensor module using six linear accelerations is mounted on the vehicle to get vehicular attitude information.

0009It would therefore be desirable to provide an attitude control system to predict attitude angle for vehicle dynamics control that includes the interdependency among the roll, pitch and yaw motions while compensating for long term maneuvers.

SUMMARY OF THE INVENTION

0010The present invention aims to estimate and predict the vehicular attitude used in a rollover control system which can prevent the vehicle from rolling over. The estimate and predicted variables are used for setting a rollover control action flag and as the feedback signals to construct the desired control forces for controlling roll stability or activate other safety devices. In detail, the rollover control action needs the information from the vehicle attitude sensing system, the available sensors, and the driving/road condition identifiers. The rollover control flag is set based on a rollover logic process. In case a positive determination of vehicle rollover is deemed from this rollover logic process, the control commands will be computed by feeding back the estimated vehicle attitude variables. The control command output is further sent to the ECU of the hardware to activate the system. In detail, the vehicle attitude sensing system uses all the sensors available for yaw stability control (including a laterlateral accelerometer, a longitudinal accelerometer, a yaw angular rate, a steering angle sensor and the wheel speed sensor signals) together with a roll angular rate sensor. The vehicle attitude is characterized by the relative Euler angles of the car body with respect to the road surface and by the global Euler angles of the car body with respect to the sea level. The vehicle attitude estimation and prediction utilize both the kinematic and the dynamic relationships derived from vehicle dynamics and vehicle dynamic models to relate the desired motion variables with the measured sensor signals.

0011In one aspect of the invention, a stability control system for an automotive vehicle includes a plurality of sensors sensing the dynamic conditions of the vehicle. The sensors may include a speed sensor, a lateral acceleration sensor, a roll rate sensor, a yaw rate sensor and a longitudinal acceleration sensor. The controller is coupled to the speed sensor, the lateral acceleration sensor, the roll rate sensor, the yaw rate sensor and a longitudinal acceleration sensor. The controller determines a global roll attitude and a global pitch attitude from the roll rate, lateral acceleration signal and the longitudinal acceleration signal. The controller determines a roll gradient based upon a past raw roll rate and current raw roll rate, the roll angular rate signal and the lateral acceleration signal, and a pitch gradient based upon a past raw pitch rate and current raw pitch rate the calculated pitch angular rate signal and the longitudinal acceleration signal. The controller determines a relative roll and relative pitch as a function of the roll gradient and the pitch gradient.

0012In a further aspect of the invention, a method of controlling roll stability of the vehicle comprises the steps of: <ul id="ul0001" list-style="none"><li id="ul0001-0001" num="0000"><ul id="ul0002" list-style="none"><li id="ul0002-0001" num="0013">measuring a roll rate of the vehicle body;</li><li id="ul0002-0002" num="0014">measuring a lateral acceleration of the vehicle body;</li><li id="ul0002-0003" num="0015">measuring the longitudinal acceleration of the vehicle body;</li><li id="ul0002-0004" num="0016">measuring the yaw rate of the vehicle body; and</li><li id="ul0002-0005" num="0017">determining relative roll angle, the relative pitch angle, the global roll and global pitch angle in response to the roll rate, the yaw rate, the lateral acceleration and the longitudinal acceleration.</li></ul></li></ul>

0018Reducing system cost is typically a goal in automotive systems. Since one of the three angular rate signals (pitch rate signal) can be predicted from the other available signals, the cost reduction of the system is possible by eliminating a pitch rate sensor. Also, vertical acceleration sensor may be eliminated to further reduce cost.

0019Other advantages and features of the present invention will become apparent when viewed in light of the detailed description of the preferred embodiment when taken in conjunction with the attached drawings and appended claims.

BRIEF DESCRIPTION OF THE DRAWINGS

0020<figref idref="DRAWINGS">FIG. 1</figref> is a diagrammatic view of a vehicle with variable vectors and coordinator frames according to the present invention.

0021<figref idref="DRAWINGS">FIG. 2</figref> is a block diagram of a stability system according to the present invention.

0022<figref idref="DRAWINGS">FIG. 3</figref> is a diagrammatic view showing the displacement (relative to road surface) of the four corners of the vehicle body along the body-fixed vertical axis.

0023<figref idref="DRAWINGS">FIG. 4</figref> is a diagrammatic view showing the two components of the relative corner displacement depicted in FIG. <b>3</b>.

0024<figref idref="DRAWINGS">FIG. 5</figref> is flow chart of determination according to the present invention.

DESCRIPTION OF THE PREFERRED EMBODIMENT

0025In the following figures the same reference numerals will be used to identify the same components. The present invention is preferably used in conjunction with a rollover control system for a vehicle. However, the present invention may also be used with a deployment device such as airbag or roll bar. The present invention will be discussed below in terms of preferred embodiments relating to an automotive vehicle moving in a three-dimensional road terrain.

0026Referring to <figref idref="DRAWINGS">FIG. 1</figref>, an automotive vehicle <b>10</b> with a safety system of the present invention is illustrated with the various forces and moments thereon during a rollover condition. Vehicle <b>10</b> has front right and front left tires <b>12</b>a and <b>12</b>b and rear right tires <b>13</b>a and left rear tires <b>13</b>b respectively. The vehicle <b>10</b> may also have a number of different types of front steering systems <b>14</b>a and rear steering systems <b>14</b>b including having each of the front and rear wheels configured with a respective controllable actuator, the front and rear wheels having a conventional type system in which both of the front wheels are controlled together and both of the rear wheels are controlled together, a system having conventional front steering and independently controllable rear steering for each of the wheels or vice versa. Generally, the vehicle has a weight represented as Mg at the center of gravity of the vehicle, where g=9.8 m/s²and M is the total mass of the vehicle.

0027As mentioned above, the system may also be used with active/semi-active suspension systems, anti-roll bar or other safety devices deployed or activated upon sensing predetermined dynamic conditions of the vehicle.

0028The sensing system <b>16</b> is coupled to a control system <b>18</b>. The sensing system <b>16</b> preferably uses a standard yaw stability control sensor set (including lateral accelerometer, yaw rate sensor, steering angle sensor and wheel speed sensor) together with a roll rate sensor and a longitudinal accelerometer. The various sensors will be further described below. The wheel speed sensors <b>20</b> are mounted at each corner of the vehicle, and the rest of the sensors of sensing system <b>16</b> are preferably mounted directly on the center of gravity of the vehicle body, along the directions x,y and z shown in FIG. <b>1</b>. As those skilled in the art will recognize, the frame from b₁, b₂and b₃is called a body frame <b>22</b>, whose origin is located at the center of Do gravity of the car body, with the b₁corresponding to the x axis pointing forward, b₂corresponding to the y axis pointing off the driving side (to the left), and the b₃corresponding to the z axis pointing upward. The angular rates of the car body are denoted about their respective axes as w_xfor the roll rate, w_yfor the pitch rate and w_zfor the yaw rate. The present invention calculations preferably take place in an inertial frame <b>24</b> that may be derived from the body frame <b>22</b> as described below.

0029The angular rate sensors and the accelerometers are mounted on the vehicle car body along the body frame directions b₁, b₂and b₃, which are the x-y-z axes of the vehicle's sprung mass.

0030The longitudinal acceleration sensor is mounted on the car body located at the center of gravity, with its sensing direction along b₁-axis, whose output is denoted as a_x. The lateral acceleration sensor is mounted on the car body located at the center of gravity, with its sensing direction along b₂-axis, whose output is denoted as a_y.

0031The other frame used in the following discussion includes the road frame, as depicted in FIG. <b>1</b>. The road frame system r₁r₂r₃is fixed on the driven road surface, where the r₃axis is along the average road normal direction computed from the normal directions of the four tire/road contact patches.

0032In the following discussion, the Euler angles of the body frame b₁b₂b₃with respect to the road frame r₁r₂r₃are denoted as θ_xbr,θ_ybrand θ_zbr, which are also called the relative Euler angles.

0033The present invention estimates the relative Euler angles θ_xbrand θ_ybrbased on the available sensor signals and the signals calculated form the measured values.

0034Referring now to <figref idref="DRAWINGS">FIG. 2</figref>, roll stability control system <b>18</b> is illustrated in further detail having a controller <b>26</b> used for receiving information from a number of sensors which may include a yaw rate sensor <b>28</b>, a speed sensor <b>20</b>, a lateral acceleration sensor <b>32</b>, a roll rate sensor <b>34</b>, a steering angle sensor <b>35</b>, a longitudinal acceleration sensor <b>36</b>, a pitch rate sensor <b>37</b> and steering angle position sensor <b>39</b>.

0035In the preferred embodiment only two axial rate sensors are used. When two of these axial rates are known, the other may be derived using other commonly available sensors.

0036That is, pitch rate sensor <b>37</b> is illustrated, it can be eliminated in the preferred embodiment.

0037In the preferred embodiment the sensors are located at the center of gravity of the vehicle. Those skilled in the art will recognize that the sensor may also be located off the center of gravity and translated equivalently thereto.

0038Lateral acceleration, roll orientation and speed may be obtained using a global positioning system (GPS). Based upon inputs from the sensors, controller <b>26</b> may control a safety device <b>38</b>. Depending on the desired sensitivity of the system and various other factors, not all the sensors <b>28</b>-<b>37</b><b>28</b>-<i><b>39</b></i>may be used in a commercial embodiment. Safety device <b>38</b> may control an airbag <b>40</b> or a steering actuator or braking actuator at one or more of the wheels <b>41</b>, <b>42</b>, <b>44</b>, <b>46</b> of the vehicle. Also, other vehicle components such as a suspension control <b>48</b> may be used to adjust the suspension to prevent rollover.

0039Roll rate sensor <b>34</b> and pitch rate sensor <b>37</b> may sense the roll condition of the vehicle based on sensing the height of one or more points on the vehicle relative to the road surface. Sensors that may be used to achieve this include a radar-based proximity sensor, a laser-based proximity sensor and a sonar-based proximity sensor.

0040Roll rate sensor <b>34</b> and pitch rate sensor <b>37</b> may also sense the roll condition based on sensing the linear or rotational relative displacement or displacement velocity of one or more of the suspension chassis components which may include a linear height or travel sensor, a rotary height or travel sensor, a wheel speed sensor used to look for a change in velocity, a steering wheel position sensor, a steering wheel velocity sensor and a driver heading command input from an electronic component that may include steer by wire using a hand wheel or joy stick.

0041The roll condition may also be sensed by sensing the force or torque associated with the loading condition of one or more suspension or chassis components including a pressure transducer in an act of air suspension, a shock absorber sensor such as a load cell, a strain gauge, the steering system absolute or relative motor load, the steering system pressure of the hydraulic lines, a tire laterally force sensor or sensors, a longitudinal tire force sensor, a vertical tire force sensor or a tire sidewall torsion sensor.

0042The roll condition of the vehicle may also be established by one or more of the following translational or rotational positions, velocities or accelerations of the vehicle including a roll gyro, the roll rate sensor <b>34</b>, the yaw rate sensor <b>28</b>, the lateral acceleration sensor <b>32</b>, a vertical acceleration sensor, a vehicle longitudinal acceleration sensor, lateral or vertical speed sensor including a wheel-based speed sensor, a radar-based speed sensor, a sonar-based speed sensor, a laser-based speed sensor or an optical-based speed sensor.

0043Steering control <b>38</b> may control the position of the front right wheel actuator <b>40</b>, the front left wheel actuator <b>42</b>, the rear left wheel actuator <b>44</b>, and the right rear wheel actuator <b>46</b>. Although as described above, two or more of the actuators may be simultaneously controlled. For example, in a rack-and-pinion system, the two wheels coupled thereto are simultaneously controlled. Based on the inputs from sensors <b>28</b> through <b>39</b>, controller <b>26</b> determines a roll condition and controls the steering position of the wheels.

0044Speed sensor <b>30</b><b>20</b>may be one of a variety of speed sensors known to those skilled in the art. For example, a suitable speed sensor may include a sensor at every wheel that is averaged by controller <b>26</b>. Preferably, the controller translates the wheel speeds into the speed of the vehicle. Yaw rate, steering angle, wheel speed and possibly a slip angle estimate at each wheel may be translated back to the speed of the vehicle at the center of gravity. Various other algorithms are known to those skilled in the art. Speed may also be obtained from a transmission sensor. For example, if speed is determined while speeding up or braking around a corner, the lowest or highest wheel speed may not be used because of its error. Also, a transmission sensor may be used to determine vehicle speed.

Connecting the Relative Attitudes with the Relative Corner Displacements

0045In operation, the method according to the present invention first correlates the relative attitude with the displacement at each corner of the vehicle. Consider a vector with x-y-z coordinates as x_b,y_b,z_bof its end point in the body frame of FIG. <b>1</b>. The z coordinator of the end point of the same vector measured in the road frame can be computed from the Euler transformation <br />z_r=−x_bsin(θ_ybr)+y_bsin(θ_xbr)cos(θ_ybr)+z_bcos(θ_xbr)cos(θ_ybr)  (1)

0046Let l be the half of the wheel track; t_fand t_rbe the distances from the center of gravity of the car body to the front and rear axles; h be the distance between the bottom of the vehicle body and the center of gravity of the vehicle along the body z-axis; θ_xbrand θ_ybrare the relative roll and pitch angles. Then in the body frame the four corners of the vehicle body where suspensions are connected with the wheel have the following coordination: <br />LF Corner: x=t_f, y=l, z=−h <br />RF Corner: x=t_f, y=−l, z=−h <br />LR Corner: x=−t_r, y=l, z=−h <br />RR Corner: x=−t_r, y=−l, z=−h  (2)

0047Let z_lf,z_rf,z_lrand z_rrbe the relative displacements of the vehicle corners at the left-front, right-front, left-rear and right-rear locations, which are measured along the direction perpendicular to the average road surface. By using the transformation in Equation (1), those corner displacements relative to the road surface can be expressed as the function of the relative roll and pitch angles θ_xbrand θ_ybr<br />z_lf=−t_fsin(θ_ybr)+l sin(θ_xbr)cos(θ_ybr)+(z_cg−h)cos(θ_xbr)cos(θ_ybr) <br />z_rf=−t_fsin(θ_xbr)−l sin(θ_xbr)cos(θ_ybr)+(z_cg−h)cos(θ_xbr)cos(θ_ybr) <br />z_lr=t_rsin(θ_ybr)+l sin(θ_xbr)cos(θ_ybr)+(z_cg−h)cos(θ_ybr)cos(θ_ybr) <br />z_rr=t_rsin(θ_ybr)−l sin(θ_xbr)cos(θ_ybr)+(z_cg−h)cos(θ_xbr)cos(θ_ybr)  (3)<br /> where z_cgis the relative displacement of the center of gravity of the vehicle with respect to the road surface, but measured along the body z-axis.

0048Referring now to <figref idref="DRAWINGS">FIG. 3</figref>, the four equations of Equation (3) pose four constraints on a set of seven variables: z_lf,z_rf,z_lr,z_rr,z_cg,θ_xbrand θ_ybr. Hence a combination of three variables can be used to compute the rest of the variables. Since θ_xbrand θ_ybrare of interest in the present invention, the possible choices are choosing three variables from z_lf,z_rf,z_lr,z_rrand z_cgto characterize θ_xbrand θ_ybr. The direct measurement of any of z_lf,z_rf,z_lr,z_rr, and z_cgis relatively expensive (for example, expensive laser distance sensors can be used). Hence, measuring any three of z_lf,z_rf,z_lr,z_rrand z_cgmay be cost prohibitive in a commercial environment with current technology. However, certain linear combinations of the four corner displacements z_lf,z_rf,z_lrand z_rrcan be related to the available sensors through dynamics. When linear combinations are related to the relative roll and pitch Euler angles θ_xbrand θ_ybr, θ_xbrand θ_ybrmay be characterized from the available sensor signals. In the following, the effort has been focused on finding those linear combinations of z_lf,z_rf,z_lr,z_rrwhich bridges between θ_xbrand θ_ybr, and the available sensor signals including the lateral and longitudinal accelerations, the roll and yaw angular rates and the wheel speed sensor signals.

The relative Attitudes Based on the Linear Combinations of the Corner Displacements

0049The linear combinations of z_lf,z_rf,z_lr,z_rr, which serve as bridges to connect θ_xbrand θ_ybrwith the available sensor signals are the following variables, which are called the relative roll and pitch gradients <maths id="MATH-US-00001" num="00001"><math overflow="scroll"><mtable><mtr><mtd><mrow><mrow><msub><mi>Θ</mi><mi>x</mi></msub><mo>=</mo><mfrac><mrow><msub><mi>Z</mi><mi>lf</mi></msub><mo>-</mo><msub><mi>Z</mi><mi>rf</mi></msub><mo>+</mo><msub><mi>Z</mi><mi>lr</mi></msub><mo>-</mo><msub><mi>Z</mi><mi>rr</mi></msub></mrow><mrow><mn>4</mn><mo>⁢</mo><mi>l</mi></mrow></mfrac></mrow><mo>⁢</mo><mstyle><mtext></mtext></mstyle><mo>⁢</mo><mrow><msub><mi>Θ</mi><mi>y</mi></msub><mo>=</mo><mfrac><mrow><msub><mi>Z</mi><mi>lf</mi></msub><mo>+</mo><msub><mi>Z</mi><mi>rf</mi></msub><mo>-</mo><msub><mi>Z</mi><mi>lr</mi></msub><mo>-</mo><msub><mi>Z</mi><mi>rr</mi></msub></mrow><mrow><mn>2</mn><mo>⁢</mo><mrow><mo>(</mo><mrow><msub><mi>t</mi><mi>f</mi></msub><mo>+</mo><msub><mi>t</mi><mi>r</mi></msub></mrow><mo>)</mo></mrow></mrow></mfrac></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>4</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths><img file="USRE40496E_D0001.tif" /><br /> Θ_xand Θ_yis related to the relative roll and pitch attitudes by manipulating the equations in (3). The final formula for the relative pitch Euler angle is <br />θ_ybr=sin⁻¹{Θ_y}  (5)<br /> and the final formula for the relative roll Euler angle θ_xbris <maths id="MATH-US-00002" num="00002"><math overflow="scroll"><mtable><mtr><mtd><mrow><msub><mi>θ</mi><mi>xbr</mi></msub><mo>=</mo><mrow><msup><mi>sin</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo>⁢</mo><mrow><mo>{</mo><mfrac><msub><mi>Θ</mi><mi>x</mi></msub><mrow><mi>cos</mi><mo>⁡</mo><mrow><mo>(</mo><msub><mi>θ</mi><mi>ybr</mi></msub><mo>)</mo></mrow></mrow></mfrac><mo>}</mo></mrow></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>6</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths><img file="USRE40496E_D0002.tif" />

0050On the other hand Θ_xand Θ_ycan be further related to the available sensor signals through dynamic equations which describe the vehicle body dynamics. Θ_xand Θ_ywill be first broken into two portions, and related to the sensor signals.

Roll and Pitch Gradients Due to Suspension and Wheel Motions

0051As shown in <figref idref="DRAWINGS">FIG. 4</figref>, a portion of the left front wheel <b>12</b>b and suspension <b>52</b> are illustrated, z_lfcan be further expressed as the sum of the two parts: the suspension stroke s_lfas measured by sensor <b>50</b> and the wheel displacement w_lfwith respect to the road surface along the direction perpendicular to the road surface. The same is true for the rest of the corner locations. The sensor <b>50</b> may measuresmeasure the change in the distance from the vehicle body to the wheel. If the four suspension strokes are s_lf,s_rf,s_lrand s_rr, and the four wheel vertical motions are w_lf, w_rf, w_lrand w_rr, then: <br />z_lf=s_lf+w_lf<br />z_rf=s_rf+w_rf<br />z_lr=s_lr+w_lr<br />z_rr=s_rr+w_rr  (7)

0052The relative roll and pitch gradients Θ_xand Θ_ymay be broken into pieces according to the suspension motion and the wheel vertical motion. The roll and the pitch gradients Θ_x-suspand Θ_y-suspdue to suspension motions s_lf,s_rf,s_lfand s_rrmay be defined as: <maths id="MATH-US-00003" num="00003"><math overflow="scroll"><mtable><mtr><mtd><mrow><mrow><msub><mi>Θ</mi><mrow><mi>y</mi><mo>-</mo><mi>susp</mi></mrow></msub><mo>=</mo><mfrac><mrow><msub><mi>s</mi><mi>lf</mi></msub><mo>+</mo><msub><mi>s</mi><mi>rf</mi></msub><mo>-</mo><msub><mi>s</mi><mi>lr</mi></msub><mo>-</mo><msub><mi>s</mi><mi>rr</mi></msub></mrow><mrow><mn>2</mn><mo>⁢</mo><mrow><mo>(</mo><mrow><msub><mi>t</mi><mi>f</mi></msub><mo>+</mo><msub><mi>t</mi><mi>r</mi></msub></mrow><mo>)</mo></mrow></mrow></mfrac></mrow><mo>⁢</mo><mstyle><mtext></mtext></mstyle><mo>⁢</mo><mrow><msub><mi>Θ</mi><mrow><mi>x</mi><mo>-</mo><mi>susp</mi></mrow></msub><mo>=</mo><mfrac><mrow><msub><mi>s</mi><mi>lf</mi></msub><mo>-</mo><msub><mi>s</mi><mi>rf</mi></msub><mo>+</mo><msub><mi>s</mi><mi>lr</mi></msub><mo>-</mo><msub><mi>s</mi><mi>rr</mi></msub></mrow><mrow><mn>4</mn><mo>⁢</mo><mi>l</mi></mrow></mfrac></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>8</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths><img file="USRE40496E_D0003.tif" /><br /> and the roll and pitch gradients Θ_x-whland Θ_y-whldue to the wheel vertical motion defined as: <maths id="MATH-US-00004" num="00004"><math overflow="scroll"><mtable><mtr><mtd><mrow><mrow><msub><mi>Θ</mi><mrow><mi>y</mi><mo>-</mo><mi>whl</mi></mrow></msub><mo>=</mo><mfrac><mrow><msub><mi>w</mi><mi>lf</mi></msub><mo>+</mo><msub><mi>w</mi><mi>rf</mi></msub><mo>-</mo><msub><mi>w</mi><mi>lr</mi></msub><mo>-</mo><msub><mi>w</mi><mi>rr</mi></msub></mrow><mrow><mn>2</mn><mo>⁢</mo><mrow><mo>(</mo><mrow><msub><mi>t</mi><mi>f</mi></msub><mo>+</mo><msub><mi>t</mi><mi>r</mi></msub></mrow><mo>)</mo></mrow></mrow></mfrac></mrow><mo>⁢</mo><mstyle><mtext></mtext></mstyle><mo>⁢</mo><mrow><msub><mi>Θ</mi><mrow><mi>x</mi><mo>-</mo><mi>whl</mi></mrow></msub><mo>=</mo><mfrac><mrow><msub><mi>w</mi><mi>lf</mi></msub><mo>-</mo><msub><mi>w</mi><mi>rf</mi></msub><mo>+</mo><msub><mi>w</mi><mi>lr</mi></msub><mo>-</mo><msub><mi>w</mi><mi>rr</mi></msub></mrow><mrow><mn>4</mn><mo>⁢</mo><mi>l</mi></mrow></mfrac></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>9</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths><img file="USRE40496E_D0004.tif" /><br /> Then <br />Θ_yΘ_y-susp+Θ_y-whl<br />Θ_xΘ_x-susp+Θ_x-whl<br />Θ_y<i>=Θ</i>_y-susp<i>+Θ</i>_y-whl<br /><i>Θ</i>_x<i>=Θ</i>_x-susp<i>+Θ</i>_x-whl  (10)

0053The relative Euler angles Θ_xbr, and Θ_ybrcan be also written as two parts: <maths id="MATH-US-00005" num="00005"><math overflow="scroll"><mtable><mtr><mtd><mrow><mrow><msub><mi>θ</mi><mi>ybr</mi></msub><mo>=</mo><mrow><msup><mi>sin</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo>⁢</mo><mrow><mo>{</mo><mrow><msub><mi>Θ</mi><mrow><mi>y</mi><mo>-</mo><mi>susp</mi></mrow></msub><mo>+</mo><msub><mi>Θ</mi><mrow><mi>y</mi><mo>-</mo><mi>whl</mi></mrow></msub></mrow><mo>}</mo></mrow></mrow></mrow><mo>⁢</mo><mstyle><mtext></mtext></mstyle><mo>⁢</mo><mrow><msub><mi>θ</mi><mi>xbr</mi></msub><mo>=</mo><mrow><msup><mi>sin</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo>⁢</mo><mrow><mo>{</mo><mfrac><mrow><msub><mi>Θ</mi><mrow><mi>x</mi><mo>-</mo><mi>susp</mi></mrow></msub><mo>+</mo><msub><mi>Θ</mi><mrow><mi>x</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>-</mo><mi>whl</mi></mrow></msub></mrow><mrow><mi>cos</mi><mo>⁡</mo><mrow><mo>(</mo><msub><mi>θ</mi><mi>ybr</mi></msub><mo>)</mo></mrow></mrow></mfrac><mo>}</mo></mrow></mrow></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>11</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths><img file="USRE40496E_D0005.tif" />

0054Since there are no restrictions in Equation (11), it is valid regardless of if the four wheels of the vehicle contact the road surface or lift from the road, as soon as the accurate characterization of the roll and pitch gradients Θ_x-suspand Θ_y-susp, and Θ_x-whland Θ_y-whlare available. Hence in the following Θ_x-susp,Θ_y-susp,Θ_x-whland Θ_y-whlmay be computed based on the available sensor signals.

Estimate the Roll and Pitch Gradients

0055From the formula in Equation (8), the roll and pitch gradients Θ_x-suspand Θ_y-suspare related to the suspension stroke. The estimation schemes are sought for computing Θ_x-suspand Θ_y-suspfrom the available sensor signals.

0056Consider in Equation (3) that the distance differences between the left side corners and right side corners are equal, that is: <br />z_lf−z_rf=z_lr−z_rr  (12)<br /> or: <br />s_lf−s_rf=s_lr−s_rr+[w_lf−w_rr−w_lf+w_rf]  (13)<br /> Since the tire deflections are much smaller than the suspension stroke, from (13) it is reasonable to say <br />s_lf−s_rf>>s_lr−s_rr  (14)<br /> or rewrite this as: <maths id="MATH-US-00006" num="00006"><math overflow="scroll"><mtable><mtr><mtd><mrow><mfrac><mrow><msub><mi>s</mi><mi>lf</mi></msub><mo>-</mo><msub><mi>s</mi><mi>rf</mi></msub></mrow><mrow><msub><mi>s</mi><mi>lr</mi></msub><mo>-</mo><msub><mi>s</mi><mi>rr</mi></msub></mrow></mfrac><mo>≈</mo><mn>1</mn></mrow></mtd><mtd><mrow><mo>(</mo><mn>15</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths><img file="USRE40496E_D0006.tif" /><br /> Hence, for any given constant weight k, we have: <maths id="MATH-US-00007" num="00007"><math overflow="scroll"><mtable><mtr><mtd><mrow><msub><mi>Θ</mi><mrow><mi>x</mi><mo>-</mo><mi>susp</mi></mrow></msub><mo>≈</mo><mfrac><mrow><mrow><mi>κ</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><msub><mi>s</mi><mi>lf</mi></msub><mo>-</mo><msub><mi>s</mi><mi>rf</mi></msub></mrow><mo>)</mo></mrow></mrow><mo>+</mo><msub><mi>s</mi><mi>lr</mi></msub><mo>-</mo><msub><mi>s</mi><mi>rr</mi></msub></mrow><mrow><mn>2</mn><mo>/</mo><mrow><mo>(</mo><mrow><mi>κ</mi><mo>+</mo><mi>l</mi></mrow><mo>)</mo></mrow></mrow></mfrac></mrow></mtd><mtd><mrow><mo>(</mo><mn>16</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths><img file="USRE40496E_D0007.tif" /><br /> In the sequential discussion, the Equation (16) may be used to describe the roll gradient Θ_x-susp.

0057Θ_x-suspand Θ_y-suspmust then be related to the available sensor signals. The following dynamic relationship which are obeyed by the car body through the Newton law described around the c.g. of the vehicle body <maths id="MATH-US-00008" num="00008"><math overflow="scroll"><mtable><mtr><mtd><mrow><mrow><mrow><msub><mi>I</mi><mi>x</mi></msub><mo>⁢</mo><msub><mover><mi>ω</mi><mo>.</mo></mover><mi>x</mi></msub></mrow><mo>=</mo><mrow><mrow><msub><mi>h</mi><mi>y</mi></msub><mo>⁢</mo><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>-</mo><mn>1</mn></mrow><mn>4</mn></munderover><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><msub><mi>F</mi><mi>yi</mi></msub></mrow></mrow><mo>+</mo><mrow><mi>l</mi><mo>(</mo><mrow><mrow><msub><mi>K</mi><mi>f</mi></msub><mo>⁢</mo><msub><mi>s</mi><mi>lf</mi></msub></mrow><mo>+</mo><mrow><msub><mi>D</mi><mi>f</mi></msub><mo>⁢</mo><msub><mover><mi>s</mi><mo>.</mo></mover><mi>lf</mi></msub></mrow></mrow><mo>)</mo></mrow><mo>-</mo><mrow><mi>l</mi><mo>⁢</mo><mrow><mo>(</mo><mrow><mrow><msub><mi>K</mi><mi>f</mi></msub><mo>⁢</mo><msub><mi>s</mi><mi>rf</mi></msub></mrow><mo>+</mo><mrow><msub><mi>D</mi><mi>f</mi></msub><mo>⁢</mo><msub><mover><mi>s</mi><mo>.</mo></mover><mi>rf</mi></msub></mrow></mrow><mo>)</mo></mrow></mrow><mo>+</mo><mrow><mi>l</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><mrow><msub><mi>K</mi><mi>r</mi></msub><mo>⁢</mo><msub><mi>s</mi><mi>lr</mi></msub></mrow><mo>+</mo><mrow><msub><mi>D</mi><mi>r</mi></msub><mo>⁢</mo><msub><mover><mi>s</mi><mo>.</mo></mover><mi>lr</mi></msub></mrow></mrow><mo>)</mo></mrow></mrow><mo>-</mo><mrow><mi>l</mi><mo>(</mo><mrow><mrow><msub><mi>K</mi><mi>r</mi></msub><mo>⁢</mo><msub><mi>s</mi><mi>rr</mi></msub></mrow><mo>+</mo><mrow><msub><mi>D</mi><mi>r</mi></msub><mo>⁢</mo><msub><mover><mi>s</mi><mo>.</mo></mover><mi>rr</mi></msub></mrow></mrow><mo>)</mo></mrow><mo>+</mo><mrow><msub><mi>K</mi><mrow><mi>anti</mi><mo>-</mo><mi>roll</mi><mo>-</mo><mi>f</mi></mrow></msub><mo>⁢</mo><mfrac><mrow><mo>(</mo><mrow><msub><mi>s</mi><mi>lf</mi></msub><mo>-</mo><msub><mi>s</mi><mi>rf</mi></msub></mrow><mo>)</mo></mrow><mi>l</mi></mfrac></mrow><mo>+</mo><mrow><msub><mi>K</mi><mrow><mi>anti</mi><mo>-</mo><mi>roll</mi><mo>-</mo><mi>r</mi></mrow></msub><mo>⁢</mo><mfrac><mrow><mo>(</mo><mrow><msub><mi>s</mi><mi>lr</mi></msub><mo>-</mo><msub><mi>s</mi><mi>rr</mi></msub></mrow><mo>)</mo></mrow><mi>l</mi></mfrac></mrow></mrow></mrow><mo>⁢</mo><mstyle><mtext></mtext></mstyle><mo>⁢</mo><mrow><mrow><msub><mi>I</mi><mi>y</mi></msub><mo>⁢</mo><msub><mover><mi>ω</mi><mo>.</mo></mover><mi>y</mi></msub></mrow><mo>=</mo><mrow><mrow><msub><mi>h</mi><mi>x</mi></msub><mo>⁢</mo><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mn>4</mn></munderover><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><msub><mi>F</mi><mi>xi</mi></msub></mrow></mrow><mo>+</mo><mrow><msub><mi>l</mi><mi>f</mi></msub><mo>⁡</mo><mrow><mo>(</mo><mrow><mrow><msub><mi>K</mi><mi>f</mi></msub><mo>⁢</mo><msub><mi>s</mi><mi>lf</mi></msub></mrow><mo>+</mo><mrow><msub><mi>D</mi><mi>f</mi></msub><mo>⁢</mo><msub><mover><mi>s</mi><mo>.</mo></mover><mi>lf</mi></msub></mrow></mrow><mo>)</mo></mrow></mrow><mo>-</mo><mrow><msub><mi>l</mi><mi>f</mi></msub><mo>(</mo><mrow><mrow><msub><mi>K</mi><mi>f</mi></msub><mo>⁢</mo><msub><mi>s</mi><mi>rf</mi></msub></mrow><mo>+</mo><mrow><msub><mi>D</mi><mi>f</mi></msub><mo>⁢</mo><msub><mover><mi>s</mi><mo>.</mo></mover><mi>rf</mi></msub></mrow></mrow><mo>)</mo></mrow><mo>-</mo><mrow><msub><mi>t</mi><mi>r</mi></msub><mo>(</mo><mrow><mrow><msub><mi>K</mi><mi>r</mi></msub><mo>⁢</mo><msub><mi>s</mi><mi>lr</mi></msub></mrow><mo>+</mo><mrow><msub><mi>D</mi><mi>f</mi></msub><mo>⁢</mo><msub><mover><mi>s</mi><mo>.</mo></mover><mi>lr</mi></msub></mrow></mrow><mo>)</mo></mrow><mo>-</mo><mrow><msub><mi>t</mi><mi>r</mi></msub><mo>(</mo><mrow><mrow><msub><mi>K</mi><mi>r</mi></msub><mo>⁢</mo><msub><mi>s</mi><mi>rr</mi></msub></mrow><mo>+</mo><mrow><msub><mi>D</mi><mi>r</mi></msub><mo>⁢</mo><msub><mover><mi>s</mi><mo>.</mo></mover><mi>rr</mi></msub></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>⁢</mo><mstyle><mtext></mtext></mstyle><mo>⁢</mo><mrow><mrow><msub><mi>M</mi><mi>s</mi></msub><mo>⁢</mo><msub><mi>a</mi><mi>y</mi></msub></mrow><mo>=</mo><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mn>4</mn></munderover><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><msub><mi>F</mi><mi>yi</mi></msub></mrow></mrow><mo>⁢</mo><mstyle><mtext></mtext></mstyle><mo>⁢</mo><mrow><mrow><msub><mi>M</mi><mi>s</mi></msub><mo>⁢</mo><msub><mi>a</mi><mi>x</mi></msub></mrow><mo>=</mo><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mn>4</mn></munderover><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><msub><mi>F</mi><mi>xi</mi></msub></mrow></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>17</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths><img file="USRE40496E_D0008.tif" /><br /> where I_xand I_yare the momentum of inertia of the car body with respect to the x and y axis respectively; M_sis the sprung mass (the mass of the car body); h_xis the c.g. height of the car body with respect to the top of the suspension; K_fand K_rare the front and rear suspension spring rates with unit N/m. K_anti-roll-fand K_anti-roll-rare the stiffnesses for the front and the rear anti-roll bar, with unit Nm/rad. D_fand D_rare the front and the rear suspension damper rates; F_xiis the ith suspension force applied to the car body along the body fixed direction b₁, and F_yiis the ith suspension force applied to the car body along the body fixed direction b₂. <br /> Define a weight: <maths id="MATH-US-00009" num="00009"><math overflow="scroll"><mtable><mtr><mtd><mrow><mi>k</mi><mo>=</mo><mfrac><mrow><mrow><msup><mi>l</mi><mn>2</mn></msup><mo>⁢</mo><msub><mi>K</mi><mi>f</mi></msub></mrow><mo>+</mo><msub><mi>K</mi><mrow><mi>anti</mi><mo>-</mo><mi>roll</mi><mo>-</mo><mi>f</mi></mrow></msub></mrow><mrow><mrow><msup><mi>l</mi><mn>2</mn></msup><mo>⁢</mo><msub><mi>K</mi><mi>r</mi></msub></mrow><mo>+</mo><msub><mi>K</mi><mrow><mi>anti</mi><mo>-</mo><mi>roll</mi><mo>-</mo><mi>r</mi></mrow></msub></mrow></mfrac></mrow></mtd><mtd><mrow><mo>(</mo><mn>18</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths><img file="USRE40496E_D0009.tif" /><br /> Since the damping rates are usually proportional to the spring rates for suspensions, it is reasonable to assume: <maths id="MATH-US-00010" num="00010"><math overflow="scroll"><mtable><mtr><mtd><mrow><mfrac><mrow><mrow><msup><mi>l</mi><mn>2</mn></msup><mo>⁢</mo><msub><mi>D</mi><mi>f</mi></msub></mrow><mo>+</mo><msub><mi>D</mi><mrow><mi>anti</mi><mo>-</mo><mi>roll</mi><mo>-</mo><mi>f</mi></mrow></msub></mrow><mrow><mrow><msup><mi>l</mi><mn>2</mn></msup><mo>⁢</mo><msub><mi>D</mi><mi>r</mi></msub></mrow><mo>+</mo><msub><mi>D</mi><mrow><mi>anti</mi><mo>-</mo><mi>roll</mi><mo>-</mo><mi>r</mi></mrow></msub></mrow></mfrac><mo>≈</mo><mi>k</mi></mrow></mtd><mtd><mrow><mo>(</mo><mn>19</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths><img file="USRE40496E_D0010.tif" /><br /> For a well balanced vehicle, the normal dead loading applied to the vehicle should not generate significant body attitude variation when the vehicle is parked on a flat road. That is, the roll and pitch attitude angles induced by the normal dead loading during flat road parking should be close to zero. For this reason, it is reasonable to assume the following holds: <br />t_rK_r=t_fK_f  (20)<br /> Similar argument can be used for suspension damping rates.

0058Through algebraic manipulation the first two equations in Equation (17) can be rewritten as the following: <maths id="MATH-US-00011" num="00011"><math overflow="scroll"><mtable><mtr><mtd><mrow><mrow><mrow><msub><mi>I</mi><mi>x</mi></msub><mo>⁢</mo><msub><mover><mi>ω</mi><mo>.</mo></mover><mi>x</mi></msub></mrow><mo>=</mo><mrow><mrow><msub><mi>h</mi><mi>y</mi></msub><mo>⁢</mo><msub><mi>M</mi><mi>s</mi></msub><mo>⁢</mo><msub><mi>a</mi><mi>y</mi></msub></mrow><mo>+</mo><mrow><mrow><mo>(</mo><mrow><msub><mi>lK</mi><mi>r</mi></msub><mo>+</mo><mfrac><msub><mi>K</mi><mrow><mi>anti</mi><mo>-</mo><mi>roll</mi><mo>-</mo><mi>r</mi></mrow></msub><mi>l</mi></mfrac></mrow><mo>)</mo></mrow><mo>⁡</mo><mrow><mo>[</mo><mrow><mrow><mi>k</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><msub><mi>s</mi><mi>lf</mi></msub><mo>-</mo><msub><mi>s</mi><mi>rf</mi></msub></mrow><mo>)</mo></mrow></mrow><mo>+</mo><mrow><mo>(</mo><mrow><msub><mi>s</mi><mi>lr</mi></msub><mo>-</mo><msub><mi>s</mi><mi>rr</mi></msub></mrow><mo>)</mo></mrow></mrow><mo>]</mo></mrow></mrow><mo>+</mo><mrow><mrow><mo>(</mo><mrow><msub><mi>lD</mi><mi>r</mi></msub><mo>+</mo><mfrac><msub><mi>D</mi><mrow><mi>anti</mi><mo>-</mo><mi>roll</mi><mo>-</mo><mi>r</mi></mrow></msub><mi>l</mi></mfrac></mrow><mo>)</mo></mrow><mo>⁡</mo><mrow><mo>[</mo><mrow><mrow><mi>k</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><msub><mover><mi>s</mi><mo>.</mo></mover><mi>lf</mi></msub><mo>-</mo><msub><mover><mi>s</mi><mo>.</mo></mover><mi>rf</mi></msub></mrow><mo>)</mo></mrow></mrow><mo>+</mo><mrow><mo>(</mo><mrow><msub><mover><mi>s</mi><mo>.</mo></mover><mi>lr</mi></msub><mo>-</mo><msub><mover><mi>s</mi><mo>.</mo></mover><mi>rr</mi></msub></mrow><mo>)</mo></mrow></mrow><mo>]</mo></mrow></mrow></mrow></mrow><mo>⁢</mo><mstyle><mtext></mtext></mstyle><mo>⁢</mo><mrow><mrow><msub><mi>I</mi><mi>y</mi></msub><mo>⁢</mo><msub><mover><mi>ω</mi><mo>.</mo></mover><mi>x</mi></msub></mrow><mo>=</mo><mrow><mrow><msub><mi>h</mi><mi>x</mi></msub><mo>⁢</mo><msub><mi>M</mi><mi>s</mi></msub><mo>⁢</mo><msub><mi>a</mi><mi>x</mi></msub></mrow><mo>+</mo><mrow><msub><mi>t</mi><mi>r</mi></msub><mo>⁢</mo><mrow><msub><mi>K</mi><mi>r</mi></msub><mo>(</mo><mrow><msub><mi>s</mi><mi>lf</mi></msub><mo>+</mo><msub><mi>s</mi><mi>rf</mi></msub><mo>-</mo><msub><mi>s</mi><mi>lr</mi></msub><mo>-</mo><msub><mi>s</mi><mi>rr</mi></msub></mrow><mo>)</mo></mrow></mrow><mo>+</mo><mrow><msub><mi>t</mi><mi>r</mi></msub><mo>⁢</mo><mrow><msub><mi>D</mi><mi>r</mi></msub><mo>(</mo><mrow><msub><mover><mi>s</mi><mo>.</mo></mover><mi>lf</mi></msub><mo>+</mo><msub><mover><mi>s</mi><mo>.</mo></mover><mi>rf</mi></msub><mo>-</mo><msub><mover><mi>s</mi><mo>.</mo></mover><mi>lr</mi></msub><mo>-</mo><msub><mover><mi>s</mi><mo>.</mo></mover><mi>rr</mi></msub></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>21</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths><img file="USRE40496E_D0011.tif" /><br /> Using the definition of Θ_x-suspand Θ_y-susp, Equation (21) can be rewritten as: <br />{dot over (w)}_x=c₀a_y+c₁Θ_x-susp+c₂{dot over (Θ)}_x-susp<br />{dot over (w)}_y=d₀a_x+d₁Θ_y-susp+d₂{dot over (Θ)}_y-susp<br />{dot over (w)}_x<i>=c</i>₀<i>a</i>_y<i>+c</i>₁<i>Θ</i>_x-susp<i>+c</i>₂<i>{dot over (Θ)}</i>_x-susp<br /><i>{dot over (w)}</i>_y<i>=d</i>₀<i>a</i>_x<i>+d</i>₁<i>Θ</i>_y-susp<i>+d</i>₂<i>{dot over (Θ)}</i>_y-susp  (22)<br /> that is, Θ_x-susp(t) and Θ_y-susp(t) obeys the 1^storder differential equations, and the coefficients c₀,c₁,c₂,d₀,d₁and d₂can be obtained by comparing Equation (21) and Equation (22). Although the analytical solution for Equation (22) are not hard to find, the solutions may be directly implemented in digital environment. On the other hand, the pitch rate signal is not measured, but an estimation of the pitch rate signal can be obtained as a function of the measured signals and the signals computed from the measured signals: <br />{dot over (w)}_y={circumflex over (θ)}_ysec({circumflex over (θ)}_x)+w_ztan({circumflex over (θ)}_x)  (23)<br /> where {circumflex over (θ)}_xand {circumflex over (θ)}_yare the estimated global roll and pitch Euler angles of the vehicle body (with respect to the sea level). The details if this are described in U.S. application Ser. No. 09/967,93809/967,038 which is incorporated by reference herein. Using the estimated pitch rate signal, (22) can be used to solve for Θ_x-susp(t) and Θ_y-susp(t) at time instant t. In the following a digital scheme will be summarized. Two variables as defined at each sampling instant: <maths id="MATH-US-00012" num="00012"><math overflow="scroll"><mtable><mtr><mtd><mrow><mrow><mrow><msub><mi>RRA</mi><mi>—</mi></msub><mo>⁢</mo><mrow><mi>RAW</mi><mo>⁡</mo><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mrow><mfrac><mn>1</mn><msub><mi>c</mi><mn>1</mn></msub></mfrac><mo>⁢</mo><mrow><msub><mover><mi>ω</mi><mo>.</mo></mover><mi>x</mi></msub><mo>⁡</mo><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></mrow></mrow><mo>-</mo><mrow><mfrac><msub><mi>c</mi><mn>0</mn></msub><msub><mi>c</mi><mn>1</mn></msub></mfrac><mo>⁢</mo><mrow><msub><mi>a</mi><mi>y</mi></msub><mo>(</mo><mi>k</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>⁢</mo><mstyle><mtext></mtext></mstyle><mo>⁢</mo><mrow><mrow><msub><mi>RPA</mi><mi>—</mi></msub><mo>⁢</mo><mrow><mi>RAW</mi><mo>⁡</mo><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mrow><mfrac><mn>1</mn><msub><mi>d</mi><mn>1</mn></msub></mfrac><mo>⁢</mo><mrow><msub><mover><mi>ω</mi><mo>.</mo></mover><mi>y</mi></msub><mo>(</mo><mi>k</mi><mo>)</mo></mrow></mrow><mo>-</mo><mrow><mfrac><msub><mi>d</mi><mn>0</mn></msub><msub><mi>d</mi><mn>1</mn></msub></mfrac><mo>⁢</mo><mrow><msub><mi>a</mi><mi>x</mi></msub><mo>⁡</mo><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></mrow></mrow></mrow></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>24</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths><img file="USRE40496E_D0012.tif" /><br /> Then at the (k+1)th sampling instant (current values), the estimates of the roll and pitch gradients {circumflex over (Θ)}_x-susp(k+1) and {circumflex over (Θ)}_y-susp(k+1) may be computed from their values in the kth sampling instant (past values) and the current and past values of RRA_RAW and RPA_RAW. The iterative formula may be expressed as the following with properly chosen coefficients e₀,e₁,f₀and f₁: <br />{circumflex over (Θ)}_x-susp(k+1)=e₀{circumflex over (Θ)}_x-susp(k)+e₁[RRA_RAW(k+1)+RRA_RAW(k)]<br />{circumflex over (Θ)}_y-susp(k+1)=f₀{circumflex over (Θ)}_y-susp(k)+f₁[RPA_RAW(k+1)+RPA_RAW(k)]  (25)

0059The wheel motion-induced roll and pitch gradients are usually much smaller than the suspension motion induced gradients due to the small tire deflections at each wheel/tire assembly. Therefore: <br />Θ_x-whl&lt;&lt;Θ_x-susp<br />Θ_y-whl&lt;&lt;Θ_y-susp  (26)<br /> or say: <br />Θ_x≈Θ_x-susp<br />Θ_y≈Θ_y-susp  (27)

0060As described above, the present invention uses Equation (27) to approximately calculate the roll and pitch gradients. The relative roll and pitch attitude angles can be computed as in Equations (5) and (6).

0061Referring now to <figref idref="DRAWINGS">FIG. 5</figref>, a flow chart summarizing the method of the present invention is illustrated. In step <b>70</b> the sensor signals of the sensor set are read. In the present example, a roll rate sensor determines the roll rate of the vehicle, a lateral acceleration sensor generates a lateral acceleration signal of the vehicle body, and a longitudinal acceleration sensor generates a longitudinal acceleration signal of the vehicle body. The yaw rate of the vehicle body is also measured. In step <b>72</b>, a calculated pitch rate signal is determined from the yaw rate, the roll rate, the lateral acceleration, and the longitudinal acceleration. In step <b>74</b> the global roll attitude and global pitch attitude are determined from the calculated pitch rate, the roll rate, the lateral acceleration, and the longitudinal acceleration. In step <b>78</b> a roll gradient is determined based upon a past roll rate and the roll rate and the lateral acceleration signal. A relative roll angle is determined in step <b>80</b> based upon the roll gradient. In step <b>82</b> a pitch gradient based upon the passivepast raw pitch rate, the calculated pitch rate, and the longitudinal acceleration is determined. In step <b>84</b> the relative pitch angle based upon the pitch gradient is determined. In step <b>86</b> a safety device is activated in response to the relative roll angle, the relative pitch angle, the global roll angle and the global pitch angle.

0062While particular embodiments of the invention have been shown and described, numerous variations and alternate embodiments will occur to those skilled in the art. Accordingly, it is intended that the invention be limited only in terms of the appended claims.