Vehicle stability control

[0002] The present invention relates generally to vehicular stability control. More particularly, the invention relates to a method for enhancing vehicle stability control.

[0003] In vehicle stability enhancement (VSE) systems, and some other chassis control systems, a significant effort during vehicle tuning may be devoted to the process of characterizing vehicle response in the yaw plane to the steering inputs. This may be accomplished by building look up tables that give steady state values of vehicle yaw rate, and sometimes lateral velocity for various steering angles and vehicle speeds. Since several values of speeds and steering angles have to be considered, the tables involve more than a hundred numbers, each of which has to be determined experimentally. This usually involves several days of testing, which has to be performed on a dry surface, thus being dependent on the weather conditions.

[0004] Another drawback of the look up tables is that linear interpolation is used between any two points defined in the table. This may lead to errors when the function approximated by piecewise linear segments is strongly nonlinear. Accordingly, it would be desirable to provide a relatively simple strategy that would permit one to determine the desired values from analytical expressions using vehicle parameter data, which are usually supplied by the vehicle manufacturer. In this way the desired values (of yaw rate and lateral velocity) could be continuously computed, or the look up table values could be determined by running a utility file on a computer. Only limited testing may be needed to verify the correctness of the model.

[0005] Therefore, it would be desirable to provide a strategy for enhancing vehicle stability control that overcomes the aforementioned and other disadvantages.

[0006] One aspect of the present invention provides a method of vehicle stability control. A rear axle cornering stiffness coefficient in a linear handling range is determined. A first understeer coefficient in a linear handling range is determined. A desired lateral acceleration is determined based on the first understeer coefficient. A second understeer coefficient is determined based on a limited magnitude of the desired lateral acceleration. A desired yaw rate is determined based on the second understeer coefficient. A desired lateral velocity is determined based on the desired yaw rate and the rear axle cornering stiffness coefficient. The rear axle cornering stiffness coefficient may be determined based on a cross-over lateral velocity. A front axle cornering stiffness coefficient in a linear handling range may be determined. The determined desired yaw rate, desired lateral velocity, and desired lateral acceleration may be transmitted to a vehicle control system.

[0007] Another aspect of the invention provides a computer usable medium, including a program, for vehicle stability control. The invention provides computer readable program code for performing the method steps described above.

[0008] The foregoing and other features and advantages of the invention will become further apparent from the following detailed description of the presently preferred embodiments, read in conjunction with the accompanying drawings. The detailed description and drawings are merely illustrative of the invention, rather than limiting the scope of the invention being defined by the appended claims and equivalents thereof.

[0009] FIG. 1 is a graph of understeer coefficients as functions of lateral acceleration for two sets of suspension parameters;

[0010] FIG. 2 is a graph of desired lateral acceleration and yaw rates determined from a linear model with saturation;

[0011] FIG. 3 is a graph of understeer coefficient from test data and a model according to the present invention;

[0012] FIG. 4 is a flow chart of a vehicle stability control algorithm made in accordance with the present invention; and

[0013] FIG. 5 is a graph of vehicle yaw rate from test data and a model according to the present invention.

[0014] The present invention may provide a simple method for determining a desired yaw rate and lateral velocity of vehicle in steady-state as functions of the steering angle, vehicle speed, and several vehicle parameters that are generally known. This could potentially eliminate, or at least greatly simplify, a costly and time-consuming process of experimentally determining the desired values of yaw rate and lateral velocity during vehicle tuning. In some applications, customer handling specifications in terms of the desired understeer coefficient under specific lateral accelerations could be directly translated into the desired values.

[0015] An example of customer specification in terms of understeer coefficient is provided in FIG. 1. Note that the understeer coefficients in the provided example increase with lateral acceleration. Note also that the understeer coefficients may be significantly affected by suspension tuning, since both curves were determined for vehicle with the same tires but different suspension parameters.

[0016] Two principle ideas provided in the following description are: (1) the cornering stiffness coefficients of the front and rear axles may be determined from simple tests, thus representing the entire vehicle influence, including the effects of suspension design; (2) the cornering stiffness coefficients and the understeer coefficient are functions of the desired lateral acceleration, which is a function of steering angle and vehicle speed.

[0017] The desired yaw rate and side-slip velocity may be in principle determined from the well-known linear bicycle model, with additional saturation at the limit of adhesion. The linear model yields the following equations for the vehicle yaw rate, Ω_d, lateral acceleration, a_yd, and lateral velocity, v_yd:

[0018] Ω_dl=v_x*δ/(L+K_u0*v_x²)  (1a)

[0019] a_ydl=v_x²*δ/(L+K_u0*v_x²)=v_x*Ω_dl  (1b)

[0020] v_ydl=Ω_dl*[b−M*a*v_x²/(C_r0*L)]  (1c)

[0021] where v_xis vehicle speed, δ is the steering angle of the front wheels (which may be computed by multiplying the measured steering wheel angle by the steer gear ratio), L is vehicle wheelbase, K_u0is the understeer coefficient in the linear range of handling, b is the distance of the vehicle center of gravity to the rear axle, a is the distance of the vehicle center of gravity to the front axle, M is vehicle mass, and C_r0is the cornering stiffness of the rear axle (both tires) in the linear range of handling; the superscript “l” refers to the (steady-state) values obtained from the linear model. Note that according to equation (1c) a vehicle speed v_xexists such that the lateral velocity v_yis identically equal to zero regardless of the steering angle. In order to account for the saturation of the tire forces at the limit of adhesion, the desired values should be saturated at the level corresponding to the maximum lateral acceleration that the vehicle can develop on dry surface, a_ymax. This yields the following: $\begin{array}{cc}{\Omega }_{\mathrm{dl\_lim}}=\{\begin{array}{c}{v}_x^{*}\delta /\left(L+K_{\mathrm{u0}}^{*}{v}_x^2\right)\mathrm{when}\left|a_{\mathrm{ydl}}\right|<a_{y\max }\\ a_{y\max }^{*}\mathrm{sign}\left(\delta \right)/v_x\mathrm{when}\left|a_{\mathrm{ydl}}\right|\ge a_{y\max }\end{array}& \left(2a\right)\end{array}$  v_ydllim=Ω_dllim*[b−M*a*v_x²/(C_r0*L)]  (2b)

[0022] The desired values of lateral acceleration and yaw rate obtained from the above equations are plotted against the measured values in FIG. 2. The linear model predicts vehicle response very well up to lateral acceleration corresponding to about 0.6*a_ymax, while the saturation function is quite accurate at the limit of adhesion. In the intermediate range, however, the model may not be accurate.

[0023] There are at least two causes for the desired yaw response determined from this simple analytical model to be inaccurate in the nonlinear range of handling. First, it is assumed in the model that both the cornering stiffness of each axle and the understeer coefficient are constants. In reality, the cornering stiffness values and the understeer coefficient may depend on the normal load transfer during cornering, which depends on the lateral acceleration. Thus they are primary functions of lateral acceleration. Secondly, the cornering stiffness values used by the model are often derived from the tire data and the known nominal load.

[0024] To improve the accuracy of the model, the cornering stiffness values used in the bicycle model should reflect suspension properties. The proposed approach of the present invention can overcome both problems. The cornering stiffness values and the understeer coefficient may be determined as functions of desired lateral acceleration. The cornering stiffness values in the linear range of handling may be determined from simple vehicle tests, thus reflecting the effects of suspension.

[0025] For the linear vehicle model, the understeer coefficient may be determined from the following equation:

[0026] K_u0=M*(b/C_f0−a/C_r0)/L=M_f/C_f0−M_r/C_r0  (3)

[0027] where a and b are distances of front and rear axle to the vehicle center of gravity, C_f0and C_r0are the cornering stiffness of front and rear axle, M_f=M*b/L and M_r=M*a/L are the front and rear mass of vehicle. In the linear model, the cornering stiffness values are assumed to be constant. In reality, they depend on the normal load transfer during cornering, which is primarily a function of lateral acceleration. This dependency is the result of the fact that the cornering stiffness per tire depends on the normal tire load. A simple and generally accepted model reflecting this relationship is as follows (Gillespie, 1992; Dixon, 1996):

[0028] C_y=C_y0*(N/N₀)*[1−k*(ΔN/N₀)]  (4)

[0029] where C_yand C_y0are cornering stiffness of one tire at an actual normal load, N, and at a nominal normal load, N₀, respectively. The actual normal load, N, is the sum of N₀and ΔN wherein ΔN is the normal load transfer. The constant, k, which may have a typical value of 0.5, is the cornering stiffness sensitivity to normal load. If the load transfer of the left tire of a given axle is ΔN, then load transfer of the right tire is −ΔN. A simple manipulation shows that the cornering stiffness for the given (e.g. front) axle is:

[0030] C_f=2*C_y0*[1−k*(ΔN/N₀)²]  (5)

[0031] Thus it is a quadratic function of the normal load transfer (a higher order polynomial may be used if more accuracy is desired). Assuming further that the roll moment distribution between front and rear axle is approximately proportional to the mass distribution, the lateral load transfer may be expressed as a function of lateral acceleration:

[0032] ΔN/N₀=(2*h_roll/t_w)*(a_y/g)  (6)

[0033] where h_rollis the height of vehicle center of gravity above the roll axis, t_wis the track width and g is acceleration of gravity. Consequently, the cornering stiffness of the front axle may be expressed as the following function of lateral acceleration:

[0034] C_f=C_f0*(1−B₂*a_y²)  (7)

[0035] Where B₂=k*[(2*h_roll)/(t_w*g)]². Similarly,

[0036] C_r=C_r0*(1−B₂*a_y²)  (8)

[0037] If the roll moment distribution is significantly different from the mass distribution, the coefficient B₂will be somewhat different for the front and rear axle. Substituting equations (7) and (8) into equation (3) yields the following equation after simple manipulation:

[0038] K_u=K_u0*[1+B₂*a_y²/(1−B₂*a_y²)]≅K_u0*[1+B₂*a_y²+B₂²*a_y⁴]  (9)

[0039] with K_u0given by equation (3). The last equality follows from a Taylor expansion of 1/(1−B₂*a_y²) under the assumption that B₂*a_y²is significantly less than 1. Thus the understeer coefficient may be represented as a polynomial in the lateral acceleration, a_y. An example of the understeer coefficient determined from the vehicle test data supplied by OEM, along with the one computed from equation (9) is shown in FIG. 3. The understeer coefficient may be represented as a polynomial in lateral acceleration and a good fit to the data may be obtained.

[0040] The lateral stiffness values for front and rear axles in the linear range of handling, C_f0and C_r0, may be determined as follows. First, the understeer coefficient in the linear range of handling, K_u0, may be determined experimentally. Usually, this value is provided by a vehicle manufacturer. This supplies the first equation, (3). The second equation may be obtained by experimentally determining the velocity at which vehicle side slip velocity is zero regardless of the steering angle (provided that vehicle remains in the linear range of handling). According to equation (1c), this characteristic velocity is:

[0041] v_x0²=b*C_r0/M_r  (10)

[0042] M_r=M*a/L is as defined previously (M is the total vehicle mass, a is the distance from center of gravity to the front axle, and L is the wheelbase). Solving equations (3) and (10) for the cornering stiffness coefficients, C_f0and C_r0, yields:

[0043] C_r0=M_r*v_x0²/b; C_f0=M_f*v_x0²/(b+K_u0*v_x0²)  (11)

[0044] The desired values of yaw rate and lateral velocity may now be computed from the equations (1a) and (1c), in which the understeer coefficient and rear cornering stiffness for the linear handling range, K_u0and C_r0, are replaced by the nonlinear functions given by equations (8) and (9), respectively. It may be desirable, however, that the desired values of yaw rate and lateral velocity be dependent only on steering angle, δ, and vehicle speed, v_x. This may be achieved when the actual lateral acceleration is replaced by the desired lateral acceleration obtained from the linear equation (1b), subjected to the suitable limitation. As a result, the understeer coefficient and the cornering stiffness become functions of steering angle and speed, but not of lateral acceleration. Since the desired lateral acceleration is larger than the actual value in the nonlinear range of handling, the coefficient B₂in equations (8) and (9) may be reduced somewhat when ay is replaced by a_ydllim(the desired lateral acceleration obtained from the linear model, equation 1b, but with the magnitude limited to a_ymax). Good results may be obtained by neglecting the 4-th order term in calculation of K_uusing a_ydllim.

[0045] By substituting the understeer coefficient, K_u, and the rear axle cornering stiffness coefficient, C_r(both coefficients being functions of steering angle δ and speed v_x) in place of K_u0and C_r0(which are constant) in equations (2a) and (2b), the desired values of yaw rate, Ω_d, and lateral velocity, v_yd, may be obtained. These desired values are the steady-state values, which describe vehicle response when both vehicle speed and the steering angle are constant or slowly-varying functions of time. For some vehicle stability enhancement systems, these values can be used as desired values. In other systems, there may be a need to represent vehicle dynamic response more accurately to better characterize the desired vehicle response during quick transient maneuvers; the process of obtaining the desired values from the steering angle and speed is sometimes referred to as a driver command interpreter.

[0046] Vehicle response in the yaw plane may be modeled as a velocity-dependent second order system, characterized by an undampened natural frequency and damping ratio. Both of these parameters are functions of vehicle speed. The parameters also (according to the linear handling model) directly depend on vehicle parameters, such as distances, mass, moment of inertia (about yaw axis), and cornering stiffness coefficients. Using the second order model, vehicle dynamic response may be obtained from the steady state values. For example, the relationship between the steady-state values and the dynamic values of yaw rate, lateral velocity, and lateral acceleration and the corresponding dynamic values in the Laplace domain may be described as follows:

[0047] Ω_ddyn(s)=Ω_d*(1+T_Ωs)/[1+(2*ζ/ω_n)*s+(1/ω_n²)*s²]  (12a)

[0048] v_yddyn(s)=Ω_d*(1+T_vys)/[1+(2*ζ/ω_n)*s+(1/ω_n²)*s²]  (12b)

[0049] a_yddyn=s*v_yddyn(s)+v_x*Ω_ddyn(s)  (12c)

[0050] where ω_nis the undampened natural frequency, ζ is the damping ratio, and s is the Laplace operand. The above equations may be written in a time domain as differential or difference equations and implemented in a microprocessor (in a manner known to those skilled in art) to generate the dynamic values from the steady state values.

[0051] The parameters of the dynamic filters used in a driver command interpreter may now be determined as functions of the nonlinear cornering stiffness parameters. The undampened natural frequency, ω_n, and the damping ratio, ζ, of the second order filter reflecting vehicle dynamics in the yaw plane may be expressed as follows:

[0052] ω_n²=(C_f*C_r*L²)/(M*I_zz*v_x²)+(C_r*b−C_f*a)/I_zz  (13)

[0053] ζ=[(C_f+C_r)/M+(C_f*a²+C_r*b²)/I_zz]/(2*v_x²*ω_n²)  (14)

[0054] where I_zzis the moment of vehicle inertia about the yaw (vertical) axis passing through the center of gravity and C_fand C_rare computed from equations (8) with ay replaced by a_ydllim.

[0055] A system for implementing the method and computer usable medium according to the present invention may include a variety of commercially available vehicle components. As recognized in the art, commercially available sensors may provide vehicle velocity, steering angle (front and rear), and other pertinent information. The vehicle manufacturer may provide such information as vehicle mass, wheelbase, understeer coefficient in the linear range of handling, and the like. Hydraulic or electrical drives may actuate parameters for braking control, suspension control, steering of front and rear wheels, and other stability control (sub)systems. The methods and determinations (e.g., calculations), including those based on equations, may be performed by a device such as a microprocessor. The microprocessor may receive input, perform calculations, and transmit output to a vehicle control system for controlling the drive(s) and, consequently, for enhancing vehicle stability. Furthermore, computer usable medium including programs, equations, and associated values may be programmed in a microprocessor memory portion (e.g., ROM, RAM, and the like) for executing functions associated with the present invention.

[0056] FIG. 4 is a flow chart of a vehicle stability control algorithm made in accordance with the present invention. The algorithm may begin by determining a cornering stiffness of the rear axle in a linear handling range (step 100). The rear axle cornering stiffness, C_r0, may be calculated from the following equation (equation 11):

[0057] C_r0=M_r*v_x0²/b

[0058] where M_ris a rear vehicle mass (M_r=M*a/L, where M is total vehicle mass, a is distance of front axle to center of gravity, L is wheelbase), b is a distance of vehicle center of gravity to the rear axle, and v_x0is a cross-over lateral velocity at which the side slip velocity is zero regardless of the steering angle (in the linear range of handling).

[0059] A first understeer coefficient in the linear handling is determined (step 101). The first understeer coefficient, K_u0, is usually provided by the manufacturer; if not, it may be determined experimentally as known in the art.

[0060] A cornering stiffness of the front axle may be determined (step 102). The front axle cornering stiffness, C_f0, may be calculated from the equation (equation 11):

[0061] C_f0=M_f*v_x0²/(b+K_u0*v_x0²)

[0062] where M_fis a front mass of vehicle and K_u0is the first understeer coefficient in the linear range of handling.

[0063] A desired lateral acceleration is determined based on the first understeer coefficient (step 103). The desired lateral acceleration, a_ydl, may be calculated from the following simplified equation (equation 1b):

[0064] a_ydl=v_x²*δ/(L+K_u0*v_x²)

[0065] where v_xis vehicle speed, δ is steering angle at the front wheels, and L is vehicle wheelbase.

[0066] The value of the desired lateral acceleration, a_ydl, may then be limited to the maximum lateral acceleration that the vehicle can develop on dry surface, a_ymax(step 104). This yields a_ydllim.

[0067] A second understeer coefficient is determined based on the limited desired lateral acceleration (step 105). The understeer coefficient, K_u, may be calculated from the equation:

[0068] K_u=K_u0*(1+B₂*a_ydllim²)

[0069] The polynomial coefficient, B₂, may be determined by the equation:

[0070] B₂=k*[(2*h_roll)/(t_w*g)]

[0071] where k is the sensitivity coefficient of tire lateral stiffness to normal load, h_rollis height of vehicle center of gravity above the roll axis, t_wis track width, and g is gravitational acceleration.

[0072] A desired yaw rate is determined based on the second understeer coefficient (step 106). The desired yaw rate, Ω_d, may be calculated from the equation: ${\Omega }_d=\{\begin{array}{c}{v}_x^{*}\delta /\left(L+K_u^{*}{v}_x^2\right)\mathrm{when}\left|a_{\mathrm{ydl\_lim}}\right|<a_{y\max }+\Delta a_y\\ a_{y\max }^{*}\mathrm{sign}\left(\delta \right)/v_x\mathrm{when}{a}_{\mathrm{ydl\_lim}}|\ge a_{y\max }+\Delta a_y\end{array}$

[0073] where the lateral acceleration value, Δa_y, may be an incremental value of about 1 m/s².

[0074] A desired lateral velocity is determined based on the desired yaw rate and the rear axle cornering stiffness coefficient (step 107). The desired lateral velocity, v_yd, may be calculated from the equation:

[0075] v_yd=Ω_d*[b−M*v_x²/(C_r*L)]

[0076] where C_r=C_r0*(1−B₂*a_ydllim²). M is the total mass of vehicle.

[0077] As an additional step, the parameters of the dynamic filters may be determined if desired from equations (13) and (14).

[0078] In FIG. 5, the desired yaw rate determined using the disclosed method is compared to the measured yaw rate. The differences are within 2 deg/s, which is within measurement errors. The described method may permit analytical determination of the desired values from known parameters with minimal effort. The model may capture major nonlinearities in vehicle handling. Additional testing may be reduced to a minimum to verify the results. The method may also be applied to vehicles with active rear steer; in this case the front steering angle, δ, in steps 103 and 106 (see FIG. 4) may be replaced by the difference (δ_f−δ_r), where δ_fis the steer angle of the front wheels and δ_rof the rear wheels.

[0079] Determining the desired yaw rate and lateral velocity/acceleration according to the present invention may provide enhanced vehicle stability control. In one embodiment, the determined desired yaw rate and lateral velocity/acceleration may be calculated by the microprocessor and then transmitted to the vehicle control system. In another embodiment, a look up table may be generated by a utility file based on the described algorithm and implemented on a computer. The vehicle control system may access the look up table as necessary. The vehicle control system may use the values determined by the microprocessor and/or look up tables to manage stability control. The vehicle control system may actuate brake control, rear steering angle, suspension characteristics, and the like to achieve enhanced stability. For example, the vehicle control system may use the determined yaw rate and lateral velocity/acceleration information to actuate hydraulic or electrical drives thereby controlling rear wheel steering.

[0080] The determinations may be achieved without the need for value tables. As such, linear interpolation need not be used between any two points defined in the table. This may avoid any errors for when the function approximated by piecewise linear segments is strongly nonlinear. In addition, the determinations could potentially eliminate, or at least greatly simplify, the process of experimentally determining the desired values of yaw rate and lateral velocity during vehicle tuning.

[0081] While the embodiments of the invention disclosed herein are presently considered to be preferred, various changes and modifications may be made without departing from the spirit and scope of the invention. For example, the described algorithm and calculations are not limited to any particular method sequence or equation. Specifically, values may be determined by a variety of strategies or equations. The disclosed equations may be exemplary of only a subset of equations useful for such determinations. Upon reading the specification and reviewing the drawings hereof, it will become immediately obvious to those skilled in the art that myriad other embodiments of the present invention are possible, and that such embodiments are contemplated and fall within the scope of the presently claimed invention. The scope of the invention is indicated in the appended claims, and all changes that come within the meaning and range of equivalents are intended to be embraced therein.