Method for Controlling the Orientation of the Rear Wheels of a Vehicle

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1. 12. (canceled)
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Abstract
A method for controlling orientation of rear wheels of a vehicle, by a computerized control system, including a module for calculation of a deflection angle for the rear wheels as a function of a deflection angle of the front wheels. In the method, when the front wheels are oriented for a period of time such that the vehicle follows a curved trajectory with an inner side and an outer side, the rear deflection angle, as determined by the calculation module, is corrected and limited to a maximum value calculated instantaneously such that a rear corner then follows a trajectory remaining within the curved trajectory previously followed by a front corner and, furthermore, at a tangent to the same.
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24 Claims
 1. 12. (canceled)
 13. A method for controlling orientation of rear wheels of a vehicle having a body of longitudinal axis and borne by orientable wheels on each side of the axis, the orientable wheels being at least one steered front wheel and at least one rear wheel, respectively, orientation of the front wheels being controlled by a driver so as to follow a path and the orientation of the rear wheels being under control of a computerized control system including a module for calculating a steering angle of the rear wheels as a function of the steering angle of the front wheels, the body of the vehicle having a substantially rectangular shape with two front corners and two rear corners overhanging in front of the front wheels and behind the rear wheels, respectively, in which method:
when the front wheels have been orientated, for a period of time, such that the vehicle follows a path that has an inside and an outside, the computerized control system determines, at successive intervals of time each corresponding to a basic position, an imaginary path equivalent to the path followed earlier by the front corner over a length corresponding to the length of the vehicle, behind a relevant basic position of the front corner and calculates, at each instant, a corrected rear steering angle such that the resulting path for the rear corner, taking into consideration the length of the vehicle, remains inside the imaginary path, wherein, to determine the imaginary path behind a basic position, the computerized control system measures, at each instant, a collection of parameters representative of displacement, including at least the current longitudinal speed of the vehicle, with a positive sign forwards, and the angles of orientation of the front wheels and of the rear wheels, with respect to the longitudinal axis of the vehicle, with a positive sign in the clockwise direction and, as the vehicle moves, divides the path followed by the front corner into a series of elementary displacements between a series of basic positions and, as the front corner enters a basic position, defines an orthonormal frame of reference for the vehicle having, as its origin, the center of gravity and two perpendicular axes, the axes being an abscissa axis corresponding to the longitudinal axis of the vehicle and an ordinates axis, and wherein, based on mean values of representative parameters stored in a memory in respect of the basic position, the control system formulates an equation, in the frame of reference of the vehicle, of an equivalent path of the front corner and, by taking it that, during a next elementary displacement, the front corner follows a forward continuation of the imaginary path and that the front steering angle is maintained, determines the predicted path of the rear corner and corrects the rear steering angle so that this predicted path of the rear corner remains inside and is at most tangential to the equivalent path of the front corner lagging behind the latter by a distance corresponding to the length of the vehicle.  View Dependent Claims (14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24)
1 Specification
The subject of the invention is a method for controlling the orientation of the rear wheels of a vehicle having two sets of orientable wheels, these being at least one steered front wheel and at least one rear wheel that can also be steered, respectively, so as to reduce the turning circle of the vehicle.
In general, a land vehicle, particularly a motor car, comprises a body bearing a cabin and resting on the ground, usually via four wheels, these being two steered front wheels controlled by the driver and two rear wheels which usually are directed along the longitudinal axis of the vehicle. In some cases, however, particularly on allterrain vehicles, it is advantageous also to be able to orientate the rear wheels of the vehicle.
This orientation of the rear wheels is determined as a function of the steering angle of the front wheels as controlled by the driver, taking into account the speed of the vehicle. This is because, at a relatively low speed, for example in a garage or a parking lot or on uneven ground, it is advantageous to reduce the turning circle by turning the rear wheels in the opposite direction to the steering angle of the front wheels which, at low speed, may be a large angle. By contrast, at higher speed, the steering angle of the front wheels is quite small and it is preferable to keep the rear wheels aligned with the vehicle for running practically in a straight line or even for achieving an “oversteer” effect by turning the rear wheels in the same direction as the front wheels.
The possibility of orientating the rear wheels therefore affords certain advantages. However, turning the rear wheels in the opposite direction to the front wheels carries the risk of causing the rear part of the vehicle to step out beyond the path followed by the front part. A situation such as this could prove dangerous when the driver is avoiding an obstacle because he cannot be sure that the rear part of the vehicle will avoid the obstacle in the same way as the front part, and it may be difficult to predict the path of the rear part, especially when the rear overhang is relatively long.
Likewise, when the vehicle is leaving a parking space with a relatively large amount of steering lock, turning the rear wheels in the opposite direction makes it easier to pull clear by reducing the turning circle but may, on the other hand, cause the rear overhang to strike an obstacle positioned alongside this parking space, for example a signpost.
It is an object of the invention to overcome such disadvantages without excessively complicating the computerized system used to control the orientation of the rear wheels.
The invention is therefore concerned, in general, with a method for controlling the orientation of the rear wheels of a vehicle having at least one steered front wheel and at least one orientable rear wheel the orientation of which is controlled by a computerized control system comprising a module for calculating a steering angle for the rear wheels as a function of the steering angle of the front wheels controlled by the driver, the vehicle having a substantially rectangular shape with two front corners and two rear corners overhanging in front of the front wheels and behind the rear wheels, respectively.
According to the invention, when the front wheels have been orientated by the driver in such a way that, for a period of time, the vehicle is following a curved path with an inside and an outside, the steering angle of the rear wheels determined by the calculation module is corrected and limited to a maximum value calculated at each instant so that the outside corner of the rear overhang, during the same period of time, follows a path that remains inside the path followed earlier by the outside corner of the front overhang and is at most tangential thereto.
In a particularly advantageous manner, while the vehicle is moving along its path, the computerized control system measures, at each instant, a collection of parameters representative of the displacement, including at least the current longitudinal speed of the vehicle, with a positive sign forwards, and the angles of orientation of the front and rear wheels with a positive sign in the clockwise direction and stores in memory, for a series of basic positions separated from one another along the path by the same elementary displacement, the mean values of said parameters calculated, over the elementary displacement, between each basic position and the previous basic position. Thus, during the elementary displacement following a basic position, the control system can, at each instant, calculate a corrected rear steering angle such that the resulting path for the rear corner, taking the length of the vehicle and the mean values of the parameters stored in memory into consideration, remains inside the path followed earlier by the front corner lagging behind the latter by a distance corresponding to the length of the vehicle.
To do this, as the front corner enters a basic position, the control system defines an orthonormal frame of reference for the vehicle having, as its origin, the center of gravity of this vehicle and having, for its abscissa axis, the longitudinal axis of the vehicle and, on the basis of the mean values of the representative parameters stored in memory in respect of said basic position, formulates the equation, in said frame of reference, of an imaginary path equivalent to the path followed earlier by the front corner over a length behind that corresponds to the length of the vehicle and, by taking it that, during the next elementary displacement, the front corner follows a forward continuation of said imaginary path and that the front steering angle is maintained, determines the predicted path of the rear corner resulting from said front angle and corrects the rear angle so that this predicted path of the rear corner remains inside and is at most tangential to the equivalent earlier path of the front corner, lagging behind the latter by a distance corresponding to the length of the vehicle.
In a preferred embodiment, the length of an elementary displacement between two basic positions is determined in such a way that, in the path of the front corner, the length of the vehicle represents an integer number of elementary displacements, and the equation of the path equivalent to the path of the front corner is formulated from the mean values of the parameters stored in memory for the same number of earlier basic positions, working back to a much earlier position lagging behind the basic position by a distance substantially equal to the length of the vehicle.
In a particularly advantageous manner, the equation for the equivalent earlier path of the front corner is formulated, for each basic position, as a function of the mean values of the lateral speed and of the yaw rate of the front corner, said mean values being calculated from the mean longitudinal speed and the mean front steering angles and rear steering angles stored in memory for the relevant basic position.
Likewise, the predicted path of the rear corner during the elementary displacement following a basic position is determined from the mean values, calculated in this way, of the lateral speed and of the yaw rate of the front corner during the previous elementary displacement.
As a preference, in the elementary displacement following a basic position, the control system likens the portion of path followed by the front corner to a continuation of the equivalent earlier path and at each instant determines the ordinate value of the rear corner in the frame of reference of the vehicle corresponding to the relevant basic position so as to calculate a rear steering angle such that said instantaneous ordinate value of the rear corner does not exceed the ordinate value of the point with the same abscissa value on the equivalent path in said frame of reference of the relevant basic position.
According to another particularly advantageous feature, the earlier path equivalent to the path followed by the front corner is an arc of a circle and, over the elementary displacement following a basic position of the front corner the control system likens the path portion followed, at each instant, by the earlier position of the front corner lagging behind its instantaneous position by the length of the vehicle, to the corresponding portion of the chord of said arc of a circle passing through the front corner and its earlier position in order, at each instant, to predict the future displacement of said earlier position and correct the rear steering angle accordingly so that the path followed by the rear corner remains separated from and is at most tangential to said chord.
According to another preferred feature, the control system formulates, for each basic position, the equation of the equivalent earlier path in the frame of reference corresponding to this basic position and keeps the same frame of reference and the same equivalent path to correct the rear steering angle in the next elementary displacement, said frame of reference and said equation of the equivalent path being readjusted in the next basic position as a function of the mean values of the parameters stored in memory in this position in order, during the next displacement, to calculate the correction for the rear angle in the new frame of reference and from the corrected equivalentpath equation.
Other advantageous features of the invention relating, in particular, to the equations used and the way in which the correction is calculated, will become more apparent in the following description of one particular embodiment, which is given by way of nonlimiting example with reference to the attached drawings.
The front wheels 11, 11′ are orientated by means, for example, of a steering rack 13, as a function of commands received, mechanically or electrically, from a steering wheel (not depicted) operated by the driver.
The orientation of the rear wheels 12, 12′ is controlled as a function of the steering angle α_{1 }of the front wheels, by a computerized control system comprising a control unit 2 which receives information corresponding to a collection of parameters representative of the displacement of the vehicle and supplied by various sensors, namely a sensor 21 that senses the degree of steering lock applied to the front wheels 11, 11′, a sensor 22 that senses the rotational speed of the front wheels in order to determine the longitudinal speed V of the vehicle, a sensor 23 that senses the yaw rate ψ, that is to say the rate at which the vehicle rotates about a vertical axis passing through a point G considered to be its center of gravity, and a sensor 24 that senses the lateral acceleration at the center of gravity.
The orientation of the rear wheels 12, 12′ is controlled by actuators 25 under the control of the control system 2 which comprises a means for calculating a steering angle α_{2 }for the rear wheels as a function of the information received and, in particular, of the steering angle α_{1 }of the front wheels, so as to reduce the turning circle as far as possible.
The rear steering angle α_{2 }is measured by sensors 26.
The various position and speed sensors may be of optical or magnetic type, for example Halleffect sensors.
The control unit 2 may be produced in the form of a microprocessor equipped with random access memory, read only memory, a central processing unit and input/output interfaces so that it can receive information from the various sensors and send instructions to the actuators 25.
Advantageously, the longitudinal speed V of the vehicle can be obtained by calculating the mean of the speed of the front wheels or of the rear wheels, which speed can be measured by the sensors of an ABS system.
In general, the vehicle body usually has a roughly rectangular shape with two front corners E_{1}, F_{1 }and two rear corners E_{2}, F_{2 }which are positioned at distances L_{1 }in front of and L_{2 }to the rear of the center of gravity G, respectively, while the front and rear wheels are positioned inboard with respect to the body 10, at distances l_{1 }and l_{2 }respectively from the center of gravity, which distances are, of course, shorter than L_{1 }and L_{2 }respectively. There is therefore a front overhang L_{1}−l_{1 }and a rear overhang L_{2}−l_{2}, which overhangs may differ in magnitude according to the type of vehicle.
In general, the driver orientates his front wheels in such a way that the front corner E_{1 }of the vehicle, positioned on the outside of the curve, avoids the obstacles, either during normal driving or when entering or leaving a parking space.
In vehicles that have just two orientable front wheels and two rear wheels directed along the longitudinal axis, the path of the outside rear corner E_{2 }always remains inside the path of the front corner E_{1}. By contrast, if the rear wheels are orientated in the opposite direction to the front wheels in order to reduce the turning circle, the path of the rear corner E_{2 }may intersect with that of the front corner E_{1 }and the rear part of the vehicle therefore runs the risk of striking obstacles that the driver had avoided with the front end.
According to the invention, the control system 2 is designed in such a way as to solve such a problem using simple means.
As mentioned above, the control system 2 at each instant measures a collection of parameters supplied by the various sensors and including, at least, the current longitudinal speed V with its sign, and the angles of orientation of the front and rear wheels, α_{1 }and α_{2 }respectively.
Furthermore, the control system calculates the means of the instantaneous values thus measured over a series of successive elementary displacements of length D in the path T1 and stores these mean values in memory for a series of basic positions each one corresponding to the end of an elementary displacement. As a preference, this elementary displacement D is chosen such that the distance L, equal to the length of the vehicle, corresponds to an integer number n of elementary displacements.
In the path T_{1}, P_{0 }denotes a basic position occupied by the front corner E_{1 }at the end of an elementary displacement D_{1 }and P_{n }denotes the lagging position occupied earlier and lagging behind the point P_{0 }by a distance L equal to the length of the body 10 of the vehicle.
Thus, before reaching a basic position P_{0 }in the path T_{1}, the front corner E_{1 }of the vehicle first of all, over a distance corresponding to the length of the vehicle, passing through a succession of basic positions P_{n}/P_{n1}, . . . , P_{2}, P_{1}, P_{0 }which are separated from one another by an elementary distance D and, for each of these positions, the control system will have stored the mean values of the parameters calculated during the previous elementary displacement.
The calculation means allow fairly short elementary displacements, for example measuring from 30 to 50 centimeters, to be performed, so that the number n of elementary displacements corresponding to the length L of the vehicle is of the order of 15 to 20, which is a value consistent with the capabilities of computerized calculation.
In the case depicted in
In order to perform the calculations, the control system advantageously uses a twowheel model of known type.
In each basic position P_{0}, the mean values of the front angle, of the rear angle and of the longitudinal speed, denoted α_{1m}, α_{2m }and V_{m }respectively, can be used to reconstruct, in the frame of reference of the vehicle defined by the axes Gx Gy, a mean lateral speed V_{ym }and a mean yaw rate ψ_{m }which are given by the equations:
V_{ym}=V_{m}*(l_{2}α_{1m}+l_{1}α_{2m})/(l_{1}+l_{2}) (1)
ψ_{m}=V_{m}*((α_{1m}−α_{2m})(l_{1}+l_{2}) (2)
From these equations it is possible to define a mean path T_{2 }consisting of a mathematical line passing as closely as possible through the various positions P_{n}, P_{n1}, . . . , P_{1}, P_{0 }occupied by the front corner E_{1 }as it moves along the path T1, along the length L of the vehicle, and the twowheel model for which makes it possible to formulate the equation in the Gx, Gy frame of reference.
This mathematical line T_{2 }can be considered to be a mean path equivalent to the path T_{1 }actually followed by the front corner E_{1 }as far as the basic position P_{0}.
From this equation formulated by the twowheel model, the control system determines the ordinate value, on the equivalent line T_{2}, of the lagging position Q_{n }which, on this equivalent path, is behind the basic position P_{0 }by a distance L equal to the length of the vehicle and therefore corresponding to the earlier position P_{n }of the front corner lying n elementary displacements earlier than the basic position P_{0}.
In practice, the ordinate value thus calculated can be written:
Y_{Q}=V_{m}/ψ_{m}−[(L_{1}+V_{ym}/ψ_{m})^{2}+(V_{m}/ψ_{m})^{2}−(L_{2}+V_{ym}/ψ_{m})^{2}]^{1/2} (3)
in which:
V_{m }is the mean longitudinal speed over the displacement D
ψ_{m }is the mean yaw rate
L_{1 }is the abscissa value for the front corner E1 in the Gx,Gy frame of reference
V_{ym }is the mean lateral speed of the front corner
L_{2 }is the abscissa value of the rear corner.
By keeping the Gx, Gy frame of reference of the vehicle corresponding to the basic position P_{0}, the twowheel model makes it possible to determine the equation for the future path T_{3}, in this frame of reference, of the rear corner E_{2 }during the next elementary displacement D′_{1 }of the front corner E_{1}. To do that, it is taken that the mean values of the parameters stored in memory at P_{0 }are kept as, therefore, are the mean longitudinal speed V_{m }and the mean yaw rate ψ_{m }indicated above.
To each instantaneous position P′ of the front corner there corresponds, in the path T_{1}, an earlier position P′_{n}, lagging behind this instantaneous position P′ by the length of the vehicle.
By approximation, the path followed, over the displacement Dn, by the lagging earlier position P′_{n }is likened to the chord P_{0}Q_{n }of the equivalent path T2.
Furthermore, because it is taken that the mean front angle α_{1m}, the longitudinal speed V_{m}, the lateral speed V_{ym }and the yaw rate ψ_{m }are kept for the displacement D′_{1}, it may be taken that the rear corner E_{2 }follows an approximate path which, in the Gx, Gy frame of reference, is a portion of an arc of a circle T_{3}.
Using these approximations, the control system can thus calculate, at each instant, a correction α′_{2 }to be applied to the rear steering angle α_{2 }so that, in the Gx, Gy frame of reference, the ordinate value for the rear corner E′_{2 }at this instant does not exceed the ordinate value for the corresponding point Q′_{n }on the chord P_{0 }Q_{n }of the equivalent path T_{2}, the arc of a circle T_{3 }thus being at most tangential to the chord P_{0 }Q_{n}.
Given that the chord P_{0 }Q_{n }is always positioned on the inside of the arc of a circle T_{2 }equivalent to the actual path T_{1 }followed by the front corner E_{1}, the rear corner E_{2 }will always remain inside this path T_{1 }and will thus avoid the obstacles that the driver had already avoided by turning the front wheels 11, 11′.
In practice, the correction α′_{2 }to be made, at each instant, to the rear angle α_{2 }can be obtained by sequentially calculating the following parameters:
a_{1}=−1/α_{1}; b_{1}=l_{1}/α_{1 }
a_{2}=Y_{P′n}/(L_{2}−L_{1}); b_{2}=−L_{1}/(L_{2}−L_{1})
a_{4}=(1+a_{1}a_{2})/(1+a_{2}^{2}); b_{4}=(b_{1}−b_{2})a_{2}/(1+a_{2}^{2})
a
_{5}
=a
_{2}
a
_{4}
; b
_{5}
=b
_{4}
a
_{2}
+b
_{2 }
a_{6}=(a_{4}−1)^{2}+(a_{1}−a_{5})^{2}−1−a_{1}^{2 }
b_{6}=L_{2}+(a_{4}−1)b_{4}+(a_{1}−a_{5})(b_{1}−b_{5})−a_{1}b_{1 }
c_{6}=b_{4}^{2}+(b_{1}−b_{5})^{2}−L_{2}^{2}−b_{1}^{2 }
A=[−b_{6}−2[b_{6}^{2}−a_{6}c_{6}]^{1/2}]/2a_{6 }
the correction to be made to the rear angle α_{2 }being:
a′_{2}=−a_{1}(L_{2}+A)/(L_{1}−A). (4)
In order to implement the method according to the invention, the control system 2, which may be of a conventional type, is modified in the way schematically depicted in
The supervisory module 3 comprises inputs receiving signals emitted by the various sensors 22, 23, 24, 26 and corresponding to the parameters representative of the displacement of the vehicle, namely:
 the current longitudinal speed V of the vehicle
 the current steering angles α_{1 }of the front wheels and α_{2 }of the rear wheels
 the sign of the longitudinal speed V which is positive when traveling in a forwards gear and negative in reverse gear.
By using a twowheel model in the way indicated above, the supervisory module 3 calculates, for each elementary displacement D between two successive basic positions, the mean values of the longitudinal speed V_{m}, of the lateral speed V_{ym }and of the yaw rate ψ_{m }of the outside front corner E_{1 }in its path T_{1}.
The mean values thus calculated are compared with limit values recorded in advance, these being respectively:
 a minimum longitudinal speed V_{min }below which the vehicle is considered to be stationary,
 a maximum speed V_{max }at which the strategy can be activated,
 a minimum yaw rate α_{0 }below which the reconstructed path is considered to be a straight line,
 a maximum front angle α_{0 }below which no limitation is to be applied to the rear steering angle.
At the end of each elementary displacement, the mean values of the parameters stored in memory in the basic position reached at that moment are therefore compared against the recorded limit values.
If the mean longitudinal speed V_{m }is above the limit V_{max}, the speed is too high for it to be possible to act upon the rear wheels and the supervisory module 3 runs the nonlimitation module 33. In the conventional way, the rear wheels in this case remain aligned with the vehicle or rather alternatively orientated slightly in the same direction as the front wheels, in order to improve roadholding.
Furthermore, if the absolute value of the current front angle is below the limit α_{0}, it is taken that the path is a straight line and once again it is the nonlimitation module 33 which is run.
If, in a basic position, the sign of the mean longitudinal speed is negative, that means that some of the preceding elementary displacement was performed in reverse gear, so the nonlimitation module 33 is run.
As long as the longitudinal speed V does not exceed the limit V_{min}, the vehicle is considered to be stationary. The speeds and the angles in the basic position at this instant are therefore initialized in order to provide the conditions for exiting the parking space.
Thus:
V_{m}=V_{min}; α_{1m}=0; α_{2m}=0.
In movement, the mean values α_{1m}, α_{2m}, V_{m }make it possible, as has already been seen, to reconstruct the mean lateral speed and the mean yaw rate which define the mean path T_{2 }equivalent to the path T_{1 }of the front corner E_{1}.
If the absolute value of the yaw rate ψ_{m }is above the limit ψ_{0}, the equivalent path T_{2 }is considered to be a circular path. The supervisory module 3 does, however, give consideration to the sign of ψ_{m }resulting from equation (2). If the mean yaw rate ψ_{m }and the mean steering angle α_{1m }during the displacement D_{1 }preceding the relevant basic position P_{0 }are of the same sign, then the supervisory module 3 runs the “cornering strategy” module 31 which calculates the correction α′_{2 }to be made to the rear angle α_{2 }in the way indicated above.
By contrast, if ψ_{m }and α_{1m }are of opposite signs, that means that, during the previous displacement D_{1}, the driver changed the direction in which he was steering and, in this case, the supervisory module 3 runs the “no limitation” module 33 which keeps the rear steering angle α_{2 }as calculated by the calculation module 21 without applying any correction to it.
The same is true if the lateral speed V_{ym }resulting from equation (1) is of the opposite sign to the front steering angle α_{1}. In this case, too, the driver has changed the direction of steering and the “no limitation” module 33 is run.
However, if the absolute value of the mean yaw rate ψ_{m }is below the limit ψ_{0}, the path is considered to be a straight line. In this case, if the lateral speed V_{ym }is of the same sign as the front steering angle α_{1}, the supervisory module 3 runs the “straight line strategy” module 31 which calculates the restriction on angle α′_{2 }in the way indicated above for the “cornering strategy” by applying the same sequential calculation (4) from the current angle α_{1 }measured at each instant and the mean values of the longitudinal speed V_{m }and the lateral speed V_{ym }in the previous displacement D_{1}, the correction α′_{2 }to be applied to the rear angle α_{2 }calculated by the calculation module 20 being given by the equation:
α′_{2}=−α_{1}(L_{2}+A)/(L_{1}−A) (4)
When the front angle α_{1 }is positive, if it turns out that the calculated correction α′_{2 }is also positive, the wheels are brought back into line with the axis of the vehicle, the corrected rear angle α_{20 }being zero. By contrast, if the correction α′_{2 }is negative, the corrected rear angle α_{20 }is equal to the maximum value of the angle α_{2 }initially calculated and of the correction α′_{2}.
Conversely, if the front steering angle α_{1 }is negative and if the calculated correction α′_{2 }is also negative, the corrected rear angle α_{20 }is zero, the wheels being brought into line with the axis of the vehicle.
If the correction α′_{2 }is negative, the corrected rear angle α_{20 }is equal to the minimum value of the initial angle α_{2 }and of the correction α′_{2}.
In all the cases indicated above in which the supervisory module 3 has to run the “no limitation” module 33, the calculated rear angle α_{2 }is kept unchanged.
The invention therefore makes it possible, without excessively complicating the control system, to correct the rear steering angle instantaneously in order to prevent the rear overhang from stepping outside the path chosen by the driver.
Of course, the invention is not restricted to the preferred embodiment described but on the contrary covers all variants thereof which fall within the claimed scope of protection and employ equivalent means.
For example, the equations for the paths T_{2 }and T_{3 }were formulated using a twowheel model of known type, but other models and other equations could be used.
Likewise, the mathematical line equivalent to the actual path is not necessarily an arc of a circle and, although it is particularly advantageous to use the chord of this arc as an approximation, other calculation means might be possible.
Furthermore, other representative parameters could be used in order to put the displacement of the vehicle into equation form.