Off road vehicle

• US 20050217906A1
• Filed: 07/13/2004
• Published: 10/06/2005
• Est. Priority Date: 01/15/2002
• Status: Active Grant

• Current Assignee: Ian James Spark
• Original Assignee: Ian James Spark
• Inventors: Spark, Ian James
• Granted Patent: US 7,464,785 B2
• Time in Patent Office: Days
• US Class Current: 180/22
• CPC Class Codes: B60G 2300/37 Vehicles having steerable wheels mounted on a vertically moving column
• CPC Class Codes: B60T 8/24 responsive to vehicle inclination or change of direction, e.g. negotiating bends using electrical circuitry or regulation means B60T8/17
• CPC Class Codes: B60T 8/246 Change of direction
• CPC Class Codes: B62D 1/12 Hand levers
• CPC Class Codes: B62D 7/1509 with different steering modes, e.g. crab-steering, or steering specially adapted for reversing of the vehicle
• CPC Class Codes: B62D 9/00 Steering deflectable wheels not otherwise provided for steering positon indicators B62D15/02

Abstract

The invention describes vehicles where the steering effect of the driver-selected wheel angles is made identical to the steering effect of positively and independently driving the driven wheels. A method of compensating for the linear portions of the slip angles of all wheels and the linear portions of the longitudinal slip of the driven wheels is described where the slip angles and longitudinal slips are deduced from the forces acting on the wheels. Methods of measuring the slip angles and longitudinal slips of all wheels by means of two dummy castors are also described. Vehicles where the steering effect of the driver-selected wheel angles is made identical to the steering effect of positively and independently braking all wheels are also described. Means of applying the above principles to vehicles with either hydrostatic or mechanical drives are also described, including simplified vehicles with only two steerable wheels.

Claims

1. A vehicle consisting of two or more wheels, at least two of which are driven wheels and at least one of which is steerable, where the axes of rotation of all wheels lie substantially in the same horizontal plane, and where the steerable wheels turn about substantially vertical axes, where the speeds of the driven wheels are independently and positively controlled so that they tend to produce a single centre of curvature for the path of the vehicle, and where all the wheel angles are positively controlled to produce a single driver-selected centre of curvature for the path of the vehicle, and where the driven wheel speeds and all the wheel angles are integrated so that the first centre of curvature is made identical to the second driver-selected centre of curvature.

2. A vehicle according to claim 1 consisting of four or more wheels, at least two of which are driven wheels and at least two of which is steerable, where the axes of rotation of all wheels lie substantially in the same horizontal plane, and where the steerable wheels turn about substantially vertical axes, where the speeds of the driven wheels are independently and positively controlled so that they tend to produce a single centre of curvature for the path of the vehicle, and where all the wheel angles are positively controlled to produce a single driver-selected centre of curvature for the path of the vehicle, and where the driven wheel speeds and all the wheel angles are integrated so that the first centre of curvature is made identical to the second driver-selected centre of curvature.

3. A vehicle according to claim 1 consisting of two or more wheels, at least two of which are driven wheels and at least one of which is steerable, where the axes of rotation of all wheels lie substantially in the same horizontal plane, and where the steerable wheels turn about substantially vertical axes, where the effective speeds of the driven wheels are independently and positively controlled so that they tend to produce a single centre of curvature for the path of the vehicle, and where all the effective wheel angles are positively controlled to produce a single driver-selected centre of curvature for the path of the vehicle, and where the driven wheel speeds and all the wheel angles are integrated so that the first centre of curvature is made identical to the second driver-selected centre of curvature, and where the effective wheel speeds are the actual wheel speeds corrected for the longitudinal slip of each wheel and where the effective wheel angles are the actual wheel angles corrected for the slip angle of each wheel.

4. A vehicle according to claim 1 consisting of four or more wheels, at least two of which are driven wheels and at least two of which is steerable, where the axes of rotation of all wheels lie substantially in the same horizontal plane, and where the steerable wheels turn about substantially vertical axes, where the actual speeds of the driven wheels are independently and positively controlled so that they tend to produce a single centre of curvature for the path of the vehicle, and where all the effective wheel angles are positively controlled to produce a single driver-selected centre of curvature for the path of the vehicle, and where the driven wheel speeds and all the wheel angles are integrated so that the first centre of curvature is made identical to the second driver-selected centre of curvature, and where the effective wheel angles are the actual wheel angles corrected for the slip angle of each wheel.

5. A vehicle according to claim 1 consisting of four or more wheels, at least two of which are driven wheels and at least two of which is steerable, where the axes of rotation of all wheels lie substantially in the same horizontal plane, and where the steerable wheels turn about substantially vertical axes, where the effective speeds of the driven wheels are independently and positively controlled so that they tend to produce a single centre of curvature for the path of the vehicle, and where all the actual wheel angles are positively controlled to produce a single driver-selected centre of curvature for the path of the vehicle, and where the driven wheel speeds and all the wheel angles are integrated so that the first centre of curvature is made identical to the second driver-selected centre of curvature, and where the effective speed of the driven wheel are the actual speed of the drive wheel corrected for the longitudinal slip of each driven wheel.

6. A vehicle consisting of four or more wheels, at least two of which are driven wheels and at least two of which is steerable, where the axes of rotation of all wheels lie substantially in the same horizontal plane, and where the steerable wheels turn about substantially vertical axes, where the effective speeds of the driven wheels are independently and positively controlled so that they tend to produce a single centre of curvature for the path of the vehicle, and where all the effective wheel angles are positively controlled to produce a single driver-selected centre of curvature for the path of the vehicle, and where the driven wheel speeds and all the wheel angles are integrated so that the first centre of curvature is made identical to the second driver-selected centre of curvature, and where the effective speeds of the driven wheels are the actual speeds of the driven wheels corrected for the longitudinal slip of each driven wheel, and where the effective wheel angles are the actual wheel angles corrected for the slip angles of each wheel.

7. A four wheeled vehicle according to claim 6 where the driver selects the centre of curvature of the path of the vehicle and the root mean square of the wheel speeds by means of a rotatable joystick and a control system adjusts the effective speeds of the four driven wheels and the effective angle of all four wheels according to the following equations.

$\frac{R}{t}=\cot$

( 90 ⁢

°

θ

/ θ

max ) = ( R X 2 + R Y 2 ) 1 / 2 / t

Where t is the track of the vehicle, θ

is the rotation of the joystick and θ

_max is the maximum rotation of the joystick.

If the driver displaces the rotatable joystick at an angle ψ

to the straight ahead position, the direction of the of curvature of the path of the vehicle will by at right angles to the direction of joystick displacement and R_X and R_Y will be given by the following equations;

R_X=R/(tan² ψ

+1)^1/2=R cos ψ


and R_Y=R tan ψ

/(tan² ψ

+1)^1/2=R sin ψ

where the driver selects the direction of the centre of curvature by displacing the joystick at right angles to this direction. The centre of curvature of the path of the vehicle is now specified by the two components R_X and R_Y. He selects the root mean square of the four wheel speeds by the amount of displacement of the joystick.

The control system then rotates the four drive wheels to the following angles;

tan φ

₁=(b=2−

R_Y)(R_X−

t/2)=tan(φ

₁′

−

α

₁)
tan θ

₂=(b/2−

R_Y)/(R_X+t/2)=tan(φ

₂′

−

α

₂)
tan φ

₃=(b/2+R_Y)/(R_X−

t/2)=tan(φ

₃′

+α

₃)
tan φ

₄=(b/2+R_Y)/(R_X+t/2)=tan(φ

₄′

+α

₄)

Where b is the wheel base of the vehicle, R_Y is the displacement of the centre of curvature forward of the centre of the vehicle and R_X is the displacement of the centre of curvature to the right of the centre of the vehicle, φ

₁,φ

₂, φ

₃ and φ

₄ are the effective angles of the front right, front left rear right and rear left wheels respectively, φ

₁′

, φ

₂′

, φ

₃′ and

φ

₄′

are the actual angles of the front right front left, rear right and rear left wheels respectively, and α

₁, α

₂, α

₃ and α

₄ t are the slip angles of the front right, front left, rear right and rear left wheels respectively.

Where the amount of displacement of the joystick d determines the root mean square of the four wheel speeds (RMSWS) according to the equation;

RMSWS=Kd=(ω

₁²+ω

₂²+ω

₃²+ω

₄²)/2

where K is an appropriate constant.

Where the individual wheel speeds are given by the equations;

ω

₁=KdR₁/RMSR=ω

₁′

exp[−

i₁] where R₁²=(b/2−

R_Y)²+(R_X−

t/2)²
ω

₂=KdR₂/RMSR=ω

₂′

exp[−

i₂] where R₂²=(b/2−

R_Y)²+(R_X+t/2)²
ω

₃=KdR₃/RMSR=ω

₃′

exp[−

i₃] where R₃²=(b/2+R_Y)²+(R_X−

t/2)²
ω

₄=KdR₄/RMSR=ω

₄′

exp[−

i₄] where R₄²=(b/2+R_Y)²+(R_X+t/2)²

And RMSR is the root mean square radius, which is given by;

RMSR=(R₁²+R₂²+R₃²+R₄²)^1/2/2=(R_X²+R_Y²+t²/4+b²/4)^1/2

where ω

₁, ω

₂, ω

₃, and ω

₄ are the effective speeds of the front right, front left, rear right and rear left wheels respectively, ω

₁′

, ω

₂′

, ω

₃′

, and ω

₄′

are the actual speeds of the front right, front left, rear right and rear left wheels respectively, R₁, R₂, R₃ and R₄ are the distances between the centre of curvature of the path of the vehicle and the front right, front left, rear right and rear left wheels respectively, and i₁, i₂, i₃ and i₄ are the longitudinal slips of the front right, front left, rear right and rear left wheels respectively.

Note that when the rotation of the joystick θ

is a maximum the radius of curvature will be zero and the direction of the displacement d of the joystick will be immaterial. It will be natural for the driver to push the joystick forward in this case to commence rotation. Pulling the joystick back will commence rotation in the opposite direction.

8. A vehicle according to claim 7 where the values of the longitudinal slip and slip angle corrected for in the control equations are deduced from the forces acting on each wheel where these forces are measured continuously by means of a triaxial load cell interposed between each wheel and the chassis of the vehicle.

9. A vehicle according to claim 8 where the value of the slip angle α

′

corrected for in the wheel angle control equations is given by the equation;

α

′

=F_x/C_α

where F_X is the cornering force acting on the wheel and C_α

is the cornering stiffness, which is given by the equation;

C_α

=(dF_x/dα

)<sub>α

→

0</sub>=K_nF_zⁿ+K_n′

Where K_n, K_n′ and

n and constants which characterise the tyre, and F_z is the vertical force acting on the wheel. In general n will lie between 0.5 and 0.8.

Thus α

′

=F_x/(K_nF_zⁿ+K_n′

)

10. A vehicle according to claim 8 where the value of longitudinal slip corrected for in the wheel speed control equations is given by:

i=F _y /C _s

where C_s is the gradient of the longitudinal force F_y versus longitudinal slip curve, and is given by;

C_s=(dF_y/di)<sub>i→

0</sub>=K_mF_z^m+K_m′

where K_m, K_m′ and

m are parameters which characterise the tyre.

Thus i′

=F_y/(K_mF_z^m+K_m′

)

11. A vehicle according to claim 10 where the forces acting on each wheel are measured by means of triaxial load cells which turn with the wheels so that the cornering force F_X, the tractive force F_Y and the vertical force F_Z are measured directly regardless of wheel angle.

12. A vehicle according to claim 10 where the forces acting on each wheel are measured by means of triaxial load cells fixed to the chassis of the vehicle so that the longitudinal force F_Y′

, the transverse force F_X′

, and F_Z exerted on the chassis by each wheel are converted to the forces relative to the wheel F_X, F_Y and F_Z according to the equations;

F_y=F_y′

cos φ

′

+F_x′

sin φ

′


F_x=F_x′

cos φ

′

−

F_y′

sin φ

′


F_z=F_z′

where φ

′

is the actual wheel angle

13. A vehicle according to claim 7 where the centre of curvature of the path of the vehicle and its rate of rotation about this centre are measured with the aid of dummy castors located at opposite ends of the vehicle, where the effective angle of each wheel can be deduced from the said centre and the actual wheel angles, and where the effective speed of each wheel can be deduced from the said centre and the rate of rotation of the vehicle about this centre.

14. A vehicle according to claim 13 where the coordinates of the centre of curvature of the path of the vehicle R_x and R_y are calculated from the equations

R_X=b/(tan φ

_R+tan φ

_F)
and R_Y=b(tan φ

R−

tan φ

_F)/2(tan φ

_R+tan φ

_F)

where R_X is the distance of the centre of curvature of the path of the vehicle to the right of the longitudinal axis of the vehicle, R_Y is the distance of the centre of curvature of the path of the vehicle forward of the transverse axis of the vehicle, b is the wheel base and φ

_R and φ

_F are the angles of the rear and front castors respectively, so that the effective wheel angles are given by the equations;

tan φ

₁=tan(φ

₁′

−

α

₁)=tan φ

_F/(1−

t(tan φ

_R+tan φ

_F)/2b)
tan φ

₂=tan(φ

₂′

−

α

₂)=tan φ

_F/(1+t(tan φ

_R+tan φ

_F)/2b)
tan φ

₃=tan(φ

₃′

−

α

₃)=tan φ

_R/(1−

t(tan φ

_R+tan φ

_F)/2b)
tan φ

₄=tan(φ

₄′

−

α

₄)=tan φ

_R/(1+t(tan φ

_R+tan φ

_F)/2b)

and the effective velocities of the wheels across the ground are given by the equations;

V₁=V_F(R_x−

t/2)cos φ

_F/R_x cos φ

₁
V₂=V_F(R_x+t/2)cos φ

_F/R_x cos φ

₂
V₃=V_R(R_x+t/2)cos φ

_R/R_x cos φ

₃
V₄=V_R(R_x−

t/2)cos φ

_R/R_x cos φ

₄

15. A vehicle according to claim 14 where the following control strategy is employed:

(1). The driver selects the desired R_x′

, R_y′ and

RMSWS. If a rotatable joy stick is used ψ

, θ

, and d are selected.

(2) The computer calculates the desired angles and speed for all four wheels.

(3). As slip angles and longitudinal slip cannot be measured yet, the computer implements the above angles and speeds.

(4). The castors now allow calculation of the effective wheel angles and wheel speeds. Slip angles and longitudinal slip are also calculated.

(5). The slip angles are now added to the actual front wheel angles and subtracted from the actual rear wheel angles in order to achieve the desired effective wheel angles. This step assumes the slip angles will not be changed by a small change in each wheel angle.

(6). The castor angles and speeds are remeasured. Note that the desired castor angles are given by the equations;

tan φ

_F=(b/2−

R_Y′

)/R_X′ and

tan φ

_R=(b/2+R_Y′

)/R_X′

(7). In an attempt to correct the effective wheel speeds to the desired wheel speed, the wheel speed error is added to the actual wheel speed. Once again it is assumed that the speed error will not be changed by this process.

(8). If the desired centre of vehicle is still not achieved, steps 4 to 7 can be repeated.

16. A simplified vehicle according to claim 7 where both left hand wheels are driven at the same speed and where both the right hand wheels are driven at the same speed where conflict between the wheel angle steering system and the wheel speed steering system is minimised by ensuring that the effective centre of curvature of the path of the path of the vehicle lies on the transverse axis of the vehicle. This is achieved by detecting the actual centre of curvature by means of the front and rear castors located midway between the front and rear driven wheels respectively and adjusting the actual wheel angles until the desired centre is achieved when both the angles and speeds of the front and rear castors are identical.

17. A simplified vehicle according to claim 16 where the total angle through which the wheels can be turned is limited to say 90 degrees. This will allow continuous variation of R_X from −

∞

to −

(b/2+t/2) and from +(b/2+t/2) to +∞

, where a value of R_X=0 can be achieved by turning the left and right driven wheels in opposite directions and driving the wheels on one side in reverse.

22. A vehicle according to claim 7 where the driven wheels are driven by purely mechanical means where the speed of each driven wheel is positively and independently controlled by one or more traditional differentials each connected in parallel with a steering differential where each steering differential is driven by either a hydraulic or electric motor where the speeds of the steering differentials are controlled by an on board computer so as to cause the speeds of the driven wheels to produce the same centre of curvature for the path of the vehicle as the driver-selected wheel angles.

23. A vehicle according to claim 22 with four wheels which are all driven where the centre of curvature of the path of the vehicle lies on the transverse axis of the vehicle so that only two wheel speeds are required so that the differential and associated parallel steering differential which is driven by the engine/gearbox is replaced by a simple right angle drive.

24. A vehicle according to claim 23 where the front differentials and tail shaft are eliminated and where the left and right front wheels are driven at the same speeds as the left and right rear wheels respectively by means of two pairs of right angle drives.

25. A vehicle according to claim 23 where the rear differentials and tail shaft are eliminated and where the left and right rear wheels are driven at the same speeds as the left and right front wheels respectively by means of two pairs of right angle drives.

26. A vehicle according to claim 22 with four wheels which are all driven where the centre of curvature of the path of the vehicle lies on the axis of the rear wheels so that four wheel speeds are required so that three differentials and three parallel steering differentials are required in order to positively drive all four wheels at speeds which will tend to produce the same centre of curvature for the path of the vehicle which is identical to the centre of curvature produce by the driver-selected wheel angles.

27. A vehicle according to claim 6 where the speed of each driven wheel is controlled by means of a speed-correcting differential which is incorporated into a speed reduction gear box close-coupled to each driven wheel, where the first input to the said differential is from the engine via the main gear box and a series of right angle drives and the second input to the said differential is provided by means of an electric or hydraulic motor which is close coupled to the said differential, where the speed of each wheel is determined by the speed of the first input which will be the same for all wheels and the speed and direction of rotation of the speed correcting electric or hydraulic motor.

28. A four wheeled vehicle according to claim 27 where the four speed-correcting hydraulic motors are each connected to a variable displacement hydraulic pump where the speed of all four pumps is proportional to the speed of the drive shafts and where the displacement and direction of pumping is controlled by varying the angle of the squash plate so that the speed of each wheel will tend to produce the same centre of curvature for the path of the vehicle where this centre is identical to that produced by the driver-selected wheel angles.

29. A six wheeled vehicle according to claim 27 with two non-driven steerable wheels at the front and four non-steerable driven wheels at the rear, where a rear axle is driven by one or more right angle drives where the four rear wheels are driven by the said axle but where each wheel speed is corrected by means of a speed-correcting differential incorporated into the speed reduction gearbox close coupled to each driven wheel where the second input to each differential is provided by a hydraulic motor also close coupled to the wheel where the speed and direction of rotation of each hydraulic motor is controlled by a variable displacement hydraulic pump where the displacement and direction of pumping is controlled to ensure that the speed of each driven wheel will tend to produce a single centre of curvature for the path of the vehicle, where the angles of all the wheels also tends to produce a single driver-selected centre of curvature for the path of the vehicle where the first centre is made identical to the second driver-selected centre.

30. A vehicle according to claim 29 where the four speed of the speed-correcting hydraulic motors is controlled by connecting them in series with a single variable displacement pump which is driven at a speed proportional to the drive axle and where the displacement and direction of flow is inversely proportional to the radius of curvature of the path of the vehicle where the displacement of each speed correcting hydraulic motors will be inversely proportional to the distance of the centre of the coupled wheel from the longitudinal axis of the vehicle.

31. A vehicle according to claim 10 where the actual angle of the wheels is corrected for the linear portion of the slip angle of each wheel so that the effective angle of each wheel will be closer to the ideal effective angle of each wheel so that the effective centre of curvature of the path of the vehicle is moved closer to the ideal centre selected by the driver, where the linear portion of the slip angle is estimated from measurements of the vertical and transverse forces acting on each wheel and the characteristics of the tyres used.

32. A vehicle according to claim 13 where the actual angle of the wheels is corrected for the slip angle of each wheel so that the effective angle of each wheel will be closer to the ideal effective angle of each wheel so that the effective centre of curvature of the path of the vehicle is moved closer to the ideal centre selected by the driver, where the slip angle of each wheel is deduced from the position of the centre of curvature of the path of the vehicle where the latter is deduced from the angles and speeds of two dummy castors located at opposite ends of the vehicle.

33. A vehicle according to claims 10 where the actual speed of the wheels is corrected for the linear portion of the longitudinal slip of each wheel so that the effective speed of each wheel will be closer to the ideal effective speed of each wheel so that the effective centre of curvature of the path of the vehicle is moved closer to the ideal centre selected by the driver, where the linear portion of the longitudinal slip is estimated from measurements of the vertical and longitudinal forces acting on each wheel and the characteristics of the tyres used.

34. A vehicle according to claim 10 where the actual speed of the wheels is corrected for the longitudinal slip of each wheel so that the effective speed of each wheel will be closer to the ideal effective speed of each wheel so that the effective centre of curvature of the path of the vehicle is moved closer to the ideal centre selected by the driver, where the longitudinal slip of each wheel is deduced from the position of the centre of curvature of the path of the vehicle and the rate of rotation of the vehicle about this centre, where the latter two parameters are deduced from the angles and speeds of two dummy castors located at opposite ends of the vehicle.

35. A vehicle according to claim 7 where the driver interface consists of two joysticks as the driver interface (replacing the rotatable joystick if specified), where the cotan of the lateral displacement of the first joystick from the central null position is proportional to the radius of curvature of the path of the vehicle and the forward displacement of the first joystick determines the root mean square wheel speed of the driven wheels, and the displacement of the second joystick is perpendicular to the direction of the centre of curvature of the path of the vehicle.

36. A vehicle according to claim 7 where the driver interface consists of a steering wheel, knob or lever and a joystick (replacing the rotatable joystick if specified) where the cotan of the rotation of the steering wheel from the central null position is proportional to the radius of curvature of the path of the vehicle, and the direction of the displacement of the joystick is perpendicular to the direction of the centre of curvature of the path of the vehicle and the amount of displacement determines the root mean square of the driven wheel speeds.

37. A vehicle according to claim 7 where R_Y is constant and where the driver interface consists of a steering wheel, knob or lever and a speed control lever or pedal (replacing the rotatable joystick if specified) where the cotan of the rotation of the steering wheel from the central null position is proportional to the radius of curvature of the path of the vehicle, and where the displacement of the speed control lever or pedal is proportional to the root mean square of the driven wheel speeds.

38. A vehicle according to claim 7 where R_Y is constant and where the driver interface consists of a single joystick (replacing the rotatable joystick if specified) where the cotan of the lateral displacement of the joystick from the central null position is proportional to the radius of curvature of the path of the vehicle, and the forward displacement of the joystick is proportional to the root mean square of the driven wheel speeds.

18. A braking vehicle consisting of four or more wheels where at least two of which are driven and at least two of which are steerable where the axes of all wheels lie substantially in the same horizontal plane and where the steerable wheels are turned about a substantially vertical axis where the speeds of the braked wheels are positively and independently controlled so that they tend to produce the same centre of curvature for the path of the vehicle, and where the angle of all wheels are positively and independently controlled so that they tend to produce a singe centre of curvature for the path of the vehicle where the first centre of curvature is made identical to the second driver selected centre of curvature.

19. A braking vehicle according to claim 18 where the individual effective wheel angles and wheel speeds are given by the following equations:

$\frac{R}{t}=\cot <br><br>\left(90<br><br>\mathrm{°<br><br>\theta <br><br>}/{\mathrm{\theta <br><br>}}_{\max }\right)={\left(R_X^2+R_Y^2\right)}^{1/2}/t$

Where t is the track of the vehicle, θ

is the rotation of the joystick or steering wheel and θ

_max is the maximum rotation of the joystick or steering wheel.

If the driver displaces the rotatable joystick at an angle ψ

to the straight ahead position, the direction of the of curvature of the path of the vehicle will by at right angles to the direction of joystick displacement and R_X and R_Y will be given by the following equations;

R_X=R/(tan² ψ

+1)^1/2
and R_Y=R tan ψ

/(tan² ψ

+1)^1/2

where the driver selects the direction of the centre of curvature by displacing the joystick or steering wheel at right angles to this direction. The centre of curvature of the path of the vehicle is now specified by the two components R_X and R_Y. He selects the root mean square of the four wheel speeds by the amount of displacement of the joystick.

The control system then rotates the four drive wheels to the following angles;

tan φ

₁=(b/2−

R_Y)(R_X−

t/2)=tan(φ

₁′

−

α

₁)
tan φ

₂=(b/2−

R_Y)/(R_X+t/2)=tan(φ

₂′

−

α

₂)
tan φ

₃=(b/2+R_Y)/(R_X−

t/2)=tan(φ

₃′

+α

₃)
tan φ

₄=(b/2+R_Y)/(R_X+t/2)=tan(φ

₄′

+α

₄)

Where b is the wheel base of the vehicle, R_Y is the displacement of the centre of curvature forward of the centre of the vehicle and R_X is the displacement of the centre of curvature to the right of the centre of the vehicle, φ

₁, φ

₂, φ

₃ and φ

₄ are the effective angles of the front right, front left, rear right and rear left wheels respectively, φ

₁′

,φ

₂′

, φ

₃′ and

φ

₄′

are the actual angles of the front right front left, rear right and rear left wheels respectively, and α

₁, α

₂, α

₃ and α

₄ t are the slip angles of the front right, front left, rear right and rear left wheels respectively.

Where the amount of displacement of the joystick or brake pedal d determines the root mean square of the four wheel speeds (RMSWS) according to the equation;

RMSWS=Kd=(ω

₁²+ω

₂²+ω

₃²+ω

₄²)^1/2/2
where RMSWS=RMSWS₀—

(desired deceleration)(time)

where RMSWS₀ is the initial value of RMSWS and K is an appropriate constant.

Where the individual wheel speeds are given by the equations;

ω

₁=KdR₁/RMSR=ω

₁′

exp[−

i₁] where R₁²=(b/2−

R_Y)²+(R_X−

t/2)²
ω

₂=KdR₂/RMSR=ω

₂′

exp[−

i₂] where R₂²=(b/2−

R_y)²+(R_X+t/2)²
ω

₃=KdR₃/RMSR=ω

₃′

exp[−

i₃] where R₃²=(b/2+R_Y)²+(R_X−

t/2)²
ω

₄=KdR₄/RMSR=ω

₄′

exp[−

i₄] where R₄²=(b/2+R_Y)²+(R_X+t/2)²

And RMSR is the root mean square radius, which is given by;

RMSR=(R₁²+R₂²+R₃²+R₄²)^1/2/2=(R²+R_y²+t²/4+b²/4)^1/2

where ω

₁, ω

₂, ω

₃, and ω

₄ are the effective speeds of the front right, front left, rear right and rear left wheels respectively, ω

₁′

, ω

₂′

, ω

₃′

, and ω

₄′

are the actual speeds of the front right, front left, rear right and rear left wheels respectively, R₁, R₂, R₃ and R₄ are the distances between the centre of curvature of the path of the vehicle and the front right, front left, rear right and rear left wheels respectively, and i_1, i_2, i₃ and i₄ are the longitudinal slips of the front right, front left, rear right and rear left wheels respectively.

20. A braking vehicle according to claim 19 where the speed of each wheel is positively controlled to achieve the driver-selected centre of curvature for the path of the vehicle by modulating the clamping torque applied to each wheel, where an on-board computer calculates the desired speed versus time program for each wheel.

21. A vehicle according to claim 20 where accelerometers on the vehicle and speed sensors on each wheel detect when the rate of deceleration selected by the driver exceeds the rate of deceleration that can be achieved with the prevailing tyre-ground interaction, where the driver selected deceleration is replaced by the maximum achievable deceleration when calculating the appropriate wheel speed-time program for each wheel.