Method of determining the height of the gravity center of a vehicle

Abstract

The height of the center of gravity of a vehicle having at least 3 axles is estimated using the slippage rate of the wheels.

Claims

1. Method of determination of the height of the gravity center HG (HGr, Hg, HGst) of a vehicle, or an element of the vehicle, wherein 3 reference points Pa, Pb and Pc are predetermined, the method comprising the steps of: a) determining the tangential forces TAn (TAax, TAbx, TAcx) at all the axles An (Aa, Ab, Ac) of the vehicle, during a first acceleration phase, b) deducing from the tangential forces TAn (TAax, TAbx, TAcx) determined in step a) the corresponding normal forces NAn (NAax, NAbx, NAcx) during the first acceleration phase, c) determining the normal force NPb at one of the predetermined reference points Pb, during a first acceleration phase, using the values determined in steps a) and b), d) repeating steps a), b), and c) at least once during a second acceleration phase, distinct from the first acceleration phase, and deducing from the preceding steps a), b), c) and d) the height of the gravity center HG (HGr, Hg, HGst) of the vehicle, or an element of the vehicle, by computing the normal forces NPa, NPb and NPc at the 3 predetermined reference points Pa, Pb and Pc, using an algorithm wherein in step a) the tangential forces TAn are determined according to the formula (1): TAnx=Q.Math.VAnx+FAnx (1) wherein: TAnx denotes the tangential force at an axle An during a braking period x, Q denotes the brake factor, VAnx denotes the braking pressure applied at an axle An during a braking period x, and FAnx denotes the driving force applied at an axle An during a braking period x.

2. Method according to claim 1, wherein the normal forces are deduced from the tangential forces in step b) according to the general formula (5): $\begin{array}{cc}{N}_{\mathrm{Anx}}=\frac{T_{\mathrm{Anx}}}{{}_{\mathrm{An}}.Math.a_{\mathrm{An}}.Math.{}_{\mathrm{An}}}& \left(5\right)\end{array}$ wherein: NAnx denotes the normal force at an axle An during a braking period x, TAnx denotes the tangential forces at an axle An during a braking period x, An denotes the number of wheels at the axle An, aAn denotes a coefficient comprised between 0 and 1, and An denotes the slippage rate of the wheels of an axle An.

3. Method according to claim 1, wherein the vehicle is a rigid vehicle having 3 axles Aa, Ab and Ac, and wherein the 3 reference points Pa, Pb and Pc correspond to the 3 axles (Aa, Ab, Ac).

4. Method according to claim 1, wherein the vehicle is an assembly of a tractor and a semi-trailer, wherein one of the reference points Pa corresponds to an axle Aa of the tractor, a second reference point Pc corresponds to an axle Ac of the semi-trailer, and wherein the third reference point Pb corresponds to the fifth wheel of the tractor.

5. Method according to claim 1, wherein the vehicle is an assembly of a tractor having two axles and a semi-trailer, wherein one of the reference point Pb corresponds to the fifth wheel of the tractor, and wherein the normal force at reference point Pb is determined in step c) according to formula (8a): $\begin{array}{cc}{N}_{{\mathrm{Pb}}^{}}=\frac{\begin{array}{c}{N}_g\mathrm{Cos}\left(\right)\left[\left(E1-E_g\right)Y_a-E_g{Y}_b\right]+\left(Y_a+Y_b\right)\\ \left[H_g\left(T_{\mathrm{Aa}}+T_{\mathrm{Ab}}+F_{\mathrm{Ab}}+F_{Aa}\right)+\left(H_{{\mathrm{Pb}}^{}}-H_g\right)T_{{\mathrm{Pb}}^{}}\right]\end{array}}{\left[\left(E_g-E1\right).Math.Y_a+E_g.Math.Y_b\right]+\left(E_{{\mathrm{Pb}}^{}}-E_g\right)\left[Y_b+Y_a\right]}& \left(8a\right)\end{array}$ wherein: Ya denotes T.sub.AbT.sub.Aa, Yb denotes T.sub.Aa2.sub.Ab, Ng denotes the normal force at the center of gravity g of the tractor, NPb denotes the normal force at the point of reference Pb, E1 denotes the distance between the first axle Aa and the second axle Ab of the tractor, TAa denotes the tangential force at the first axle Aa, TAb denotes the tangential force at the second axle Ab, TPb denotes the tangential force at the reference point Pb, Aa denotes the slippage rate at the first axle Aa, Ab denotes the slippage rate at the second axle Ab, Eg denotes the distance between the gravity center g of the tractor and its first axle Aa, Hg denotes the height of the gravity center g of the tractor, EPb denotes the distance between the reference point Pb and the first axle Aa of the tractor, HPb denotes the height of the reference point Pb, and a denotes the inclination angle of the slope.

6. Method according to claim 1, wherein the vehicle is an assembly of a tractor having three axles and a semi-trailer, wherein one of the reference point Pb corresponds to the fifth wheel of the tractor, and wherein the normal force at the fifth wheel is determined in step c) according to formula (8b): $\begin{array}{cc}{N}_{{\mathrm{Pb}}^{}}=\frac{N_g\mathrm{Cos}\left(\right)Z_1-H_g\left(T_{\mathrm{Aa}}+T_{\mathrm{Ab}}+T_{\mathrm{Ac}}-T_{{\mathrm{Pb}}^{}}\right)Z_2}{Z_3+\left(E_g-E_{{\mathrm{Pb}}^{}}\right)Z_2}& \left(8b\right)\end{array}$ wherein:
Z1={E.sub.gT.sub.AaX.sub.AcX.sub.Ab2.sub.Aa(E1E.sub.g)T.sub.AbX.sub.Ac(E.sub.gE.sub.Pb)2.sub.AaT.sub.AbX.sub.Ab}
Z2={2.sub.AaT.sub.AbX.sub.Ac+X.sub.AbT.sub.AaX.sub.Ac+2.sub.AaT.sub.AcX.sub.Ab}
Z3={2.sub.Aa(E1E.sub.g)T.sub.AbX.sub.Ac+(E.sub.gE.sub.Pb)2.sub.AaT.sub.AcX.sub.AbE.sub.gT.sub.AaX.sub.AcX.sub.Ab} where: NPb denotes the normal force at the point of reference Pb, Ng denotes the normal force at the center of gravity g of the tractor, E1 denotes the distance between the first axle Aa and the second axle Ab, Eg denotes the distance between the gravity center g of the tractor and its first axle Aa, Hg denotes the height of the gravity center g of the tractor, TAa denotes the tangential force at the first axle Aa of the tractor, TAb denotes the tangential force at the second axle Ab of the tractor, TAc denotes the tangential force at the third axle Ac of the tractor, TPb denotes the tangential force at the reference point Pb, Aa denotes the slippage rate at the first axle Aa of the tractor, Ab denotes the slippage rate at the second axle Ab of the tractor, Ac denotes the slippage rate at the third axle Ac of the tractor, EPb denotes the distance between the reference point Pb and the first axle Aa of the tractor, XAb denotes AbAb, wherein Ab is the number of wheels at the axle Ab, and XAc corresponds to the terms AcAb wherein Ac is the number of wheels of the axle Ac.

7. Method according to claim 1, wherein the algorithm used in the determination of the height of the gravity center HG in step e) comprises the following formula $\begin{array}{cc}\mathrm{HG}=\frac{d_1\left(N_{\mathrm{Pb}2}-N_{\mathrm{Pb}1}\right)+d_2\left(N_{\mathrm{Pc}2}-N_{\mathrm{Pc}1}\right)+}{m\left({}_2-{}_1\right)}& \left(10\right)\end{array}$ wherein: HG denotes the height of the gravity center, d1 denotes the distance between the reference point Pa and the reference point Pb, d2 denotes the distance between the reference point Pa and the reference point Pc, NPb2 denotes the normal force at the reference point Pb during the second braking phase, NPb1 denotes the normal force at the reference point Pb during the first braking phase, NPc2 denotes the normal force at the reference point Pc during the second braking phase, NPc1 denotes the normal force at reference point PC during the first braking phase, m denotes the mass of the vehicle, 2 denotes the acceleration during the second braking phase, 1 denotes the acceleration during the first braking phase, and denotes a corrective value.

8. Method according to claim 7, wherein is selected from 0 for a rigid truck and the term HPb (TPb2-TPb1), for a vehicle comprising a tractor and a semi-trailer, wherein: HPb denotes the height of the reference point Pb, when Pb corresponds to the fifth wheel, TPb2 denotes the tangential force at the reference point Pb during the second braking phase, and TPb1 denotes the tangential force at the reference point Pb during the first braking force.

9. A-Method of determination of the height of the gravity center HG (HGr, Hg, HGst) of a vehicle, the vehicle comprising a tractor having at least three axles (Aa, Ab, Ac) and a semi-trailer, wherein 3 reference points Pa, Pb and Pc are predetermined, the method comprising the steps of: a) determining the tangential forces TAn (TAax, TAbx, TAcx) at all the axles An (Aa, Ab, Ac) of the vehicle, during a first acceleration phase, b) deducing from the tangential forces TAn (TAax, TAbx, TAcx) determined in step a) the corresponding normal forces NAn (NAax, NAbx, NAcx) during the first acceleration phase, c) determining the normal force NPb at one of the predetermined reference points Pb, during a first acceleration phase, using the values determined in steps a) and b), d) repeating steps a), b), and c) at least once during a second acceleration phase, distinct from the first acceleration phase, and deducing from the preceding steps a), b), c) and d) the height of the gravity center HG (HGr, Hg, HGst) of the vehicle, or an element of the vehicle, by computing the normal forces NPa, NPb and NPc at the 3 predetermined reference points Pa, Pb and Pc, using an algorithm wherein in step a) the tangential forces TAn are determined according to the formula (1): TAnx=QVAnx+FAnx (1) wherein: TAnx denotes the tangential force at an axle An during a braking period x, Q denotes the brake factor, VAnx denotes the braking pressure applied at an axle An during a braking period x, and FAnx denotes the driving force applied at an axle An during a braking period x.

10. Method of determination of the height of the gravity center HG (HGr, Hg, HGst) of a vehicle, or an element of the vehicle, wherein 3 reference points Pa, Pb and Pc are predetermined, the method comprising the steps of: a) determining the tangential forces TAn (TAax, TAbx, TAcx) at all the axles An (Aa, Ab, Ac) of the vehicle, during a first acceleration phase, b) deducing from the tangential forces TAn (TAax, TAbx, TAcx) determined in step a) the corresponding normal forces NAn (NAax, NAbx, NAcx) during the first acceleration phase, c) repeating steps a) and b) at least once during a second acceleration phase, distinct from the first acceleration phase, and deducing from the preceding steps a), b), and c) the height of the gravity center HG (HGr, Hg, HGst) of the vehicle, or an element of the vehicle, by computing the normal forces NPa, NPb and NPc at the 3 predetermined reference points Pa, Pb and Pc, using an algorithm wherein in step a) the tangential forces TAn are determined according to the formula (1): (1) TAnx=QVAnx+FAnx wherein: TAnx denotes the tangential force at an axle An during a braking period x, Q denotes the brake factor, VAnx denotes the braking pressure applied at an axle An during a braking period x, and FAnx denotes the driving force applied at an axle An during a braking period x.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) FIG. 1: Tangential and normal forces in a rigid vehicle having 3 axles, during an acceleration period.

(2) FIG. 2: Tangential and normal forces in a combination of a tractor having 2 axles and a semi-trailer, during an acceleration period.

(3) FIG. 3: Tangential and normal forces in a combination of a tractor having 3 axles and a semi-trailer, during an acceleration period.

DETAILED DESCRIPTION

(4) The method of the present invention, according to an aspect thereof, comprises a first step of determining the tangential forces TAn at the axles An of a vehicle v.

(5) In general way, a tangential force is determined at a given axle An, during a braking period x, according to the formula (1):
TAnx=Q.Math.VAnx+FAnx(1)
Wherein
TAnx denotes the tangential force at an axle An during a braking period x,
Q denotes the brake factor,
VAnx denotes the braking pressure applied at an axle An during a braking period x,
FAnx denotes the driving force applied at an axle An during a braking period x.
In case an axle does not correspond to a driving axle, then the corresponding driving force FAn equals 0.

(6) For a rigid truck having 3 axles Aa, Ab and Ac, the tangential forces at each of these axles can be independently determined according to the following formulae (1a), (1b) and (1c):
TAax=Q.Math.VAax+FAax(1a)
TAbx=Q.Math.VAbx+FAbx(1b)
TAcx=Q.Math.VAcx+FAcx(1c)
Wherein
TAax, TAbx and TAcx denote the tangential force at each of the corresponding axle Aa, Ab and Ac, during a braking period x,
Q denotes the brake factor,
VAax, VAbx and VAcx denote the braking pressure applied at each of the corresponding axle
Aa, Ab, and Ac, during a braking period x,
FAax, FAbx, and FAcx denote the driving force applied at each of the corresponding axle Aa,
Ab, and Ac, during a braking period x.

(7) In general, for a rigid vehicle having n axles, wherein n=3, 4, 5 or more, formula of type (1) can be independently applied to each of the n axles.

(8) For a combination of a tractor and a semi-trailer, the above formula (1) is applied to all the axles of the vehicle, including those of the semi-trailer.

(9) The present method, according to an aspect thereof, further comprises a step of determining the normal forces NAn at each of the n axles of the vehicle. The normal forces NAn may be for example deduced from the corresponding tangential forces TAn determined above. To this extend, the slippage rate w of the wheels w, may be used according to the formulae (4):

(10) $\begin{array}{cc}w=\frac{\mathrm{Sv}-\mathrm{Sw}}{\mathrm{Sv}}& \left(4\right)\end{array}$

(11) Wherein

(12) w denotes the slippage rate of a wheel w,

(13) Sv denotes the absolute speed of a vehicle v,

(14) Sw denotes the speed of the wheel w on the ground.

(15) On a given axle An, the slippage rate w may be different from one wheel to another. The present method can consider either of the following alternatives:

(16) The slippage rate An of an axle An corresponds to the average of the slippage rate w of all its wheels.

(17) The slippage rate An of an axle An corresponds to the lowest slippage rate w of all its wheels.

(18) The slippage rate An of an axle An corresponds to the highest slippage rate w of all its wheels.

(19) A normal force NAn at a given axle An may be determined, during a braking period x, according to the general formula (5):

(20) $\begin{array}{cc}{N}_{\mathrm{Anx}}=\frac{T_{\mathrm{Anx}}}{{}_{\mathrm{An}}.Math.a_{\mathrm{An}}.Math.{}_{\mathrm{An}}}& \left(5\right)\end{array}$

(21) Wherein

(22) NAnx denotes the normal forces at an axle An during a braking period x,

(23) TAnx denotes the tangential forces at an axle An during a braking period x,

(24) An denotes the number of wheels at the axle An,

(25) aAn denotes a coefficient comprised between 0 and 1,

(26) An denotes the slippage rate of the wheels of an axle An.

(27) An preferably denotes the average slippage rate of the wheels of a given axel An and TAn preferably corresponds to the cumulative tangential forces at said axle An. However, the calculus may be applied to each wheel individually.

(28) The coefficient a represents the adherence of a given wheel on the ground. A coefficient equal to 0 means no adherence at all, whereas a coefficient equal to 1 means a perfect adherence. In practice the coefficient a may be comprised between 0.2 and 0.8. For the purpose of the determination of the height of the gravity center according to the present method, according to an aspect thereof, it is considered that all the wheels of the vehicles have the same coefficient a.

(29) For a rigid vehicle having 3 axles Aa, Ab, and Ac, the normal forces at each of these axles may thus be independently determined according to the formulae (5a), (5b) and (5c):

(30) $\begin{array}{cc}{N}_{\mathrm{Aax}}=\frac{T_{\mathrm{Aax}}}{{}_{\mathrm{Aa}}.Math.a.Math.{}_{\mathrm{Aa}}}& \left(5a\right)\\ N_{\mathrm{Abx}}=\frac{T_{\mathrm{Abx}}}{{}_{\mathrm{Ab}}.Math.a.Math.{}_{\mathrm{Ab}}}& \left(5b\right)\\ N_{\mathrm{Acx}}=\frac{T_{\mathrm{Acx}}}{{}_{\mathrm{Ac}}.Math.a.Math.{}_{\mathrm{Ac}}}& \left(5c\right)\end{array}$

(31) Wherein

(32) NAax, NAbx and NAcx denote respectively the normal force at the Axles Aa, Ab and Ac, during a braking period x,

(33) TAax, TAbx and TAcx denote respectively the tangential forces at each of the axles Aa, Ab, and Ac during a braking period x,

(34) denotes the number of wheels at an axle An,

(35) a denotes the coefficient of adherence above discussed.

(36) Aa, Ab, Ac denote respectively the average slippage rate of the wheels of the corresponding axles Aa, Ab, and Ac.

(37) In case the rigid truck has 4 or more reference points, corresponding to its 4 or more axles, formula of type (5) can independently be applied to each of the axles.

(38) For a combination of a tractor and a semi-trailer, the normal force at one of the reference point may not be directly determined as above-discussed. For example in case one of the reference point Pb is the fifth wheel, then the corresponding normal force NPb can be computed based on the tangential and normal forces previously determined at the axles An of the tractor. Where the tractor has 2 axles, the normal forces NPb at the reference points Pb may be determined according to the formula (8a) below:

(39) $\begin{array}{cc}{N}_{{\mathrm{Pb}}^{}}=\frac{\begin{array}{c}{N}_g\mathrm{Cos}\left(\right)\left[\left(E1-E_g\right)Y_a-E_g{Y}_b\right]+\left(Y_a+Y_b\right)\\ \left[H_g\left(T_{\mathrm{Aa}}+T_{\mathrm{Ab}}+F_{\mathrm{Ab}}+F_{Aa}\right)+\left(H_{{\mathrm{Pb}}^{}}-H_g\right)T_{{\mathrm{Pb}}^{}}\right]\end{array}}{\left[\left(E_g-E1\right).Math.Y_a+E_g.Math.Y_b\right]+\left(E_{{\mathrm{Pb}}^{}}-E_g\right)\left[Y_b+Y_a\right]}& \left(8a\right)\end{array}$

(40) Wherein

(41) Ya denotes T.sub.Ab.sub.Aa

(42) Yb denotes T.sub.Aa2.sub.Ab

(43) Ng denotes the normal force at the center of gravity g of the tractor

(44) NPb denotes the normal force at the point of reference Pb

(45) E1 denotes the distance between the first axle Aa and the second axle Ab of the tractor

(46) TAa denotes the tangential force at the first axle Aa

(47) TAb denotes the tangential force at the second axle Ab

(48) TPb denotes the tangential force at the reference point Pb

(49) Aa denotes the average slippage rate of the wheels of the first axle Aa,

(50) Ab denotes the average slippage rate of the wheels of the second axle Ab,

(51) Eg denotes the distance between the gravity center g of the tractor and its first axle Aa

(52) Hg denotes the height of the gravity center g of the tractor

(53) EPb denotes the distance between the reference point Pb and the first axle Aa of the tractor

(54) HPb denotes the height of the reference point Pb

(55) denotes the inclination angle of the slope

(56) It has to be noted that the normal force NPb determined at the fifth wheel of the tractor corresponds to the normal force of the semi-trailer onto the tractor.

(57) Similarly, the term TPb corresponds to the tangential force at the fifth wheel, and results from the action of the semi-trailer.

(58) In a general way, the tangential force TPb can be determined according to the following formula (9)
T.sub.Pb=F.sub.An+T.sub.AnP.sub.TR.Math.Sin()+m.Math.(9)

(59) Wherein

(60) FAn denotes the driving forces at all the An axles,

(61) TAn denotes the tangential forces at all the An axles,

(62) PTR denotes the weight of the tractor

(63) m denotes the mass of the vehicle

(64) denotes the acceleration

(65) denotes the slope of the ground

(66) Where the tractor has 2 axles, TPb can be determined according to the following formula (9a)
T.sub.Pb=F.sub.AaF.sub.Ab+F.sub.Aa+T.sub.Aa+T.sub.AbP.sub.TR.Math.Sin()+m.Math.(9a)

(67) Wherein

(68) TPb denotes the tangential force at the reference point Pb, corresponding to the fifth wheel

(69) FAa denotes the driving force at the front axle Aa,

(70) FAb denotes the driving force at the second axle of the tractor Ab,

(71) TAa denotes the tangential force at the front axle Aa,

(72) TAb denotes the tangential force at the second axle of the tractor Ab,

(73) m denotes the mass of the vehicle,

(74) denotes the acceleration,

(75) denotes the slope of the ground, and

(76) PTr denotes the weight of the tractor

(77) The normal forces NPa at the reference point Pa, and NPc at the reference point Pc, are still determined according to the formula (5) above.

(78) In case the tractor has 3 axles, the following formula (8b) can be used:

(79) $\begin{array}{cc}{N}_{{\mathrm{Pb}}^{}}=\frac{N_g\mathrm{Cos}\left(\right)Z_1-H_g\left(T_{\mathrm{Aa}}+T_{\mathrm{Ab}}+T_{\mathrm{Ac}}-T_{{\mathrm{Pb}}^{}}\right)Z_2}{Z_3+\left(E_g-E_{{\mathrm{Pb}}^{}}\right)Z_2}& \left(8b\right)\end{array}$

(80) Wherein
Z1={E.sub.gT.sub.AaX.sub.AcX.sub.Ab2.sub.Aa(E1E.sub.g)T.sub.AbX.sub.Ac(E.sub.gE.sub.Pb)2.sub.AaT.sub.AbX.sub.Ab}
Z2={2.sub.AaT.sub.AbX.sub.Ac+X.sub.AbT.sub.AaX.sub.Ac+2.sub.AaT.sub.AcX.sub.Ab}
Z3={2.sub.Aa(E1Eg)T.sub.AbX.sub.Ac+(E.sub.gE.sub.Pb)2.sub.AaT.sub.AcX.sub.AbE.sub.gT.sub.AaX.sub.AcX.sub.Ab}

(81) Where:

(82) NPb denotes the normal force at the point of reference Pb

(83) Ng denotes the normal force at the center of gravity g of the tractor

(84) E1 denotes the distance between the first axle Aa and the second axle Ab

(85) Eg denotes the distance between the gravity center g of the tractor and its first axle Aa

(86) Hg denotes the height of the gravity center g of the tractor

(87) TAa denotes the tangential force at the first axle Aa of the tractor,

(88) TAb denotes the tangential force at the second axle Ab of the tractor

(89) TAc denotes the tangential force at the third axle Ac of the tractor

(90) TPb denotes the tangential force at the reference point Pb

(91) Aa denotes the average slippage rate of the wheels of the first axle Aa of the tractor

(92) Ab denotes the average slippage rate of the wheels of the second axle Ab of the tractor

(93) Ac denotes the average slippage rate of the wheels of the third axle Ac of the tractor

(94) EPb denotes the distance between the reference point Pb and the first axle Aa of the tractor

(95) XAb denotes AbrAb, wherein Ab is the number of wheels at the axle Ab

(96) XAc corresponds to the terms AcAc wherein Ac is the number of wheels of the axle Ac

(97) In this specific case, the term TPb can be determined according to the formula (9b):
R.sub.T=F.sub.Aa+F.sub.Ab+F.sub.Ac+T.sub.Aa+T.sub.Ab+T.sub.Ac+m.Math.P.sub.TR.Math.Sin()(9b)

(98) Wherein

(99) FAc denotes the driving force at the third axle Ac,

(100) TAc denotes the tangential force at the third axle Ac,

(101) And wherein the other numerical references have the same meaning as above.

(102) Formulae (8a) and (8b) are specific examples on how the normal forces NPb can be determined at the reference point Pb. However, any other method allowing to determine the normal force at the reference point Pb, during braking phases, can be used as alternative.

(103) The values of normal forces, obtained for each of the 3 reference points, are determined according to the steps above-discussed during at least 2 distinct acceleration phases, wherein the acceleration of the vehicle is preferably not the same. Thus, at least 2 sets of values are obtained corresponding to a first acceleration phase and a second acceleration phase.

(104) It is preferable that the steps are repeated while the slope remains unchanged.

(105) The 2 sets of values are computed according to the following formula (10) to determine the height of COG:

(106) $\begin{array}{cc}\mathrm{HG}=\frac{d_1\left(N_{\mathrm{Pb}2}-N_{\mathrm{Pb}1}\right)+d_2\left(N_{\mathrm{Pc}2}-N_{\mathrm{Pc}1}\right)+}{m\left({}_2-{}_1\right)}& \left(10\right)\end{array}$

(107) Wherein

(108) HG denotes the height of the gravity center G of the vehicle if it is a rigid vehicle, or the height

(109) of the gravity center of the semi-trailer in case of a combination of a tractor and a semi-trailer.

(110) d1 denotes the distance between the first reference point Pa and the second reference point Pb,

(111) d2 denotes the distance between the first reference point Pa and the third reference point Pc,

(112) NPb2 denotes the normal force at the second reference point Pb during the second braking phase,

(113) NPb1 denotes the normal force at the second reference point Pb during the first braking phase,

(114) NPc2 denotes the normal force at the reference point Pc during the second braking phase,

(115) NPc1 denotes the normal force at reference point Pc during the first braking phase,

(116) m denotes the mass of the vehicle,

(117) 2 denotes the acceleration during the second braking phase,

(118) 1 denotes the acceleration during the first braking phase,

(119) denotes a corrective value.

(120) The corrective value may be equal to 0 for a rigid vehicle. Thus, in the particular case of a rigid vehicle having 3 axles, corresponding to the 3 reference points Pa, Pb and Pc, the height of the gravity center HGr is determined according to the formula (10a):

(121) $\begin{array}{cc}{\mathrm{HG}}_r=\frac{d_1\left(N_{\mathrm{Pb}2}-N_{\mathrm{Pb}1}\right)+d_2\left(N_{\mathrm{Pc}2}-N_{\mathrm{Pc}1}\right)}{m\left({}_2-{}_1\right)}& \left(10a\right)\end{array}$

(122) Wherein

(123) HGr denotes the height of the gravity center Gr of the rigid vehicle,

(124) d1 denotes the distance between the first reference point Pa, corresponding to the first axle Aa, and the second reference point Pb, corresponding to the second axle Ab,

(125) d2 denotes the distance between the first reference point Pa, corresponding to the first axle Aa, and the third reference point Pc, corresponding to the third axle Ac,

(126) NPb2 denotes the normal force at the second reference point Pb during the second braking phase,

(127) NPb1 denotes the normal force at the second reference point Pb during the first braking phase,

(128) NPc2 denotes the normal force at the reference point Pc during the second braking phase,

(129) NPc1 denotes the normal force at reference point Pc during the first braking phase,

(130) m denotes the mass of the vehicle,

(131) 2 denotes the acceleration during the second braking phase,

(132) 1 denotes the acceleration during the first braking phase,

(133) The corrective value may be equal to the terms HPb(TPb2TPb1) for a combination of a tractor and a semi-trailor. Thus, in this particular case, the height of the gravity center HGst of the semi-trailer will be determined by the formula (10b):

(134) $\begin{array}{cc}{\mathrm{HG}}_{\mathrm{st}}=\frac{\begin{array}{c}{d}_1^{}\left(N_{{\mathrm{Pb}}^{}2}-N_{{\mathrm{Pb}}^{}1}\right)+d_2^{}\left(N_{{\mathrm{Pc}}^{}2}-N_{{\mathrm{Pc}}^{}1}\right)+\\ H_{{\mathrm{Pb}}^{}}\left(T_{{\mathrm{Pb}}^{}2}-T_{{\mathrm{Pb}}^{}1}\right)\end{array}}{m\left({}_2-{}_1\right)}& \left(10b\right)\end{array}$

(135) Wherein

(136) d1 denotes the distance between the first axle of the tractor Aa and the reference point Pb, corresponding to the fifth wheel of the tractor,

(137) d2 denotes the distance between the first axle of the tractor Aa and the reference point Pc, corresponding to a rear axle of the semi-trailer,

(138) NPb2 denotes the normal force at the reference point Pb during the second braking period,

(139) NPb1 denotes the normal force at the reference point Pb during the first braking period,

(140) NPc2 denotes the normal force at the reference point Pc during the second braking period,

(141) NPc1 denotes the normal force at the reference point Pc during the first braking period,

(142) HPb denotes the height of the reference point Pb,

(143) TPb2 denotes the tangential force at the reference point Pb during a second braking phase,

(144) TPb1 denotes the tangential force at the reference point Pb during a first braking phase,

(145) m denotes the mass of the vehicle,

(146) 2 denotes the acceleration during the second braking phase,

(147) 1 denotes the acceleration during the first braking phase,