A Comparative Study of Equivalent Modeling for Multi ...

A Comparative Study of Equivalent Modeling for Multi-axle Vehicle Yubiao Zhang, Yanjun Huang*, Hong Wang, Amir Khajepour Department of Mechanical and Mechatronics Engineering, University of Waterloo, ON, N2L3G1, Canada Abstract: In this paper, equivalent modeling methods of a multi-axle vehicle are presented and compared. Firstly, for the sake of comparison, a single-track model of a three-axle and a two-axle vehicle is developed, and then existing equivalent modeling derivations are presented and discussed. Next, the proposed model based dynamic equivalence of force/moment at the centre of gravity (CG) is introduced and optimized. It represents the approximately equivalent steady state and transient response of the yaw rate and side slip angle, which allows different cornering stiffness on the central and rear axle. Finally, to demonstrate how the proposed method is advantageous to the other equivalent models available in the literature, different simulation cases are compared in the dimension of time-domain, eigenvalues characteristics and frequency-domain. Furthermore, the proposed method is extended to any multi-axle vehicle configurations and a general expression is formulated. Keywords: multi-axle vehicle, vehicle modeling and dynamics, optimization

1. Introduction In order to transport a larger volume of freights via a relatively fixed infrastructure, one way to increase transportation efficiency is to employ longer vehicles with more axles [1]. These vehicles are often highly specialized and customized to perform particular tasks like logistic, military, long-distance coaches, and public transportation. With different handling properties, but driving on existing infrastructure, the safety and stability problems of multi-axle vehicles continue to be an active research area among researchers and automobile manufacturers [2-4]. A single-track model is well known and widely used in traditional two-axle vehicles with the yaw rate and lateral velocity (or sideslip angle at CG) as the degrees of freedom to predict the handling performance and stability both in time and frequency domain analyses[2,4,5]. An equivalent two-axle model of multi-axle vehicles can be used to easily study such vehicles’ handling and stability characteristics. Also, compared to the nonlinear model or even original multi-axle model, it is more computational efficient to design the active modelbased control systems[2,6-8], for instance differential braking, torque vectoring and active steering. As a result, several equivalent models have been developed in the literature. There are three major attempts existing in the literature to develop the equivalent model of three-axle vehicles. William [9,10] presents the basic notions of the equivalent wheelbase and the understeer coefficient of three-axle vehicles when compared to the two-axle model. It is obtained directly from the steady-state response of the state space model. Furthermore, Williams [11] extends the concept to vehicles with any arbitrary number of axles. He also develops a unified way to study handling of multi-axel vehicles by the equivalent wheelbase and understeer coefficient. The general expression of the characteristic equation with steady-state vehicle dynamics is also extended to the n-axle vehicle. Ding and Guo [12] apply William’s work to develop the 2M (twice-multiple) equivalent approach to analyze both the effect of vehicle body roll and n-axle handling on vehicle dynamics. However, the sideslip angle is not considered from the above work. Winkler and Gillespie [13-15] generate an equivalent three-axle vehicle wheelbase from a virtual front axle and rear axle, which is consistent with the handling characteristics of the dynamics equations. The method is only applicable to the vehicle with one steering front-axle and two driving rear-axles. The basic concept behind their approach is to move the equivalent front axle forward and move the equivalent rear axle rearward, subjecting to assumption that the two equivalent axles is free of slipping. The equivalent parameters such as distance and cornering therefore, are

found. In Johannes’s thesis[16], it explains the equivalent modeling approach in detail. In addition, Winkler [14] further presents a comprehensive study on the steady-state turning of multi-axle vehicles by using Pacejka’s [17] handling diagram methodology. A generalized expression for an arbitrary number of non-steered axles is achieved. However, this method is limited to the situation of a steady-state low-speed turn. Given that the normal load is equally shared between the tandem axles in the rear and all tires have the same cornering stiffness, Ellis [18] proposes a method of equating the action of a tandem axle bogie (central and rear axle) to a single equivalent axle acting in the midway of the central axle and rear axle by adding an augmented moment to produce a compensation for the equivalent two-axle model. Based on a similar steady-state development, the equivalent wheelbase using Ellis’s method is shown to yield a compatible connection to the general three-axle development with simplifying assumptions consistent with Williams and Winkler and Gillespie’s derivation. However, this method is limited by the same cornering stiffness and an equally shared load. The accuracy of the equivalence is questionable when these conditions change. In this paper, a new model is derived from dynamic equalization, which means the forces and moment at CG are always equivalent between the two models. An over-constrained optimization problem is formulated and the stiffness of the equivalent rear wheels and equivalent distance from CG to equivalent rear axle are found via solving the optimization problem. As a statement, the method generates an approximately equivalent model not completely equivalent since it minimizes the errors between the proposed model and the original model. It promises not only an approximately equivalent steady state response but also an approximately equivalent transient response. In addition, a high accuracy of sideslip angle response is guaranteed. The proposed method’s efficiency is verified through simulations in several scenarios to show both transient and steady state behavior in time and frequency domains. The proposed approach can be used to more conveniently study three-axle vehicles dynamics and stability. In this way, it is feasible to build a unified model and a holistic controller for vehicles with different numbers of axles. In Sec. 2, a three-axle vehicle model and its equivalent model are developed. In Sec. 3, the existing equivalent models in the literature are presented and discussed. Sec. 4 leads to the proposed method of equivalent modeling and explains how to find the equivalent parameters using optimization. In Sec. 5, simulation results are presented and compared to demonstrate the advantages of the proposed method. Sec. 6 formulates a general expression for any multi-axle vehicle and Sec. 7 concludes the paper.

2. Vehicle Single-track Model A fixed third axle is added to the classic two-axle single-track model as described in Figure 1. Fancher [19] found that the yaw resisting moment from the distance between dual tires was far less significant when compared to the moment that from the tandem spread. Hence, the effect of dual tires used on the tandem rear axles is neglected and thus the single-track model extended to multi-axle vehicle is used.

Figure 1. Derivation of single track model of three-axle vehicle and its equivalent model

The single-track model is widely used in the literature and is not explained here in detail. The equations for lateral dynamics and yaw motion are directly given by:

m(v   u )  Fyf  Fyc  Fyr

(1)

I z  Fyf l f  Fyclc  Fyr lr

(2)

Assuming the slip angles are small, tire lateral forces have a linear relationship to the tire slip angle. The lateral force of a displaced axle i can be expressed as

v   li  C i ( i  u ) Fyi   C (  v   li )   i i u

(i  f ) (3)

(i  c, r ;  c   r  0)

In addition, to obtain the sideslip angle on CG, one can easily derive it from

  tan 1

v u

(4)

In Equation (1), (2), (3) and (4), we have:

m, I z

- vehicle mass and the moment of inertia about z axis,

u, v,  ,  - vehicle longitudinal speed, lateral speed, yaw rate and body slip angle at vehicle’s CG, Fyf , Fyc , Fyr - lateral force of front, central and rear tires, l f / lc / lr / l -distance from CG to front/central/rear axle, the distance between front axle to rear axle,

C f / C c / C r - front/central/rear wheels cornering stiffness,

 f /  c /  r - front/central/rear wheels steering angle. By substituting (3) to (1) and (2), the yaw dynamics can be represented by the following state space model:

C  C c  C r   f     mu     l C l C l C    f  f c c r r  x Iz 

l f C f  lcC c  lr C r   C f    2 mu      mu l f 2C f  lc 2C c  lr 2C r      l f C f   x  I I zu  z 

1 

   f   

(5)

B

A

Figure 1 also shows an equivalent model of a two-axle vehicle. It is assumed a single equivalent rear axle can be used to represent two original axles. To describe the classic two-axle single-track model (i.e. the bicycle model), one can easily formulate it in the state-space form, C  Ceq   f     mu     l C l C     f  f req eq  x Iz 

l f C f  lreq Ceq   C f   2  mu      mu 2 2   l C l C    l f C f  f  f req eq     x I zu  Iz 

1 

Aeq

   f   

(6)

Beq

lreq , Ceq - distance from CG to the equivalent rear axle and the equivalent cornering stiffness representing tires of the central and rear axle.

3. Existing Equivalent Modeling There are three equivalent two-axle models in the literature that have been proposed for multi-axle vehicles. These models are reviewed below. 3.1 Daniel E.Williams Derivation In this two-axle single-track model, the steady-state vehicle yaw rate response is fully determined by two parameters: understeer coefficient and wheelbase. The vehicle dynamic response is assumed only dependent on these two parameters. Williams [10] derives an equivalent rear axle directly from the steady state response of the linear model when compared with the two-axle model. The understeer coefficient and wheelbase are characterized analogous with that of the two-axle vehicle. In the steady state, the time derivatives are zero and the steady-state can be solved in terms of the steering angle. The equivalent understeering coefficient and equivalent wheelbase for the three-axle model are defined. 

Steady yaw rate w.r.t steering angle (two-axle model)

m(lreq Ceq  l f C f )  ss u ,  , K eq  2  f leq  K eq u leq C f Ceq

(7)

where K eq is known as the understeering coefficient of the two-axle vehicle. The sign of its value depends on the cornering stiffness of the tires and their position with respect to the CG. 

Steady yaw rate w.r.t steering angle (three-axle model)

u[(l f  lc )C f C c  lC f C r ]  ss  2 2  f (l f  lc ) C f C c  l C f C r  (lr  lc ) 2 C r C c  mu 2 (lcC c  lr C r  l f C f )

(8)

Rewrite the equation (8) as the form

 ss u  2 2 2 C C ( l  l )  C C l  C C ( f  f c f c  f r  r  c l r  lc ) (l f  lc )C f C c  lC f C r

(9)



m(lcC c  lr C r  l f C f ) (l f  lc )C f C c  lC f C r

leq

u2

K eq

Therefore, in Williams’s method, it is made to be equivalent that

leq 

C f C c (l f  lc )2  C f C r l 2  C r C c (lr  lc )2

K eq 

(l f  lc )C f C c  lC f C r

m(lreq Ceq  l f C f ) leq C f Ceq



, lreq  leq  l f

m(lcC c  lr C r  l f C f ) (l f  lc )C f C c  lC f C r

(10)

(11)

The equivalent concerning stiffness is calculated by

Ceq 

ml f C f m(leq  l f )  K eq C f leq

(12)

3.2 Winkler–Gillespie Derivation Consider the three-axle vehicle and an equivalent model illustrated in Figure 2, in Winkler–Gillespie’s derivation [13-15], the vehicle is assumed to be in a steady-state turn at a very low speed such that the lateral acceleration can be treated as zero. The idea behind is to find equivalent positions of the virtual front and rear axles so that the equivalent model operate with zero slip angles and generate zero lateral forces at very low speeds. It is defined that  is the distance from central and rear axle to the mid-point between them and the geometric wheelbase lg is the distance between the tandem center and real front axle.

Figure 2. Derivation of the equivalent wheelbase of a three-axle vehicle in a steady-state low-speed turn In a steady-state turn at very low speed, the requirements of static equilibrium of lateral forces and yaw moment lead to

F

 Fyf  Fyc  Fyr  0

(13)

  Fyf (lg  )  2 Fyr   0

(14)

y

M

c

Tire lateral forces have a linear relationship to the tire slip angle. The lateral force of a displaced tire expressed as

a b  b Fyf  C f ( ); Fyc  C c ( ); Fyr  C r ( ) R R R

i

can be (15)

Explained by Winkler–Gillespie [14] in detail, R is the low-speed turn radius, a is the distance ahead of the front tire at which an imaginary tire, steered to the same angle as the real front tire, would have to be located to experience zero side slip. b is the distance aft of the center of the rear suspension where an imaginary, nonsteering tire would have to be located to experience zero slip. With substitution and simplification, the virtual distance a , b , the equivalent wheelbase and cornering stiffness can be achieved by a

2C  2  (C c  C r )(lg  )  2C r (  b) ; b  r C f (lg  ) 2C r   (C c  C r )(lg  )

leq  a  lg  b; Ceq  C c  C r Furthermore, if

C c  C r , it can be simplified to

(16)

(17)

leq  lg 

 2 Ceq  2  ; Ceq  C c  C r lg C f lg

(18)

3.3 J.R. Ellis Derivation Instead of deriving two virtual equivalent axles, Ellis replaces the tandem axle bogie with an equivalent axle in the midway of the central and rear axles and an augmented moment is added to produce a compensation for the equivalent two-axle model [18]. However, it is only appropriate in the special cases where the tires are equally loaded on to the central and rear axles. Ellis shows that when both rear axles have equal cornering stiffness, the yaw augmented moment M resist produced is M resist 

2C r  2 u

(19)

Extending Ellis’ development, modifying the state-space form in Equation (6), one can achieve the corresponding equivalent wheelbase and cornering stiffness. Fortunately, similar to the development of the Winkler–Gillespie method, same results are achieved in the three-axle steady-state model [10,14].

4. The Proposed Method 4.1 Equalization based on CG force/moment To summarize the previous work, it can be seen that Williams’s derivation is based on an equivalent steady-state yaw rate and the understeering coefficient is also equal to that of the two-axle model. However, the side slip angle of CG, as another important control variable in stability control system is neglected from this equalization. Winkler– Gillespie’s derivation is based on the assumption of a steady-state low speed and the lateral acceleration is approximately equal to zero. The limitation is that the equalization is only admissible in the low-speed situation and the stability control design is quite restricted using this model. Similar drawbacks arise in J.R. Ellis’s derivation. The proposed method here is based on dynamic equalization, which means the forces and moment at CG are identical between two models. The method promises both approximately equivalent steady state response and transient response. Two equivalent parameters lreq and Ceq are introduced when deriving the model. From the model of three-axle and two-axle, the equations for lateral dynamics and yaw motion can be expressed by: 

The lateral motion of three-axle model and two-axle model

m(v   u )  C f ( f 

m(v   u )  C f ( f  

v lf u

v lf u

) +C c (

)  Ceq (

v   lc v   lr )  C r ( ) u u

v   lreq u

)

The yaw motion of three-axle model and two-axle model v lf v   lc v   lr I z  l f C f ( f  )  lcC c ( )  lr C r ( ) u u u v lf v   lreq I z  l f C f ( f  )  lreqCeq ( ) u u

(20) (21)

(22) (23)

where the right hand sides of Equation (20) and Equation(21) actually represent the summative lateral force at CG, and the right hand sides of Equation (22) and Equation(23) represent the summative yaw moment at CG.

From the fundamental vehicle dynamics theory, the forces and moment at CG determines the vehicle motions. In order to make motion of the three-axle model and the two-axle model be identical, we attempt to equalize the lateral forces and yaw moment at CG of two models. This means operate the right hand side of Equation (20) and Equation (21) to be equal, so does to Equation (22) and Equation (23). Assuming that the front wheels stiffness and the distance from CG to front axle remain unchanged, so the equivalent rear wheels stiffness Ceq and equivalent distance from CG to equivalent rear axle lreq are the only two unknown parameters need to be solved. By eliminating term with l f and C f , we obtain Ceq (

v   lreq

lreq Ceq (

u v   lreq u

 v   lc v   lr Ceq  C c  C r  )  C r ( )    u u   lreq Ceq  lcC c  lr C r  v   lc v   lr  lreq 2Ceq  lc 2C c  lr 2C r  )  lcC c ( )  lr C r ( )   u u 

)  C c (

(24)

Continuing eliminating the terms with vehicle state variables of v ,  and u , three independent equations are deduced shown as Equation (24). But we only need to solve two unknowns lreq and Ceq , and this becomes an overconstrained problem. Therefore, we pursue to find an optimal solution that minimizing the error between three-axle full model and the proposed model. Defining that lreq represents as x-axis and Ceq represents as the y-axis and plot the functions of Ceq = f (lreq ) derived from Equation (24). A generic three-axle vehicle is used and plotting results are shown in Figure 3. Enlarging the intersections from the left figure, an optimization area is obtained in the right figure.

Figure 3. Equivalent parameters of lreq and Ceq 4.2 Optimization of equivalent parameters As shown in Figure 4, to find the optimal lreq and Ceq , the above over-constrained problem is formulated as a simple convex optimization problem as follows,

5

x 10

p2(x2,y2)

p1(x1,y1)

2.4 2.38 2.36

p(x,y)

y

2.34 2.32 2.3 2.28

p3(x3,y3)

2.26 2.24 1.32

1.34

1.36

1.38

1.4

1.42

x Figure 4. Problem formulation Minimize : f ( x, y ) where, f ( x, y )  pp1  pp2 2

p1 ( x1 , y1 )  (

p3 ( x3 , y3 )  (

2

3

 pp3   ( x  xi ) 2  ( y  yi ) 2 2

i 1

lcCc  lr Cr l 2C  l 2C , Cc  Cr ) ; p2 ( x2 , y2 )  ( c c r r , Cc  Cr ) Cc  Cr Cc  Cr

lc 2Cc  lr 2Cr (lcCc  lr Cr ) 2 , ) lcCc  lr Cr lc 2Cc  lr 2Cr

Constraint:

x  0; y  0. Note: The objective function f ( x, y ) is an Euclidean norm, or

l2 -norm, which means a convex function. The

constraint is also a convex set [20]. The problem is then solved by making the gradient of f ( x, y )

 2(3x  x1  x2  x3 )  0 f ( x, y)      2(3 y  y1  y2  y3 )  0

 x *  ( x  x  x ) / 3    1 2 3   y * ( y1  y2  y3 ) / 3

The optimal point of lreq and Ceq is found as

p*  (

x1  x2  x3 y1  y2  y3 , ). 3 3

5. Comparison and Discussion 5.1 Step-steer Manoeuver To compare the propose model with the three existing equivalent models in the literature, the responses of the models to a step-steer (i.e. front wheels steering angle = 2°) input to a three-axle vehicle at different speeds (20km/h, 60km/h and 90km/h) are presented. In the simulations, it is assumed that the cornering stiffness of central and rear axles are identical.

The simulation results of the 3-axle vehicle and the equivalent models are shown in Figure 5. As seen, the results show the proposed method has over-performed all the previous methods and especially it has a better response of slip angle. At a low speed (20km/h), all methods have a good response tracking to the original model. This is because the lateral acceleration is very small and W-G/ J.R. Ellis’s derivation is based on a steady turn at a very low speed. However, when the speed is increased to 60km/h and 90km/h, the error in sideslip angle increases in other equivalent models, while the proposed one holds a very good tracking on both yaw rate and side slip angle. One may note that the Williams’s derivation results in a good tracking only in yaw rate response, and it precisely shows how the method is developed from equivalent steady state yaw rate. 20 yaw rate(deg/s)

yaw rate(deg/s)

8 6 4 2 0

0

2

4 t(sec)

6

side slip angle(deg)

side slip angle(deg) yaw rate(deg/s) side slip angle(deg)

10

5

5 0

0

2

4 t(sec)

6

8

1 0.5

0

2

4 t(sec)

6

8

0.3

Original three-axle model D. E.Williams derivation W–G/J.R. Ellis derivation The proposed method

0.5

0

2

4 t(sec)

6

8

0

2

4 t(sec)

6

8

0

0.2 -0.5

0.1 0

(a)

1

0

0.5

0.4

1.5

0

15

10

0

8

2

0

15

-1

0

2

4 t(sec)

(a)

(b) 20km/h at step steer

(b)

60km/h at step steer

(c)

90km/h at step steer

6

8

(c)

0.5 Figure 5. Yaw rate1 and side slip angle response w.r.t different speeds ( C c  C r  1.4e5 N/rad ) t(sec)

1

5.2 Different Combinations of Cornering Stiffness 0.5

In the following simulations, the equivalent models are compared in step-steer and lane change manoeuvers with 0 0 cornering 2 4 6 In this 8 case, the speed of the vehicle is assumed to be 90 km/h and 70 km/h for the stepdifferent stiffness. t(sec) steer and lane change maneuvers, respectively. The results are shown in Figure 6. As seen, the proposed model can accurately predict the yaw rate and side slip angle of the original three-axle model. The additional manoeuver of single lane change presents a similar phenomenon.

10 5

0

2

4 t(sec)

6

-0.5 -1

yaw rate(deg/s) side slip angle(deg)

0

2

4 t(sec)

6

8

2

Original three-axle model D. E.Williams derivation W–G/J.R. Ellis derivation The proposed method 0

0.5 t(sec)

1

-20

6

8

0

2

4 t(sec)

6

8

0

2

4 t(sec)

6

8

1

-1

0.5 0 -0.5 -1

0

2

(a)

4 6 t(sec) (b) 90km/h at step steer( C

 8e4 N/rad; C r  1.4e5 N/rad )

(b)

90km/h at step steer( C

 1.4e5 N/rad; C r  8e4 N/rad )

(c)

70km/h at single lane change( C

(a)

1

0.5

4 t(sec)

0

-2

0

-40

0

side slip angle(deg)

0

0

10

1 side slip angle(deg)

side slip angle(deg)

0.5

-1.5

20

0

8

yaw rate(deg/s)

15

0

20

30 yaw rate(deg/s)

yaw rate(deg/s)

20

8

(c)

c

c

c

 8e4 N/rad; C r  1.4e5 N/rad )

Figure 6. Yaw rate and side slip angle response

1

5.3 Discussion on Changing Tires Stiffness 0.5

In practice, it is known that the cornering stiffness depends on the normal tire load, which particularly happens at turnings. In this section, it is discussed that how the proposed model is influenced by the changes in cornering 0 The obtained three equations of two equivalent parameters are repeated here for convenience. stiffness. 0 2 4 6 8

Ceq  C c  C r (25) lreq Ceq  lcC c  lr C r 2 2 2 lreq Ceq  lc C c  lr C r Consider the central and rear wheels cornering stiffness changed from a specific range from minimum to maximum C r [C r min , C r max ] . For simplicity, if we assume that the central wheels cornering stiffness has a linear t(sec)

relationship to rear wheels cornering stiffness,

C c  C r and by substituting Equation (26) to Equation(25), and then elimination by substitution, we have,

lreq 

l 2  lr 2  lc  lr , lreq 2  c 1  1 

From Equation (27) it can be seen that the equivalent parameter makes sense that

(26)

(27)

lreq is independent to the tires stiffness, which

lreq is a coefficient of distance parameters of lc and lr . Figure 7 shows lreq in Equation (27) as a

function of  . As it can be seen in the figure, the changes in equivalent parameter

lreq is small with the range of 

from 0 to 4. Generally, the commercial vehicle are equipped with the same tires, some multi-axle vehicles use dual tires. The ratio of the central wheels cornering stiffness to rear wheels cornering stiffness is therefore assumed to be

around 1, 0.5 or 2. But the vehicle loads condition/distribution affects the cornering stiffness on each axle. This ratio  is strictly correlated with the ratio of the vertical loads on the wheels when all other conditions are the same (same tires, same inflation pressure, etc.). With comprehensive consideration, plotting  from 0 to 4 in Figure 7 provides enough evidence showing that ratio  has little impacts on deriving equivalent parameter

lreq .

1.7 lreq=(*lc+lr)/(1+ )

1.6

lreq=(*lc 2+lr2)/(*lc+lr)

lreq

1.5 1.4 1.3 1.2 1.1

0

0.5

1

1.5

2

2.5

3

3.5

4



Figure 7. Equivalent

lreq w.r.t changing ratio of C c and C r ( C c  C r )

Since the ratio of C c and C r is fixed for any given vehicle, we study the variation of Ceq as a function of

lreq

when  is chosen to be 1, 0.5 and 2 in Figure 8. The results in Figure 8 demonstrate how the equivalent model is influenced by the changes in cornering stiffness. Results show that when C r [C r min , C r max ] , changes from 60000 to 120000, the equivalent small range of

lreq stayed at the same range and it could always find the optimal value from this

lreq . In addition, the equivalent Ceq reasonably change with respect to the changing of C r , which

makes sense according to the vehicle’s dynamic characteristics. 5

5

2.5

5

x 10

x 10

x 10

C r  120000

C r  120000

3.6

C r  120000

1.6

3.1

C r  90000

3.6

Ceq

3.1

2.6

1.34

1.36 1.38 lreq (=1)

1.4

Cer=(1+ )Cr ler*Cer=(lc*+lr)*Cr ler2*Cer=(lc 2*+lr2)*Cr

2.1

1.22

2.6

2.1

C r  60000 1 1.32

C r  90000

1.2

1.5

1.6 1.2

C r  90000

Ceq

x 10

Ceq

5

Ceq

2

1.24 1.26 lreq (=2)

C r  60000 0.8

1.44

1.46 1.48 lreq (=0.5)

1.5

C r  60000 1.6 1.2 1.22 1.24 1.26 1.28 1.3 lreq (=2)

Figure 8. Equivalent parameters of lreq and Ceq with changing cornering stiffness

1.28

1.3

5.4 Root Locus and Frequency Responses In order to investigate the driving stability during straight line with the possible disturbance such as wind or uneven surface, we consider the steering angle to be equal to zero in Equation (5) and (6). Under the assumption of constant velocity, one arrives at the linear homogenous state space equations x  Ax - original three-axle model

(28)

x  Aeq x - equivalent two-axle model

(29)

The system eigenvalues (poles) are the eigenvalues of matrix A / Aeq , solved by the characteristic equation

det( sI  A)  0

(30)

The linear system modelled with such an equation is known to be asymptotically stable if all the eigenvalues of matrix A / Aeq have negative real parts. We define the matrix A / Aeq eigenvalues in the form of: s1,2  d  id

(31)

where i, d , d are the imaginary unit, damping coefficient and damped natural frequency, respectively. Furthermore, the damping ratio  and un-damped natural frequency  n can be expressed as

 

d d  d2 2

, n  d 2  d2

(32)

It indicates an unstable system if the damping ratio  is negative. The eigenvalues distribution (root locus) and damping characteristics are evaluated for the vehicle parameters of velocity and rear cornering stiffness. We used the other parameters of the vehicle in Section 5.1. Considering the vehicle velocity increases from 20km/h to 120km/h, Figure 9(a) depicts the root locus (system eigenvalues) with respect to a series of speed points. The arrow shows the increasing direction of the speed. The un-damped natural frequency and damping ratio are plotted as functions of speed in Figure 9(b). Similarly, Figure 10 plots root locus and damping characteristics w.r.t the increasing cornering stiffness of rear wheels, and the central wheels cornering stiffness is equal to C r . Results suggest the vehicle is mathematically stable of all derived models. It is worth noting that the proposed method has a preferable coincidence in root locus and damping characteristics as well as in time-domain responses.

10

25

8 6

Natural frequency(rad/s)

Original three-axle model D. E.Williams derivation W–G/J.R. Ellis derivation The proposed method

2

The proposed method

15 10 5 20 1

-2 -4 -6 -8 -10 -70

D. E.Williams derivation W–G/J.R. Ellis derivation

20

0

Damping ratio

Imag Axis

4

Original three-axle model

-60

-50

-40 -30 Real Axis (a) Root locus

-20

-10

40

60

80

100

120

100

120

0.9 0.8 0.7

0

20

40

60 80 Vehicle velocity(km/h)

(b) Natural frequency and damping ratio

Figure 9. Root locus and damping characteristics w.r.t velocity Natural frequency(rad/s)

8 6 4

Imag Axis

2 Original three-axle model D. E.Williams derivation W–G/J.R. Ellis derivation The proposed method

0

Damping ratio

-6

-8

-7.5 -7 Real Axis

(a) Root locus

D. E.Williams derivation W–G/J.R. Ellis derivation The proposed method

0 0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4 5

-4

-8.5

Original three-axle model

5

x 10

1

-2

-8 -9

10

-6.5

-6

-5.5

0.9

0.8

0.7 0.6

0.7

0.8 0.9 1 1.1 1.2 Rear cornering stiffness(N/rad)

1.3

1.4 5

x 10

(b) Natural frequency and damping ratio

Figure 10. Root locus and damping characteristics w.r.t rear wheels cornering stiffness After steady-state response analysis, stability analysis, we now turn to frequency domain analysis and this is necessary to analyze and compare the transient response in vehicle dynamics. Figure 11 shows the frequency responses of yaw rate and sideslip angle at the speed of 60km/h. Again, in frequency-domain, the results indicate that the responses from proposed method almost completely overlap to where are from the original three-axle model. But the other two conventional derivations somehow fail to match it in frequency-domain.

-10

20

Magnitude (dB)

Magnitude (dB)

10 0 -10

Original three-axle model

-20

D. E.Williams derivation W–G/J.R. Ellis derivation

-20

-30

-40

The proposed method -50 45

Phase (deg)

Phase (deg)

-30 0

-45

-90

0

-45

-90 -1

0

10

1

10

2

10

10

-2

-1

10

0

10

Frequency (Hz)

10

1

10

2

10

Frequency (Hz)

(a) Yaw rate

(b) Side slip angle

Figure 11. Frequency responses of yaw rate and sideslip angle

6. General Formulation for Any Multi-axle Vehicle Following the obvious similarities in the development of the two-axle and three-axle equalization, the concept now is extended to account for a multi-axle vehicle of one front axle and any number of rear axles. Assuming that the vehicle has one front axle and rear (n  1) axles, li (i  1, 2, , n  1) is distance from CG to i th rear axle, C i (i  1, 2,

, n  1) is the total wheels cornering stiffness of i th rear axle. For convenience, the lateral and yaw

motion of such a multi-axle vehicle is written by

v lf

v   li ) u u i 1 n 1 v lf v   li I z  l f C f ( f  )   li C i ( ) u u i 1

m(v   u )  C f ( f 

n 1

) + C i (

(33) (34)

Next, we make the total force/moment of the right hand side of Equation (21) and Equation (23) equal to the force/moment items in Equation (33) and Equation (34). With elimination, the following expression is obtained.

Ceq (

v   lreq

lreq Ceq (

u v   lreq u

n 1  C  C i   eq n 1  v   li i 1  )   C i ( )  n 1 u   i 1   lreq Ceq   li C i  n 1 v   li  i 1  )   li C i ( ) n 1   u 2 2 i 1 lreq Ceq   li C i  i 1 

The problem is transfer to find an optimal point of lreq and Ceq expressed by the known parameters Referring to the same procedures in Section 4.2, the optimization problem is formulated as Minimize : f ( x, y )

(35)

li and C i .

where, f ( x, y )  pp1  pp2 2

2

3

 pp3   ( x  xi ) 2  ( y  yi ) 2 2

i 1

n 1

 l C

p1 ( x1 , y1 )  ( in11

i

 C i 1

n 1

i

n 1

,  C i ) ; p2 ( x2 , y2 )  (

 li 2Ci

i 1

i

i 1 n 1

 C i 1

The optimal point of lreq and Ceq is readily found as

p*  (

i

n 1

l

n 1

2 i C i ( li C i ) n 1 , ni 11 ). ,  C i ) p3 ( x3 , y3 )  ( in11 2 i 1  liCi  li Ci i 1

2

i 1

x1  x2  x3 y1  y2  y3 , ). 3 3

Therefore, the general expression of lreq and Ceq for any axle-vehicle with of one front axle and n  1 rear axles is formulated as:

lreq 

x1  x2  x3 y  y2  y3 , Ceq  1 3 3

(36)

Substituting lreq and Ceq into Equation (6), the general formulation of equivalent two-axle model is achieved.

7. Conclusion and Future Work Different equivalent modeling methods that describe the fundamental lateral dynamics of a three-axle vehicle were comparatively studied. Major previous work, including D. E.Williams, Winkler-Gillespie, and J.R. Ellis, were discussed and an approximately equivalent model based on CG force\moment was developed and optimized. Through different simulation cases, the model presented in this paper performed more accurately compared to the other methods, and particularly for slip angle response. This is due to the fact that the proposed model is based on a dynamics equivalence of CG forces and moment. Moreover, the impact of changing cornering stiffness at the rear axle on the equivalent parameters lreq and Ceq was investigated and vehicle stability of eigenvalues and frequencydomain characteristics were discussed and compared. Using concept of the proposed method, it is possible to extend the work to multi-axle vehicles with any arbitrary number of front and rear axles. A general formulation is therefore derived for the multi-axle vehicle with one front axle and any number of rear axles. To point it out, the resulting approximately equivalent model can be expediently used to study the dynamics and stability characteristics of any multi-axle vehicle without formulation for every case. Moreover, this simplification provides a great benefit on model-based controller design just as readily as the common two-axle single-track model.

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Appendix TABLE I Parameters for a generic three-axle vehicle Parameter 𝒎 𝒍𝒇 𝒍𝒓 𝒍𝒄 𝒍 𝑰𝒛 𝑪𝜶𝒇 𝑪𝜶𝒄 𝑪𝜶𝒓

Description Vehicle mass Distance of front axle from CG Distance of rear axle from CG Distance of central axle from CG Distance of front axle from rear axle Yaw moment of inertia Front axle cornering stiffness Central axle cornering stiffness Rear axle cornering stiffness

Unit kg m m m m kg/m2 N/rad N/rad N/rad

Value 2200 1.6 1.65 1 3.25 3000 100000 depends on the case depends on the case

A Comparative Study of Equivalent Modeling for Multi-axle Vehicle Yubiao Zhang, Yanjun Huang*, Hong Wang, Amir Khajepour Department of Mechanical and Mechatronics Engineering, University of Waterloo, ON, N2L3G1, Canada Manuscript ID: NVSD-2017-0129

Dear editors and reviewers, On behalf of all authors, I greatly appreciate your quick reply on my paper. Please find the only-one-point rebuttal to your comments/concerns. It is in blue after the point. Best regards, Response to Reviewer 3 Reviewer #3: In my opinion, some significant corrections must still be made in Eqs. (20-23). Considering Eq. (3), which is now correct, I think that the signs in front of all terms with the stiffness coefficients C c , C r and Ceq should be changed in the equilibrium equations (20-23). Please, check them carefully. Since this is clearly an oversight, I think that there is no need of replying to the reviewer after the correction, unless the authors believe that I am wrong. R: Thanks for your careful comments again. I added negative signs in brackets of the equilibrium equations (20-23). I believe they are correct now.