An interval uncertain optimization method for vehicle ...

Applied Mathematical Modelling xxx (2014) xxx–xxx

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An interval uncertain optimization method for vehicle suspensions using Chebyshev metamodels q Jinglai Wu a, Zhen Luo a, Yunqing Zhang b,⇑, Nong Zhang a a b

School of Mechanical and Mechatronic Engineering, University of Technology, Sydney, NSW 2007, Australia National Engineering Research Center for CAD, Huazhong University of Science & Technology, Wuhan 430074, China

a r t i c l e

i n f o

Article history: Received 9 April 2013 Received in revised form 19 November 2013 Accepted 7 February 2014 Available online xxxx Keywords: Uncertain optimization Suspensions Chebyshev metamodel Interval arithmetic

a b s t r a c t This paper proposes a new design optimization framework for suspension systems considering the kinematic characteristics, such as the camber angle, caster angle, kingpin inclination angle, and toe angle in the presence of uncertainties. The coordinates of rear inner hardpoints of upper control arm and lower control arm of double wishbone suspension are considered as the design variables, as well as the uncertain parameters. In this way, the actual values of the design variables will vary surrounding their nominal values. The variations result in uncertainties that are described as interval variables with lower and upper bounds. The kinematic model of the suspension is developed in software ADAMS. A high-order response surface model using the zeros of Chebyshev polynomials as sampling points is established, termed as Chebyshev metamodel, to approximate the kinematic model. The Chebyshev meta-model is expected to provide higher approximation accuracy. Interval uncertain optimization problems usually involve a nested computationally expensive double-loop optimization process, in which the inner loop optimization is to calculate the bounds of the interval design functions, while the outer loop is to search the optimum for the deterministic optimization problem. To reduce the computational cost, the interval arithmetic is introduced in the inner loop to improve computational efficiency without compromising numerical accuracy. The numerical results show the effectiveness of the proposed design method. Ó 2014 Elsevier Inc. All rights reserved.

1. Introduction The kinematic characteristics of vehicle suspension systems have main effect on vehicles’ handling performance. Since the positions of hardpoints of the suspension system will largely determine its kinematic characteristics, the coordinates of these hardpoints are commonly considered as design variables to optimize the suspension system performance. Due to the complex of suspension system, the models are usually developed using some commercial software, which may reach a comparable level of accuracy in engineering. However, the computation is expensive for complex models in optimization. So the approximation or metamodeling methods are often used in engineering optimization to save the computational cost [1] for real-world problems.

q

This article belongs to the Special Issue: Topical Issues on computational methods, numerical modelling & simulation in Applied Mathematical Modelling.

⇑ Corresponding author. Tel.: +86 27 8754 3973; fax: +86 27 8754 3670. E-mail address: [email protected] (Y. Zhang). http://dx.doi.org/10.1016/j.apm.2014.02.012 0307-904X/Ó 2014 Elsevier Inc. All rights reserved.

Please cite this article in press as: J. Wu et al., An interval uncertain optimization method for vehicle suspensions using Chebyshev metamodels, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.02.012

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J. Wu et al. / Applied Mathematical Modelling xxx (2014) xxx–xxx

Building a metamodel normally involves two steps: (1) employing design of experiments to sample the computer simulation, and (2) selecting an approximation model to represent the data and fit the model with the sample data [2]. Over the past, a number of metamodels have been developed, such as the polynomial regression model [3], Kriging model [2], radial basis functions (RBF) [4–6], and multivariate adaptive regression splines (MARS) [7]. Jin et al. [8] studied four popular metamodeling techniques, including the polynomial regression, Kriging model, MARS, and RBF based on multiple performance criteria. The results showed that the polynomials regression model has advantages such as efficiency, transparency, and conceptual simplicity over the other metamodels. Although the approximation accuracy of traditional quadratic polynomial models is not good enough for strong non-linear problems, it can be improved by using higher order polynomial model. Therefore, the high order polynomial will be used in this paper. As aforementioned, as the first step in the construction of a metamodel, the choice of sampling points is of great importance. The larger the number of sampling points, the higher the possibility to describe the unknown process, but also the higher computational cost [9]. Thus, one of the issues of the sampling methods is how to obtain unknown information in terms of a limited number of sampling points with reasonable computational cost and numerical accuracy. One of the traditional sampling methods is the Design of Experiments (DOEs), which includes a family of methods, such as the Factorial Design (FD) [3], Central Composite Design (CCD) [10], Pseudo-Monte Carlo Sampling (PMCS) [11], and Quasi-Monte Carlo Sampling (QMCS) [11–13]. Amongst these methods, FD and CCD belonging to the classical experimental designs [9] have been widely used to construct polynomials models. The level of FD and CCD are usually selected as uniform grid in the whole design space, which may not be optimal. Thus, we use the zeros of Chebyshev polynomials [14] as the level of FD to implement a new experimental test design, to obtain higher accuracy. So far the majority of works in vehicle dynamics are based on the assumption that all parameters of vehicle systems are deterministic. However, a number of real-world problems are too complex to be defined deterministically due to the lack of the sufficient information. Actually the uncertainties are inherent in loads, parameters, material properties, fraction tolerance, boundary conditions and geometric dimensions [15] in the whole life cycle of design, manufacturing, service and aging. The deterministic assumption may lead to designs which cannot satisfy the expected performance goal or even unfeasible designs. Hence, there is an increasing demand to consider the impact of uncertainties quantitatively in the optimization of vehicle systems due to unavoidable variability and uncertainty, in order to enhance vehicle performance and safety. The Reliable-Based Design Optimization (RBDO) and the Robust Design Optimization (RDO) represent two major paradigms for the design optimization under uncertainty [16]. RBDO methods are characterized by the use of analytical techniques to find a particular point in the design space, which is related to the probability of the system failure, defined by a limit state function. This point is often referred to the most probable point (MPP) or the design point [17]. RDO that can improve the quality of a product, by minimizing the effect of variations without eliminating the causes, is another paradigm for designs under uncertainties. Du et al. [16] propose an integrated framework for design optimization under uncertainty that takes both the robustness of the objective and the probability of the constraints into account, so the RDO and RBDO can be achieved simultaneously. Some studies of suspension systems based on RBDO or RDO have been presented in references. The work of [18] proposed a RBDO method considering the design variables as random variables, and the reliability of the suspension performance was quantified by the kinematic and compliance characteristics. The [19] studied a robust design methodology, in which a multi-objective evolutionary algorithm (MOEA) was applied to a passive suspension system of a linear quarter car. Robustness indexes have been analytically derived using the first-order Taylor approximation, thus allowing the robustness of each objective function to be integrated into the design process. Kim et al. [20] presented a robust design optimization for suspension systems, taking into account the kinematic behaviors influenced by bush compliance uncertainty. The variances of design goals are obtained through sampling the RBF metamodels, and a sequential approximation optimization technique is used to solve a robust design problem for the suspension system. Besides the RBDO and RDO, some other uncertain methods are also proposed for the analysis of vehicle dynamics, such as the polynomial chaos method [21], Monte Carlo method [22,23] and so on. All the previous RBDO and RDO methods require the accurate probability information for the uncertain variables. However, it is generally expensive and time-consuming, and sometimes even impossible to get sufficient information to determine exact probability distribution functions, due to the complexity of engineering problems. Furthermore, Beb-Haim and Elishakoff [24] have shown that even small variations deviating from the real distributions may cause relatively large errors to the probability in the feasible region of the design space, and then may result in unreliable results of the optimization. As a result, the probabilistic methods for engineering problems with uncertainties may experience difficulty due to the absence of complete information. Recently, non-probabilistic methods [25,26] have provided alternative and useful supplements to the probability methods. In particular, the interval method [22,23] has experiencing popularity, because it makes it possible to effectively express the uncertainties for uncertain-but-bounded parameters. Interval uncertain methods only require the lower and upper bounds of the uncertainty parameters, without necessarily knowing the precise distribution function. The determination of lower and upper bounds for an uncertain variable will be much easier than the identification of a precise probability distribution. For vehicle suspensions, the positions of hardpoints, spring stiffness and damping rate may vary around their nominal values due to production tolerances and wear, ageing, etc. [23]. The bounds of hardpoints positions can be extracted easier than the statistical information.



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Several interval-based methods have been proposed. For instance, Luo and his co-workers [27] proposed a mathematical definition to describe the non-probabilistic reliability index using the multi-ellipsoid convex model for structure optimization problems. The uncertain optimization results in a nested double-loop procedure, to which the computational cost is prohibitive. To reduce computational cost of the nested optimization, Kang et al. [28] studied the RBDO of non-deterministic structures using both the stochastic and uncertain-but-bounded variations. The inner loop was eliminated by using the linearization-based approach, leading to a more efficient optimization than the original double-loop optimization. However, the linearization requires that the uncertain-but-bounded variations are relatively small or moderate, because the linear model has a lower numerical accuracy. This paper aims to propose a new design optimization method for improving the kinematic characteristics of vehicle suspension systems, which includes a Chebyshev metamodel for the implementation of the approximation, and the interval arithmetic is applied to eliminate the inner loop optimization. The proposed method is expected to improve the computational efficiency without compromising the numerical accuracy of the nested double loop optimization.

2. Analysis of double wishbone suspensions 2.1. The kinematic analysis of double wishbone suspensions The characteristics of wishbone suspensions can be mathematically divided into two groups, namely, kinematic performance and dynamic performance [20]. The kinematic performance of suspensions has large influence on the vehicles’ handling, while the dynamic performance influences the ride comfort. Since the positions of hardpoints obviously influence the kinematic performance rather than dynamic performance, only the kinematic characteristics of suspensions will be optimized in this paper. Compared to the Macpherson suspension, the double wishbone suspension has better kinematic performance but relatively a more complicate structure, making the optimization of the mechanical structure more difficult. Fig. 1 shows the model of a double wishbone suspension, mainly including tie rod, knuckle, absorber, upper control arm, and lower control arm. In this study, the kinematic analysis model is developed using the commercial software ADAMS/Car. The simulation condition for the wheel stroke is a bump of 50 mm and a rebound of 50 mm. With the suspension model, the kinematic characteristics, described with the camber angle, caster angle, kingpin inclination angle, and toe angle, are shown as Fig. 2. The camber angle is the angle between the vertical axis of the wheels used for steering and the vehicle z-axis (Fig. 2(a)). If the top of the wheel is farther out than the bottom, the camber angle would be positive, otherwise it will be negative. Caster

Fig. 1. The model of double wishbone suspension.


4


z

z caster angle

camber angle

kingpin z inclination angle

toe angle front

kingpin axis kingpin axis

(a)

(b)

(c)

(d)

Fig. 2. camber angle, caster angle, kingpin inclination angle, and toe angle.

is the angle to which the kingpin axis is tilted forward or rearward from vertical, as viewed from the side (Fig. 2(b)). If the kingpin axis is tilted backward (that is, the top pivot is positioned farther rearward than the bottom pivot), then the caster is positive; if it is tilted forward, then the caster is negative. Kingpin inclination angle is measured in degrees from the center line kingpin to vertical, as viewed from the front or rear (Fig. 2(c)). The angle of the two front wheels or rear wheels relative to each other and the car as seen from above is the toe angle (Fig. 2(d)). These angles are changed with the wheel stroke, and they can be obtained through the numerical simulation. The number of wheel simulation step is set to 50 in the ADAMS model, so the output data is provided at 51 symmetrical points in the wheel travel from 50 mm to 50 mm. The plots of these characteristic indexes with respect to the wheel travel are shown from Figs. 3 to 6, respectively. It is noted that appropriate constraints to these indexes are required in order to improve the handling performance, which will be described in the next subsection. 2.2. The optimization model of suspensions The axis direction of inner revolution joint of upper control arm and lower control arm (Fig. 1) influences the kinematic performance of the suspension, so the coordinates of rear inner hardpoint of upper control arm (point A) and rear inner hardpoint of lower control arm (point B) are chosen as design variables. As the x-directional coordinates of the two hardpoints have little influence on the axis direction of revolution joint, only the y- and z-directional coordinates are considered as design variables. The relevant design vector is denoted as u ¼ ½u1 u2 u3 u4 T ¼ ½yA zA yB zB T . The initial values of the design variables are set as u ¼ ½490 560 450 185T mm, and the range of design variables are set as ±60 mm in terms of the initial values. As aforementioned, the design variables contain uncertainties, which make the actual values to fluctuate around their nominal values. It is assumed that the uncertain range of each design variable is ±3 mm from its nominal value, and then the design vector can be expressed as interval vector as:

½u ¼ u þ ½Du; ½Du ¼ ½½3; 3 ½3; 3 ½3; 3 ½3; 3T :

ð1Þ

If the design variables are considered as interval variables, the response of kinematic performance will not be a curve but an interval belt, as shown from Figs. 3 to 6. To improve the kinematic performance, the maximum variation in the transient responses over the wheel travel is required to be minimized. In details, the maximum variation of camber angle, caster angle, and kingpin inclination angle is required to be no more than 2°, 1°, and 3° over the wheel travel, respectively, and the toe angle is no less than 0.5° when the wheel travel is set to 50 mm. To minimize the variation of camber angle and kingpin inclination angle, the uncertain optimization problem is defined as follows:

Fig. 3. The camber angle.



5

Fig. 4. The caster angle.

Fig. 5. The kingpin inclination angle.

Fig. 6. The toe angle.

8 Minimize : > > > > > > > > > > >
c ðuÞ ¼ max u2½u;t f 1 ðu; tÞ min u2½u;t f 1 ðu; tÞ 6 2 ; > > 1 > > > c ðuÞ ¼ max u2½u;t f 2 ðu; tÞ min u2½u;t f 2 ðu; tÞ 6 1 ; > > 2 > < c ðuÞ ¼ max 3 3 3 u2½u;t f ðu; tÞ min u2½u;t f ðu; tÞ 6 3 ; > > subject to : > 4 > c4 ðuÞ ¼ min u2½u f ðu; tÞ t¼50 P 0:5 ; > > > > > > > > > > lb 6 ½u ¼ u þ ½Du 6 ub; 50 6 t 6 50; > > > > > > : : lb ¼ ½430 500 390 125T ; ub ¼ ½550 620 510 245T ;

ð2Þ

where u ¼ ½u1 u2 u3 u4 T is the vector including design variables. t denotes the wheel travel from 50 to 50 mm, lb and ub are the lower and upper bounds of the design vector. f 1 ðu; tÞ; f 2 ðu; tÞ; f 3 ðu; tÞ; and f 4 ðu; tÞ are the camber angle, caster angle, kingpin inclination angle, and toe angle, respectively, which are the functions with respect to the design variables and wheel travel. c1(u), c2(u), c3(u), and c4(u) represent the variation of camber angle, caster angle, kingpin inclination angle, and Please cite this article in press as: J. Wu et al., An interval uncertain optimization method for vehicle suspensions using Chebyshev metamodels, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.02.012

6


the minimum value of toe angle, respectively, when the wheel stroke equals to 50 mm, shown from Figs. 3 to 6 a1 and a3 and are the weighting coefficients which are both set to 0.5 in this paper, indicating two equally weighted individual objectives in the design. Since the suspension model is built in software, it is difficult to get the explicit expressions for camber, caster, kingpin inclination and toe, and the simulation of the complex model is time-consuming. 3. Chebyshev metamodel of double wishbone suspensions In this Section, the approximation using Chebyshev polynomials theory is presented firstly, and then a new Chebyshev metamodel is proposed, using the zeros of Chebyshev polynomials as the sampling points. Two test functions will be sued to showcase the higher accuracy of the proposed sampling points than traditional sampling method. Lastly, the Chebyshev metamodel for the double wishbone suspension is constructed using the proposed method. 3.1. Chebyshev polynomials approximation theory If a function f(x) belongs to C[a, b], which means f(x) is continuous in [a, b], there will be a polynomials p(x) which converges to the function f(x) on [a, b], that is

kf ðxÞ pðxÞk1 < e;

x 2 ½a; b:

ð3Þ

This expression is validity for any e > 0, and the theorem was proposed by Weierstrass [29]. Let Pn(x) denote the set of polynomials of degree not bigger than n. For every nonnegative integer n there exists a unique polynomial pn ðxÞ in Pn(x), such that

kf ðxÞ pðxÞk1 P kf ðxÞ pn ðxÞk1 ¼ En ðf Þ;

x 2 ½a; b;

ð4Þ

where pðxÞ 2 Pn ðxÞ. pn ðxÞ can be regarded as the best uniform difficult to obtain pn ðxÞ when the degree of polynomials n > 2.

approximation of degree n to f(x) in [a, b]. However, it is The truncated Chebyshev series are very close to the best uniform approximation polynomials [14], so we employ them to approximate the original function. The function f(x) can be approximated by the truncated Chebyshev series of degree n as follows:

f ðxÞ pn ðxÞ ¼

n X 1 f0 þ fi C i ðxÞ; 2 i¼1

ð5Þ

where fi are the constant coefficients, and Ci(x) denote the Chebyshev polynomials

C i ðxÞ ¼ cos ih;

ð6Þ

where h ¼ arccosðð2x ðb þ aÞÞ=ðb aÞÞ 2 ½0; p. To simply the notes, we define x 2 ½1; 1 and other domain can be transformed to this domain via the linear transformation. The Chebyshev polynomials can also be denoted as x 2 ½1; 1, where the recurrence formula is expressed as

C 0 ðxÞ ¼ 1; C 1 ðxÞ ¼ x;

ð7Þ

C n ðxÞ ¼ 2xC n1 ðxÞ C n2 ðxÞ;

n P 2:

Based on the orthogonality of Chebyshev polynomial, the constant coefficients fi and fj1 ;...;jk can be calculated through the following formula [30]

fi ¼

2

Z

p

1 1

f ðxÞC i ðxÞ 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx ¼ p 1 x2

Z p

f ðcos hÞ cos ihdh;

i ¼ 0; 1; 2; . . . ; n:

ð8Þ

0

The integrals for Eq. (8) can be calculated using numerical integral methods, such as the Mehler integrals [30] (or called Gauss–Chebyshev quadrature [31]) that may produce high accuracy, so it will be used in this paper, and then Eq. (8) can be transformed to

fi ¼

2

p

Z

1 1

m m f ðxÞC i ðxÞ 2 pX 2X pffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx f ðxj ÞC i ðxj Þ ¼ f ðcos hj Þ cos ihj ; m j¼1 p m j¼1 1 x2

ð9Þ

where m denotes the order of the integral formula, and xj, which denotes the interpolation points in the integral formula, are the zeros of the Chebyshev polynomials of degree m, shown as follows

xj ¼ cos hj ;

hj ¼

2j 1 p ; m 2

j ¼ 1; 2; . . . ; m:

ð10Þ

For multi-dimensional problems, the Chebyshev polynomials are the tensor product of each one-dimension polynomials. A k-dimensional Chebyshev polynomials of x 2 ½1; 1k is given by


7


C n1 ;...;nk ðx1 ; . . . ; xk Þ ¼ C n1 ðx1 Þ . . . C nk ðxk Þ ¼ cos n1 h1 . . . cos nk hk :

ð11Þ

The function f(x) can also be approximated as truncated Chebyshev series of degree n as

f ðxÞ

n X

...

j1 ¼0

n p X 1

2

jk ¼0

fj1 ;...;jk C j1 ;...;jk ðxÞ;

ð12Þ

where p expresses the number of subscript equals to zero, such as jl = 0 (l = 1,. . . , k), and fj1 ;...;jk is the coefficient vector with the length of (n + 1)k, calculated by the following equation

fi1 ;...;ik ¼

k Z p 2

p

Z p ...

0

ðf ðcos h1 ; . . . ; cos hk Þ cos i1 h1 . . . cos ik hk Þdh1 . . . dhk :

ð13Þ

0

Use the Mehler integrals for multi-dimensional integrals, we have

fi1 ;...;ik

k X m m X 2 ... f ðcos hj1 ; . . . ; cos hjk Þ cos i1 hj1 . . . cos ik hjk : m j ¼1 j ¼1 1

ð14Þ

k

For the k-dimensional problem, the interpolation points will be the tensor product of the interpolation points in each dimension, as follows:

X ¼ X1 . . . Xk ;

ð15Þ k

where Xi is the interpolation points of each dimension; the number of interpolation points is s = m . Thus, the construction process of Chebyshev approximation polynomials involves two steps: (1) to calculate the values of the evaluated function at the interpolation points; (2) to use Eq. (14) to calculate the coefficients and then develop the approximation polynomial using Eq. (12). To make a good trade-off between the accuracy and efficiency, the following relationship m = n + 1 is usually adopted in Eqs. (12) and (14), in this case the number of coefficients equals to the number of interpolation points. More information about the Chebyshev polynomial approximation theory can be found in the reference [30], which has shown the Eq. (12) has higher accuracy than truncated Taylor series expansion polynomial. 3.2. Chebyshev metamodel As described in last section, the process of constructing Chebyshev approximation polynomial is the same as that in constructing a metamodel. If we use the interpolation points in Eq. (15) as the sampling points, and the least squares method to calculate the coefficients, this process would be the traditional RSM [3]. Transform Eq. (9) to the following expression

f ðxÞ

n X

...

j1 ¼0

n p X 1 jk ¼0

2

fj1 ;...;jk C j1 ;...;jk ðxÞ ¼ bT a ¼: ^f ðxÞ;

ð16Þ

where the coefficient vector

b ¼ ½ b1

T

bs ¼ ½ ð1=2Þk f0;...;0

ð1=2Þp fj1 ;...;jk

T

ð16 aÞ

fn;...;n ;

and the polynomial basis vector

a ¼ ½ a1

T as T ¼ ½ C 0;...;0 ðxÞ C j1 ;...;jk ðxÞ C n;...;n ðxÞ ; and

s ¼ ðn þ 1Þk :

ð16 bÞ

The coefficient vector is calculated through the least squares method, as follows: 1

T

b ¼ ðAT AÞ AT Y; where Y ¼ ½ f ðx1 Þ f ðxs Þ ;

ð17Þ

where Y is the vector consisting of function values evaluated at the sampling points that are the interpolation points, determined by Eq. (15), and the matrix A 2 Rs s is composed by the values of polynomial basis vector at the sampling points by

3 a1 ðx1 Þ as ðx1 Þ 6 . .. .. 7 7 A¼6 . . 5; 4 .. 2

a1 ðxs Þ

ð18Þ

as ðxs Þ

where x1. . .xs denote the sampling points. The expression of Eq. (16) is respect to the Chebyshev series, and it can also be expressed with respect to the Taylor series

" ~f ðxÞ ¼ cT b; where b ¼ ½ b 1

T

bs ¼

k Y i¼1

x0i

k Y

#T xni

ð19Þ

;

i¼1

where the polynomial basis vector, and the coefficient vector c ¼ ½ c1

cs T can calculated by


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J. Wu et al. / Applied Mathematical Modelling xxx (2014) xxx–xxx 1

c ¼ ðBT BÞ BT Y;

ð20Þ s s

where the matrix B 2 R

2 6 B¼6 4

is determined by

3 b1 ðx1 Þ bs ðx1 Þ .. .. .. 7 7 . . . 5: b1 ðxs Þ

ð21Þ

bs ðxs Þ

If the same sampling points x1. . .xs are used in Eqs. (18) and (21), Eqs. (16) and (19) will provide the same approximation model, which can be proved as follows.From Eqs. (6) and (7), we know that the polynomial basis vector a can be obtained through a linear transformation from vector b, such as

b ¼ Da;

ð22Þ

s s

where D 2 R is the linear transformation matrix between the two vectors a and b. Substituting Eq. (22) to Eqs. (18) and (21), we can obtain

B ¼ ADT :

ð23Þ

Substitute x = x1,. . . , xs into the two basis vectors a and b in the above equation, sequentially, then we can obtain the matrix form of the transformation, as follows:

2 6 B¼6 4

3 2 32 a1 ðx1 Þ as ðx1 Þ b1 ðx1 Þ bs ðx1 Þ D11 7 6 6 . .. .. 7 .. .. .. 7 ¼ 6 .. 76 . . 54 .. . . . 5 4 . D1s a1 ðxs Þ as ðxs Þ b1 ðxs Þ bs ðxs Þ

3 Ds1 .. .. 7 T 7 . . 5 ¼ AD : Dss

ð23Þ

Substituting Eq. (17) into Eq. (16), we have T

^f ðxÞ ¼ bT a ¼ ððAT AÞ1 AT YÞ a ¼ YT AðAT AÞ1 a:

ð24Þ

Substituting Eq. (20) into Eq. (19) results in the following equation T

1

~f ðxÞ ¼ cT b ¼ ððBT BÞ1 BT YÞ b ¼ YT BðBT BÞ1 b ¼ YT ACT ððACT ÞT ACT Þ Ca ¼ YT ACT ðCT Þ1 ðAT AÞ1 C1 Ca 1 ¼ YT AðAT AÞ a ¼ ^f ðxÞ:

ð25Þ

Therefore, the two approximation models shown in Eqs. (16) and (19) are equivalent, which means the sampling points determine the polynomial model rather than the basis vector for the polynomial model. Since the zeros of Chebyshev polynomials are used as the sampling points, we called this polynomial approximation model as Chebyshev metamodel. It should be noted that the Chebyshev metamodel may be expressed in different types in line with the basis vector, and here we choose the Taylor series-based expression shown in Eq. (19) due to its simple format. The only difference between the Chebyshev metamodel and the standard RSM is the sampling points, because the experiment levels are uniform distributed in the design space in traditional RSM, while the zeros of Chebyshev polynomials are used as the sampling points for the proposed method. For example, the sampling points of uniformly distributed three levels in the design space [1, 1] will be [1, 0, 1], but the three sampling points of the Chebyshev metamodel will be pffiffiffi pffiffiffi ½ cosðp=6Þ cosðp=2Þ cosð5p=6Þ ¼ ½ 3=2 0 3=2 . The proposed sampling method may obtain higher approximation accuracy, especially for high order polynomial approximations. Two test functions, namely, the Multi function [32] and Schaffer’s function [33], are employed to validate the accuracy of the Chebyshev metamodel. We introduce two indexes, the maximum error and mean error, to describe the accuracy of the approximation model, defined as follows:

emax ¼

~jÞ ~jÞ maxðjy y meanðjy y ; and emean ¼ ; maxðjyjÞ maxðjyjÞ

ð26Þ

~ represent the values of the original funcwhere emax and emean are the maximum error and mean error, respectively. y and y tion and values of the regression polynomial in Eq. (19). The test points in Eq. (26) can be selected as grid points with 40 equidistant nodes for each dimension. Multi function: g ¼ x sinð4pxÞ y sinð4py þ pÞ þ 1, where x; y 2 14 ; 14 . The plot of Multi function is shown in Fig. 7. The order of approximation polynomials is changed from 1 to 6, using the zeros of Chebyshev polynomials and uniform grid as the sampling points, respectively. Fig. 8 shows the maximum error and mean error changed with the order of the approximation polynomial. pffiffiffiffiffiffiffiffiffi sin2 ð x2 þy2 Þ0:5 Schaffer’s function: g ¼ 0:5 þ , where x; y 2 ½1; 1. 2 2 2 ð1þ0:001ðx þy ÞÞ

The plot of Schaffer’s function is shown in Fig. 9. Fig. 10 shows the maximum error and mean error changed with the order of the approximation polynomial. Please cite this article in press as: J. Wu et al., An interval uncertain optimization method for vehicle suspensions using Chebyshev metamodels, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.02.012


9

Fig. 7. Multi function.

Fig. 8. Error of approximation polynomials for Multi function.

Fig. 9. Schaffer’s function.

Figs. 8 and 10 show that both the maximum error and mean error of Chebyshev zeros based sampling are smaller than the uniform grid sampling for each order of approximation polynomials. It is noted that the error shown in Fig. 8 and 10 is a logarithm error, which has been shown in the legend of the figure (log(emax), log(emean)), so the smaller value of y-axis denotes the lower error.


10


Fig. 10. Error of approximation polynomials for Schaffer’s function.

3.3. Chebyshev metamodel for double wishbone suspensions Here the Chebysheve metamodel for the double wishbone suspensions including the camber angle, caster angle, kingpin inclination angle, and toe angle of are established through the above theory. From Section 2, it can be seen that these kinematic performance indexes are the function of both design variables u and wheel travel t. To simplify the description, regularizing the design variables and wheel travels to [1, 1] through a linear transformation, denoted by n and T

n¼

þ uÞ 2u ðu 2t ðt þ tÞ and T ¼ ; t t u u

ð27Þ

where t ¼ 50 and t ¼ 50 are the bump and rebound of the wheel travel. We use the third order polynomial with respect to u to construct the metamodel, so the sampling points of each dimensional design variable will be Xi ¼ cosð½ p=8 3p=8 5p=8 7p=8 Þ. Since there are 4 design variables, the total number of sampling points would be 44 = 256. Running the kinematic model of suspensions at all the sampling points, 256 groups of data about the indexes with respect to the wheel travel (51 symmetrical discrete wheel travel nodes) will be obtained. Here 256 sample points may take some time in the numerical implementation. However, after sampling, a Chebyshev metamodel is built to replace the original complicated model in a more efficient manner, while ensure the approximation accuracy of the surrogate model. First, the Chebyshev metamodel with respect to the design variables n at each discrete wheel travel node are constructed. We use the camber angle as the example to illustrate the process of constructing the metamodel. Using Eqs. (19) and (20) at each wheel travel node, the Chebyshev metamodel of camber angle for each discrete wheel travel node can be built as

f 1 ðuðnÞ; T l Þ ¼

3 X i1 ¼0

...

3 X fi11 ;i2 ;i3 ;i4 ðT l Þni11 ni22 ni33 ni44 ;

ð28Þ

i4 ¼0

2 where Tl is the given lth wheel travel node, where T l ¼ 1 þ 50 l (l ¼ 0; 1; . . . ; 50), and so fi11 ;i2 ;i3 ;i4 ðT l Þ corresponds to 51 coefficients. In the above Chebyshev metamodel for each node, the coefficients are 51 discrete values in the dimension of wheel travels, so we can use these discrete values as a set of sampling points to build another metamodel of the coefficients with respect to the wheel travel. Since the range of the wheel travel is relatively large, higher order polynomial will be employed to construct the metamodel. In this paper, we use the 4th order polynomials to approximate the coefficients, leading to another metamodel, given as follows:

fi11 ;i2 ;i3 ;i4 ðTÞ ¼

4 X fi11 ;i2 ;i3 ;i4 ;q T q ;

ð29Þ

q¼0

where q is the power of the wheel travel T, and fi11 ;i2 ;i3 ;i4 ;q are the coefficients calculated using the least-square method, according to the sampling coefficients. Introducing the wheel travel as variable in Eq. (28), and substituting Eq. (29) into Eq. (28), the completion for the metamodel of the camber angle is finally expressed as a continuous function: Please cite this article in press as: J. Wu et al., An interval uncertain optimization method for vehicle suspensions using Chebyshev metamodels, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.02.012


f 1 ðu; TÞ ¼

3 X i1 ¼0

...

3 X 4 X ðfi11 ;i2 ;i3 ;i4 ;q T q Þni11 ni22 ni33 ni44 :

11

ð30Þ

i4 ¼0 q¼0

As described previously, the development of the final metamodel for the camber angle actually contains two sub metamodeling processes: a metamodel for the camber angle at each wheel travel node, and then another metamodel for the coefficients in the expression of the camber angle with respect to the wheel travel. The two processes are decoupled, which may reduce the possibility of ill conditioned in the numerical calculation. The flow chart of constructing the metamodel is shown in Fig. 11, which mainly contains 4 steps: (1) compute 256 sampling points, which are the tensor product of each dimensional sampling points, namely, the zeros of 4th order Chebyshev polynomial; (2) run the kinematic model of double wishbone suspensions at each sampling points and output the indexes with respect to the wheel travel; (3) construct the metamodel of the camber angle at each wheel travel node; (4) build the metamodel of the coefficients of the camber angle with respect to the wheel travel and then obtain the final expression of camber angle metamodel. Similarly, the metamodels for caster angle, kingpin inclination angle, and toe angle can be developed via the same process.

4. Optimization for vehicle suspensions under uncertainties 4.1. Double-loop optimization strategy From Eq. (2), we can find that the optimization model includes a nested double loop process, which leads to time-consuming optimization. The outer loop calls the objective and constraints, which are evaluated by inner optimization iteratively. Since the performance index of suspension kinematic characteristics is a function of the maximum and minimum values over the intervals, including the variation of camber angle, caster angle, kingpin inclination angle, and toe angle, which is difficult to apply many well-developed gradient-based optimization algorithms to the problem [20]. On the other

Input

n = 3, m = 4, k = 4, [ u, u ]

k

Produce the sampling points

( 2 j − 1) π ; u θj = j

=

uq + uq

2m 2 j = 1, 2,..., m; q = 1, 2,..., k q

+

uq − uq 2

cos θ j ;

Calculate the camber angle through kinematic model at the sampling points

f 1 ( u s ,Tl ) ; s =1,...,256; l = 0,1,...,50 Construct the meta-model with respect to design variables through Eq. (28) 3

3

i1 = 0

i4 = 0

f 1 (u,Tl )= ∑ ...∑ f i1,i ,i ,i (Tl ) ξ1i1ξ 2i2 ξ3i3 ξ 4i4 1 2 3 4

Construct the complete meta-model through Eqs. (29) and (30) 4

fi11,i2 ,i3 ,i4 (T )= ∑ fi11,i2 ,i3 ,i4 ,qT q q =0

3

3

4

f 1 (u,T )= ∑ ...∑∑ f i1,i ,i ,i ,q ξ1i1ξ 2i2 ξ3i3 ξ 4i4 T q i1 = 0

i4 = 0 q =0

1 2 3 4

Fig. 11. The flow chart for constructing the metamodel.


12


hand, we seek the global optimal solution and avoid multiple local minima, so some heuristic techniques with strong global searching ability will be used. This study employs the Multi-Island Genetic Algorithm (MIGA) [34,35] to solve the outer loop optimization problem. The MIGA is similar to the conventional GA, consisting of two processes: the first is the selection of individuals for the production of the next generation, and the second is the manipulation of the selected individuals to form the next generation by crossover and mutation operations [36]. However, in MIGA, the population is divided into several sub-populations and the migration operation is added. Each sub-population evolves independently to optimize the same objective function. The migration occurs every M generation, and copies of the individuals which are the best N% of the island populations are allowed to migrate. The MIGA may make the solution converge to the global optimum faster than conventional genetic algorithms. The outer optimization is implemented by optimization software tool iSIGHT. The constraints c1, c2, c3 and c4 in Eq. (2) contain the maximization and minimization operations which have to be calculated through the inner optimization process. Based on Eq. (2), the inner loop contains seven parallel optimization problems, which are used to maximize and minimize the camber angle, caster angle, and kingpin inclination angle over the wheel travel, respectively, as well as to minimize the toe angle corresponding to the wheel travel of 50 mm. The inner optimization model is described as follows:

Find maximize subject to

T

u ¼ ½ u1 u2 u3 u4 ; t; f i ðu; tÞ; i ¼ 1; 2; 3; ~ 36u6u ~ þ 3; u

ð31aÞ

50 6 t 6 50; u2

u3

u4 T ; t;

Find

u ¼ ½ u1

minimize subject to

f i ðu; tÞ; i ¼ 1; 2; 3; ~ 36u6u ~ þ 3; u

ð31bÞ

50 6 t 6 50; u ¼ ½ u1

minimize

f 4 ðu; 50Þ; ~ 36u6u ~ þ 3; u

subject to

u2

u3

u4 T ;

Find

ð31cÞ

~ denotes the vector including the nominal values of design variables produced by the outer loop through each cycle. where u Since the explicit expression of objectives in Eq. (31) has been obtained in Section 3, many optimization algorithms based on gradient information can be employed. In this paper, the inner loop optimizations are completed by the optimization package, which actually includes the SQP [37] in MATLAB software. The flow chart of the nested double-loop approach is shown in Fig. 12. In each iteration of outer loop, the design point is produced and is transferred to the inner loop as the initial design point. Then the inner loop including seven optimization processes is implemented to calculate the maximum and minimum values of camber angle, caster angle, kingpin inclination angle, and toe angle. When the inner optimizations finished, the inner objectives are transferred to outer loop to evaluate the objective and constraints in the outer loop. Repeating the previous process until the outer optimization accomplished, the final optimization results can then be obtained. 4.2. Interval arithmetic based optimization strategy It is computationally expensive to perform the double-loop optimization, as the inner optimizations have to be completed in each cycle of the outer optimization. In this part, the interval arithmetic is introduced to replace the inner loop optimization, which can greatly reduce the computational cost. The interval arithmetic can be used to estimate the range of a function. However, interval arithmetic usually leads to a larger numerical overestimation. As a result, another issue in this section is how to reduce the overestimation when the interval arithmetic is employed. An uncertain real number can be considered as a real interval number containing all the possible values of the uncertain real. The interval number can be defined as

½x ¼ ½x; x ¼ fx 2 R : x 6 x 6 xg;

ð32Þ

where x and x denotes the lower bound and upper bound of interval number [x], respectively. Interval arithmetic operations are defined on the real set R, so that the interval result closes to all possible real results. The width of interval [x] is defined as

wð½xÞ ¼ x x:

ð33Þ

With the two real intervals [x] and [y], the four arithmetic operations are defined as

½x ½y ¼ fx y : x 2 ½x; y 2 ½yg for 2 fþ; ; ; g:

ð34Þ


13


Starting point u0, k=0

Starting point

Calculation for updating design variables uk based on Eq. (2) MIGA k=k+1 No

k

max f 1 ( u k , t ) min f 1 ( u k , t )

Converge ?

max f 2 ( u k , t )

(uk , t ) max f ( u k , t ) min f 3 ( u k , t ) min f 4 ( u k ,50 ) min f

Yes End Outer loop

2

0

, l=0

Calculation for updating design variables ul based on Eq. (31) SQP l=l+1 No

3

Converge ? Yes End Inner loop

Fig. 12. The flow chart of the double-loop approach.

It is easy to use interval arithmetic operations to get the range of solution, but interval arithmetic will lead to a larger overestimation, if no particular scheme is introduced to control this problem. Considering a function f from Rn to Rm, the interval function [f] from IRn to IRm will be an inclusion function for f if

8½x 2 IRn ; f ð½xÞ ½f ð½xÞ:

ð35Þ

For a large class of functions f, one purpose of interval analysis is to provide inclusion functions [f], which can be evaluated to ensure that the result is not too large in a reasonable and quick manner [38]. To make the result sharper, the high-order Taylor series expansion can be used. If the function f is (n + 1)-order partially differentiable on opening set containing the interval [x], we can obtain the nth order Taylor inclusion function [38] as follows:

½fT n ð½xÞ ¼ f ðxc Þ þ f 0 ðxc Þ½Dx þ . . . þ

1 ðnÞ 1 f ðxc Þ½Dxn þ ½f ðnþ1Þ ð½xÞ½Dxnþ1 ; n! ðn þ 1Þ!

ð36Þ

where xc is the midpoint of [x], and ½Dx ¼ ½x xc is the interval of [x]. Using Eq. (36) to calculate the function will ensure a sharper interval than the direct evaluation of the interval function. If the function changes obviously in the range of an interval variable, the Taylor inclusion function will still produce large overestimation. In this research, the metamodel has larger variation in the dimension of the wheel travel, because the wheel travel interval is relatively large. So the interval bisection method is used to overcome this difficulty. An interval [x] can be divided into two subintervals which are ½xL ¼ ½x; xc and ½xR ¼ ½xc ; x, called left interval and right interval. The inclusion function [f]([x]) is calculated as the union of inclusion function [f]([xL]) and [f]([xR])

½fT n ð½xÞ ¼ ½fT n ð½xL Þ [ ½fT n ð½xR Þ:

ð37Þ

The interval bisection can be terminated when the range of inclusion function changes very small. The details algorithm of the interval bisection method is described in Fig. 13 where the e in Fig. 13 denotes the error limit, which can be set as a small positive number. To show the capacity of the present algorithm in the control of overestimation, we consider a random input u = [520530400150]T to calculate f1, using the conventional interval arithmetic, the proposed interval algorithm, and the optimization algorithm, respectively. The results are given as:

½f1 int ¼ ½2:7713; 1:508; ½f1 tb ¼ ½1:1055; 0:3451;

ð38Þ

½f1 opt ¼ ½1:1054; 0:3466; where ½f1 int and ½f1 tb denote the results obtained by using the conventional interval arithmetic, and the proposed algorithm combing the Taylor inclusion function and interval bisection algorithm, respectively, and ½f1 opt is the optimization result. We can find that the conventional interval arithmetic produces larger overestimation, while the proposed interval algorithm is close to the optimization result, so in this sense, we can conclude that the proposed internal algorithm can control the overestimation. Please cite this article in press as: J. Wu et al., An interval uncertain optimization method for vehicle suspensions using Chebyshev metamodels, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.02.012

14


Input

f ,[x ], e

[F0 ] = ⎡⎣ f T ⎤⎦ ([x ]) n

[x L ] = [x, xc ], [x R ]= [xc , x ] Τ [x] = ⎡⎣[x L ] [x R ]⎤⎦ [F ] = ⎡ f ⎤ ([x ]) ⎡ f ⎤ ([x ]) 1 ⎣ T⎦ L ⎣ T⎦ R [F0 ] = [F1 ] e= F − F , F − F n

1

No

n

0

1

0 ∞

e≤ε Yes Output

[F1 ]

Fig. 13. Algorithm of interval bisection.

Considering the uncertain variables ½ u1 u2 u3 u4 as interval variables, the constraints and objective of the optimization model (Eq. (2)) can be calculated through the interval arithmetic, and the optimization model can be transformed to

8 minimize > > > > > > > > > > >
> > > > > c2 ðuÞ ¼ wðf 2 ð½u; ½tÞÞ 6 1 > > > < c ðuÞ ¼ wðf 3 ð½u; ½tÞÞ 6 3 : 3 > subject to : > 4 > > c4 ðuÞ¼ ð½u; 50Þ P 0:5 > > > > > > > > > > lb 6 ½u ¼ u þ ½Du 6 ub; ½t ¼ ½50; 50 > > > > > > : : lb ¼ ½ 430 500 390 125 T ; ub ¼ ½ 550 620 510 245 T

ð39Þ

Therefore, the inner loop optimization of the original uncertain optimization has been eliminated, and as a result the nested double-loop optimization has been changed to a single-loop optimization. The flowchart of interval arithmetic-based optimization is shown in Fig. 14. It is noted that the range of uncertainties of design variables are small. Only the wheel travel uses the interval bisection algorithm while the design variables are calculated directly through the Taylor inclusion function.

4.3. Optimization results The kinematic performance of the double wishbone suspension is evaluated with the conventional nested double-loop optimization approach and the proposed interval arithmetic-based approach, respectively. The parameters of MIGA are set as: 100 populations in each island, 10 islands and 10 generations, so there are 1000 populations in each generation. The objective in average value and best value changing with the generation are shown in Figs. 15 and 16, respectively. The final optimization results are shown in Table 1. The bold number in Table 1 shows that the constraint is violated. From the results we can see that the optimized suspension satisfies the constraints by minimizing the objective function. The objective function evaluated using the interval arithmetic-based method is equal to that calculated by the double-loop approach, but the constraints are relatively different. This is because the interval arithmetic-based method will produce a wrap effect of the original function, which means the maximal value produced by interval arithmetic is slightly larger than optimization. Correspondingly, the minimal value of the interval arithmetic is slightly smaller than the minimum of optimization. Thus, the interval arithmetic-based optimization method can satisfy constraints more rigorously. The computational time denotes that the interval method is more efficient, because it only requires less than one third of time compared to the double-loop method. With respect to the wheel travel, Figs. 17–20 show the variations of the camber angle, caster angle, kingpin inclination angle, and toe angle, respectively, by considering the initial conditions, the doubleloop optimization solutions and interval arithmetic based optimization solutions. The figures are plotted via the scanning Please cite this article in press as: J. Wu et al., An interval uncertain optimization method for vehicle suspensions using Chebyshev metamodels, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.02.012


15

Starting point u0, k=0 Calculation the objectives and constraints using Taylor inclusion function and interval bisection method Calculation for updating design variables uk based on Eq . (39) k=k+1 No

Converge ? Yes End

Fig. 14. Flowchart of interval optimization.

Fig. 15. The average objective.

Fig. 16. The best objective.


16


Table 1 Optimization results.

Initial Double-loop Interval

u1 (mm)

u2 (mm)

u3 (mm)

u4 (mm)

c1 (°)

c2 (°)

c3 (°)

c4 (°)

obj (°)

Time (’)

490.0 518.9 541.9

560.0 525.7 524.4

450.0 394.2 393.0

185.0 128.0 128.0

2.10 0.40 0.38

0.49 0.94 1.00

2.46 0.47 0.48

1.48 0.44 0.44

2.28 0.43 0.43

– 265 81

Fig. 17. Comparison of camber angle.

Fig. 18. Comparison of caster angle.

Fig. 19. Comparison of kingpin inclination angle.



17

Fig. 20. Comparison of toe angle.

method using the kinematic model of double wishbone suspensions. The plots show a great improvement of the suspension kinematic performance. The two different optimization approaches can provide very close optimization results. The maximum variation of the camber angle over the wheel travel reduces to 0.40° and 0.38° from 2.10° with the same initial conditions, by using the double-loop method and the interval method, respectively. The maximum variation of the caster angle over the wheel travel increases to 0.94° and 1.00°, under the same initial value 0.49°, by using the double-loop method and the interval method. However, both the optimization results satisfy the constraints. At the same time, the maximum variation of the kingpin inclination angle over the wheel travel is reduced to 0.47° and 0.48° from the initial value 2.46°, by using the double-loop method and the interval method, respectively. The toe angle when the wheel travel equals to 50 mm increase to 0.44° for both the optimization methods, while the initial value is 1.48° which violated the constraint. 5. Conclusions In this paper, a new design optimization method has been proposed for improving the kinematic performance of vehicle suspensions under parameter uncertainties. The positions of rear inner hardpoints of the upper control arm and lower control arm are considered as design variables, as they largely influence the suspension kinematic performance. At the same time, the design variables are also regarded as the uncertain parameters, due to the production tolerances and wear, ageing, etc. A double wishbone suspension kinematic model is developed using the software ADAMS, which involves the kinematic characteristics such as the camber angle, caster angle, kingpin inclination angle, and toe angle. A Chebyshev metamodel is proposed by using the zeros of Chebyshev polynomials as the sampling points, which can effectively reduce the computational cost of the original simulation model with higher numerical accuracy. Two benchmark mathematical testing examples are used to evidence that the zeros of Chebyshev polynomial-based sampling strategy can produce higher approximation accuracy than the uniform grid sampling strategy. Two optimization methods with uncertainties are presented: the first is the conventional nested double-loop optimization approach, and the second is the proposed interval arithmetic-based optimization method. In the former, the outer loop evaluates the objective function by satisfying the constraints via the MIGA, while the inner loop calculates the interval objective and constraints at each cycle of the outer loop. In the later, the Taylor inclusion function and interval bisection method are applied to control the overestimation, when the interval arithmetic is applied to eliminate the inner loop optimization. The optimization results show that the suspension kinematic performance is greatly improved relative to the initial conditions. The interval arithmetic-based optimization method is more efficient than the conventional double-loop optimization method. Hence, the present uncertain optimization method can be applied to more complicate mechanical systems. Acknowledgments This study is partially supported by the National Natural-Science-Foundation of China (NSFC) (11172108, 51175197, and 51105229), and the Chancellor’s Research Fellowship, The University of Technology, Sydney, Australia (2032063). References [1] G.G. Wang, S. Shan, Review of metamodeling techniques in support of engineering design optimization, J. Mech. Des. 129 (2007) 370–380. [2] T.W. Simpson, T.M. Mauery, J.J. Korte, F. Mistree, Kriging models for global approximation in simulation-based multidisciplinary design optimization, AIAA J. 39 (2001) 2233–2241. [3] R.H. Myers, D. Montgomery, Response Surface Methodology: Process and Product Optimization Using Designed Experiments, John Wiley and Sons Inc., Toronto, 1995.


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