1-s2.0-s0094114x08000505-main

Available online at www.sciencedirect.com Mechanism

Mechanism and Machine Theory 44 (2009) 42–56

www.elsevier.com/locate/mechmt

andMachine Theory

Kinematic and sensitivity analysis and optimizationof planar rack-and-pinion steering linkages

A. Rahmani Hanzaki a,*, P.V.M. Rao b, S.K. Saha b

a Mechanical Engineering Department, Shahid Rajaee University, Lavizan, Tehran 16788, Iranb Mechanical Engineering Department, Indian Institute of Technology Delhi, Hauz Khas, New Delhi 110 016, India

Received 10 May 2007; received in revised form 19 February 2008; accepted 26 February 2008Available online 15 August 2008

Abstract

In this paper, the combined kinematic and sensitivity optimization of a rack-and-pinion steering linkage is performed.This steering linkage is the most common steering system used in passenger cars. Although, the steering linkage hasreceived a lot of attention for the minimization of the steering errors, no attempt has been made so far to investigatethe sensitivities of optimum dimensions relative to variation of link lengths. The kinematic optimization of the linkageis carried out using three homogenous design parameters. The objective of the proposed optimization is to minimize max-imum steering error during cornering. This is followed by a sensitivity analysis to predict how the steering error is affectedby manufacturing tolerances, assembly errors, and clearances resulting due to wear. Since the optimized kinematic error isvery sensitive to the variations of the linkage parameters, the kinematic and post-optimal sensitivity optimization of thesteering linkage is performed in an integrated manner. The methodology proposed in this work helps the designers ofrack-and-pinion steering linkage to choose the linkage parameters whose maximum steering error (MSE) and sensitivityare minimum.� 2008 Published by Elsevier Ltd.

Keywords: Rack-and-pinion steering linkage; Steering error; Kinematic optimization; Post-optimal sensitivity; Cognates

1. Introduction

Among the steering linkages, rack-and-pinion steering linkage is the most widely used in passenger cars. Itconsists of two steering arms, two tie rods, and a rack. The linkage has two common configurations, namely,central take-off and side take-off, as shown in Fig. 1. In central take-off (CTO) configuration, the tie rods andthe rack are connected at the middle of the rack as depicted in Fig. 1a, while in side take-off (STO) configu-ration, these connections are at the rack ends as shown in Fig. 1b. Each of the above configurations can beeither trailing or leading type, as shown in Fig. 2a and b, respectively [1].

0094-114X/$ - see front matter � 2008 Published by Elsevier Ltd.

doi:10.1016/j.mechmachtheory.2008.02.014

* Corresponding author. Tel.: +98 91 23252729; fax: +98 22970052.E-mail addresses: [email protected] (A. Rahmani Hanzaki), [email protected] (P.V.M. Rao), [email protected]

(S.K. Saha).

mailto:[email protected]



(a) Trailing (b) Leading

Front motion Front motion

Fig. 2. Trailing and leading type of rack-and-pinion steering linkages.

Steering arm Steering arm Tie rod Tie rod

Rack Rack Pinion

Wheel(a) Central take-off (CTO) (b) Side take-off (STO) Kingpin Kingpin

Pinion

Fig. 1. Rack-and-pinion steering linkage and its configurations.

Wt

Wb

I Oθ θ

Fig. 3. Ackermann condition for a vehicle when turning.

A. Rahmani Hanzaki et al. / Mechanism and Machine Theory 44 (2009) 42–56 43

In order to provide pure rolling to the road wheels and to reduce wear and tear of the tires, a steering link-age must handle the vehicle so that it follows Ackermann principle (see Fig. 3). This principle states that dur-ing low speed cornering when free from lateral inertia forces, the verticals drawn from the centers of the wheelsshould meet at the center of bend, i.e., point O of Fig. 3. For a two-wheel steering vehicle, this point must lieon the common axis of the rear wheels [1]. Referring to Fig. 3, the relation between the inner wheel angle, hI,and the outer wheel angle according to Ackermann principle, hOA, is given as

hOAðhIÞ ¼ tan�1 1

cot hI þ W t

W b

¼ tan�1 1

cot hI þ 1=wb

ð1Þ

where wb = Wb/Wt is the normalized expression of the wheel base, Wb, with respect to wheel track, Wt. Inreality, Eq. (1) is never satisfied for every radius of orientations. Hence, there are efforts to synthesize the link-age so that Ackermann principle is satisfied for any orientation of the wheels as closely as possible. In order todo that, it is necessary to obtain the angle hO for a given value of hI. Hence, an appropriate kinematic model ofthe steering linkage is essential. In addition, the kingpin inclination and caster angles that provide complianceto the steering linkage with the suspension system have little influence on the motion transmission of the steer-ing linkage. As a result, the real rack-and-pinion steering linkage, which is spatial in nature, can be modeled asa planar linkage for the investigation of Ackermann condition. Such a simplification of the steering system hasbeen also used by other researchers, e.g. [2,3].

44 A. Rahmani Hanzaki et al. / Mechanism and Machine Theory 44 (2009) 42–56

Error optimization studies in steering linkages have been attempted by many researchers. Zarak and Town-send [3] optimized the STO configuration as shown in Fig. 1b, where they considered the distance between theinner and outer wheel turning centers as the steering error and minimized it for different rack travels. Duringoptimization, they used four parameters that were normalized with respect to the wheel track. Felzien andCronin [4] investigated the minimization of steering errors. They considered an integrated McPherson suspen-sion and steering linkage model, and minimized the weighted sum of the squares of steering errors. Simionescuand Smith [5] discussed Watt II function generating cognates, and showed that STO configuration of the steer-ing linkage has infinite number of cognates, of which one is the CTO configuration. Simionescu and Smith [2]used three parameters, namely, a normalized link length/a link length ratio and two angles, in the case ofCTO/STO configuration to optimize the steering errors of the linkages. The choice of the design parametersused by them, some of which are in terms of angles, is not appropriate however for link-length sensitivity anal-ysis. Hence, the kinematic optimization carried out in this paper uses three homogeneous design parameters,namely, those which have the units of length. Moreover, manufacturing tolerances, assembly errors, and clear-ances resulting due to wear, which are inherent to any real steering linkage, may affect the objective functionvalue significantly. Hence, it is important to perform a post-optimal sensitivity analysis, in addition to kine-matic optimization. Furthermore, a method based on rack-and-pinion steering linkage cognates given in [5] isused to generalize the steering optimization. Using this methodology, the steering linkage can be optimizedonce and the results are extended to desired rack-and-pinion steering linkage whether CTO or STO withany rack length. To the best of the authors’ knowledge, no work concerning the sensitivity of the error forthe optimized dimensions of a rack-and-pinion steering system has been reported so far. In addition, the sen-sitivity minimization, which is carried out in this paper as a part of optimization process, is also new in thecontext of rack-and-pinion steering linkage design. Such minimization of both the objective function andthe sensitivity has been considered important in the literature of robust optimal design [6,7]. In summary,the contributions of the paper are

� A simple generalized methodology for the optimization of a rack-and-pinion linkage for both CTO andSTO is proposed.� Post-optimal sensitivity of the steering linkage is performed.� The steering linkage is optimized for minimum steering error as well as for minimum sensitivity.� A methodology for multi-objective optimization problem is also presented, where the number of design

parameters is reduced by one, thus increasing the efficiency and speed of the optimization.

This paper is organized as follows: Section 2 shows that the optimized results for a CTO linkage can be usedto find the optimized parameters of a corresponding STO linkage, consequence that any STO configurationhas a CTO cognate configuration. Section 3 presents a generalized kinematic modeling of a planar six-barrack-and-pinion steering linkage, which would then be used in Section 4 to optimize the steering linkage understudy. Post-optimal sensitivity analysis is then carried out in Section 5, which is further used in Section 6 tooptimize the steering linkage that has minimum sensitivity in addition to minimum steering error. The pro-posed methodologies are illustrated in Section 7, followed by the conclusions in Section 8.

2. Cognates

The side take-off (STO) configuration of the steering linkage, shown in Fig. 1b, is more common in passengercars. This linkage noted AB0C1C2D0E in Fig. 4, has infinite number of cognates [5], meaning that the input-out-put link behaviors of all the cognate linkages are same. For example AB00 C01C02D00E whose links are parallel tothe original linkage is one of them. Continuing in a similar manner, as shown in Fig. 4a, linkage ABCDE is alsoa cognate, which is nothing but the central take-off (CTO) configuration of the steering linkage introduced inFig. 1a [5]. Note from Fig. 4b that another CTO cognate for the STO linkage, i.e., AB0C1C2D0E of Fig. 4b, isA1B1C2D0E, where A1E = AE–Lr [5]. As a result, if the link lengths of the STO linkages shown in Fig. 4a aredivided by AE–Lr, the link lengths of their CTO cognate, ABCDE of Fig. 4a, will be obtained. Since in functiongenerating cognates, the angular variables of similar links of cognates are equal, the results obtained for theCTO configuration can be utilized for the analysis of STO configuration cognates as well. It also implies that

D’

B’

A

A1

E

C2 C1B1

La

Lt

Wt

Lr

La

r

Lt

(b) Different arranging of the cognates

D B

D′ B′

AE

C

C2 C1

La

Wt

Lr

la

Lt

lt

D″

γ

γ

θ

B″C′1C′2

Lt

(a) A set of the cognates of the steering linkage

Fig. 4. Cognates of rack-and-pinion steering linkage.


the STO linkages are as sensitive as their CTO cognate. Hence, to extend the results obtained for a CTO con-figuration to its STO cognates, the following simple method can be used.

The real linkage link lengths except rack length are divided by (Wt–Lr), while the wheel base and wheeltrack, i.e., Wb and Wt, are divided by Wt. This normalization is opposed to that in previous works by otherresearchers like Simionescu and Smith [2] and Zarak and Townsend [3], where the normalization is done withrespect to wheel track, Wt. By this normalization, the CTO linkage, namely, ABCDE of Fig. 4a is obtained.Moreover, the normalized parameters, La, Lt, H, Wb, and Wt, are denoted as la, lt, h, wb, and wt, respectively.After optimization, the optimum la, lt, and h are denormalized1 to find La, Lt, and H of the optimized STOlinkage for a certain rack length. It is obvious that a CTO configuration can be treated in the same waydue to the fact that it is same as a STO linkage except its rack length, Lr, is zero. Note that Lr is definedas the distance of the two tie rod ends connected to the rack for the kinematic problem.

3. Kinematic modeling

As explained in Section 2, the CTO configuration shown in Fig. 1a is considered here for the purpose ofkinematic modeling and analysis. The CTO steering linkage is kinematically a six-bar linkage [8] as shownin Fig. 5. In this figure, the links are denoted by #1, . . . , #6 and the joints by 1, . . . , 7. Kutzbach criterionillustrates that the DOF of this linkage is one. Moreover, vectors, l1 and l5 denote the steering arms of lengthla, whereas vectors, l2 and l4, represent the tie rods of length lt. Furthermore, at the end of every link, a two-dimensional coordinate system is considered that is fixed to its previous link.

From Fig. 5,

1 De

l1 þ l2 ¼ ðwt=2þ bÞiþ hj ð2Þ

where wt is the wheel track, b is the rack displacement, h is the distance from the front wheel axis to the rackaxis, and i and j are the unit vectors along X1 and Y1, respectively. Eq. (2) can be written in terms of its scalarcomponents as

normalizing here means converting the normalized parameters to real link lengths by multiplying the normalizing factor.

1

2

3

4

5

#1

#2

#3

#4

#5

X1

Y1

Xθβ

θθ

θ

θ

θ

2

X3

X4

X5

X6

l1l2

l5

l4

hwt

+b2

3, 4 5 5

6

71

O1

O7

6

2

Fig. 5. Six-bar planar rack-and-pinion steering linkage for the CTO configuration.


lach1 þ ltch12 ¼ wt=2þ b and lash1 þ ltsh12 ¼ h ð3; 4Þ
where c and s stand for cosine and sine functions, respectively, h12 � h1 + h2, and the normalized wheel track,wt, is unity. Referring to the derivations shown in Appendix A, the initial angle of the left steering arm denotedas h10 is obtained from Eq. (A.9) by using Eqs. (A.7) and (A.8). This is also equal to h60. If the range of theinner wheel rotation is considered as 40�, which is quite practical for most of the passenger cars, the results canbe obtained for the range of hI = 0–40�. Note that, for the trailing configuration, as shown in Fig. 2a,h1 = h10 � hI, and for the leading configuration which is shown in Fig. 2b, h1 = h10 + hI when turning left. Fur-thermore for straight ahead configuration, the initial value of h6, i.e., h60, is equal to h10. To evaluate angle h6
in an arbitrary position of the mechanism, the expression of the rack displacement is calculated first as

b ¼ lach1 � wt=2�ffiffiffiffiDp

ð5Þ
where
D � l2ach2

1 � k1 þ 2lahsh1 and k1 � �l2t þ l2

a þ h2 ð6Þ
In Eq. (5), + sign should be used. This is followed by the derivation of an expression similar to Eq. (2) using the
right side of the steering linkage, Fig. 5, which would provide the values of h6 given by Eqs. (A.10) and (A.11).Pressure angle, which plays important role in force transmission efficiency of a linkage, is defined as the

angle between the velocity vector of the driven link and the driver link direction. The pressure angle at joint2 of the steering linkage of Fig. 5 is shown by b2 and is the angle between the perpendicular to the steeringarm, i.e., #1, and the tie rod, namely, #2. According to the above definition, the pressure angle at joint 3is the same as h3. The pressure angles at joints 4 and 5 are found similar to those at joints 3 and 2, respectively.

4. Steering error optimization

In this section, optimization of the steering linkage with CTO configuration is carried out to minimize thesteering error, dhO, which is defined as

dhO ¼ jhO � hOAj ð7Þ
where hO is the actual angle made by the outer front wheel during steering maneuver and hOA is the correctangle for the same wheel based on the Ackermann principle given by Eq. (1). Referring to Fig. 5, the angle hO
is the variation of h1 or h6 when turning left, depending on if the steering linkage is trailing or leading, respec-tively. The objective here is to minimize the maximum value of the steering error, abbreviated as MSE duringentire rack travel [2], where wt is unity. The following constraints are considered:

(I) The initial assemblage of the linkage must be feasible by the link lengths, which implies that D10 of Eq.(A.8) should be non-negative, i.e., D10 P 0.

(II) The input link should be able to rotate at least 40�. In other words, when hI-max = 40�, D of Eq. (6)should be non-negative, i.e., D P 0.

(III) The rack displacement must be unidirectional while satisfying condition II meaning that jbij > jbi�1 j, fori = 1, . . ., n, discrete positions of the rack, where bi is the solution of Eq. (5) corresponding to the inputangle, hIi, at the ith step.


(IV) The output link should also be able to rotate in correspondence with the rack travel, b. In other words,D6 of Eq. (A.11) must satisfy D6 P 0.

The kinematic optimization problem is now formally posed as

Minimize MSE; w � max �jhOðhIÞ � hOAðhIÞj ð8aÞSubject to D10 P 0; D P 0; D6 P 0; and ð8b–dÞ

jbijP jbi�1j i ¼ 1; 2; . . . ; n ð8eÞ

Variations of the objective function, Eq. (8a), with respect to la, lt, and h are shown in Fig. 6a and b, as a sam-ple, where la, lt, and h represent the normalized steering arm length, tie rod length and the distance from thefront wheel axis to the rack axis as already pointed out. In these figures, there are two groups of points lying

(a) MSE, , vs. h and lt (trailing, wb =1.4, la =0.14)

0

0.2

0.4

0.30.4

0.50.6

0.7

0

5

10

15

20

h

l t

MSE

(deg

)

(b) MSE, , vs. lψ a and lt (trailing, wb =1.4, h =0.1)

0.1

0.2

0.3

0.4 0.30.4

0.50.6

0.7

0

5

10

15

20

l t

la

MSE

(deg

)

ψ

Fig. 6. Variation of maximum steering error.


on the horizontal planes where correspond to 20� and 10�. The points whose MSEs are 20� indicate the con-figurations with MSEs more than 20�, which are not acceptable. Hence, their MSEs are replaced by 20�. Sim-ilarly, the flat surface corresponding to MSE = 10� is plotted to show the non-feasible or non-practical valuesof linkage parameters. In other words, the link lengths associated to MSEs of 10� in Fig. 6a and b is not fea-sible, even if it is feasible, it is impractical as the input link cannot rotate by desired angle of hI-max = 40�. Afterstudying the plots, the following observations can be made:

(i) MSEs along a curved line of Fig. 6a and b are so close to each other that choosing a set of link lengths asoptimum values is not acceptable in practice. Hence, additional constraints like space limitation, racktravel, sensitivity, etc. can be considered during optimization.

(ii) The slope of the MSE function, w, around the optimum points is very high and different on their rightand left neighborhoods, which specify high sensitivity of the function, w. Since undesirable variations ofthe design parameters of the steering linkage, such as manufacturing tolerances of the link lengths, clear-ances in the joints, body deformation, and adjustment of the toe angle of the front wheels. usually exist,an appropriate sensitivity analysis with respect to the design parameters is important.

Keeping the above two observations in mind, optimization is carried out using the optimization toolbox ofMATLAB in this stage. The results for wb = 1.4 are shown in Figs. 7 and 8, and closely match the results ofSimionescu and Smith [2].

In order to verify the correctness of the results, a number of optimum points were analyzed and validatedby the methodology given in [8].

5. Post-optimal sensitivity analysis

For sensitivity study, two main definitions are found in the literature; Sharfi and Smith [9] define it as thevariation of the output with respect to a small variation of the input, whereas in [10], it is defined as the ratio ofthe variation of the objective function to a small variation of the design parameters at optimum point. Thelatter is also referred to ‘‘post-optimal sensitivity” and adopted here. In this context, Knappe [11] used partialderivatives to study the influence of parameter deviations on the output. He obtained the total output devi-ation by adding the deviations due to individual parameters, then used these results to assign the practical tol-erances. Sharfi and Smith [9] formulated tolerances and clearances corresponding to the allowable outputerror using partial derivatives for a 10-bar linkage. Chakraborty [12] made a probabilistic model of mecha-nisms considering the tolerances on link lengths and joint clearances. Dhande and Chakraborty [13] treated

0 0.1 0.2 0.3 0.4

0.35

0.4

0.45

0.5

0.55

0.6

1

2

3

4

h

l* t

0 0.1 0.2 0.3 0.40.65

0.7

0.75

0.8

0.85

0.9

0.95

1

1

2

3

4

h

MS

E (

deg

)

(a) Optimum tie rod length (b) Maximum steering error,

Fig. 7. Optimum tie rod length and corresponding maximum steering error for trailing CTO configuration, wb = 1.4, and la of (1)la = 0.14, (2) la = 0.16, (3) la = 0.19, and (4) la = 0.24.

0 0.1 0.2 0.3 0.40.55

0.6

0.65

0.7

0.75

0.8

1

23

4

h

l* t

0 0.1 0.2 0.3 0.4

0.35

0.4

0.45

0.5

1

2

3

4

h

MS

E (

deg

)

(a) Optimum tie rod length (b) Maximum steering error,

Fig. 8. Optimum tie rod length and corresponding maximum steering error for leading CTO configuration, wb = 1.4, and la of (1)la = 0.14, (2) la = 0.16, (3) la = 0.19, and (4) la = 0.24.


the tolerances and clearances of a four-bar function generating linkage as optimization parameters, andobtained the optimum values using the allowable output deviation as a constraint. Note that, for an optimi-zation problem with continuous differentiable objective function, the condition for optimality is that the firstderivatives of the objective function are zero, as indicated in Fig. 9a. In such cases, the second derivatives lead-ing to the Hessian matrix are used to analyze the post-optimal sensitivity of the output function with respect tothe variations of design variables [10]. In the case of an objective function whose gradient is not continuousnear the optimum point, as illustrated in Fig. 9b, the first derivatives in the neighbourhood can be used tostudy the sensitivity. As mentioned in item (ii) of Section 4, post-optimal sensitivity analysis is importantto investigate the influence of the variation of the design parameters, namely, la, lt, and h, on the variationof maximum steering error (MSE), w of Eq. (8a). Here, sensitivity is defined as the ratio of the variation ofthe objective function, w, to the small variations in the design parameters la, lt, and h.

In the present case, the optimum point is a cusp, as evident from the plots of Fig. 6, i.e., the first derivativesof the objective function, w, with respect to the design parameters, la, lt, and h, at the optimum point are notdefined. Hence, the derivatives in the positive and negative neighborhoods of the optimal point are usedinstead in the post-optimal sensitivity analysis.

For this purpose, the following six-dimensional sensitivity vector is introduced:

Dw � wlaþ wla� wltþ wlt� whþ wh�� T ð9Þ

where wlaþ � ow=ola for ðla � l�aÞ > 0 and wla� � ow=ola for ðla � l�aÞ < 0, in which l�a is the optimum value ofla, etc.

x

F(x) ψ

design parameter

Optimum point Optimum point (a cusp)

+ variation of design parameter

Fig. 9. Possible configurations of an optimum point (a) smooth and (b) non-smooth objective function.


Fig. 10 shows the sensitivity plots obtained using the first partial derivatives of the objective function for thetrailing CTO configuration, where Richardson’s extrapolation algorithm [14] is used here to ensure that thederivatives are computed reliably.

0 0.1 0.2 0.3 0.44

5

6

7

8

9

10

11

12

1

2

3

4

h

MS

E v

aria

tion

(%

) / le

ng

th v

aria

tion

(%

)

0 0.1 0.2 0.3 0.410

15

20

25

30

1

2

3

4

hM

SE

var

iatio

n(%

) / l

eng

th v

aria

tion

(%)

(a) Positive neighborhood of la (b) Negative neighborhood of la

0 0.1 0.2 0.3 0.420

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70

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thva

riat

ion

(%

)

0 0.1 0.2 0.3 0.460

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) / le

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thva

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(%

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(c) Positive neighborhood of lt (d) Negative neighborhood of lt

0 0.1 0.2 0.3 0.40

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MS

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thva

riat

ion

(%

)

0 0.1 0.2 0.3 0.40

5

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35

1

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4

h

MS

E v

aria

tion

(%

) / le

ng

thva

riat

ion

(%

)

(e) Positive neighborhood of h (f) Negative neighborhood of h

Fig. 10. Sensitivity plots of the steering error for trailing CTO configuration with wb = 1.4, and la of (1) la = 0.14, (2) la = 0.16, (3)la = 0.19, and (4) la = 0.24.


Now, sensitivity with respect to positive or negative variation of every parameter, i.e., in positive and neg-ative neighborhoods, is provided in the plots in term of percentage of the MSE variation to percentage oflength variation. Hence, the percentage of MSE variation due to just the length variation is achieved bymultiplying the sensitivity into the percentage of the length variation. The sensitivity analysis for the leading

0 0.1 0.2 0.3 0.40

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0 0.1 0.2 0.3 0.45

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hM

SE

var

iatio

n(%

) / le

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th v

aria

tion

(%)

(a) Positive neighborhood of la (b) Negative neighborhood of la

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tio

n (

%)

0 0.1 0.2 0.3 0.440

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tio

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(c) Positive neighborhood of lt (d) Negative neighborhood of lt

0 0.1 0.2 0.3 0.40

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1234

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h

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(e) Positive neighborhood of h (f) Negative neighborhood of h

Fig. 11. Sensitivity plots of the steering error for leading CTO configuration with wb = 1.4, and la of (1) la = 0.14, (2) la = 0.16, (3)la = 0.19, and (4) la = 0.24.


configuration of the steering linkage which was optimized in Fig. 8 is done and the results are presented inFig. 11.

The key observation from the plots is: sensitivity in positive and negative neighborhoods of every designparameter is different, as shown in Figs. 10 and 11. Hence, this point should be seriously taken into accountwhile assigning manufacturing tolerances, etc. This means that plus and minus tolerances to the link lengthsshould not preferably be the same. For example, sensitivity in positive neighborhood of lt, Fig. 11c, is almosttwice of that in negative neighborhood of the parameter, Fig. 11d, for a leading configuration. Thus, the plustolerance of tie rod length should be tighter than its minus tolerance.

6. Sensitivity and kinematic optimization

Since the optimized sets of the design parameters based on only kinematic consideration are infinitive, asreported in Section 4, and very sensitive, as reported in Section 5, it is preferable to optimize a function thattakes into account both the kinematics and sensitivity aspects, i.e., providing a kinematically optimized valuesthat are least sensitive. Such multi-objective optimization can be treated in two ways: (a) according to [15], anew objective function can be defined as the weighted sum of the two objective functions, namely, the MSEand sensitivity and (b) one of the objective functions is considered as a constraint, while the second one is trea-ted as the objective function [16]. Here, however, a new approach is proposed to attempt optimization in twostages. According to this approach, the problem is firstly optimized for one of the objective functions, i.e.,kinematic optimization, and among all the solutions generated in this stage, the algorithm searches for theoptimum one based on the second objective function, i.e., sensitivity optimization. Therefore, the problemis started with two design variables, namely, la and h at every stage of sensitivity calculation. The thirdone, i.e., lt, is obtained from MSE optimization. The three variables are utilized to calculate sensitivity. Forthe problem at hand, the second objective function is defined as

Minimize Sensitivity S ¼ w1ðwlaÞ þ w2ðwlt

Þ þ w3ðwhÞ ð10aÞ

for which the corresponding constraints are considered as

Lal 6 La 6 Lau; and H l 6 H 6 Hu ð10b–cÞMSE 6 allowable limit ð10dÞMinimum allowable value 6 rack stroke 6 maximum allowable value ð10eÞMaximum pressure angle ðbÞ 6 Allowable pressure angle ð10fÞUser-defined constraints ðif anyÞ ð10gÞ

In Eq. (10a), wla, wlt

and wh are the sensitivities of the linkage w.r.t. la, lt, and h, respectively. Since, everyone of these variables has two values, which in positive and negative neighborhoods of the parameter, e.g., la,the bigger one is chosen as the sensitivity. Moreover, in Eq. (10a), w1, w2, and w3 are the weighting factors.Furthermore, Lal, Lau, Hl, and Hu are the lower and upper limits of La and H, respectively, and Eq. (10d) con-strains the MSE to allowable MSE, Eq. (10e) constrains rack stroke between minimum and maximum allow-able values. Next constraint is pressure angle constraint, being important for force transmission efficiency. Asreported in [2], between pressure angles defined in Section 3, b2 is usually maximum at the maximum steeringangle, which can be depicted from Fig. 5. Also the maximum pressure angle of about 60� is considered suitablefor good force transmission efficiency and to avoid jamming the linkage. This constraint is taken into accountin Eq. (10f). Additional constraints, like space limitation, can also be included, but are not considered here.Now, the optimization methodology is shown in the flowchart of Fig. 12.

A MATLAB code based on the proposed methodology has been written. It has three components, namely,a main program and two function programs, each of kinematic optimization and sensitivity calculations. Outof three normalized design variables, i.e., la, lt, and h, two, namely, la and h, are kept constant for MSE opti-mization and are entered in the kinematic optimization program. As discussed in Section 4 and shown in Figs.7 and 8, the output of this part is optimum lt for the given constant values of la and h. These three design vari-ables are then utilized to calculate sensitivity of the linkage. Finally, the link lengths of the optimized linkageare achieved by multiplying la, lt, and h into the normalization factor (Wt–Lr).

Enter initial parameters La, Lr, H, Wt, and Wb

Normalize La and H by(Wt - Lr)

Kinematics optimization, eqs. (8a-8e): Find optimum lt

Normalize Wb and Wt

by Wt

End

Sensitivity objective function calculation

Find optimum La, Lt, and H by denormalizing optimum la, lt, and ht , respectively

Are theconstraints satisfied?

andIs the objective

function minimum?

Chooseanother set of design variables

according to optimization

method

Yes

No

Fig. 12. Combined optimization flowchart.


7. A case study

The kinematics and post-optimal sensitivity optimization for the rack-and-pinion steering linkage based onthe concepts outlined above is illustrated with the help of an STO trailing linkage used in a commercial hatch-back car. Its actual and normalized link lengths are given in the first and second columns of Table 1. The actualvalues are approximate, as obtained from real model measurement and not from manufacturer’s blueprint. Inthe third column, the kinematically optimized link lengths, the MSE, and the sensitivity of the system withrespect to design variables are given. At this stage, sensitivity is not considered during optimization, i.e., theoptimization is based on methodology presented in Section 4. Note here that the real dimensions of theSTO configuration appearing in the 1st column of Table 1 is first normalized and shown in the 2nd column.The resulting linkage is then optimized and denormalized to the optimized STO, whose dimensions appearin the 3rd column of the table. As it is expected, La and H are unchanged and only Lt is changed to optimumLt. The optimum dimensions of the linkage considering both MSE and sensitivity based on methodology pre-sented in Section 6 are shown in the 4th column of the table, where w1, w2, and w3 are considered equal to 0.33.The limits of Eq. (10b–c), namely, Lal, Lau, and others are as also shown in Table 1. To solve this constrainedoptimization problem, the method of Multiplier (MOM) [15] is applied to the problem to penalize the con-straints, and Simplex direct search method [15] is used to find the optimum point. In addition, the proper initialvalues should be found for the optimization problem to converge to optimum without violating the constraints.

For this purpose, the bracketing method [15] is applied to the objective function over the defined span tofind suitable starting point. The optimum results show good improvement in sensitivity almost without sac-rificing the optimized steering error. Finally, the kinematic steering error and sensitivity of the error due tovariation of every linkage parameters, namely, La, Lt, and H, is verified by applying 1% variation of every

Table 1Dimensions of a STO steering linkage before and after optimization

Initialdimensions

Normalizeddimensions

Kinematic optimumdimensions

Kinematic andsensitiveOptimum dimensions

Steering arm length La = 110 mm la = 0.2048 La = 110 mm La = 170.5 mmTie rod length Lt = 256 mm lt = 0.4767 Lt = 266.2 mm Lt = 212.3 mmDistance from rack axis to wheel

axisH = 177 mm h = 0.3269 H = 177 mm H = 15 mm

Wheel base Wb = 2175 mm wb = 1.7901 Wb = 2175 mmWheel track Wt = 1215 mm wt = 1 Wt = 1215 mmMax. error, dhO 3.9� 0.7� 0.75�Rack length, Lr 678 mmSensitivity, S, of Eq. (10a) 59.32% 28.27%MSE variation (%) due to 1% +ve Lavariation 11.39% 0.42%MSE variation (%) due to 1% �ve Lavariation 27.1% 0.48%MSE variation (%) due to 1% +ve Ltvariation 44.6% 34.5%MSE variation (%) due to 1% �ve Ltvariation 112.5% 83.2%MSE variation (%) due to 1% +ve H variation 40.2% 0.91%MSE variation (%) due to 1% �ve H variation 16.32% 1.9%

Allowable Ackermann error = 0.75�; Lal = 100 mm; Lau = 300 mm; Hl = 0; Hu = 300 mm; minimum rack stroke = 100 mm; maximumrack stroke = 140 mm; maximum pressure angle = 56�.


parameter in separate stages of kinematic calculation and find the steering error variation. The results closelymatch with those obtained from the optimized results. It proves that the optimization is successful and itsresults are reliable.

8. Conclusions

In this paper a combined kinematic and sensitivity optimization of a rack-and-pinion steering linkage arepresented for the first time. Since the variations of the linkage parameters cannot not be avoided in practice,attention to the sensitivity analysis and optimization is considered essential. The kinematic optimization ofthe steering linkage is carried out using three homogenous design parameters all having the unit of length asa requirement for post-optimal sensitivity analysis. A simple but generalized methodology based on rack-and-pinion steering linkage cognates is proposed to analyze the central take-off configurations. The resultsare applicable to side take-off configuration of the steering linkage as well. The analysis of the linkage showsthat the system is very sensitive with respect to variations of the links lengths close to optimum and are differentleft and right of the optimum point. This has implications in assigning tolerances to the links lengths. Nextobservation is that the sensitivities of the steering error with respect to the variations of different design param-eters vary very widely. Hence, it should be taken into account that some alignments during maintenance,namely toe angle by changing the tie rod length, can increase the steering error extensively. For optimizationof both the kinematic steering error and sensitivity to link length variation, a two-level multi-objective optimi-zation was proposed. The advantage of the proposed method is that it reduces the number of design variables byone, thus increasing the stability and efficiency of the optimization. The proposed methods were successfullyapplied to the rack-and-pinion steering linkage of a real vehicle, and the numerical results provided in the paper.

Appendix A

The analytical method used to solve the nonlinear position problem of the central take-off rack-and-pinionsteering linkage is as follows:

Eqs. (3) and (4) are first rearranged as

ltch12 ¼ ðwt=2þ bÞ � lach1 and ltsh12 ¼ h� lash1 ðA:1Þ


Squaring and summing the expressions, in Eq. (A.1) one obtains the following:

l2t ¼ ðwt=2þ bÞ2 þ h2 þ l2

a � ðwt þ 2bÞlach1 � 2lahsh1 ðA:2Þ
Now, denoting, k1 � �l2
t þ l2a þ h2, k2 � [k1 + (wt/2 + b)2] /la, and

z1 � tanh1

2; sh1 �

2z1

1þ z21

and ch1 �1� z2

1

1þ z21

ðA:3Þ

Eq. (A.2) becomes

½k2 þ ðwt þ 2bÞ�z21 � 4hz1 þ ½k2 � ðwt þ 2bÞ� ¼ 0 ðA:4Þ

which is a quadratic equation in z1. The solution, z1, is given by

z1 ¼2h�

ffiffiffiffiffiD1

p

k2 þ ðwt þ 2bÞ ðA:5Þ

where

D1 � 4h2 � k22 þ ðwt þ 2bÞ2 ðA:6Þ

For the initial condition where the solution, z1, is denoted as z10, is solved from Eq. (A.5) for b = 0. Thus,Eqs. (A.5) and (A.6) are converted to:

z10 ¼2h�

ffiffiffiffiffiffiffiD10

p

k2 þ wt

ðA:7Þ

in which

D10 � 4h2 � k22 þ w2

t ðA:8Þ
The joint angle, h1, is finally obtained from Eq. (A.5) as
h1 ¼ atan2ðsh1; ch1Þ ðA:9Þ

where ‘atan2’ is the function used in any computer programming language to provide unambiguous angle re-sults to tan�1. Similar steps can be followed for other side of the linkage of Fig. 5 to find h6 = atan2(sh6, ch6)where sh6 and ch6 are obtained from the value of z6 which is solved as

z6 ¼2h�

ffiffiffiffiffiD6

p

k3 þ ðwt � 2bÞ ðA:10Þ

where

D6 � 4h2 � k23 þ ðwt � 2bÞ2 and k3 � ½k1 � ðwt=2� bÞ2�=la ðA:11Þ

Note that + and � signs in Eqs. (A.5), (A.7) and (A.10) correspond to the two different assembly positionsof the mechanism, where only + sign is applicable for the steering linkages.

References

[1] J. Reimpell, H. Stoll, J.W. Betzler, The Automotive Chassis: Engineering Principles, second ed., SAE International, ButterworthHeinemann Publisher, Oxford, 2001.

[2] P.A. Simionescu, M.R. Smith, Initial estimates in the design of rack-and-pinion steering linkages, ASME J. Mech. Des. 122 (2) (2000)194–200.

[3] C.E. Zarak, M.A. Townsend, Optimal design of rack-and-pinion steering linkages, ASME J. Mech. Transm. Automat. Des. 105(1983) 220–226.

[4] M.L. Felzien, D.L. Cronin, Steering error optimization of the McPherson strut automotive front suspension, Mech. Mach. Theory 20(1) (1985) 17–26.

[5] P.A. Simionescu, M.R. Smith, Applications of Watt II function generator cognates, Mech. Mach. Theory 35 (2) (2000) 1535–1549.

[6] H. Han, T. Park, Robust optimal design of multi-body systems, Multibody Syst. Dyn. 11 (2004) 167–183.


[7] C. Zang, M.I. Friswell, J.E. Mottershead, A review of robust optimal design and its application in dynamics, Comp. Struct. 83 (4-5)(2005) 15–326.

[8] A. Rahmani Hanzaki, S.K. Saha, P.V.M. Rao, Analysis of a six-bar rack-and-pinion steering linkage, in: Proceedings of SAEINDIAInternational Mobility Engineering Congress, 2005, pp. 103–108.

[9] O.M.A. Sharfi, M.R. Smith, A simple method for the allocation of appropriate tolerances and clearances in linkage mechanisms,Mech. Mach. Theory 18 (2) (1983) 123–129.

[10] P.E. Gill, W. Murray, M.H. Wright, Practical Optimization, Academic Press, London, 1988, pp. 320-323.[11] L.F. Knappe, A technique for analyzing mechanism tolerances, Mach. Des. (1963) 155–157.[12] J. Chakraborty, Synthesis of mechanical error in linkages, Mech. Mach. Theory 10 (1975) 155–165.[13] S.G. Dhande, J. Chakraborty, Analysis and synthesis of mechanical error in linkages – a stochastic approach, ASME J. Eng. Ind. 95

(3) (1973) 672–676.[14] S. Yakowitz, F. Szidarovszky, An Introduction to Numerical Computations, Macmillan Publishers, New York, 1990, pp. 214–216.[15] K. Deb, Optimization for Engineering Design, Algorithms and Examples, Prentice-Hall of India, New Delhi, 1995, pp. 162–163.[16] A.B. Koteswara Rao, Analysis and Design of Hexaslide Manipulators for Machine Tools Application, Ph.D. Thesis, Mechanical

Engineering Department, Indian Institute of Technology Delhi, 2004, pp. 64–70.

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