4bar Synthesis

Mechanism and Machine Theory 37 (2002) 1487–1504 www.elsevier.com/locate/mechmt

Optimum synthesis of the four-bar function generator in its symmetric embodiment: the Ackermann steering linkage P.A. Simionescu *, D. Beale Department of Mechanical Engineering, Auburn University, 201 Ross Hall, Auburn, AL 36849, USA Received 29 October 2001; received in revised form 16 July 2002; accepted 16 July 2002

Abstract The problem of optimum synthesis of the planar four-bar function generator is investigated and the practical case of the Ackermann steering linkage considered as an example. The reduced number of design parameters of this symmetric four-bar linkage allowed inspecting the design space of various types of objective functions through 3D representations, and their properties suggestively highlighted. For practical purposes, the numerical results were summarized in a set of parametric design-charts useful to the automotive engineer in conceiving the steering linkage of a new vehicle. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Four-bar mechanism; Optimum synthesis; Ackermann linkage; Steering error

1. Introduction Synthesis of function generators is a common mechanism design problem. It requires finding the geometric parameters for which the input–output function of the mechanism best approximates a given function. Without considering the graphical methods that are of low precision and ineffective for multiple loops and for spatial mechanisms, synthesis methods can be broadly classified into two categories: exact point approach and optimization techniques [1–3]. In its classic formulation, optimum synthesis requires minimizing the deviation between an imposed function and the actual input–output function of the mechanism, in the presence of some dimensional or functional constraints (usually maintaining practical link-length ratios, avoiding

*

Corresponding author. Tel.: +1-334-844-5867; fax: +1-334-844-5865. E-mail addresses: [email protected] (P.A. Simionescu), [email protected] (D. Beale).

0094-114X/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 9 4 - 1 1 4 X ( 0 2 ) 0 0 0 7 1 - X

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collision between elements and ensuring an acceptable motion transmission efficiency, i.e., maintaining the pressure angles in the joints less than certain limits).

2. Optimum synthesis of mechanisms for the generation of functions Suppose that a function f ðuÞ is to be generated by a mechanism over the interval ½ui ; uf , for a given input-link stroke Du ¼ uf  ui and a given output-link stroke Dw ¼ wf  wi (where ‘‘i’’ stands from initial position and ‘‘f’’ stands from final position). In order to ensure a linear dependence between the variable u and the input angle u, the following relation should be fulfilled: uðuÞ ¼ ui þ K1 ðu  ui Þ:

ð1Þ

Similarly, a linear dependence between the function value f ðuÞ and the output angle w requires that: wðf ðuÞÞ ¼ wi þ K2 ðf ðuÞ  f ðui ÞÞ;

ð2Þ

where K1 and K2 are termed scale factors and are given by the following formulas: K1 ¼

uf  ui Du ¼ Du uf  ui

and K2 ¼

wf  wi Dw : ¼ f ðuf Þ  f ðui Þ Df ðuÞ

ð3Þ

For a certain point uj in the interval ½ui ; uf  over which f ðuÞ is to be reproduced 1, the deviation between the imposed function wðf ðuÞÞ and the actual function generated by the mechanism wðuÞ is given by: dwðuj Þ ¼ wðuðuj ÞÞ  wðf ðuj ÞÞ;

ð4Þ

where wðuðuj ÞÞ must be determined through kinematic analysis of the mechanism. Without specifying at this point how the design variable are chosen, a mean-square-norm based objective function that can be employed in the synthesis of a mechanism has the form: n n 1X 1X ½wðuðuj ÞÞ  wðf ðuj ÞÞ2 ¼ ½dwðuj Þ2 ; ð5Þ F01 ð  Þ ¼ n j¼1 n j¼1 where n is the number of design points in the interval ½ui ; uf , not necessarily equally spaced. As compared to the more commonly known Euclidean norm, the mean-square-norm has the benefit that it is less dependent on the number of design points n [4]. A closer coincidence between the imposed function and the input–output function of the mechanism it is ensured by employing a maximum-norm based objective function: F02 ð  Þ ¼ max jwðuðuj ÞÞ  wðf ðuj ÞÞj ¼ max jdwðuj Þj

ð6Þ

in that at the optimum point the extreme negative and positive errors, calculated with relation (4), will result equal [5,6]. The value of the function is also easy interpretable and can be used for comparisons and quick evaluations of the performance of the mechanism solution, without the need to determine the actual output error of the mechanism through kinematic analysis. However, 1

We make here the observation that u1 and un must not be the ends of the interval [ui ; uf ].

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F02 suffers from the fact that it is not differentiable, being less suitable for minimization using a gradient-based searching algorithm (later in this paper a remedy to the nondifferentiability of the maximum-norm objective functions will be proposed). This is why F01 , which is differentiable, continues to be the most preferred objective function in doing optimum mechanism synthesis [2,6].

3. Four-bar function generator synthesis In case of the four-bar planar mechanism (Fig. 1a), alternative optimum synthesis methods have been proposed [2,7,8], like solving the over-determined systems of equations of constraint for a minimum sum-of-squares of the residuals dBCj : n X ðdBCj Þ2 ¼ ðBC  BC j Þ2 ; ð7Þ j¼1

where n is the number of design points, BC is the coupler length and BC j the variable distance between joints B and C given by the known formula: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2 Þ ð8Þ BC j ¼ ðxBj  x Cj Þ2 þ ðyBj  yCj with xBj ¼ ABcosuðuj Þ  AD=2;

yBj ¼ ABsinuðuj Þ;

x Cj

yCj ¼ CDsinwðuj Þ:

¼ CDcoswðuj Þ þ AD=2;

ð9Þ

FreudensteinÕs equation [9], that can be derived through algebraic manipulation from the same Eqs. (7)–(9), have been also used in defining over-determined systems of equations that were solved for minimum residual error [10]. Since the input–output function of the four-bar linkage is scaling invariant, the link lengths are usually normalized with respect to the ground link length AD or with the input link length AB, or these lengths are imposed fixed values during synthesis. In Fig. 1b a modified four-bar mechanism is shown that has an extra degree of freedom; this feature allows the input and output links to be positioned exactly in accordance with the imposed

Fig. 1. Schematic of a planar four bar mechanism shown in a current position j (a), and the corresponding modified 2DOF mechanism with variable-length coupler used in defining the objective function F03 , F04 and F05 (b).

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function. For j ¼ 1; . . . ; n design points, a maximum-norm based objective function to be minimized, describing the condition that the distance between the joint centers B and C of this 2DOF mechanism is to vary as little as possible during its working range (and therefore ensuring that the corresponding four-bar mechanism will best approximate the given function) is: F03 ð  Þ ¼ max jBC  BC j j ¼ max jdBCj j:

ð10Þ

This resembles solving the over-determined system of equations of constraint for a minimum residual error. It is to be noticed that for a number of design-points n less than or equal to 5 (the maximum number of design variables in case of planar four-bar function generator synthesis) the system of equations becomes determined and the transition to the zero-order exact point synthesis is made. (It is briefly mentioned here that higher order synthesis can also be applied to the mechanism, that is, to impose, together with the position of the output link, the velocity, and even the acceleration values in some design points [1]). The idea of using in the synthesis process of a modified mechanism with a variable-length coupler was introduced in the fifties by Artobolevsky et al. [11]. Because an associated mechanism with an extra degree of freedom is used in defining the objective function, it was termed in [12], in an attempt to describe a more general approach, the method of increasing the degree of freedom of the mechanism. So far it has been successfully applied to synthesize four-bar, Stephenson II and Stephenson III, and Watt II planar mechanisms, and spatial four-bar and slider-crank mechanisms [3,7,8,11–13]. The main disadvantage of employing an objective function of the type (10) is that the global minimum corresponds to a degenerate mechanism, having some of the elements of zero length (this was first noticed in [3] and it will be suggestively highlighted through 3D representations of the objective function in the following paragraph). Considering a four-bar linkage normalized with respect to the ground-link (AD ¼ 1), the global minimum will correspond to zero-length input and output link mechanisms (AB ¼ CD ¼ 0). In the case of a normalized four-bar linkage with AB ¼ 1, the global minimum will correspond to a mechanism having AD ¼ CD ¼ 0. The same degenerate solution is obtained when solving for minimum residuals the over-determined system of equations (7). However, if FreudensteinÕs equation is used instead, this degeneracy is avoided because the lengths of the input and output links appear in the denominator. Therefore, by dividing the expression of the objective function F03 with the length CD of the output link: F04 ð  Þ ¼ max jðBC  BC j Þ=CDj ¼ max jdBCj =CDj

ð11Þ

the convergence to degenerate mechanism solutions will be avoided, since for CD approaching zero the functionÕs value approaches infinity. F04 has the advantage over an objective function F03 that its feasible domain is fully continuous and has a smoother variation, therefore reducing to some extent its nonlinearity. The main disadvantage of the objective function F04 resides in the fact that its value is not a direct measure of the output error, and therefore cannot be used for quick evaluation of the performance of the mechanism obtained. Consequently, a kinematic analysis is required to determine the input–output function of the mechanism and, further on, the actual output error. Based on kinematic considerations, it has been proven that between the coupler-length variation dBCj and the output error dwj the following approximate relation exists [8,11]:

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dwj ffi dBCj =ðCDcosl Cj Þ;

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ð12Þ

where l Cj is the pressure angle in the joint C of the 2DOF associate mechanism calculated for a given design position j. The pressure angle is defined as the angle between the velocity of the jointcenter and the reaction force inside the joint, which, for mechanism performing relatively slow motions, coincides with the axis of the coupler. For planar four-bar linkages, the pressure angle is the complement of the more commonly used transmission angle, which however does not have a direct equivalent in case of spatial four-bar mechanisms. A more appropriate objective function to be minimized, that combines the favorable properties of F04 with a good estimate of the output error dwj , will be: F05 ð  Þ ¼ max jðBC  BC j Þ=ðCDcosl Cj Þj ¼ max jdBCj =ðCD  cos l Cj Þj:

ð13Þ

The pressure angle l Cj can be determined in several ways, the simplest being to subtract 90° from the angle occurring between vectors BC j and DC j : l Cj ¼ 90°  arctan jBC j DC j j=jBC j  DC j j

ð14Þ

since the instantaneous velocity vector, noted V in Fig. 1b, is perpendicular to the output link vector CD. In the following, a new proof of the above relation (12) will be given based on the elasticcoupler mechanism in Fig. 2. If an external moment Mj is applied at the output link while maintaining the input link fixed, the angular error dwj can be brought to zero through the deformations of the coupler (considered linearly elastic of rate ke , while the other elements are considered infinitely rigid). Since the work generated by the moment Mj is equal to the deformation energy of the coupler, the following relation holds: Z dw Z BCj ke dBCj dBC ¼ MðwÞdw ð15Þ BC

0

which results in: 0:5ke dBC2j ¼ 0:5Mmax j dw;

ð16Þ

Fig. 2. Four-bar mechanism with elastic coupler used to demonstrate the approximate relation between the output error dwj and the coupler-length variation dBCj .

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where Mmax j is the final value of the moment Mj for which dwj becomes zero. This equation, together with equation (17), derived from the free-body-diagram of joint C, with Fej the reaction force in this joint: Mmax j =CD ¼ Fej cos l Cj ¼ ke dBCj cos l Cj

ð17Þ

results, after eliminating ke , in the same relation (12). Fig. 2 reminds of the minimum-energy deformation synthesis methods [14]. Here however it is used only for the proof of the approximate relation (12). Because in a synthesis problem the pressure angle is determined for the deformed coupler (the input and output link in the imposed positions), it is obvious that relation (12) is an exact one only for l Cj ¼ lCj . Therefore, the closer to the minimum the better the approximation will be for both the pressure angle and the output error, which means that l C can be used with confidence in checking the maximum value of the pressure angle during the optimization process, and appropriate penalty functions can be defined using this parameter. It is to be observed that the above equations (10)–(17) are directly applicable to the synthesis of the spatial four-bar function generators (when appropriate expressions for the coordinates of joints B and C must be used), case in which the simplifications resulting from the avoidance of the direct use of the input–output function of the mechanism are even more significant than in the case of the planar mechanism [4].

4. The Ackermann linkage synthesis The Ackermann steering linkage (see Figs. 3 and 4) that will be further investigated is a symmetric four-bar function generator, a consequence of the identical steering requirements for the turning of the vehicle to the left and to the right. All real steering mechanisms are complex spatial linkages and the parameters defining their geometry are quite numerous. However, from a kinematic standpoint, when side-slipping and tire elasticity are neglected, the turning geometry of a real vehicle is not affected by change of scale. For this reason it is possible to utilize a single vehicle size parameter viz. the wheelbase vs. wheel track ratio, and dimensionless link lengths, which by re-scaling result in the required dimensions of a practical steering mechanism. This

Fig. 3. Ackermann linkage approximated as planar four-bar mechanism displayed in the reference position corresponding to the straight-ahead motion of the vehicle (a), and the associated 2DOF mechanism with variable-length coupler similar to the one in Fig. 1b (b).

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Fig. 4. Variants of steering control used in conjunction with Ackermann linkages (a), and schematic of a four-wheel vehicle during turning (b)––for the position shown, relation (18) is exactly verified (according to [4] Ackermann steering linkage cannot ensure more than one exact point during turning).

normalization of linear dimensions reduces by one the total number of geometric parameters. By further neglecting the kingpin inclination and camber angles, and considering the Ackermann linkage as a planar mechanism, important simplifications can be achieved, resulting in a reduction of the total number of design parameters to only 2. The reduced number of design parameters will allow visualizing the shape of the various objective functions introduced before, and their properties suggestively highlighted. Some empirical recommendations of how to synthesize the Ackerman steering linkage are available in the literature [15–18]. The first analytical design of the planar Ackermann linkage was performed by Wolfe [16] who combined the synthesis equation of the four-bar linkage and the condition of correct turning of the vehicle into an approximate relationship in two geometric parameters of the steering mechanism. This approach led him to consider as design variables the normalized steering knuckle arm length and the wheelbase/wheel track ratio. Parameter normalization was also used by Rao [17] in tackling the design of the same linkage. Using an exact point approach, the author produced a number of design charts, which, however, appear difficult to use mainly because of the impractical correlation chosen between the different geometrical parameters of the mechanism. One of the main requirements of the steering mechanism of a vehicle is to give to the steerable wheels a correlated turning, ensuring that the intersection point of their axis lies on the extension of the rear wheel axis (point Q in Fig. 4b), which is equivalent to the following relation (the condition of correct turning, due to Ackermann): hOA ðhI Þ ¼ arctan

1 ; cot hI þ 1=ðWb =Wt Þ

ð18Þ

where hI is the turning angle of the inner wheel, hOA is the ideal turning angle of the outer wheel, while Wb and Wt are the wheelbase and the wheel track of the vehicle respectively. In the above

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expression (18) it is usual to approximate the front wheel track Wt to the kingpin track AD as will be done in the following. The deviation between the desired and the real turning angle provided to the wheels by the steering mechanism is called the steering error. Because the steering knuckle maximum pivoting angle hImax is limited by the final drive performances or by possible wheel–axle or wheel–body interference, it is best to evaluate the steering error at the outer wheel, taking the steering knuckle from the inside of the turn as input element: dhO ðhI Þ ¼ hO ðhI Þ  hOA ðhI Þ;

ð19Þ

where hO is the turning angle of the outer wheel and hOA is the ideal turning angle calculated with relation (18). This definition of the steering error is in accordance with the fact that the steering control is achieved by actuating one of the steering knuckles, which can be considered as input link; there are however vehicles where the Ackerman linkage is controlled via hydraulic cylinders acting upon the tie rod (Fig. 4a). The normalized planar Ackermann linkage shown in Fig. 3a is defined by two parameters: the initial angle u0 and the length l of the steering knuckle arm. The distance between the two ground joints A and D is given unit value and consequently, all the linear dimensions of the mechanism are normalized with respect to Wt since AD ffi Wt . Following the considerations in the previous paragraph, the 2DOF associated mechanism with a variable-length coupler to be used in synthesis will look like the one in Fig. 3b. A maximum norm objective function equivalent to F05 in (13) will have in this case the form:      BC  BC   dBC     j  j ð20Þ F5 ðl; u0 Þ ¼ max   ¼ max  :  l cos l Cj   l cos l Cj  In the above relation the distance BC j between joints B and C corresponding to a current position j ¼ 1; . . . ; n is given by the same Eq. (8), with the difference that: xBj ¼ l cosðp  u0  hOA ðhIj ÞÞ  0:5; x Cj

¼ l cosðu0  hOj Þ þ 0:5;

yCj

yBj ¼ l sinðp  u0  hOA ðhIj ÞÞ;

¼ l sinðu0  hOj Þ;

ð21Þ

and the current turning angle hIj of the inner wheel is: hIj ¼ jhImax =n

ð22Þ

while hOA ðhIj Þ is the theoretical angle imparted to the outer wheel; notice that j ¼ 1; . . . ; n ensures that hI > 0, thus avoiding the singularity in equation (18). The reference length of the tie-rod BC in relation (20) is obtained by considering the steering knuckle arms in the straight-ahead position of the wheels, that is hI ¼ hO ¼ 0: BC ¼ 1 þ 2l cos u0 :

ð23Þ

According to Lukin et al. [18] the length of the steering knuckle arm l varies from vehicle to vehicle, usually ranging between 0.14 and 0.18 of the wheel track (Wt ), while the wheelbase/wheel track ratio (Wb =Wt ) ranges between 1.4 and 2.4. A plot of the objective function F5 in the case of a vehicle having the size ratio Wb =Wt ¼ 1:9 and for n ¼ 60 is shown in Fig. 5d. The maximum turning angle of the inner wheel was considered hImax ¼ 40°, a common value for front-wheel drive automobiles (in case of rear wheel drive

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Fig. 5. Graphic representations generated for Wb =Wt ¼ 1:9, hImax ¼ 40° and n ¼ 60 of the objective function F1 (equivalent to F01 ) (a) and of the objective function F2 (equivalent to F02 ) (c), to be compared with the objective function F6 (mean-square norm of the approximate steering error) (b) and with the objective function F5 (d). Also given are the graphs of the square of F2 (e) and square of F5 (f) which no longer experience tangent and curvature discontinuities around minima.

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vehicles and agricultural tractors, this angle may be larger, up to 50–55°). In Fig. 5b is given a plot of an approximate-steering-error mean-square-norm based objective function denoted F6 . For clarity, the upper parts of all graphs in Fig. 5 have been trimmed out together with the occasional discontinuities that might occur for l Cj ¼ 90°. The wider continuity of the approximate-steeringerror based objective functions F5 (maximum norm) and F6 (mean-square norm) is visible when comparing their graphs with the similar graphs of exactly calculated output-error based objective functions F2 (Fig. 5c) and F1 (Fig. 5a). An extended feasibility domain is beneficial to the optimization process, ensuring the convergence of the search even for starting points distant from the minimum. The graphs in Fig. 5c and d show clearly the tangent and curvature discontinuity near the optimum of all maximum-norm objective functions, which is likely to impair the convergence capabilities of first and second order optimization algorithms (particularly in the final stages of the search). As Fig. 5e and f show, a remedy to the tangent and curvature discontinuity of the maximum-norm based objective functions is to square the respective functions. For example, the function represented in Fig. 5e would have the following expression:  2  2 ð24Þ F22 ðu; lÞ ¼ max jhO ðhIj Þ  hOA ðhIj Þj ¼ max hO ðhIj Þ  hOA ðhIj Þ ; where hIj ¼ jhImax =n and j ¼ 1, n. This can be termed square-max norm and the objective functions employing this type of norm are supposed to behave better when minimized using searching subroutines that calculate the gradient by finite differences. For comparison purposes in Fig. 6a and b are given the plots of the objective functions F3 equivalent to F03 (showing that its global optima correspond to degenerate mechanisms with l ¼ 0), and of the objective F4 equivalent to F04 (that does not experience this degeneracy). Visible from both Figs. 5 and 6 is that there are two minimum steering error domains, one occurring for the angles u0 around 110° (which is the trailing Ackermann linkage) and the other for the angle u0 around 290° (which is the leading Ackermann linkage). It may also be seen from Fig. 5 that variation of the steering knuckle arm length l has a relatively small effect upon the steering error,

Fig. 6. Graphic representations generated for Wb =Wt ¼ 1:9, hImax ¼ 40° and n ¼ 60 of the objective function F3 , equivalent to F03 (the global optima of which correspond to degenerate mechanism solutions with l ¼ 0) (a), and of the objective function F4 , equivalent to F04 (which does not experience this degeneracy, being however not defined for l ¼ 0) (b).

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Fig. 7. Plots of the objective functions F1 (a) and F2 (b) generated for Wb =Wt ¼ 1:9, hImax ¼ 40° and n ¼ 60 an increased domain of the variable l, showing of shift in the global minimum domain (note that orientation of the z-axis is reversed).

and therefore its value can be chosen based on other criteria, like the maximum possible reaction forces occurring in the joints B and C or space limitation imposed by other neighboring parts of the chassis and engine. If however larger values of the length of the steering-knuckle arm are investigated, a shift of minimum domains will occur as can be seen from Fig. 7. This is in accordance with a recent finding of Yao and Angeles [19], who investigated the global-minima of an objective function equal to the sum of squares of the residuals of FreudensteinÕs equations of the planar Ackermann linkage. The global-optimum mechanism is not directly applicable due to impractical link-length ratios, and therefore in the referred paper [19] it is suggested using instead of a kinematically equivalent Watt II linkage, derived with the aid of its correspondent focal six-bar linkage. It is to be expected that the mechanism solutions obtained through optimization will be dependent on the type of objective function employed (mean square or maximum norm based) and of the number n and spacing of the design-points. The distribution of the design points has notable influence upon the results obtained only if they are in small number, comparable to the maximum number of exact points that can be imposed to the mechanism. This aspect will not be discussed here since it relates more to exact-point synthesis [1], and only the case of equally spaced design points will be considered. In Fig. 8 it is given the variation with the number of design points n of the optimum angle u0 obtained by minimizing the objective functions F1 and F2 . In case of the objective function F1 there is a smooth, monotonic variation of the optimum angle u0 towards an asymptotic value, an approximation of which can be found (with some computational effort), by running the optimization for n very large. In case of the objective function F2 , this variation is nonmonotonic, having the same trend towards a limit value corresponding to n approaching infinity. However, this limit value can be calculated precisely without resorting to a very large number of function evaluations, by making use of the following substitution: min ðdhOj Þ ¼ min f ðhI Þ ¼ min jdhO ðhI Þj

j¼1;1

ð25Þ

and performing a one-dimensional minimization of the function f ðhI Þ ¼ jdhO ðhI Þj over the interval ð0; hImax .

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Fig. 8. Variation with the number of design points n of the optimum angle u0 obtained when minimizing objective function F1 (a) and F2 (b), in case of the trailing linkage with l ¼ 0:16 and for Wb =Wt ¼ 1:9 and hImax ¼ 40°.

A number of numerical experiments have been carried out with the objective functions defined so far: F1 , F2 , F3 and F4 and of their mean-square and square-max norm counterparts, including variants in which the maximum norm was calculated by searching for the maximum of the function f ðhI Þ using BrentÕs algorithm [20]. The values of the optimum angle u0 calculated for Wb =Wt ¼ 1:9, hImax ¼ 40° and n ¼ 60 together with the corresponding maximum steering errors (determined by minimizing the same function f ðhI Þ for 0 6 hI 6 hImax ) and the relative computation effort are gathered in Table 1. It is confirmed that mean-square-norm based objective functions ensure a faster convergence of the searching algorithm than maximum-norm objective functions. The results also show some Table 1 Comparison between the optimum solutions obtained for various objective functions in case of Wb =Wt ¼ 1:9, hImax ¼ 40° and l ¼ 0:16 Objective function (j ¼ 1; . . . ; 60)

Trailing Ackermann linkage

Leading Ackermann linkage

Optimum u0

Max. error

CPU

Optimum u0

Max. error

CPU

109.76569 109.73893 109.74116

0.69358 0.68799 0.68846

1.000 1.545 4.545

295.88067 295.89347 295.89253

0.60770 0.60961 0.60947

1.000 2.000 3.454

Max norm of: BCj =l or BCj Approximate dhOj Exact dhOj Exact dhO (min dhO determined iteratively)

109.18607 109.15783 109.15899 109.15921

0.54755 0.53428 0.53405 0.53401

3.000 4.000 7.000 5.454

295.16230 295.17160 295.17114 295.17115

0.42054 0.41953 0.41939 0.41939

2.545 4.454 7.000 6.000

Square max norm of: BCj =l or BCj Approximate dhOj Exact dhOj Exact dhO (min dhO determined iteratively)

109.18607 109.15783 109.15899 109.15921

0.54755 0.53428 0.53405 0.53401

2.454 4.454 6.454 5.000

295.16230 295.17160 295.17115 295.17115

0.42054 0.41953 0.41939 0.41939

2.454 4.000 6.454 5.454

Mean-square norm of: BCj =l or BCj Approximate dhOj Exact dhOj

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convergence-speed improvements in case of the square-max-norm based objective functions, while the results are (almost) identical with those obtained using the maximum norm based objective functions (the identical results listed in the table for these two functions are due to the fact that the square-max-norm based objective functions have been actually scaled with a factor of 100). Also visible from Table 1 is a convergence increase in case of maximum or square-max-norm based objective functions with the maximum error calculated iteratively. One possible strategy that can help further reducing the computation effort is to begin the search with a mean-squarenorm based objective function and shift towards the end of the search to a maximum-norm based objective function that ensure more precise mechanism solutions. It is evident from Table 1 that the optimum mechanisms derived by minimizing the maximum-norm and square-norm based objective functions, ensure the best precision. The fluctuations visible on the graph in Fig. 8b can be also exploited in forcing an exploratory effect of the search, by first considering a small number of design points, that can be increased as the search advances, with the possibility of resorting to a precise evaluation of the maximum output error close to the optimum.

5. Ackermann linkage design recommendations One important functional requirement that has to be considered in the design of any mechanism is the insurance of a good motion-transmitting efficiency and avoiding joint jamming, which is equivalent to limiting the maximum deviation from 0° of the pressure angle in the joints. For the Ackermann linkage, this translates into the requirement that the pressure angle lC , or the approximate angle l C calculated with relation (14), does not exceed some maximum value, usually 45–50°. If a self-return of the output link is ensured due to the action of gravity or of other active forces, even larger deviations of the pressure angles can be considered acceptable. This is the case of the real Ackerman steering linkage, where the caster and camber angles have a self-aligning effect upon the steering wheels. Therefore a maximum value of the pressure angle of 65° was considered acceptable, and was included as inequality constraint in the optimization process. By minimizing with respect to u0 a square-max norm version of objective function F1 completed with the constraint lC 6 65° using BrentÕs algorithm, the 3D parametric design charts in Figs. 9 and 10 have been generated. These charts permit determining a proper initial angle u0 of the steering knuckle arm in the case of a vehicle with a known Wb =Wt ratio and for an imposed maximum turning angle hImax and a given steering knuckle arm normalized length l. The design chart in Fig. 9 should be used when selecting the parameters of a trailing Ackermann linkage, while the one in Fig. 10 should be used when designing a leading Ackermann linkage. Also given are performance charts showing the expected maximum steering error dhO and maximum pressure angles lC in the joints the corresponding steering mechanism will ensure. It is to be expected that the optimum parameters selected using the proposed charts will not require modifications in the case of the real steering mechanism, with nonzero kingpin inclination and camber angles. The designer will however need to calculate the exact maximum pivoting angle of the outer wheel (necessary for further dimensioning of the steering control mechanism) that might differ to some extent from the planar simplified mechanism.

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Fig. 9. Design charts useful for selecting the optimum angle u0 of a trailing Ackermann linkage (a), accompanied by performance charts showing the maximum steering error (b) and the maximum pressure angle in the joints (c) that can occur during operation.

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Fig. 10. Design charts useful for selecting the optimum angle u0 of a leading Ackermann linkage (a), accompanied by performance charts showing the maximum steering error (b) and the maximum pressure angle in the joints (c) that can occur during operation.

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6. Sensitivity analysis of the Ackermann linkage It has been shown earlier that variations of the steering knuckle arm length l have little influence upon the maximum steering error of the mechanism, and any manufacturing errors will have negligible influence upon the mechanismÕs performance. The angle u0 of the steering knuckle arm might also experience variations about the prescribed value, due to incorrect front wheel toeangle adjustment (presumably less than 1°) and to lesser extent due to manufacturing and assembling tolerances. The design charts in Figs. 9 and 10 show that, if within the above limits, they have little effect upon the steering error. There are however practical situations in which the geometric parameters of the steering mechanism experience significant modifications. These are the cases when the vehicle to be equipped with an Ackermann steering linkage is to have an adjustable wheelbase or wheel track. This requirement is usual for agricultural tractors and the like, the wheels of which must match the available space between various crop rows. It is therefore important to know how to select the main design parameter u0 such that the steering performance will be satisfactory in any circumstances. There are also cases when the designer may deliberately choose to have an increased maximum positive steering error in the sense of equation (19) and therefore it is of interest to know the direction in which to modify the parameter u0 , as provided by the proposed design charts, in order to achieve this. According to [21], an enlarged positive steering error will cause a reduction of the minimum turning radius of the vehicle and, correspondingly, a better maneuverability in narrow spaces. To answer these questions, the following function has been graphically studied:

max dhOj for jmax dhOj j P jmin dhOj j;

ðj ¼ 1; . . . ; nÞ; ð26Þ F2 ðl; u0 Þ ¼ min dhOj for jmin dhOj j P jmax dhOj j; where hOj is calculated as for the case of objective function F2 (which is equal to the maximum norm of an approximate steering error), for the same number of points n ¼ 60 and maximum turning angle hO max ¼ 40°, such that jF2 j is identical to F2 . Plots of this new function F2 generated for the case of a vehicle having the normalized length of the steering knuckle arm l equal to 0.16 and for trailing and leading Ackermann linkages are given in Fig. 11a and b. The same

Fig. 11. Variation with Wb =Wt and with u0 of the maximum steering error (positive or negative) in case of a trailing Ackermann linkage (a) and of a leading Ackermann linkage (b) with l ¼ 0:16 and hImax ¼ 40°.

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graphs, but for an exact calculated steering error, are available in [22] and show that the differences are almost undetectable. By comparing Fig. 11a and b, one can also observe that the leading Ackermann linkage of a given vehicle is slightly less sensitive to Wb =Wt variations than the trailing linkage is. The same graphs show that for a vehicle having an adjustable wheelbase or wheel track it is better to choose the optimum angle u0 corresponding to the maximum achievable Wb =Wt ratio. In this case, the steering error of a mechanism with same length and angle u0 of the steering knuckle arm will be positive for lower Wb =Wt ratios, a more favorable case than a negative steering error. 7. Conclusions A review of the most common type of objective used in mechanism synthesis have been carried out. For the simple case of the planar Ackermann steering linkage the properties of these objective functions have been investigated via 3D representations and numerical experiments. The calculation of the maximum absolute error in maximum norm based objective functions has been solved for the first time as a one-dimensional optimization problem, and some benefits of this approach highlighted. A new norm named square-max norm has been proposed to be used in defining objective functions, which has all the benefits of the known maximum norm based objective functions, and in addition ensures tangent and curvature continuity near the optimum. The influence of the number of design points n upon the optimum solution in the case of maximum norm based objective functions suggested starting the optimization with a small n and increase its value as the search approaches optimum; the benefits are an increased exploratory effect of the search and a reduction of the overall computation effort. Also beneficial can be the successive use of different objective functions during optimization, however with the disadvantage of increasing the problem preparation time. Finally a number of 3D parametric design charts have been proposed that can help the automotive engineer in determining the optimum geometry of the Ackermann steering linkage of a vehicle. Performance charts of the maximum steering error and pressure angle in the joints have also been provided, together with a study of the sensitivity of the mechanism error upon parameter variation. This sensitivity analysis is particularly useful when a selection has to be made between a trailing or a leading Ackermann linkage, or when designing an adjustable wheelbase or wheel track vehicle. References [1] A.G. Erdman, G.N. Sandor, S. Kota, Mechanism Design: Analysis and Synthesis, Prentice Hall, 2001. [2] A.G. Erdman (Ed.), Modern Kinematics. Developments in the Last Forty Years, Wiley, New York, 1993. [3] C.H. Suh, A.W. Mecklenburg, Optimal design of mechanisms with the use of matrices and least squares, Mechanism and Machine Theory 8 (1973) 479–495. [4] P.A. Simionescu, Contributions to the Optimum Synthesis of Linkage Mechanisms with Applications, Doctoral Dissertation, Polytechnic University of Bucharest, 1999. [5] L. Markus, The synthesis of mechanisms as a minimax principle of the optimal selection of parameters, Proceedings of the First International Symposium on Mechanisms and Computer Aided Design Methods, Bucharest, A (1973) 422-429.

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