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March 29, 2018 | Author: Bkksmk Muhammadiyah Kramat | Category: Sphere, Steering, Gear, Mechanics, Euclidean Geometry
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PII: Mech. Mach. Theory Vol. 33, No. 5, pp. 535±549, 1998 # 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0094-114X/98 $19.00 + 0.00 S0094-114X(97)00072-4 A FUNCTION GENERATING DIFFERENTIAL MECHANISM FOR AN EXACT SOLUTION OF THE STEERING PROBLEM ANTONIO CARCATERRA{ Dipartimento di Meccanica e Aeronautica, via Eudossiana 18, 00184, Roma, Italy WALTER D'AMBROGIO Dipartimento di Energetica, 67040 Roio Poggio (AQ), Italy (Received 18 September 1996) AbstractÐThe paper presents an original solution to the problem of correct steering. It is obtained by modifying the well-known six-bar linkage with the addition of a di€erential mechanism between the steering wheel and the original linkage. The di€erential mechanism, coupled to a higher pair, resolves the single input coming from the steering wheel into two outputs satisfying the condition of correct steering. Forces transmitted by the higher pair are evaluated through a static analysis. A numerical simulation, with practical criteria for the selection of relevant design variables, concludes the paper. # 1998 Elsevier Science Ltd. 1. INTRODUCTION The problem of vehicle steering has never ceased to draw the attention of researchers, as demonstrated by papers on the subject being published at an almost constant rate. Perhaps this continual interest is justi®ed by the fact that the steering linkage is one of the most widespread mechanisms, being built in as many samples as the number of produced cars and trucks. Many classical solutions are described in Ref. [1], from the approximate one obtained with the Ackermann±Jeanteaud or Panhard four-bar linkages, to the theoretically exact solutionÐthough practically a€ected by clearances in the jointsÐobtained with a centralized six-bar linkage devised at the end of the last century, and known as Bernardi double-four-bar linkage from the name of its inventor, to the Bourlet steering mechanism. The most widely adopted devices are based on the Ackermann steering linkage, optimized to yield a minimum displacement of the velocity pole of the front wheels from the rear axle. Several papers have been devoted to the analysis and synthesis of this type of mechanism, regarded by Wolfe [2] as a planar four-bar linkage, by Dvali and Aleksishvili [3] as a spherical four-bar linkage and by Arday®o and Qiao [4] as a spatial six-bar Watt chain, and, more recently, to its optimum design through nonlinear programming techniques, discussed by Zanganeh et al. [5]. The reason discouraging the use of mechanisms performing a theoretically exact steering function is that they need a great number of links and joints, giving rise to a cumulative error, due to clearances, which is comparable to the error obtained with the Ackermann linkage. Moreover, these mechanisms look quite di€erent from the simple steering linkage, so that their introduction would require large modi®cations in the car architecture: something that car manufacturers certainly dislike. The paper deals with a modi®cation of the steering mechanism in order to obtain an exact solution of the steering problem. This can be performed by introducing a variable length coupler in the four-bar linkage or a variable size ternary link in the Watt chain. The modi®cation can be regarded as a possible small correction to an already well-designed steering linkage. Since the length variation must occur as a consequence of the rotation of the steering wheel, the mechanism must retain a single degree of freedom. Moreover, the length variation should be such that {To whom all correspondence should be addressed. 535 each cam pair should produce a quite large lift and transmit the whole force necessary to turn the corresponding wheel. such as the six-bar Watt chain shown in Fig. in traditional steering mechanisms. together with numerical simulations. and c1. 2. A better solution is the use of a di€erential mechanism resolving the single input given by the rotation of the steering wheel into the two outputs that are needed to turn each front wheel of the right amount: the correct relationship can be obtained by enforcing a constraint through a higher pair. the axes of the front wheels intersect at a point P0 lying on the rear axis. 2(b). . After introducing the basic idea of the paper. Hence the length variation can be derived as a function of the turning angle. D'Ambrogio the rotations of the front wheels satisfy the requirement for a correct steering. thus a€ecting the law of motion and yielding no advantage with respect to the Ackermann steering linkage. The condition can be expressed as: a f …c1 . 2(a)]. However. the two mirrorimage four-bar linkages m1 and m2 transform the displacement s into turning angles c1 and c2 that approximately satisfy condition (1). 1. This would rapidly bring to wear damage of the cam pair. Equivalently. 1. the condition of correct steering requires that for any turning angle. In fact. Then. Condition (1) is only approximately met by steering mechanisms of the Ackermann type [Fig. with an appropriate design. c2 are the turning angles of the front wheels. c2 † ˆ cot c1 ÿ cot c2 ÿ ˆ 0. Let: Fig. this could be accomplished by using a cam-follower chain to drive independently each front wheel. In this case. The analysis will include practical considerations on how to choose the primary dimensions of the mechanism. BASIC IDEA As known. only a fraction of the force necessary to turn the wheels acts directly on the higher pair. a is the distance between the pivot centers of the front wheels. a general procedure to design a function generating di€erential mechanism will be discussed and then applied to the particular case of the modi®cation of the Ackermann linkage.536 A. Carcaterra and W. 1. the rotation b is converted into a displacement s by a rack-and-pinion mechanism. …1† p where p is the wheel base of the vehicle. as shown in Fig. as shown in Fig. The condition of correct steering. 2. (b) decomposition into two mirror-image four bar linkages driven by a rack and pinion mechanism. Due to mirror-image symmetry: g1 …s† ˆ ÿg2 …ÿs†: …3† . (a) Six-bar steering mechanism. c1 ˆ g1 …s† c2 ˆ g2 …s† …2† be the transfer functions of m1 and m2.Exact solution of the steering problem 537 Fig. the displacements s1 and s2 must be such that: f …g1 …s1 †.538 A. 3. 3(b)] converting the input rotation b of the carrier into the two displacements s1 and s2. as shown in whose exploded view is shown in Fig. the prodi€erential mechanism with cylindrical gears. Steering mechanism: (a) the rack and pinion mechanism is replaced by a di€erential gear. the di€erential mechanism must transform the input b into two outputs s1. two planet wheels (P) with axis . (However. c2 to satisfy condition (1). (b) scheme of the di€erential gear. In the modi®ed mechanism of Fig. D'Ambrogio Fig. 3(a). SOLUTION USING A DIFFERENTIAL GEAR Here. di€erential mechanism with bevel gears. consists of a ®xed a carrier (S) with axis a±a. s2 satisfying Equation (4). 4. 3. g2 …s2 †† ˆ 0: …4† Consequently. the steering gear G consists of a di€erential mechanism [Fig. The di€erential gear. reference will be made to a cedure can be easily adapted to a Appendix A). For the turning angles c1. spherical housing (B) of radius RB. Carcaterra and W. Let O and P be the intersections of c±c and d±d with the spherical surface. reference meridian corresponding to the position of O when the front wheels are in the straight position. when b = 0. The carrier is free to rotate around axis a±a with respect to the housing. …6† . 4. the coordinates of P. Exploded view of the di€erential mechanism. If the point lies on the mean surface of the housing. C2) with axis a±a. A ®xed spherical coordinate system can be considered with a polar axis a±a. the latitude and the longitude (see Fig. The coordinates of point O are: Wo ˆ 0 jo ˆ b: …5† Assuming that for b = 0 the arc OP forms an angle E with the reference meridian. and two gears (C1. respectively. r = RB and the only signi®cant coordinates are W and j.Exact solution of the steering problem 539 Fig. Both the planet gears are linked to the carrier by pin-joints with axis c±c. and origin at the centre C of the housing: the coordinates r. c±c orthogonal to the carrier axis a±a. are: WPo ˆ arctan…tan d cos E† jPo ˆ arctan…tan d sin E†. W and j of a given point represent the distance from C. one planet gear is coupled to the ®xed sphere by a spherical cam pair. where an eccentric stud with axis d±d located at an angle d from c±c slides within a groove machined on the inner surface of the sphere. In order to obtain the desired relationship between the rotation of the two gears C1 and C2. 5). If the carrier is rotated through an angle b about a±a point O describes an arc on the equatorial circle and subtends angle b. The absolute rotation of the gears is the sum of gC1r. gP † ˆ 0 …12† which amounts to an implicit relationship between b and gP. 3(b)]. select RW. g2 …bRW ÿ gP cot aRW †† ˆ 0.540 A. is: s1 ˆ gC1 RW ˆ bRW ‡ gP cot aRW s2 ˆ gC2 RW ˆ bRW ÿ gP cot aRW …10† where RW is the pitch radius of the pinion. gC2: gC1 ˆ b ‡ gP cot a gC2 ˆ b ÿ gP cot a: …9† Each of the gears transmits the motion to a pinion that is part of a pinion±rack pair.e F…b. the displacements of the two racks. gP can be computed through Equation (12) in order that condition (4) is satis®ed. while for b $0. gC2r. gP(b)) and jP(b. yielding: f …g1 …bRW ‡ gP cot aRW †. 2bmax . g2(s2)) = 0. and b (frame motion). which in turn drive the steering arms. E 2. D'Ambrogio Fig. for any given input angle b. gP(b)) of the groove that allows one to satisfy the condition of correct steering are obtained. Coordinate system on the spherical housing. Carcaterra and W. Then. . In summary. . …11† i. gP and gC1. b. 5. the solution can be determined according to the following algorithm: 1. The above expressions can be introduced into condition (4). point P is constrained to slide along the groove s so that OP must rotate of an angle gP around axis c±c. …8† a being the pitch angle of the gears [Fig. so that the di€erential gear de®nitely yields the following relationship between the kinematic variables b. The relationship among the coordinates of P and the variables b and gP is as follows:  WP ˆ arctan…tan d cos…gP ‡ E†† : …7† jP ˆ arctan…tan d sin…gP ‡ E†† ‡ b The relative rotation gCr of the gears C1 and C2 with respect to the carrier is related to the rotation of the planet gear by: gC1r ˆ gP cot a gC2r ˆ ÿgP cot a. f(g1(s1). By denoting with s1 and s2. by introducing b and gP(b) into Equation (7). the parametric equations WP(b. Therefore. a. loop on b = 0 . 1. for any input angle b. due to the forces transmitted to the planet wheel by the wheels C1 and C2. dgC2r are the virtual rotations of the crown gears with respect to the carrier. that singular positions of the mechanism are avoided and. possibly. …14† where CP = RBn. dgC2r can be computed from Equation (8) and substituted into Equation (15). Though these forces may exhibit a slight variation with the position of the links connected to each wheel. compute gP such that F(b. …17† where f is the friction coecient between the groove and the stud. It is obtained. that no overload occurs on the higher pair. Moreover. …18† …19† . The moment of the contact force S about the planet axis c±c is given by: MS ˆ …CP  S†  c. with the help of Fig. where S and N can be evaluated at any point of s. 4): n t b = tn c be the unit vector normal to the sphere at P the unit vector tangent to the groove s in P and opposed to the slip velocity w of the stud the unit vector normal to n and t the unit vector parallel to the planet axis c±c The contact force S between the groove and the stud can be expressed as: S ˆ Nb ‡ Ft. FW2 are the forces that must be applied to the racks to turn the tyres with respect to the road. 4.e. dgC1r. and to ensure that its value remains limited. i. compute WP=arctan(tan d cos(gP+E)) jP=arctan(tan d sin(gP+E)) + b end loop.Exact solution of the steering problem 541 2. for the sake of simplicity it will be assumed that FW1=FW2=FW=constant. The moment about the planet axis c±c. thereby yielding: …MS ÿ 2FW RW cot a†dgP ˆ 0 ) MS ˆ 2FW RW cot a: …16† By expanding MS one obtains: MS ˆ ‰CP  …Nb ‡ Ft†Š  c ˆ ˆ NRB ‰n  …t  n†Š  c ‡ FRB …n  t†  c ˆ ˆ NRB ‰…n  n†t ÿ …t  n†nŠ  c ÿ FRB b  c ˆ ˆ NRB t  c ÿ FRB b  c ˆ NRB …t  c ÿ f b  c†. …15† where dgC1r. …13† i. 3(b) indicating the signs: MS dgP ÿ FW1 RW dgC1r ‡ FW2 RW dgC2r ˆ 0.gP) = 0 2.2. can be computed by applying the principle of virtual work to the di€erential gear in the motion relative to the carrier. as the sum of a normal component parallel to b and a friction component parallel to t. i. Let (see Fig. By combining Equations (16) and (17) it follows: Nˆ and: 2FW RW cot a .e. and FW1.e. RB …t  c ÿ f b  c† p S ˆ N 1 ‡ f 2. STATIC ANALYSIS The aim of the analysis is to evaluate the force transmitted by the higher pair formed by the stud and the groove. one has: sin…jP ÿ b† ˆ 0 ) sin…gP ‡ E† ˆ 0 sin WP ˆ 0 ) cos…gP ‡ E† ˆ 0: …27† Therefore. gP(b)) and jP(b. de®ned by the spherical coordinates WP(b. i. respectively. to derive useful design information.e. An in®nitesimal arc ds can be described in the local coordinates by: dx ˆ RB djP dy ˆ RB dWP dz ˆ 0 …21† and the unit vector t can be expressed as tˆ where ds ˆ dx dy i‡ j. from Equation (7): @jP cos…gP ‡ E† @gP ˆ ‡ 1 6ˆ 0 for any gP ‡ E: 2 @b 1 ‡ ‰tan d sin…gP ‡ E†Š @b …28† The derivative @WP/@b in Equation (26) vanishes under the same condition expressed by the ®rst of Equation (27): @WP sin…gP ‡ E† @gP ˆÿ ˆ 0 ) sin…gP ‡ E† ˆ 0: 2 @b 1 ‡ ‰tan d cos…gP ‡ E†Š @b Thus tc = 0 only if sin(jP ÿ b) and @WP/@b are zero. the unit vector c can be expressed in the form: c ˆ ÿsin…jP ÿ b†i ÿ sin WP j ‡ cos WP k …25† @jP @WP ÿ sin WP : @b @b …26† so that t  cA ÿ sin…jP ÿ b† Singularities occur when t  c ˆ 0: From Equation (7). sin(jP ÿ b) and sin WP in Equation (26) can never vanish at the same time. In fact. ˆ RB @b ds ds …23† dy @WP : A @b ds …24† that is dx @jP A @b ds On the other hand. Let i. The dot product t  c must be evaluated at any point P of the groove s. the meridian and the normal to the sphere at P. y and z. D'Ambrogio The foregoing computation can be performed numerically once the groove s is completely de®ned in order to verify that the load on the higher pair is below some prescribed value. that is when t is orthogonal to c. The derivative @jP/@b in Equation (26) is always di€erent from zero. A local cartesian coordinate system can be selected with origin in P and axes xyz oriented as the parallel. i. j and k be the unit vectors corresponding to x. ds ds …22† q p dx2 ‡ dy2 ˆ RB dj2P ‡ dW2P . by setting: Nˆ 2FW RW cot a RB t  c …20† It can be seen that N 4 1 as t  c 4 0. gP(b)).542 A. However. a simpli®ed analysis can be conducted by neglecting the friction force. and dx @j db ˆ RB P  @b ds ds dy @WP db  . if: …29† . Carcaterra and W.e. the condition that must be ful®lled to avoid singularities is: ÿp < gP ‡ E < 0 for E < 0 0 < gP ‡ E < p for E > 0. the length l of the two tie-rods A1B1. …31† which becomes an important design constraint. 5. the distance h between the front axle and the translating bar B1B2 connecting the tie-rods. O2A2. . 6): the length r of the two levers O1A1. implying that the point P must never cross the meridian on which the point O lies. …30† Recalling that when the front wheels are in the straight ahead position gP=0. the distance between the front axleÐexpressed as a fraction f of the wheel base pÐand the intersection I between the extensions of the levers when the wheels are in the straight-ahead position. .1.Exact solution of the steering problem sin…gP ‡ E† ˆ 0 4 gP ‡ E ˆ lp. Rotation of the front wheels corresponding to the maximum steering angle and primary dimensions of the six-bar steering linkage. . This yields the maximum value of the steering angle of the outer wheel: p ˆ arcsin min . It is assumed that p = 2.6 m. 22. 21. 6. MODIFICATION OF A SIX-BAR STEERING LINKAGE The above ideas are implemented on a previously designed six-bar steering linkage that is modi®ed with the addition of a di€erential gear in order to satisfy the condition of correct steering for any required turning angle. 543 l ˆ 0.3 m and the minimum required steering radius. Rmin st =6 m (see Fig. These dimensions Fig. . a = 1. …32† cmax 1 Rst while the corresponding rotation of the inner wheel should be p ˆ arctan min cmax 2 Rst coscmax ÿa 1 …33† 5. A2B2. 6). Primary dimensions of the traditional steering linkage The primary dimensions of the six-bar steering linkage are represented by (see Fig. and f = 0.3 m. Carcaterra and W. are selected as: r = 0. they imply low speed and consequently low tyre wear. In fact. gP …b ‡ 2p††. Design of the additional di€erential gear The design parameters for the additional di€erential gear are the angle E between the arc OP0 and the reference meridian on the sphere. Steering error of the six-bar linkage: di€erence between the actual and the exact value of the steering angle of the inner wheel versus the steering angle of the outer wheel. 7 as function of c1. D'Ambrogio Fig.01 m. obtained in correspondence with cmax 1 . ÿbmax < b < bmax : …35† . the pitch radius RW of the pinion is given by the ratio of smax and the rotation bmax SW of the steering wheel corresponding to the maximum steering angle: RW ˆ smax bmax SW …34† For a typical value of bmax SW 34p. and eventually the transmission ratio t between the rotation of the steering wheel bSW and the rotation b of the carrier. gP …b†† 6ˆ WP …b ‡ 2p. with the pinion linked directly to the steering wheel. moreover. A ®rst design constraint on t can be identi®ed as follows. To avoid self-intersections. h = 0. the angle d between the eccentric stud axis d±d and the planet axis c±c.544 A. then it increases for larger angles: the trend is considered acceptable since large steering angles are encountered with low probability during the ordinary use of the vehicle.2. By assuming that the bar B1B2 is driven from the steering wheel through a rack and pinion mechanism.3 m. it is not advisable to couple the steering wheel directly to the carrier if self-intersections of the groove s are to be avoided. the pitch angle a of the gears C1 and C2. The maximum displacement smax of the bar B1B2. Such self-intersections could create problems in the coupling between the stud and the groove.78 thereby yielding the steering error Ðdi€erence between the actual value and the correct value of c2Ðrepresented in Fig. The error is very low for small steering angles. RW is found using the primary dimensions listed above as RW30. l = 0. 5. it is required that: WP …b.3 m. and. can be computed from the primary dimensions of the mechanism through Equation (2). 7. Adding the two equations. Parametric representation of the groove s.e. Fig. Recalling condition (31). Subtracting the second equation from the ®rst. for instance. 8. it is obtained: s1 ‡ s2 ˆ 2bRW ) RW ˆ …s1 ‡ s2 †max …s1 ‡ s2 †max . …38† ÿp=2 < gP < p=2. In fact. this requires: E ˆ 2p=2.Exact solution of the steering problem 545 A sucient condition for this is that the total rotation of the carrier 2bmax does not exceed 2p. in order to keep the mechanism far enough from singularities. The choice of E can be made so that the singular positions are located symmetrically with respect to the straight-ahead position of the front wheels. ÿp=3  gP  p=3 i:e: j gP j  p=3: …40† This can be translated into a condition on the pitch angle a. bmax < p ) t ˆ b 1 < : bSW 4 …36† The pitch radius of the pinion can be computed from Equation (10). …39† so that Equation (31) yields in all cases and. it is obtained: s1 ÿ s2 ˆ 2gP cot aRW . the design constraint on gP can be made more strict by requiring. i. …41† .  2bmax 2p …37† max where smax can be computed through Equation (2) once cmax and cmax are determined 1 and s2 1 2 using Equations (32) and (33). the rotation of the planet can be computed from Equation (10). . the contact force has been converted to nondimensional units after dividing by the force that should be applied to the bar B1B2 in the traditional linkage. As a consequence. as p/2 ÿ a. Figure 10(a) and (b) shows the normal component of the contact force vs the carrier rotation.e. d was selected as 1. Although the angle d can be readily selected as the pitch angle of the planets. D'Ambrogio i. gP ˆ s1 ÿ s2 j s1 ÿ s2 jmax p )j gP jmax ˆ  : 2RW cot a 2RW cot a 3 Since the only quantity still to be determined is a. pRW 2 3 …s2 ÿ s1 †max …42† …43† which yields the condition on the pitch angle of the gear and leads to the design of the di€erential gear.546 A. although the weight increase is not substantial. The design constraint on a was a < 818. 9.e.5 ÿ 2(p/2 ÿ a). i. 6. It should be noted that the contact force remains limited and varies slightly with respect to its average value. r. leaving substantially unaltered the traditional Ackermann linkage. accordingly a was selected as 708. while Fig. A numerical simulation was carried out using the previously selected values of Rmin st . i. Figure 8 shows the parametric representation of the groove s in the spherical coordinates W and j. such as 1.5(p/2 ÿ a). 9 shows the planet wheel rotation vs the carrier rotation. CONCLUSIONS The relevant features of the solution proposed in the paper can be summarized as follows: . the employment of this mechanism does not a€ect the basic architecture of the front part of the vehicle. Planet rotation vs carrier rotation. l. 308 and E as 908. to obtain a smaller contact force on the groove. it can be desirable to chose a higher value.e. Moreover. h and f. Fig. Carcaterra and W. The exact steering is obtained only by modifying the steering box. Of course the devised solution introduces additional cost and weight to the steering mechanism. usually designed to host a simple and space-saving Ackermann steering linkage. it follows that:   3 j s1 ÿ s2 jmax 2 pRW ) a  arctan cot a  . The correct steering is obtained by introducing a relatively small number of additional kinematic pairs. Finally. can be extended without relevant theoretical modi®cation by considering it as a spatial six-bar Watt chain. With respect to a cam-follower mechanism. and consequently. Normal component of the contact force on the groove for: (a) positive. 10. directly driving each front wheel. . the errors introduced by clearances are limited. with respect to the traditional Ackermann linkage. 7).Exact solution of the steering problem 547 Fig. the use of a di€erential mechanism leads to a signi®cant reduction of the wear damage of the cam mechanism. . it can be noticed that the previously developed kinematic synthesis of the spherical cam mechanism. The proposed solution can be extended to a four-wheel steering vehicle simply by using two connected gear steering di€erential mechanisms similar to that described in this paper. . The comparison with the six-bar steering linkage shows substantial improvements for large steering angles (see for instance Fig. . . and (b) negative carrier rotation. applied to the Ackermann steering linkage regarded as a planar mechanism. in Italian. 1971. 22. b is the rotation of the carrier and gP is the rotation of the planet gear with respect to the carrier. N. Scheme of the cylindrical di€erential gear. A. where the link CO represents the carrier. D. Front view of the cylindrical di€erential gear. Torino. 167±175. 1987. 4. A. 5.. Mechanisms and Machine Theory.548 A. and KecskemeÂthy. Wolfe. Costruzioni Automobilistiche. Zanganeh. K. Il Veicolo.. and Aleksishvili. 6.. 1959. Levrotto & Bella.. 6. Transactions of ASME Journal of Engineering and Industry. D'Ambrogio REFERENCES 1. Italy. 2524±2528. pp. . 1960. Chap. Pollone.. Arday®o. 10±14. R. D. Carcaterra and W. A1. G. 1995. Re and Ri are the pitch radii of the ring and sun gears. while Rp is the pitch radius of the planet gear.. D. On the optimum design of a steering mechanism. A2. Angeles. 315±319. Dvali. The rotations ge and gi of the ring and sun gears are given by:  ge ˆ b ‡ gP  Rp =Re …A1† gi ˆ b ÿ gP  Rp =Ri Fig. J. W. Fig. and Qiao. 1. 3. Analytical design of an Ackermann steering linkage. APPENDIX A Design Of A Cylindrical Di€erential Gear The epicyclic train is schematically shown in Figs A1 and A2. 2. E. Proceedings of the 9th World Congress on the Theory of Machines and Mechanisms. R. Mechanism and Machine Theory. The parametric equations of the groove can be established as follows (see Fig. it is: s1 ˆ ge RW ˆ …b ‡ gP  Rp =Re †RW s2 ˆ gi RW ˆ …b ÿ gP  Rp =Ri †RW …A2† where RW is the pitch radius of both pinions. an implicit relation between b and gP can be found similar to Equation (12): Fc …b. . A2). where a stud P placed at a distance r (<Rp) from the planet gear axis is constrained to slide within a groove s machined on the plate surface. In the ®xed reference frame xy. the coordinates of P are:  xP ˆ …Re ÿ Rp †cos b ‡ r cos…gP ‡ E† yP ˆ …Re ÿ Rp †sin b ‡ r sin…gP ‡ E† …A4† …A5† The pairs (b. By introducing the above expressions into the condition of correct steering [Equation (4)]. The position of the stud P ®xed with the planet wheel can be expressed in complex form as: CP ˆ CO ‡ OP ˆ …Re ÿ Rp †ejb ‡ rej…gP ‡E† where E is the initial orientation of OP with respect to CO. gP) that satisfy the implicit relation (A3) can be introduced into relation (A5) to yield the parametric equations of the groove s that solves the problem. with x aligned with the initial position of the carrier.Exact solution of the steering problem 549 and assuming that both are connected to rack and pinion mechanisms that drive the steering arms. the planet gear can be linked to a ®xed plate by a cam pair. gP † ˆ 0 …A3† To enforce this condition. 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