Acta Polytechnica


DOI:10.14311/AP.2020.60.0252
Acta Polytechnica 60(3):252–258, 2020 © Czech Technical University in Prague, 2020

available online at https://ojs.cvut.cz/ojs/index.php/ap

KINEMATIC SYNTHESIS OF CENTRAL-LEVER STEERING
MECHANISM FOR FOUR WHEEL VEHICLES

Santiranjan Pramanika, b, ∗, Sukrut Shrikant Thipsec

a Symbiosis International (Deemed University), Faculty of Engineering, Department of Mechanical Engineering,
Lavale, Mulshi, Pune, Maharashtra 412115, India

b College of Military Engineering, Department of Mechanical Engineering, Old Mumbai - Pune Hwy, Dapodi,
Pimpri-Chinchwad, Maharashtra 411031, India

c The Automotive Research Association of India, Survey No.102, Paud Road, Rambaug Colony, Kothrud, Pune,
Maharashtra 411038, India

∗ corresponding author: santiranjan_pramanik@rediffmail.com

Abstract. A central lever steering mechanism has been synthesized to obtain five precision points for
a four-wheel vehicle using Hooke and Jeeves optimization method. This compound mechanism has
been studied as two identical crossed four-bar mechanisms arranged in series. The optimization has
been carried out for one crossed four-bar mechanism only instead of the entire mechanism. The number
of design parameters considered for the optimization is two. The inner wheel has been considered to
rotate up to 52 degrees. The steering error, pressure angle and mechanical advantage of the proposed
mechanism have been compared with those achieved by the Ackermann steering mechanism. The
proposed mechanism has less steering error, more favourable pressure angle and increased mechanical
advantage. The method of compounding the mechanism is also applicable when the central lever is
offset from the longitudinal axis of the vehicle.

Keywords: Ackermann steering, four-wheel vehicle, Hooke and Jeeves method, compound steering
mechanism, unsymmetrical mechanism, steering error, crossed four-bar mechanism.

1. Introduction
There are several types of steering mechanisms used
for four-wheel vehicles. There are symmetric four-
bar mechanism, centre-lever mechanism and rack-and-
pinion type mechanism. Fahey and Huston [1] consid-
ered an eight-bar mechanism for steering automobiles.
They suggested this mechanism after modifying a lead-
ing Ackermann steering mechanism. This eight-bar
mechanism eliminated the divergent end behaviour [1]
of the existing Ackermann mechanism and the maxi-
mum steering error was reduced to 0.03 degree. The
progressive deviation in the steering error curve with
the increasing rotation of the inner wheel nearing the
end range is referred here as divergent-end behaviour.
However, there were two small and disproportionate
links in the eight-bar mechanism and hence the mech-
anism was unpractical.

Simionescu and Smith [2] found out initial estimates
for the design of centre-lever steering linkages. They
considered both leading and trailing mechanisms for
which they produced parametric design charts.

De-Juan et al. [3] considered kinematic synthesis of
six-bar steering linkages for both leading and trailing
configurations. They considered mixed-leading and
mixed-trailing configurations as well. In their work,
it was found that the steering error curve had five
precision points for leading configuration only.

De-Juan et al. [4] considered three types of leading
steering mechanisms and optimized these considering

steering error and transmission angles and compared
them.
Bajpai et al. [5] explained optimization of a func-

tion of several variables by Hooke and Jeeves method,
which was a direct method in which it was not neces-
sary to differentiate the function with respect to the
variables.

Zhao et al. [6] proposed a geared five-bar steer-
ing mechanism with noncircular gear sectors capable
to exactly satisfy Ackermann’s formula. However,
manufacture imperfections and backlash can produce
steering error. They suggested that this mechanism
can be applied to light carriages.
Peterson and Kornecki [7] considered a design of

the steering mechanism of a wide power frame used
to harvest tree fruits. The unit was collapsible to
half the width for the purpose of transportation. The
ratio of the wheel track to wheelbase was 1.55 while
harvesting fruits and 0.77 while being transported on
the road.
A crossed four-bar steering mechanism that in-

cluded two spur gears has been considered in ref-
erence [8]. There, Hooke and Jeeves method has been
used to minimize the objective function that com-
prised of steering error only. The present mechanism
has been considered to consist of two such crossed
four-bar mechanisms to eliminate the spur gears. The
mechanism provides five precision points and is a
trailing mechanism. The kinematic synthesis has been
necessary for crossed four-bar mechanism only and

252

https://doi.org/10.14311/AP.2020.60.0252
https://ojs.cvut.cz/ojs/index.php/ap


vol. 60 no. 3/2020 Kinematic synthesis of central-lever steering mechanism. . .

there are two design parameters, thus simplifying the
calculation.
The aim of the present work is to carry out kine-

matic synthesis of a centre lever steering mechanism
in a simplified approach. The kinematic synthesis of
the offset centre lever steering mechanism has been
considered here for the first time. It was found that
the offset centre lever steering mechanisms can be
designed easily if the central hinge is not imposed to
be located on the longitudinal axis of the vehicle.

2. Crossed four-bar steering
mechanism

Figure 1 shows a crossed four-bar steering mechanism.
The vehicle has been turned towards right. AGFB is
the crossed four-bar mechanism. The inner wheel is
rotated about centre I and the outer wheel has been
rotated about centre A. When the inner wheel rotates
by angle FBE, then the outer wheel rotates by angle
GAD. As the steering arm BF is rotated counter
clockwise, the steering arm AG is rotated clockwise.
Since these are rotating in opposite directions the use
of spur gears has become necessary.

3. Condition of correct steering
The steering is assumed geometrically correct if the
axes of rotation of the wheels intersect at the instanta-
neous centre of rotation of the vehicle relative to the
ground. Figure 2 shows that when the inner wheel
rotates by an angle γ, then the outer wheel rotates
by an angle α. The steering position is correct as all
four wheels rotate about a common centre O. From
this geometry the following equation can be written:

cot α− cot γ =
t

w
(1)

4. Optimization method
The mechanism shown in Figure 3 has two parameters.
These are angle KAG (β) and steering arm length
AG (r). The inner wheel has been rotated by angle
φ (angle FBE) and then the rotation of the outer
wheel (angle GAD) has been found out. The correct
rotation of the outer wheel has been found out using
an equation (1). Then the steering error has been
calculated as follows.
The initial straight-ahead position has two equal

angles KAG and FBH. The two steering arms AG
and FB are equal. The length of the coupler has been
found as

GF =
√

(d− 2r sin β)2 + (2r cos β)2 (2)

Angle BAE = tan−1
(
BEV

AEH

)
(3)

where
BEV = r sin

(π
2
−β + φ

)
(4)

and
AEH = d−r cos

(π
2
−β + φ

)
(5)

Angle DAE = cos−1
{
r2 + (AE)2 − (GF)2

2 ×r ×AE

}
(6)

where
AE =

√
(BEV )2 + (AEH)2 (7)

Angle DAB = Angle DAE − Angle BAE (8)
The rotation of the outer wheel is given by

Angle GAD =
π

2
−β − Angle DAB (9)

The correct angle of rotation of the outer wheel is
given by

Angle GADcorrect = cot−1
(

cot φ +
t

w

)
(10)

The steering error is given by

Error = Angle GAD − Angle GADcorrect (11)

The objective function is given by

Obj fun =
∑

Error2 (12)

The limits of the above sum are φ from zero
to 52 degree. The design variables are β and
r.
Hooke and Jeeves optimization method [5] has

been applied to minimize the objective function (12).
We use, as initial estimate, r = 1.9374 and angle
β = 13.5407° obtained by an analytical method in
another study [9]. A vehicle with a track to wheel-
base ratio 0.326 has been considered, for which the
final estimate has been made. Using the concept of
compound mechanisms, the track to wheelbase ra-
tio has been taken as 0.163. By optimization, we
found the length of the steering arm 1.933 units and
inclination angle β as 13.521 degree, where the dis-
tance AB is 10 units. The steering arm angles are
β = 13.521 degree. We have shown the steering error
curve in Figure 4. The maximum steering error of the
proposed compound mechanism has been reduced to
0.044 degree by the through optimization. The steer-
ing error of the proposed mechanism is less compared
to that of the Ackermann steering mechanism of the
benchmark vehicle [10].
The pressure angle at any joint is defined

as the angle between the output link and the
direction of the force applied by the input link.
The pressure angles at joint D and E are found out
as follows:

Angle (ADE) = cos−1
{
r2 + (GF)2 − (AE)2

2 ×r ×GF

}
(13)

The pressure angle at joint D is given by

µ1 =
{

Angle (ADE) −
π

2

}
(14)

253



Santiranjan Pramanik, Sukrut Shrikant Thipse Acta Polytechnica

Figure 1. Crossed four-bar steering mechanism including gears.

Figure 2. Geometry of correct steering.

Figure 3. Geometry of a crossed four bar mechanism.

254



vol. 60 no. 3/2020 Kinematic synthesis of central-lever steering mechanism. . .

Figure 4. Comparison of steering error.

Now the length DB is given by

DB =
√
{d−r · cos(DAB)}2 + {r · sin(DAB)}2

(15)
The angle (BED) is given by

Angle (BED) = cos−1
{
r2 + (DE)2 − (DB)2

2 ×r ×DE

}
(16)

The pressure angle at joint E is given by

µ2 =
{π

2
− Angle(BED)

}
. (17)

The pressure angle of the proposed steering mecha-
nism has been compared with that of the conventional
Ackermann steering of the benchmark vehicle [10]. As
Figure 5 shows, the pressure angle of the proposed
compound steering mechanism is less than that of the
Ackermann steering mechanism.
Mechanical advantage of a mechanism is de-

fined as the ratio of the torque produced in the
output link and the torque provided in the in-
put link. This has been calculated [9] from the
product (cos µ1 × cos µ2).

The mechanical advantages of the two mechanisms
have been compared in the Figure 6 and it has been
found that the proposed mechanism has a higher me-
chanical advantage.

5. Compound six member
mechanism

A compound six-member mechanism has then been
considered, comprising of two mirrored crossed four-
bar mechanisms. The compound mechanism has a
track to wheelbase ratio 0.326, which is twice the
ratio of 0.163. The track to wheelbase ratio 0.163 for
a vehicle is not feasible, but using ratio 0.163 after
compounding effect is suitable as the vehicle has the
said ratio 0.326 which is used in benchmark vehicle12
M, a bus by Ashok Leyland [10] which is the same as
above. The front wheel track of this vehicle is 2020 mm
which is shown as 20 units in Figure 7. Hence one unit

is 101 mm in this figure. The wheelbase of the vehicle
is 6200 mm. The length of the arms CE, BD, BF
and AG is 1.933 units, which are equal to 195.2 mm.
The length of the coupler GF and DE is 994 mm.

The crossed four bar mechanism is such that two
such chains can be series connected as shown in the
Figure 7. Here, the central lever BFD can be moved
backwards in order to increase the space for installing
the engine and the gearbox. The paper under refer-
ence [8] uses spur gears to bring the mechanism behind
the front axle and the use of gears may produce more
steering error. So the mechanism suggested in the
present paper has two benefits. The first benefit is the
availability of more space and the second one is the
elimination of the spur gears. The Figure 8 shows the
modification of the mechanism to increase the avail-
able space. The present paper shows the usefulness
of crossed four bar chain. In this figure, the central
hinge joint is shifted backwards by the distance OG,
which is 520 mm, for example. Now the distance GF
and AG is 1136 mm. The arms AB, GC, GD and EF
are all 219.6 mm in length. The length of the coupler
DE and BC is 1118 mm each. The length of the link
CD is 286.8 mm.
In case there is such a constraint that the hinge

point G cannot be placed on the longitudinal axis
of the vehicle, then this can be accommodated by
modifying the mechanism as shown in the Figure 9.
The kinematic synthesis for such an offset centre lever
mechanism is difficult by the presently available meth-
ods. But the concept of compounding the mechanism
is very useful to devise an offset centre lever steering
mechanism.

In Figure 9, it has been shown that the central hinge
G has been located by distances x and y. This shift
is not done intentionally but is required when there is
a constraint that the central hinge cannot be placed
on the longitudinal axis. Here, the distances x and y
have been chosen arbitrarily as an example. The links
AB and GC are equal in size. Also, the links EF and
GD are equal in length. For a non-zero value of the
distance x, the links GC and GD are unequal. In this

255



Santiranjan Pramanik, Sukrut Shrikant Thipse Acta Polytechnica

Figure 5. Comparison of pressure angles.

Figure 6. Comparison of mechanical advantages.

Figure 7. Compound six bar mechanism.

256



vol. 60 no. 3/2020 Kinematic synthesis of central-lever steering mechanism. . .

Figure 8. Modification of the mechanism to increase space.

Figure 9. Offset center lever steering mechanism.

figure, for example, x is 200 mm and y is 500 mm. The
distance AO and FO are equal to half of the front
track, which is 2020 mm. Using Pythagorean theorem,
we find the distances AG and GF as 1309.2 mm and
951.9 mm respectively. The arms EF and GD are
184 mm in length. But the arms AB and GC each
are 253.1 mm in length. The length of the coupler
BC is AG× 994/1010 = 1288.5 mm. The length of
the coupler DE is GF × 994/1010 = 936.8 mm. The
length of the link CD is calculated as follows.
The angle CGD is given by

tan−1
(

Y

OA + X

)
+ tan−1

(
Y

OF −X

)
+ 2β =

tan−1
(

500
1010 + 200

)
+ tan−1

(
500

1010 − 200

)
+

+ 2 × 13.521 =
22.452 + 31.686 + 27.042 = 81.18° (18)

From the triangle CGD the following equation can
be written

CD2 = GC2 +GD2−2×GC×GD×cos(CGD) =
= 1842 + 253.12 − 2 × 184 × 253.1 × cos(81.18°) =

83 634.26 (19)

Therefore, the length of the link CD is 289.2 mm.

The steering error curve for mechanisms shown
throughout Figure 7 to Figure 9 should be same as
all of these are obtained by compounding two crossed
four-bar mechanisms having similar proportions. The
steering error curve has been shown in Figure 10,
which shows that by compounding effect, the steering
error has been reduced from that of a single crossed
four bar mechanism that employs two equal spur or
helical gears.

6. Conclusions
It has been shown that two crossed four-bar chains
can be added in series to obtain a centre lever steering
mechanism. The kinematic synthesis of this centre
lever steering mechanism has been found to be easy in
the present work. The mechanism can be suitably mod-
ified to provide more space for the placement of the
engine, gearbox and other parts of the vehicle. This
mechanism is more accurate than the conventional
Ackermann steering mechanism and also the crossed
four bar steering mechanism. This mechanism has
a low pressure angle and high mechanical advantage
compared to the Ackermann steering mechanism.
Also, it can accommodate a constraint like the re-

quirement of the placement of the central hinge at
some offset from the longitudinal axis of the vehicle.
In this case, two crossed four-bar chains are not iden-

257



Santiranjan Pramanik, Sukrut Shrikant Thipse Acta Polytechnica

Figure 10. Comparison of steering error.

tical as one is smaller in size and the other is bigger
in size because of the offset. The steering arms are
unequal in length. But both have been designed for
equal value of the wheel track by a wheel base ratio
0.163.
In the case of a wide track vehicle mentioned in

reference [7], the steering mechanism has to be made
by adding four numbers of crossed four-bar chains each
having an imaginary track to wheelbase one-fourth of
that of the expanded vehicle while harvesting fruits.
While this vehicle has to be transported on the road,
it can be made collapsible by using two crossed four-
bar chains in series by discarding two such four-bar
chains.

References
[1] S. O. Fahey, D. R. Huston. A Novel Automotive
Steering Linkage. Journal of Mechanical Design
119(4):481 – 484, 1997. doi:10.1115/1.2826393.

[2] P. A. Simionescu, M. R. Smith. Initial Estimates in
the Design of Central-Lever Steering Linkages . Journal
of Mechanical Design 124(4):646 – 651, 2002.
doi:10.1115/1.1505853.

[3] A. De-Juan, R. Sancibrian, F. Viadero. Optimal
synthesis of function generation in steering linkages.
International Journal of Automotive Technology
13(7):1033 – 1046, 2012. doi:10.1007/s12239-012-0106-4.

[4] A. De-Juan, R. Sancibrian, F. Viadero. Optimal
synthesis of steering mechanism including transmission
angles. In Proceedings of EUCOMES, pp. 177 – 183. 2008.

[5] A. C. Bajpai, L. R. Mustoe, D. Walker. Advanced
Engineering Mathematics. John Wiley and Sons,
London, 1990.

[6] J. S. Zhao, X. Liu, Z. J. Feng, J. S. Dai. Design of an
Ackermann type steering mechanism. Journal of
Mechanical Engineering Science 227(11):2549 – 2562,
2013. doi:10.1177/0954406213475980.

[7] D. L. Peterson, T. S. Kornecki. Steering mechanism for
wide-track vehicles. Applied Engineering in Agriculture
2(1):16 – 17, 1986. doi:10.13031/2013.26699.

[8] S. Pramanik, S. S. Thipse. Kinematic synthesis of a
crossed four-bar mechanism for automotive steering.
International Journal of Vehicle Structures and Systems
9(3):169 – 171, 2017. doi:10.4273/ijvss.9.3.07.

[9] S. Pramanik. Design and Development of Innovative
Crossed Four-bar Steering Mechanism for Automobiles.
Ph.D. thesis, Symbiosis International University, Pune,
India, 2019.

[10] Wikipedia. Ashok Leyland 12M.
https://en.wikipedia.org/wiki/Ashok_Leyland_12M,
2019. Accessed: April 2020.

258

http://dx.doi.org/10.1115/1.2826393
http://dx.doi.org/10.1115/1.1505853
http://dx.doi.org/10.1007/s12239-012-0106-4
http://dx.doi.org/10.1177/0954406213475980
http://dx.doi.org/10.13031/2013.26699
http://dx.doi.org/10.4273/ijvss.9.3.07
https://en.wikipedia.org/wiki/Ashok_Leyland_12M

	Acta Polytechnica 60(3):252–258, 2020
	1 Introduction
	2 Crossed four-bar steering mechanism
	3 Condition of correct steering
	4 Optimization method
	5 Compound six member mechanism
	6 Conclusions
	References