CONTRIBUTION

Kinematic Analysis and Design ofSuspension-steering Mechanisms for Vehicles

Prof. Nobuyuki IWATSUKI
Dr. Eng, Tokyo Institute of Technology

Aiming to understand suspension and steering mechanisms for vehicles, kinematic analysis of the MacPherson
strut suspension mechanism and Ackermann-Jeantaud scheme steering mechanism which are frequently used in
front engine-front drive small cars is explained based on the essential spatial geometry and vector analysis. Motion
performance of vehicles due to the mechanisms is quantitatively evaluated with the fluctuation of turning center andthe transmissibility of steering mechanism. The process and results of the optimum design of the mechanisms basedon the criteria of the motion performance are then explained.
Key Words: steering mechanism, suspension mechanism, spatial linkage, kinematics, mechanism design,
optimization

1. Introduction
Vehicle wheels are supported by a suspension
mechanism to absorb the vibrations caused by unevenness
in the travel surface. A steering mechanism rotates
front wheels around the vertical axis to turn the vehicle.
Especially in FF (front-engine, front-wheel drive)
vehicles, a complex spatial link mechanism where a wheel
driving mechanism is incorporated in the suspension
and steering mechanisms is packed in confined spaceand can be viewed as an art work of mechanical device.
The suspension and steering mechanisms have also
progressed over the long history of vehicles, and canbe considered to be almost technically completed. But
higher-performance electric power steering devices andthe rise of electric vehicles have recently been creating
the potential for vehicle drive system innovations and new
four-wheel independent steering methods. New designsof the suspension and steering mechanisms will then be
required.
New designs call for learning from the past by
analyzing past mechanisms to thoroughly understand theirdesigns. But ever-advancing vehicle design information
is a corporate trade secret not accessible in detail. The
author is a university professor specializing in linkmechanisms and machine elements, and with previously
little interest or knowledge in vehicle mechanisms. But
when examining suspension and steering mechanisms,
the author found that many vehicle guidebooks andmanuals1)-5) showed many diagrams and photos without
dimensions and specifications and did not explain

JTEKT ENGINEERING JOURNAL English Edition No. 1019E (2023)

the quantitative evaluations of suspension or steering
performance in detail. On a more practical level, there
was not enough information that could be included inteaching materials used to teach or instruct students incourses such as mechanical kinematics.
In this article, the kinematic analysis and thequantitative evaluation of motion performance of thesuspension and steering mechanisms which are frequently
used in commercial vehicles are explained based on the
kinematics of the spatial linkage and the optimum design
to improve the motion performance is then tried.
As mentioned, suspension and steering mechanisms
have a long history of research and development.
Therefore, each car builder accumulates expertiseindividually. The kinematic analysis presented in thisarticle is just a kinematic analysis of a spatial linkage
with rigid links. It is based on spatial geometry andvector analysis, and is not completely new method.
However, since the black-box kinematic analysis withcommercial software is recently becoming mainstream,
it must be important to provide a geometry-based
explanation conveying the essence of kinematic analysis.
The author hopes that this article gives readers the better
understanding of spatial linkage to apply to developmentof new mechanisms in future.

3

Kinematic Analysis and Design of Suspension-steering Mechanisms for Vehicles

2. Suspension and Steering Mechanisms
2. 1 Suspension and Steering Mechanism Types
Axle suspension and independent suspension are thetwo main types of vehicle wheel suspension systems.
Most passenger cars use independent suspension 1)–6).
The various types of independent suspensions available
include trailing-arm, MacPherson strut, double wishbone
and multi-link types. Compact passenger cars commonly
use MacPherson strut suspensions since they have few
parts and enable compact designs. Steering systems
commonly transmit steering wheel rotation via a steering
shaft, convert it into the left-right linear motion of arack bar, and rotate the left/right front wheels around thevertical axis via tie rods. The linear motion of the rackbar is obtained with a pinion-rack mechanism or a ballscrew. Most of today’s passenger cars use power assisted
steering systems. Electric power steering systems use an
assist motor to rotate the pinion or ball screw nut1)–6).

2. 2 Suspension and Steering Mechanisms to be
Analyzed

5

f ＝1

⑴

where F is the mechanisms’ degree-of-freedom, N is thenumber of links (including fixed links), f is the degreeof freedom of a kinematic pair, and J f is the number of

4

Kingpin axis
Wheel
Hub carrier

Rack bar
Inner ball joint
Tie rod

Wheel bearing

Vehicle body
Outer ball joint
Suspension arm
Knuckle arm
Ball joint

Fig. 1 Suspension and steering mechanisms to be analyzed
S⑥StrutStrut displacement
P

Wheel drive inputThis article deals with the MacPherson strut suspension
mechanism commonly used in FF compact vehicles,
along with the steering mechanism that converts the linear
motion of the rack bar to rotate the front wheels via the tie
rods. The motion conversion mechanism used to convert
the rotation of the steering shaft to the linear motion of
the rack bar is not discussed.
Figure 1 shows an overview of the mechanisms to beanalyzed. Only the right-front wheel of an FF vehicle isshown in the figure. The hub carrier contains the wheel
bearing that supports the drive shaft. It is supported by astrut composed of a spring and shock absorber, and by asuspension arm extending from the vehicle body. Steering
is done by the linear displacement of the rack bar rotating
the hub carrier around the kingpin axis via the tie rod andknuckle arm. The middle shaft keeps the wheel rotation
speed constant during steering via two constant velocity
joints.
Figure 2 shows Fig. 1 represented as a mechanism
diagram. The diagram shows only the right half of thesuspension and steering mechanisms, ignoring the wheel
drive system (the mechanism from the middle shaft to thewheel bearing). The suspension and steering mechanisms
can be assumed as a spatial link mechanism. The
degrees-of-freedom of the spatial link mechanism can becalculated using Grübler’s formula as

F ＝ 6（ N − 1 ）− R（ 6 −f ）J f ,Constant
velocity joint

Strut with spring andshock absorber

Middle shaft
Drive shaft

CV

Steering input
P

R

CV

⑤Hub carrier Sterring
motionS Knuckle arm③Rack bar ④Tie rod①Vehicle body
R

Q

P

S

S

②Suspension arm
WheelFig. 2 Mechanism diagram of suspension and steering
mechanisms

kinematic pairs having degree of freedom f . Figure 2
shows that there are two prismatic pairs (P) and a revolutepair (R) with f = 1, and four spherical pairs (S) with f =3,
therefore J 1 = 3, J 3 = 4 and N =6. Substituting these values
into Eq. ⑴ yields:

F ＝ 6（ 6 − 1 ）−［（6 − 1）3＋（6 − 3 ）4]＝ 3 ,⑵

indicating that the mechanisms are spatial linkmechanisms with three degrees of freedom. Two of the
inputs are the linear displacement of the rack bar, andthe angular displacement of the suspension arm or the
linear displacement of the strut. The other one degreeof freedom is the rotation of the tie rod around the axis
connecting the spherical pairs (ball joints) at both endsof the tie rod, which does not affect the motion of thesuspension or steering. So this suspension-steering
mechanism is a spatial link mechanism with two degreesof-freedom, in which the input displacement of thesuspension and the steering input are given independently.

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Kinematic Analysis and Design of Suspension-steering Mechanisms for Vehicles

3. Kinematic Analysis of Suspension andSteering Mechanisms
3. 1 Previous ResearchzKinematic analysis of MacPherson strut suspensionsteering mechanism has a long history with a highly
extensive literature 7)–12). The research initially used a
simplified model with coinciding strut and kingpin axes.
This model later gave way to a detailed three-dimensionalmodel, providing precise analysis. Analysis work often
derives and solves closed-circuit vector equationsof spatial mechanisms, and is difficult to understandgeometrically. Researchers are also recently starting toanalyze commercial multibody systems and solving forboth motion and dynamics. However, such analysis isbecoming a black box process.
This article therefore endeavors to explain the analysis
procedure in a way that makes the geometry easy tounderstand.

3. 2 Analysis Procedure
Figure 3 is a mechanism diagram showing only thestrut, hub carrier and tie rod of Fig. 2. A static coordinate
system O-xyz is fixed to the vehicle body, and a moving
coordinate system RWC -ngf is set with the center ofthe front wheel as its origin and the axle as the n axis.
Kinematic pairs are distinguished by two subscript lettersthat define each pair’s position vector. For example, thespherical pair that connects the tie rod and knuckle armon the hub carrier is written SOB . Its position vector in thestatic coordinate system is written POB . The values given
as mechanism constants are the tie rod length L T , the
RWC -ngf coordinate system position vectors pSU , pOB andpKP of the spherical pair SSU at the strut top end, sphericalpair SOB and kingpin spherical pair SKP , as well as the unit
direction vector u S from the strut bottom end Q to strut
top end SSU .
As described in Section 2, the mechanism to beanalyzed has two independent inputs. If these two inputs
are assumed as a linear displacement of the rack bar
and an angular displacement of the suspension arm, theposition vector, P IB , of spherical pair SIB connecting therack bar and tie rod, and the position vector, P KP , of the
kingpin spherical pair SKP can be easily calculated.
The first step is to determine the strut displacement.
The position vector of spherical pair SSU at the strut top
end is given by the equation below in the RWC -ngf
coordinate system.

ｐSU ＝ ku S ＋ｐSB

SSU

⑶

PSU

x

PSU

Strut

y
O

uS
P

PSB
PWC

PIB
V
SIB

PH
PKP
POB

f Hub carrier

fH
Q
PSB

POB
SOB

rOB

LT

g
RWC

Wheel center
gH
n
nH

H
PKP
SKP
Kingpin
Knuckle arm

Rack bar
Tie rod

Suspension arm

Fig. 3 Positions of kinematic pairs in the suspension andsteering mechanisms

Expressing the distance between SSU and SKP in theO-xyz and RWC -ngf coordinate systems yields |p SU –pKP | = |PSU – PKP |. Substituting Eq. ⑶ into this equationand solving the quadratic equation for k yields:k ＝− u S・（ pSB −pKP ）＋［u S・（ pSB −pKP ）］2−（ pSB −pKP ）・（ pSB −pKP ）＋（ PSU −PKP ）・（ PSU −PKP ）. ⑷The kingpin axis-direction unit direction vectors in the
RWC -ngf and O-xyz coordinate systems are given by thefollowing equations.

uK ＝p SU − p KP
| p SU − p KP |, UK ＝

PSU − PKP
| PSU − PKP |⑸

The hub carrier acquires the SOB displacement androtates around the kingpin axis. With H as the foot of theperpendicular descending from SOB to the kingpin axis,
the position vector in the RWC -ngf coordinate system isgiven asp H ＝ℓu K ＋ p KP .

⑹

The condition of orthogonality yields:（ p H −p OB ）・u K ＝ 0 .

⑺

Substituting Eq. ⑹ into Eq. ⑺ and solving for ℓ yields:
・u K .
ℓ＝（ p OB − p KP ）⑻

The position vector of H in the O-xyz coordinate
system is given by:

PH ＝ℓUH ＋ PKP ,⑼and the distance between SOB and H is given by:

r OB ＝ （ p OB − p H ）・（ p OB − p H ）.

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5Kinematic Analysis and Design of Suspension-steering Mechanisms for Vehicles

Viewing rotation around the kingpin axis as rotation bya revolute pair at H, the two links of SIB – SOB – H are thetwo linkages of spherical pair – spherical pair – revolutepair. If the positions and postures of the revolute pair andspherical pairs at both ends are provided for these two
linkages, the position of a spherical pair in between canbe calculated13). In other words, the spherical pair SOB isgeometrically defined as the intersection of the spherecentered at SIB with radius LT (the tie rod length) and thecircular ring with radius rOB centered at H (Fig. 3).
To find this point of intersection, consider a spatial
plane containing the circular ring with radius rOB centeredat H. A new moving coordinate system H-nH gH fH with
H as its origin is set in this plane. The fH axis is set in
the uK direction. Let the nH axis direction be the common
normal to the fH axis and a line that passes through SIB
and runs parallel to the fH axis. Now we find point V, theother foot of this common normal. If the position vectorof V in the O-xyz coordinate system is given by:

PV ＝ mUK ＋ PIB .

・UK ＝ 0 .
（ PH −PV ）PH − PV, JH ＝ K H × I H .
| PH − PV |

KH ］

H
fH

c

fH

rOB
SOB

Fig. 4 Position of outer ball joint

coordinate system is therefore given geometrically by:（ c − p）
r OB cospOB , H ＝ r OB sin（ c − p）
0

c ＝ cos−1⒀⒁

Using the unit direction vectors of Eqs. ⒁, thecoordinate transformation matrix for rotating from the
H-nH gH fH coordinate system to the O-xyz coordinate
system is given by the formula below.

JH

LHV
rT

⑿

From the position vectors obtained above, the unit
direction vectors in the fH, nH and gH axis directions canbe written in the O-xyz coordinate system as:T［T H ］＝［ I HV⒃,

where,

So substituting Eq. ⑾ into Eq. ⑿ yields:m ＝（ P H − PIB ）
・UK .

rOB

rT

⑾

The condition of orthogonality yields:

KH ＝U H , I H ＝

gH
S'OB

⒂

Figure 4 shows V, SOB and H in the nH gH plane in themoving coordinate system H-nH gH fH.
The position vector of S OB in the H- n H g H f H

2
2
rOB
＋L VH
−rT2 ,
2 rOB L VH

・（ PV −PIB ）,r T ＝ L T2 −（ PV −PIB ）・（ PH −PV ）.
L VH ＝ （ PH −PV ）⒄

Using Eqs. ⒂ and ⒃, the position vector of SOB in theO-xyz coordinate system can be calculated as
T
POB ＝［T H ］
pOB , H ＋PH .

⒅

For three points on the hub carrier (SOB , H, andSKP ), the position vectors p OB , p H , and p KP in the RWC ngf coordinate system and the position vectors P OB ,P H , and P KP in the O-xyz coordinate system have now
been found. From these position vectors, the coordinatetransformation matrix for rotating from the RWC -ngf
coordinate system to the O-xyz coordinate system isgiven as
T
i KP
T
［T ］
＝［ I KP

TT
J KP KKP ］
,
j KP

⒆

T

kKP
where I KP , J KP , and K KP are the nKP , gKP and fKP axis
unit direction vectors as seen in the O-xyz coordinate
system and iKP , jKP and kKP are the nKP , gKP and fKP axis
unit direction vectors as seen in the RWC -ngf coordinate
system, for a new moving coordinate system fixed in
DSKP SOB H, having its origin at SKP , having the direction
normal to DSKP SOB H as the fKP axis, the SKP SOB direction
as the nKP axis, and the direction orthogonal to these axes

6

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Kinematic Analysis and Design of Suspension-steering Mechanisms for Vehicles

as the gKP axis. These unit direction vectors can now be
calculated using the equations below.

I KP ＝i KP ＝

POB − PKP
| POB − PKP |, KKP ＝

I KP ×（PH − PKP）

, JKP ＝K KP × I KP ,| I KP ×（PH − PKP）|

i KP ×（p H − p KP）
p OB − p KP
, k KP ＝, j KP ＝ k KP × i KP
| p OB − p KP || i KP ×（p H − p KP）|⒇

The origin of moving coordinate system RWC -ngf (thecenter of the front wheel, RWC ), and the position vectorof the strut bottom end Q in the O-xyz coordinate system
are given by:
T
PWC ＝ PH −［T ］
pH ,T
PSB ＝［T ］
p SB ＋ PWC .

angle of the left and right wheels are considered to be
zero during straight-line travel. PRP in the table is the rack
bar’s center position, P SA is the position of the revolutepair at the base of the suspension arm, K SA is the unit
direction vector of the revolute pair in the rotational axis
direction, W is the distance between the centers of the left
and right wheels, and L is the distance between the front
and rear wheel axles during straight-line travel. LT is thetie rod length and L S is the suspension arm length, as
determined dependently on other values in the table. The
rack bar in this example is positioned in front of the front
wheel axle.
The mechanism’s main constants are the position
vectors as seen in the moving coordinate system RWC ngf on the hub carrier. They are given by the equations
below, from the position vectors given in Table 1 as seenin the O-xyz coordinate system during straight-line travel.

The strut length is given by:

q ＝ （ PSU − PSB ）・（ PSU − PSB ）.
The motions of the suspension-steering mechanism
have now all been found for the case in which two inputs(the translational displacement of SIB and the displacementof SKP from the angular displacement of the suspensionarm) have been provided for the suspension-steering
mechanism. The calculations above are entirely analyticaland free from any numerical approximations. While notcovered in this article, the velocity, acceleration, angularvelocity and angular acceleration of each link or pair canbe analytically calculated by time differentiation of all
the equations. Analytical application of inverse dynamics
analysis could also make it possible to find a mechanism’s
drive force/drive torque or the forces and moments acting
on each pair when the mechanism is performing thedesired motion while subject to an external force.

4. Examples of Kinematic Analysis andMotion Performance Evaluation
4. 1 Example of Analysis
This section provides an analysis of a mechanism
having the specifications listed in Table 1. The values
shown are actual measurement values for a mechanism ina given compact FF vehicle. They represent items such as
the position vectors in the O-xyz coordinate system fixed
to the vehicle body, for kinematic pairs in the suspensionsteering mechanism of the right front wheel when thesteering amount is zero (during straight-line travel).
When the left and right mechanisms are completely
symmetrical, the origin O of the O-xyz coordinate systemis the midpoint of the centers of the left and right front
wheels during straight-line travel, the +y direction is the
direction of forward vehicle travel, and the +x directionis the vehicle’s right direction. The camber angle and toe

JTEKT ENGINEERING JOURNAL English Edition No. 1019E (2023)

p OB ＝POB , 0 −PWC , 0 , p KP ＝PKP , 0 −PWC , 0 ,p SB ＝PSB , 0 −PWC , 0 , uS ＝

PSU − PSB , 0.
| PSU − PSB , 0 |The analysis procedure of Section 3. 2 was used
to analyze the motion of the mechanism having thespecifications of Table 1. The analysis range is frommaximum left-turn steering, to straight-line travel, andto maximum right-turn steering. In other words, the rackbar displacement range is –s max s s max (s = 0 during
straight-line travel).
Figure 5 shows how the left and right steering angles(hL , hR ) and camber angles (uL , uR ) vary in relation torack bar displacement, s , when the suspension arm isfixed as it is during straight-line travel. The steering angle
is defined as the angle between the x axis and the xy
plane projection of the vector of the n axis of the moving
coordinate system RWC -ngf fixed at the center of thefront wheel. The camber angle is defined as the angle
between the z axis and the yz plane projection of thevector of the f axis. The n axis of the moving coordinate
system fixed at the center of the left front wheel is set sothat the vehicle’s right direction is positive. The g axis is
set so that the vehicle’s forward direction is positive. So
during straight-line travel, the moving coordinate systemsof the left and right front wheels RWC -ngf have identicalpostures. The maximum rack bar displacement smax givenby the calculation results in a steering angle with anabsolute value of hmax = 45°.
The steering angles of Fig. 5 (a) decrease
monotonically relative to rack bar displacement, s .
Using hL = hR = 0° for s = 0, we can see that hL (–s )= –hR (s ) and hR (–s ) = –hL (s ). The camber angles ofFig. 5 (b) are a characteristic of the MacPherson strut
suspension mechanism. As shown, they vary with rack7

Kinematic Analysis and Design of Suspension-steering Mechanisms for Vehicles

bar displacement, s , with the wheels tilting in the turning
direction. Figure 5 (c) plots the difference between theright and left steering angles. This left-right differenceis generated in accordance with rack bar displacement,s , and illustrates the performance of a feature called
the Ackermann-Janteau scheme 1)–6) that enables lines
extending from the axles of the left and right front wheels
to intersect.
Table 1 Dimensions of a suspension and steering mechanism
to be analyzed（Unit：mm, except KSA ）P WC, 0 （730.0，0.0，0.0）
P SA （350.0，220.0，−68.0）T
P SU （560.0，−57.0，470.0）T KSA （0.0324，0.9947，−0.0972）TP SB , 0 （590.0，−5.0，54.0）T W
1 460.0
P KP , 0 （680.0，17.0，−86.0）T L
2 500.0
P RP
LT（0.0，77.0，48.0）T
414.38T
P IB , 0 （280.0，77.0，48.0） LS
387.86
P OB , 0 （690.0，120.0，6.0）T

0.4
0.2
y, ｍ

T

Figure 6 plots the circles in the center of the width
direction of the mechanism and wheels, showing plotsof four postures (s /s max = –1, –0.5, 0, 0.5, 1). The whitecircles in the graph are the spherical pairs at the tie rod
ends (S IB and S OB ), the red circles are the wheel centers(R WC ), and the blue circles are the kingpins (S KP ). As
shown, the camber angle increases and the wheel center
position fluctuates as the vehicle turns.
Figure 7 shows the effect of the suspension arm
angular displacement w on the right wheel steering angle
and camber angle when s = 0 and s = s max . Dq in thegraph is the strut length variation. The graph shows thatthe camber angle increases as the strut extends, and thendecreases from strut contraction due to centrifugal force
once the turn outer ring has been reached. The graph also
shows that the effect on the steering angle is small during
straight-line travel.

1.0

0.2

0.5

0.4
1.0

0.0

0.5

0.0

1.0

s/smaxuR , u L °

0.6

0.8

1.0

0.8 0.6 0.4 0.2

0.0
x, m

0.2

0.4

0.6

0.8

1.0

0.0

0.4
1.0

uL

0
5
10
1.0

uR

Fig. 6 Skeleton diagram of the mechanism while steering(b) Camber angles
0.5

0.0

0.5

1.0

2.0

s/smax
0.4(c) Difference ofsteering angles

0.3
0.2

1.0

0.1
0.0
1.0

0.5

0.0

0.5

1.0

s/smaxFig. 5 The analyzed steering and camber angles

Dh R , Du R °｜hR - h｜L / hmax

0.4

0.2

10
5

0.2

60
Dh（
R s ＝0）Dh（
R s ＝ smax）
Du（
R s ＝0）Du（
R s ＝ smax）

30

Dq
0.0

0

1.0

30

2.0
10

5

0

5

Dq , mm

0.5

0.0
x, m

0.2

hR / hmax

1.0
1.0

0.8 0.6 0.4 0.2

0.4

hL / hmax

0.5

z, ｍ

hR / hmax , hL / hmax(a) Steering angles

0.0

60
10

w°

Fig. 7 Effect of suspension input

8

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Kinematic Analysis and Design of Suspension-steering Mechanisms for Vehicles

4. 2 Motion Performance Evaluation
This section uses the analysis results to evaluate
the suspension and steering mechanism’s motion
performance.

⑴ hR (0) = 0⑵ hR (smax ) = –hmax⑶ The first and second derivatives of hR (s ) arecontinuous during left and right turns.

4. 2. 1 Turning Center Fluctuation
Turning center fluctuation is the first item forevaluation. Figure 8 shows wheel motion during an
ideal turn, which has been covered by several different
literatures. Assuming a camber angle of zero and looking
at the xy plane, the vehicle’s turning center is defined as
the point C where lines extending from the axles of thesteered front wheels intersect a line extending from therear wheel axle. Being able to steer the two front wheels
in this way is held to prevent skidding. But the one degreeof freedom of the steering mechanism makes it impossible
to perfectly execute this type of ideal steering, resulting inturning center position fluctuation. In other words, lines
extending from the axles of the two front wheels intersect
a line extending from the rear wheel axle at CL and CR as
shown in Fig. 9. The midpoint of these two intersection
points is defined as the virtual turning center C, and the
distance between the coordinate system’s origin O (themidpoint between the two front wheel centers) and C isdefined as the turning radius R . As shown in the equation
below, let the turning center fluctuation assessment be
defined as the sum of the absolute values of the angles(DhL and DhR ) formed by the central axis of each front
wheel and the line segment OC.
j ＝ | DhL | ＋ | DhR |Before evaluating the mechanism, the ideal steering
angle first needs to be considered. Given a rightfront wheel steering angle hR (s ) within the rack bar
displacement range 0 d s d s max (right turn), the left
front wheel steering angle hL (s ) can be determined.
The x coordinate of turning center C in Fig. 8 is given
geometrically by:

x TC ＝

W
2−L cot h（.
R s）

When the straight line passing through the axle of theleft front wheel passes through point C, the left front
wheel steering angle hL (s ) is given by:＝− tan−1h（L s）

L
W −L cot h（R s）

For a left turn, hL (–s ) = –hR (s ) and hR (–s ) = –hL (s )can be used as described previously. So hR (s ) should be
set in a form enabling smooth steering ranging from a left
turn, to straight-line travel, to a right turn. The conditions
for hR (s ) are set as follows:

JTEKT ENGINEERING JOURNAL English Edition No. 1019E (2023)yhL
L（ W/2, 0）

W/2
O

hR

W/2
x

R（W/2, 0）

L
C（xTC ,  L ）Rear axle

Turning center

Fig. 8 Ideal steering motionhL
L（ W/2, 0）

y

W/2
O

hR

W/2
x

DhL

R（W/2, 0）DhR
Turning radius R

L

Turning centerC（xTC ,  L ）C（
R xR ,  L ）Rear axle
C（
L xL ,  L ）Fig. 9 Actual steering motion

If hR (s ) were given as a linear function of s forexample, Condition ⑶ would clearly not be satisfied. So
the function will be given in the following form:＝ C 0 ＋ C 1 s ＋ C 2 e ksh（R s）

where C 0, C 1 and C 2 are unknown coefficients and k isa constant. The exponential function term has been added
to provide variation in the rate of change with respect tos . The constant term C 0 has been added on the other end
as an adjustment used to satisfy the condition hR (0) = 0.
Conditions ⑴ and ⑵ above yield the equations below.

C0 ＝

hmax ＋ C 1 s max

e k s max − 1，C 2 ＝ − C 0

So C 0, C 1 and C 2 are given by linear functions of C 1.
Condition ⑶ now yields the equation below (details
omitted).

d
d
h（
＝
h（
L 0）
R 0）
ds
ds

9

Kinematic Analysis and Design of Suspension-steering Mechanisms for VehiclesSo first derivative continuity has been established. Now
from the condition of second derivative continuity, using

d2
d2
h（
＝ 2 h（
R 0）
L 0）
2
ds
ds
yields the equation below by solving for C 1 (details again
omitted).
−b ＋

C1 ＝

b 2 − 4 ac,
2awhere,a ＝W（ 1＋k d ）2，b ＝ k（ 2W k de ＋ 2We ＋L k d ），−s max2c ＝ k e（We ＋L ），d ＝ek s max

−1

，e ＝

−hmax

e

k s max−1

4. 2. 2 Motion Transmissibility Evaluation
This section evaluates whether steering can be executedwithout any statical problems caused by either turning
center fluctuation or rack bar linear displacement, s . When

1.0

0.0

1.0
1.0

d2 hR /ds2/ hmax,
d2 hL /ds2/ hmax

hR / hmax , hL / hmax(a) Steering angles
0.5

hL / hmax

0.5

hR / hmax
0.5

0.0

0.5

So a steering angle function that satisfies Conditions ⑴to ⑶ for a given k has been found.
Figure 10 shows the calculation results for the ideal
steering angle function when k = 5.0. As shown, thederivative is continuous. The turning center fluctuationassessment here is always j = 0. Comparing these results
to Fig. 5 reveals a large difference between the left andright steering angles.
Now we’ll evaluate steering performance for theexample analysis of Fig. 5. Figure 11 (a) plots the turning
radius. Figure 11 (b) plots Dh—the difference between
the hL (s ) values of Fig. 5 and the hL (s ) values calculated
from Eq.
using the hR (s ) results of the example
analysis. Figure 11 (c) shows turning center fluctuationassessment j (Eq. ). Figures 11 (b) and 11 (c) showthat turning center fluctuation increases as the absolutevalue of the steering angle increases.

1.0

100

(c) 2st derivatives

50
0
50
100
1.0

d2 hL /ds2/ hmaxd hR /ds / hmax
2

2

0.5

10
15
20
1.0

0.4

(b) 1st derivatives｜hR - h｜L / hmaxd hR /ds/ hmax,
d hL /ds/ hmax05d hR /ds/ hmax
d hL /ds/ hmax
0.5

0.0

0.5

0.0

1.0

s/smaxs/smax0.5

1.0(d) Difference of angles

0.3
0.2
0.1
0.0
1.0

0.5

0.5

0.0

s/smax1.0

s/smaxFig. 10 Ideal steering angle function
80

5.0(a) Turning radius

j

R/W

3.0

40
20
0
1.0

(c) Fluctuation of turning center

4.0

60

RMS value

2.0
1.0

0.5

0.0

0.5

0.0
1.0

1.0

0.5

90

(b) Error to ideal steering

0.05

aR , a L °Dh/ hmax

0.10

0.00
0.05
0.10
1.0

0.5

0.0
s/smax

0.0

0.5

1.0

s/smaxs/smax0.5

1.0(d) Pressure angles

60

aR

aL
30
0
1.0

0.5

0.0

0.5

1.0

s/smaxFig. 11 Motion performance evaluation of the analyzed mechanism

10

JTEKT ENGINEERING JOURNAL English Edition No. 1019E (2023)

Kinematic Analysis and Design of Suspension-steering Mechanisms for Vehicles

suspension mechanism strut displacement is ignored andthe kingpin axis is considered a revolute pair at point H,
the steering mechanism of Fig. 3 can be considered aspatial slider crank composed of a rack bar, tie rod andknuckle arm. So the pressure angle a between the tie rod
and knuckle arm can be evaluated. From Fig. 3, a can becalculated as

position of the spherical pair SIB (xIB , 0) and the y positionof the spherical pair SOB (y OB , 0) have high sensitivity
and increase or decrease in inverse correlation to the two
objective functions jRMS and amax.
10.0(a) Fluctuation of turning center

8.0

a ＝ cos−1 U K ×| POB − PH |POB − PIB・| POB − PIB |.

jRMS, °x OB

POB − PH

Figure 11 (d) plots the change in pressure angles
between the left and right tie rods and knuckle arms of theexample analysis. As shown, the pressure angles increaseas the absolute value of the steering angle increases.

JTEKT ENGINEERING JOURNAL English Edition No. 1019E (2023)

y RB = y IB

4.0
2.0

x IB

z OB

z RB = z IB

y OB
0.0
40

20

0

20

(b) Maximum pressure angle

x OB

70

amax, °

65
60

40

D p , mm75

5. Optimum Design
An attempt was made to derive the optimum designusing the mechanism performance indexes covered in
Section 4. The design covers the steering mechanism
only, leaving the MacPherson strut suspension mechanism
specifications unchanged. Specifically, the mechanism’s
performance was improved by changing the rack bar
position, rack bar length, tie rod length, and the positionof the spherical pair connecting the tie rod and knuckle
arm on the hub carrier. But to satisfy the straight-line
travel condition applicable when rack bar displacements = 0, the six component variables of the kinematic pair
position vectors PIB , 0 and POB , 0 for the straight-line travel
state of Table 1 are handled as design variables.
The ideal steering angle function of Eqs.
andexplained in Section 4 could be set as the objectivefunction for optimization, and the generated steering
angle functions could be optimized to match it. Butthat approach would be difficult and not enable motion
transmissibility to be evaluated at the same time.
Instead, jRMS is used as the turning center fluctuation
objective function for optimization and a max as the
motion transmission characteristic objective function for
optimization. jRMS is the RMS value of evaluation index
j within the rack bar’s permissible movement range, andamax is the maximum value of pressure angle a between
the tie rod and knuckle arm. Since lower values are moredesirable for both of these variables, optimization iscarried out to minimize the objective function.
The first step is to estimate how the objective functions
change in response to minute adjustments to these design
variables. Figure 12 shows how objective functions
jRMS and amax are affected by minute changes made to
the Table 1 value of each design variable independently(the p of Dp is the applicable design variable).
While naturally these plots only show results in theneighborhood of the Table 1 values, they show that the x

6.0

z RB = z IB

y OB

55

x IB

z OB

y RB = y IB

50
45
40

20

0

20

40

D p , mmFig. 12 Variation of performance evaluation criteria due todesign variables

So to cover both of the objective functions, the six
design variables are optimized using the equation below(the linear sum of both functions) as the objectivefunction.
U（ x IB , 0 , y IB , 0 , z IB , 0 , x OB , 0 , y OB , 0 , z OB , 0 ）＝ w 1 jRMS ＋w 2 a max

The downhill simplex method 14) is used as the
optimization method, with suitable side constraints
imposed on the design variables to account for the
space occupied by the mechanism. The s max value isalso corrected each time to keep the maximum steering
angle at h max = 45°, since changing the mechanism
specifications could result in different maximum steering
angle h max values for the same rack bar maximum
displacement s max value.
Figure 13 shows the optimization result for (w 1, w 2)
= (1.0, 0.0), which covers only turning center fluctuation.
Figure 13 (a) shows the turning center fluctuation index
j, Fig. 13 (b) shows the pressure angles a, and Figs. 13(c) and (d) are skeleton diagrams. As shown in the graph,
there is significant reduction in index j. But the maximum
pressure angle is slightly larger than the initial value, andthe motion transmission characteristic deteriorates despite
an improvement in turning performance.

11

Kinematic Analysis and Design of Suspension-steering Mechanisms for Vehicles

5.0
4.0

(a) Fluctuation of Turning center

j

3.0
2.0

RMS value

1.0
0.0
1.0

0.5

0.0

0.5

1.0

s/smax90aR , a L °Figure 14 shows the optimization result for (w 1, w 2) =(1.0, 12.0). The graphs show that the resulting mechanism
managed to improve on the initial values for both theturning center fluctuation index j and the pressure angles
a, indicating that the multi-objective optimization was
successful. The resulting mechanism has a longer rackbar but a shorter tie rod. A longer rack bar might notnecessarily be beneficial as problems from its elastic
deformation could be foreseen. Although handled by
the side constraints imposed on the design variables,
spherical pair S OB is positioned fairly close to the wheel
center. So the space occupied by the mechanism that was
not accounted for by this article should possibly also be
added to the objective functions for optimization.

(b) Pressure angles

60

aR

30
aL
0
1.0

0.0

0.5

0.5

1.0

s/smax
0.4

5.0
4.0

(a) Fluctuation of Turning center

0.2
y, m

j

3.0
2.0
1.0

RMS value

0.0
1.0

0.5

0.0

0.5

0.2

1.0

(c) x-y planes/smaxaR , a L °

90

0.4
0.0

(b) Pressure angles

0.4

0.6

0.8

1.0

0.6

0.8

1.0

0.4

aR

30

0.2

0.0

0.5

0.5

1.0

s/smax
0.4

z, m

aL
0
1.0

0.2

x, m

60

0.0
0.2
(d) x-z plane

0.2
y, m

0.0

0.4
0.0

0.2

0.4
x, m

0.0

Fig. 14 Optimization result 2
0.2(c) x-y plane
0.4
0.0

0.2

0.4

0.6

0.8

1.0

x, m
0.4

z, m

0.2
0.0
0.2
(d) x-z plane
0.4
0.0

0.2

0.4

0.6

0.8

x, m

Fig. 13 Optimization result 1

12

1.0

6. Conclusion
To gain an understanding of the workings and motionsof vehicle suspension and steering mechanisms, thisarticle has presented the example of a MacPherson strut
suspension mechanism and a steering mechanism driving
a hub carrier using a rack bar and tie rod. The kinematic
analysis of these mechanisms as a spatial link mechanismwith 2 degrees-of-freedom was explained. By using theanalysis, the motion performance evaluation was carriedout and the optimum design with the evaluated quantities
as objective function was tried.
While the kinematic analysis method is not new, asimple explanation of it has been provided in the formof vector analysis based on the essence of the spatial
geometry determining the mechanism motion. ThisJTEKT ENGINEERING JOURNAL English Edition No. 1019E (2023)

Kinematic Analysis and Design of Suspension-steering Mechanisms for Vehicles

analysis procedure has not used any numerical analysis
or other approximation analysis. So all the motion
displacements derived in this article could be analyticallytime-differentiated to enable velocity or acceleration
analysis. The acceleration could then be used with inverse
dynamics analysis to determine all the dynamic forces
and moments acting on each link or pair in a mechanism.
The author hopes that these analyses are applied to future
design.
The optimum design example presented in this article
yielded a mechanism with superior performance relative
to the initial mechanism. However, the occupied spaceof the mechanism is taken into account only by givingside constraints for the design variables, the actual
occupied space is therefore expected to be added to theobjective functions. But this addition may diminish thebest performance currently obtained because the original
initial mechanism was designed from actual vehicle
measurement values and must already be close to the
optimum solution.
The author hopes this article will be able to provide
some help to readers working on mechanical analysis anddesign.

10) DANIEL A. MANTARAS, PABLO LUQUE, CARLOS
VERA: Development and validation of a threedimensional kinematic model for the McPherson steeringand suspension mechanisms, Mechanism and Machine
Theory, 39 (2004) pp. 603-619.
11) Jyun NANGO, Mitsuhito NAGAO and Hiroshi KONNO,
Kinematic Analysis of MacPherson Strut Mechanism,
Transactions of JSME (Ser. C), 74-747 (2008) pp. 28162825.
12) K. VIKRANTH REDDY, MADHU KODATI, KISHENC H AT R A , S A N D I PA N BA N DYO PA D H YAY: A
comprehensive kinematic analysis of the double
wishbone and MacPherson strut suspension systems,
Mechanism and Machine Theory, 105 (2016) pp. 441470.
13) Takahiro ARUGA and Nobuyuki IWATSUKI: Kinematic
Analysis of Spatial Closed-loop Link Mechanisms
Based on Spatial 3-link Prime Structures, Proc. of the 4thInternational Jc-IFToMM Symposium, (2021) pp. 1-8.
14) NELDER, J. A, MEAD, R.: A simplex method forfunction minimization, The Computer Journal, Vol. 7,
Issue 4, (1965) pp. 308-313.

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6)

7)

8)

9)

Teruo ITSUKI: Automotive Engineering (3rd Edition),
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Automobile, Natsumesha (2021) (in Japanese).
Yoji IIJIMA: Basic Knowledge of Automobile
Suspension, Nikkan Kogyo Shinbun (2018) (in Japanese).
Yasuo TAKAHASHI: Basic Knowledge of Automobile
Mechanisms (Continued) -Step up Edition-, Nikkan
Kogyo Shinbun (2022) (in Japanese).
Masato ABE: Automotive Vehicle Dynamics - Theory
and application – (2nd Edition), Publishing Office, Tokyo
Denki University (2012) (in Japanese).
CRONIN, D.L.: MacPherson Strut Kinematics,
Mechanism and Machine Theory, 16-6 (1981) pp. 631644.
FELZIEN, M.L., CRONIN, D.L.: Steering Error
Optimization of the MacPherson Strut Automotive Front
Suspension, Mechanism and Machine Theory, 20-1
(1985) pp. 17-26.
KANG, H.Y., SUH, C.H.: Synthesis and Analysis of
Spherical-Cylindrical (SC) Link in the McPherson
Strut Suspension Mechanism, Trans. ASME, Journal of
Mechanical Design, Vol. 116 (1994) pp. 599-606.

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