IEEJ Journal of Industry Applications
Vol.10 No.3 pp.310–316
DOI: 10.1541/ieejjia.20004677

Paper

Model-based Filter Design for Triple Skyhook Control
of In-Wheel Motor Vehicles for Ride Comfort
Tomonori Suzuki∗a) Student Member, Masahiro Mae∗
Takuma Takeuchi∗∗ Non-member,
Hiroshi Fujimoto∗
Etsuo Katsuyama∗∗ Non-member

Student Member
Senior Member(Manuscript received May 7, 2020, revised Dec. 11, 2020)
J-STAGE Advance published date : Feb. 19, 2021

Vibration suppression control in vehicles is important for ride comfort. In vehicle electrification, the use of anin-wheel motor as an actuator for vibration suppression has attracted attention as an alternative to active suspension systems. However, a model-free conventional controller design method cannot suﬃciently suppress vibration
at the most significant frequencies (4 Hz to 8 Hz) because of mechanical resonance. In this study, we analyze thefrequency response of an experimental vehicle. Using the identified frequency response of the vehicle, we design apre-compensation filter based on the desired vibration suppression performance characteristics. The vibration suppression performance improved with the experimental vehicle at target frequencies from 4 Hz to 8 Hz.
Keywords: vibration suppression control, vehicle motion control, electric vehicle, in-wheel motor, model-based design, ride comfort

1.

Introduction

In recent years, electric vehicles (EVs) have attracted muchattention as a result of increasing environmental awareness.
EVs have high motion performance because of their motor
characteristics. Electric motors have three significant advantages over internal combustion engines: their torque response
is several hundred times faster and more accurate than that of
the internal combustion engine, their output torque can be directly measured from the motor current value, and distributed
small motors enable flexible arrangements (1) . By utilizingthese features eﬀectively, advanced vehicle motion control
becomes possible (2) (3) .
In particular, the distributed arrangement of motors enables
various drivetrain configurations in EVs. One such configuration is an in-wheel motor (IWM) system in which motors
are mounted in each wheel. The IWM system has the advantages of reducing the total weight of a drivetrain and increasing the freedom of interior layout design (4) . From a control
perspective, torque-vectoring enables higher motion performance, and the shorter driveshaft removes low-frequency resonances which create a substantial limitation in control (5) (6) .
To improve motion performance and ride comfort, several
vibration suppression control methods that lie in several Hz
bands have been proposed (7)–(9) . However, the IWM systemincreases the unsprung weight and causes deterioration ofride comfort over 4 Hz (10) (11) . To solve this problem, studies(a) Experimental vehicle.

on vibration suppression for the unsprung vertical axis modeand the longitudinal axis mode that lay in a higher frequency
band (10 Hz and above) have been conducted (12)–(14).
These vibration suppression control methods at high bandwidths use the high torque responsiveness of the IWM system. As described above, extensive research using the IWM
system has been carried out to suppress vibration in a wide
frequency range (15) .
Katsuyama et al. proposed triple skyhook (tSH) control,
which is a vertical vibration suppression control method (16) .
This method has the potential of suppressing an extensive
range of frequency vibration but uses a lowpass filter (LPF)with a low cutoﬀ frequency to stabilize the feedback control system. The low-cutoﬀ-frequency LPF limits the gain
and delays the phase of the target frequency 4 Hz to 8 Hz andworsens the vibration suppression performance. Therefore,
it is desired to stabilize the control system without using a
low-cutoﬀ-frequency LPF.
In this paper, we analyze vehicle vibration using a quartercar model and identify the frequency response of an experimental vehicle, which is described in section 2. In section 3,
we reveal the problem of the conventional method. In sectiona) Correspondence to: Tomonori Suzuki. E-mail: tomonorisuzuki@ieee.org∗
The University of Tokyo
5-1-5, Kashiwanoha, Kashiwa, Chiba 277-8561, Japan∗∗
Toyota Motor Corporation
1200, Mishuku, Susono, Shizuoka 410-1193, Japanc 2021 The Institute of Electrical Engineers of Japan.
⃝(b) IWM unit.

Fig. 1. Experimental vehicle with IWM units

310Model-based Filter Design for tSH Control for Ride Comfort（Tomonori Suzuki et al.）Fig. 2. Quarter-car model
Table 1. Vehicle model parameter definitions
Symbol Meaning
m1
ks
kt
z0
z2
θ
r

Symbol Meaning

Quarter unsprung mass
Spring stiﬀness
Tire stiﬀness
Road displacement
Sprung mass displacement
Angle of instant suspension rotation
Radius of tire

m2
cs
ct
z1
Fd
Fc
T

Fig. 4. Frequency response of the quarter-car model

Quarter sprung mass
Damping coeﬃcient
Tire damping coeﬃcient
Unsprung mass displacement
Driving force
Suspension reaction force
Motor torqueFig. 3. Suspension reaction force
Fig. 5. Frequency response from motor torque to acceleration of sprung mass4, we propose the design method of a pre-compensation filter
for suppressing the mechanical resonance excitation, whichcauses instability, and improving the vibration suppression
performance. We can design the pre-compensation filter withdesired frequency characteristics using a sensitive function.
As an example, we designed the pre-compensation filter tosuppress the vibration at 4 Hz to 8 Hz, which is most significant for drivers (17) . In section 5, we conducted experiments
and validated the eﬀectiveness of the proposed model-based
filter design method.
2.

m2 z2 s2 = (cs s+ks )(z1 −z2 ) + Fc , · · · · · · · · · · · · · · · · · (1)m1 z1 s2 = −(cs s+ks )(z1 −z2 ) + (ct s+kt )(z0 −z1 ) − Fc .
· · · · · · · · · · · · · · · · · · · · (2)The transfer function model from motor torque to sprungmass acceleration of the quarter-car model is given as shownin (3), and its Bode diagram is shown in Fig. 4.
Using general passenger vehicle parameters, this transfer
function has two resonances around 1.3 Hz and 10 Hz andone anti-resonance around 9.9 Hz.
We use the grey box modeling in the system identification.
From the result of the quarter-car model analysis, the frequency response of the experimental vehicle can be fitted tothe sum of two resonance modes.
The first resonance comes from sprung mass and suspension stiﬀness. The second resonance comes from unsprung mass and tire stiﬀness. When the passengers’ weightchanges, the first resonance frequency changes slightly.
However, the second resonance frequency does not change.
Though details will be described in section 4, the second
resonances cause instability in the feedback control system.
The resonance which is suppressed by the filter is unaﬀectedagainst passengers’ weight change.
2.3 Suspension Reaction ForceThe suspension
reaction force Fc is defined as the vertical element ofModel and Experimental Vehicle Analysis

2.1 Experimental Vehicle with IWM
We use the experimental vehicle shown in Fig. 1(a). The experimental vehicle has four IWM units shown in Fig. 1(b) and is equippedwith four vertical acceleration sensors on the sprung mass
above each wheel. The rear left part of the experimental vehicle is analyzed as a quarter-car model described in the following sections.
2.2 Vehicle Vibration Model
The vehicle vibration
is analyzed by the quarter-car model as shown in Fig. 2. Thismodel consists of a quarter sprung mass, a quarter unsprung
mass, suspension, and a tire. The definition of each variable
is shown in Table 1. Fc is a suspension reaction force, which
is described in section 2.3 and Fig. 3.
The motion equations of the quarter-car model are given as
follows:
311

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Model-based Filter Design for tSH Control for Ride Comfort（Tomonori Suzuki et al.）z¨2s2 (m1 s2 + ct s + kt )=· · · · · · · · · · · · · · · · · · · · · (3)43
Fc m1 m2 s + {(m1 + m2 )c s + m2 ct }s + {(m1 + m2 )k s + m2 kt + c s ct }s2 + (c s kt + ct k s )s + k s kt
"
!
0.7
0.3
−3
+ 2
e−35×10 s · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (4)P(s) = 6 × 10−3 s2 2
s + 6.786s + 127.9 s + 32.04s + 2852
suspension internal forces caused by the driving force.
Fig. 3 shows the lateral view of the simplified kinematic
model of a vehicle with suspension reaction forces. The arm
between the sprung and unsprung masses has a structure thatrotates around the instant center of rotation (ICR). The driving force working on the tire in the longitudinal direction
causes a moment in the arm about the ICR. The vertical force
that balances the moment of the driving force is then generated.
Assuming that the angle between the ground and ICR of
the arm is θ, the magnitude of the driving force is Fd , the slip
ratio is a constant, and the suspension reaction force Fc in thevertical direction can be calculated as follows:
Fc = Fd tan θ = rT tan θ.· · · · · · · · · · · · · · · · · · · · · · · · · (5)Fig. 6. Concept of triple skyhook control (Dotted lines
express virtual elements of the control.)

2.4 System Identification of Experimental Vehicle

with IWM
We identify the frequency response from therear left wheel motor torque to the corresponding sprung vertical acceleration using a multi-sine input (18) . When the multisine input is applied to the rear left wheel motor, the remaining wheels’ motor torques are set to 0. The identified frequency range is from 0.1 Hz to 10 Hz, which is the torquecontrol bandwidth of the experimental vehicle. The identified frequency response and its fitting line are displayed inFig. 5. This frequency characteristic has two resonances andone anti-resonance. The characteristic is the same as thequarter-car model. The frequency response function (FRF)
is fitted using deadtime and the sum of several modes. The
deadtime is estimated as 35 ms, and the resonance frequencies are estimated as 1.8 and 8.5 Hz. The total FRF P(s) isexpressed in equation (4).
3.

Fig. 7. Overall feedback control system block diagram

to compare the filter eﬀect corresponds to the LPF. An HPF
does not change the gain or phase over its bandwidth and does
not aﬀect LPF equivalent pre-compensation filter design. We
describe the conventional and proposed pre-compensation filter in section 3.2 and 4.1 in detail.
The transfer function of the quarter-car model from z1 to z2
with this input is expressed as follows:z2 =Vibration Suppression Control

3.1 Triple Skyhook Control
The tSH control feeds
back the vertical displacement, velocity, and acceleration of
the sprung mass (16) .
The transfer function of the quarter-car model from z1 to z2
is shown as follows:cs s + ks
z2 =z1 .· · · · · · · · · · · · · · · · · · · · · · · · · · (6)m2 s 2 + c s s + k s

cs s + ks
z1 .
(m2 +α2 D(s))s2 + (c s +α1 D(s))s + (k s +α0 D(s))· · · · · · · · · · · · · · · · · · · · (8)The control input can change the three parameters ofthe denominator in equation (8). When α2 = αm2 , α1 =αc s , α0 = αk s , the tSH control method multiplies z2 by
1/(1 + αD(s)) compared to the model without control. Physically, this means that the sprung mass is hooked with threevirtual elements: spring, damper, and inerter (19) , as shown in
Fig. 6.
The overall feedback control system block diagram is displayed in Fig. 7. tSH control with a pre-compensation filter
D(s) is applied to the plant.
3.2 Conventional Model-free Filter Stabilization
If
we apply the tSH control without any pre-compensation filter to the vehicle plant, the controller excites the mechanicalresonance. The conventional tSH control uses a low cutoﬀ
frequency such as 3 Hz LPF as a filter D(s) to suppress themechanical resonance. However, a low cutoﬀ frequency LPF
delays the phase at 4 Hz to 8 Hz. Fig. 8 shows the Nyquist
plots of the identified plants with tSH control with LPF.
In Nyquist plots, the sensitive function corresponds to the

Since only the second derivative of the sprung mass displacement z̈2 is measured, the control input is calculated fromthat.
m2 s 2 + c s s + k sz̈2 D(s),· · · · · · · · · · · · · · · · · · (7)s2where α is the control gain, and D(s) is a pre-compensation
filter. The pre-compensation filter consists of LPF for stabilization and HPF for eliminating integrated drift and roadsurface gradient eﬀect in the conventional method. The lowcutoﬀ-frequency LPF worsens vibration suppression performance. In this paper, we do not use an HPF to make it easierFc = −α312

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Model-based Filter Design for tSH Control for Ride Comfort（Tomonori Suzuki et al.）

Step 4 Lead the phase of the target frequency at 4 Hz to
8 Hz to lower the sensitive function.
Step 5 Tune the tradeoﬀ of the gain and phase based on thesensitive function.
The transfer functions of the filter components are expressed as follows:
LPF (e.g. 2nd order Butterworth LPF)
DLPF (s) =w2c,· · · · · · · · · · · · · · · · · · · · · · · (9)2
s + 2wc s + w2c

where wc is a cutoﬀ frequency.
Notch filter:
Dnotch (s) =Fig. 8. Nyquist plots of plants with tSH control withLPF

s2 + 2dζwn s + w2n,· · · · · · · · · · · · · · · · · · (10)s2 + 2ζwn s + w2nwhere d is a width parameter, ζ is a depth parameter, andwmax is the center angular frequency.
Phase lead filter:
Dlead (s) =Ts + 1,· · · · · · · · · · · · · · · · · · · · · · · · · · · · (11)aT s + 1

where
1 − sin φmax
, · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (12)
1 + sin φmax
1
, · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (13)
T= √
awm
a=

φmax is the maximum lead phase, and wm is the center angularfrequency.
This procedure is applicable for all wheels independently.
However, the adjustment is also assumed to be needed formutual interference between each wheel.
4.2 Application of Filter Design for Experimental Vehicle
This section shows a filter design example for theexperimental vehicle. Fig. 10(a) shows the Nyquist plots ofthe plants with the tSH control with no filters (Step 1). Thiscase is unstable around 15 Hz and 43 Hz. Because the system identification bandwidth is 10 Hz, we lower the gain over
10 Hz using a high-cutoﬀ-frequency LPF for gain stabilization and model error suppression (Step 2). In this case, weapply the second-order Butterworth LPF, whose cutoﬀ frequency is 10 Hz. Next, Fig. 10(b) shows the Nyquist plotsof the plants with the tSH control with only a high-cutoﬀfrequency LPF. This case is unstable because of the resonance at 9.5 Hz, which is under the system identification
bandwidth. This resonance comes from unsprung mass andtire stiﬀness. Therefore the resonance frequency does notchange against the passenger weight change. A notch filter
should stabilize this identified resonance by the gain (Step3). The notch filter strength has a tradeoﬀ between gain stability around 9.5 Hz and phase delay instability under 9.5 Hz.
We can now stabilize the tSH control. Finally, we design thephase lead filter to lower the sensitive function (Step 4). In
this example, we lead the phase most at 4 Hz to 8 Hz, which
is significant for the driver’s ride comfort (17) . The stronger the

Fig. 9. Filter design procedure

inverse of the distance between the Nyquist plots and a(−1, 0j) point. The phase delay at 4 Hz to 8 Hz move the plots
close to the (−1, 0j) point and worsens the sensitive function.
High sensitive function means that the controller cannot suppress the vibration disturbance very well at that frequency.
The conventional method uses the high-pass filter (HPF) toeliminates the eﬀect of road slopes. This paper does not use
it to make it easier to compare the LPF eﬀect. Because HPF
does not change gain nor phase over the bandwidth, there is
no eﬀect of HPF over bandwidth, such as 0.6 Hz.
4.

Proposed Filter Design Method for tSH Control Method

4.1 General Filter Design Method
We design precompensation filters based on the identified frequency response fitting data. The design procedure is described below,
and its overview is shown in Fig. 9.
Step 1 Apply the tSH control method without filters to thevehicle plant.
Step 2 Lower the gain over the identified bandwidth(greater than 10 Hz) using the high-cutoﬀ-frequency
LPF.
Step 3 Stabilize the gain of the identified resonance at
10 Hz using notch filters.

D(s) =

s2 + 2 × 2.6 × 0.19 × 66s + 662
0.055s + 1
632· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (14)222
2
0.17 × 0.055s + 1
s + 2 × 63s + 63
s + 2 × 0.19 × 66s + 66
313

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Model-based Filter Design for tSH Control for Ride Comfort（Tomonori Suzuki et al.）(a) Step 1: Nyquist plots of the plants with tSH control with no filter.

(b) Step 2: Nyquist plots of the plants with tSH control with LPF.

(c) Step 3: Nyquist plots of the plants with tSH control with LPF and notch filter.

(d) Step 4, 5: Nyquist plots of the plants with tSH control with LPF, notch
filter, and phase lead filter.

Fig. 10. Nyquist plots of filter design procedurephase lead filter is, the lower the sensitive function at 4 Hz to
8 Hz is. However, It is needed to tune the parameters in a
tradeoﬀ with gain instability over the phase lead filter bandwidth. The transfer function of the designed filter is shownin (14).
The Nyquist diagram of the plants with the conventional
LPF and the proposed filter is displayed in Fig. 11. The pointof 6 Hz of the proposed filter, which is in the vibration suppression target frequency, is farther from the −1 point thanthe conventional LPF. This indicates the low-sensitive function at 6 Hz, as shown in Fig. 12. There exists a tradeoﬀ between the sensitive function of the target frequency (4 Hz to8 Hz) and other frequencies such as 35 Hz in this case. Thus,
it is necessary to tune the sensitive function based on the desired vibration suppression frequency characteristics.
5.

Fig. 11. Nyquist plots of plants with tSH control withfilterExperimental Validation

The experimental vehicle with the tSH control with theconventional LPF and the proposed filter was run on a roadwith a random surface at 60 km/h. The designed controller isapplied for the rear left wheel motor. The conventional LPF
is a first-order LPF with a cutoﬀ frequency of 3 Hz, and the
proposed filter is designed as discussed in the previous section. The gain α of the tSH control is set to α = 1.5.
The power spectral density (PSD) of the verticalacceleration of the sprung mass is shown in Fig. 13. The 4 Hz
to 8 Hz vibration is the most significant for drivers (17) . As in
the filter design, the vibration at the target frequency 4 Hz
to 8 Hz of the proposed filter is lower than that of the conventional LPF. There exists a tradeoﬀ between 4 Hz to 8 Hz
and 25 Hz to 30 Hz. 25 Hz to 30 Hz vibrations are less critical than 4 Hz to 8 Hz for drivers. Although 25 Hz to 30 Hz
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Model-based Filter Design for tSH Control for Ride Comfort（Tomonori Suzuki et al.）vehicle. As a result, it is possible to increase the cutoﬀ frequency of the LPF and improve the vibration suppression performance at the target frequency. The match between the sensitive function and the experimental PSD results validates the
proposed model-based filter design method. In the experimental results, the proposed method improved the vibration
suppression performance at 4 Hz to 8 Hz, which is most significant for driver ride comfort. In the filter design process,there exists a tradeoﬀ between several frequencies. By identifying the driver’s wide range of vibration sensitivity, we can
optimize overall frequency characteristics. In this paper, themutual interference between each wheel is not considered.
The integrated filter design of all four wheels will be futurework.
Acknowledgment
This work was partially supported by JSPS KAKENHI
Grant Number JP18H03768.

Fig. 12. Sensitive function of plants with tSH control
and filter

References
(1)

(2)

(3)

(4)
(5)

Fig. 13. PSD of acceleration of sprung mass
(6)

vibrations are also perceived as noise, the noise was not a
problem in the sensory test. It may be the need to suppress the
25 Hz to 30 Hz noise in the future when other vehicle noises
get more silent.
The diﬀerence of the exciting frequency around 30 Hz between the design process and the experimental result comesfrom phase modeling error. There exists phase modeling error over 10 Hz because a low-dimensional fitting model isused. However, the phase diﬀerence does not matter in thedesign process because the high bandwidth gain is loweredby the low pass filter as a whole.
The experimental results show the validation of the modelbased design of the pre-compensation filter for realizing desired frequency characteristics.
6.

(7)

(8)

(9)

( 10 )( 11 )( 12 )

Conclusion

This paper proposed the model-based design of a precompensation filter for the tSH control method. The conventional tSH control method uses a low-cutoﬀ-frequency
LPF to stabilize the system. However, a low-cutoﬀ-frequency
LPF delays the phase at the target frequency of 4 Hz to 8 Hz,
which worsens the vibration suppression performance. In
the proposed model-based design method, we can design apre-compensation filter to realize the desired frequency characteristics based on the identified frequency response of the( 13 )( 14 )( 15 )( 16 )( 17 )

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Model-based Filter Design for tSH Control for Ride Comfort（Tomonori Suzuki et al.）( 18 )( 19 )

Hiroshi Fujimoto (Senior Member) received the Ph.D. degree in the
Department of Electrical Engineering from the University of Tokyo in 2001. In 2001, he joined the Department of Electrical Engineering, Nagaoka University of Technology, Niigata, Japan, as a research associate. From 2002 to 2003, he was a visiting scholar in
the School of Mechanical Engineering, Purdue University, U.S.A. In 2004, he joined the Department ofElectrical and Computer Engineering, Yokohama National University, Yokohama, Japan, as a lecturer andhe became an associate professor in 2005. He is currently an associate professor of the University of Tokyo since 2010. He received the Best Paper
Awards from the IEEE Transactions on Industrial Electronics in 2001 and2013, Isao Takahashi Power Electronics Award in 2010, Best Author Prizeof SICE in 2010, the Nagamori Grand Award in 2016, and First Prize Paper Award IEEE Transactions on Power Electronics in 2016. His interests
are in control engineering, motion control, nano-scale servo systems, electric vehicle control, motor drive, visual servoing, and wireless motors. He
is a senior member of IEE of Japan and IEEE. He is also a member of theSociety of Instrument and Control Engineers, the Robotics Society of Japan,
and the Society of Automotive Engineers of Japan. He is an associate editor of IEEE/ASME Transactions on Mechatronics from 2010 to 2014, IEEE
Industrial Electronics Magazine from 2006, IEE of Japan Transactions on
Industrial Application from 2013, and Transactions on SICE from 2013 to2016. He is a chairperson of JSAE vehicle electrification committee from2014 and a past chairperson of IEEE/IES Technical Committee on Motion
Control from 2012 to 2013.

human exposure to whole-body vibration Part 1” (1997)
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M.C. Smith: “Synthesis of mechanical networks: the inerter”, IEEE Transactions on Automatic Control, Vol.47, No.10, pp.1648–1662 (2002)

Tomonori Suzuki (Student Member) received the B.S. degree fromThe University of Tokyo in 2019. He is currently
working towards an M.S. degree in the Departmentof Advanced Energy, Graduate School of Frontier
Sciences, the University of Tokyo. His interests arein control engineering, motion control, and vibration
suppression control. He is a student member of theInstitute of Electrical and Electronics Engineers andthe Institute of Electrical Engineers of Japan.

Masahiro Mae (Student Member) received the B.E. and M.S. degreefrom The University of Tokyo in 2018 and 2020, respectively. He is currently working towards the Ph.D.
degree in the Department of Electrical Engineering
and Information Systems, Graduate School of Engineering, The University of Tokyo. His research interests are in control engineering, precision motion control, multirate control and multi-input multi-output
systems. He is a student member of the Institute ofElectrical and Electronics Engineers.

Etsuo Katsuyama (Non-member) received the B.S. degree in Tokyo
Institute of Technology in 1997. He joined Unisia
Jecs in Japan in 1997. From 2007 to 2010, he joined
Toyota Motorsport GmbH in Germany as a chassis
engineer of formula 1. He currently joins Toyota Motor Corporation in Japan as a chassis engineer of passenger vehicles. He received the Arch T. Colwell
Merit Award from the Society of Automotive Engineers in 2015, and the Outstanding Technical Paper
Awards from the Society of Automotive Engineers ofJapan in 2013 and 2018.

Takuma Takeuchi (Non-member) received M.S. degree in The University of Tokyo, Japan in 2017. In 2017, He joined
Toyota Motor Co., Ltd., Japan. His research interest
includes vehicle motion control.

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