Narrow tilting dynamic

IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 19, NO. 4, AUGUST 2014 1309

Switching Dynamic Modeling and Driving StabilityAnalysis of Three-Wheeled Narrow Tilting Vehicle

Hiroki Furuichi, Jian Huang, Member, IEEE, Toshio Fukuda, Fellow, IEEE, and Takayuki Matsuno

Abstract—Traffic congestion is one of the main problems moderncities face, exacerbated by widespread use of normal four-wheeledtransportation. A promising solution to this challenge is to developnarrow electric vehicles and apply them in daily transportation.These kinds of narrow vehicles need less parking and lane spaceand cause no air pollution. In this study, we developed a new con-ceptual narrow tilting vehicle (NTV) that has one front wheel andtwo rear wheels. All the three wheels can tilt to improve the stabilityof NTV during turning. To fully describe the dynamic behaviors,a new switching dynamic model of NTV was derived. This modelconsiders several NTV states including normal running, tempo-rary running with one rear wheel not on the ground, and totallyfalling down. Based on this model, a simulation platform is estab-lished, which is useful for testing different control methods and forinvestigating dangerous driving situations. A new active tilt con-troller is proposed based on a tilt dynamic model linearized aroundan equilibrium on which the lateral acceleration is zero. The con-troller is easily realized from the point of view of engineering andthe closed-loop system stability was also investigated. The drivingstability problem of NTV was discussed using peak-to-peak analy-sis method. Stability condition of input steering angle and angularvelocity was obtained, which may be a useful tool for designingthe steer-by-wire controller in the future. All the proposed model,control method, and stability conditions were verified through sim-ulations and experiments.

Index Terms—Active tilt control, driving stability, modeling,narrow tilting vehicle (NTV), peak-to-peak gain analysis.

Manuscript received February 20, 2013; revised June 29, 2013; accepted July25, 2013. Date of publication September 16, 2013; date of current version April25, 2014. Recommended by Technical Editor B. Shirinzadeh. This work wassupported by the International Science and Technology Cooperation Program ofChina “Precision Manufacturing Technology and Equipment for Metal Parts”under Grant 2012DFG70640, in part by the International Science and Technol-ogy Cooperation Program of Hubei Province “Joint Research on Green SmartWorking Assistance Rehabilitant Robot” under Grant 2012IHA00601, in partby the Program for New Century Excellent Talents in University under GrantNCET-12-0214, and the Fundamental Research Funds for the Central Univer-sities under Grant 2013ZZGH007. (Corresponding author: J. Huang.)

H. Furuichi was with the Department of Micro-Nano Systems En-gineering, Nagoya University, Nagoya 464-8603, Japan. He is nowwith Nissan Motor Company, Ltd., Yokohama 220-8686, Japan (e-mail:[email protected]).

J. Huang is with the Key Laboratory of the Ministry of Education forImage Processing and Intelligent Control, School of Automation, HuazhongUniversity of Science and Technology, Wuhan 430074, China (e-mail:[email protected]).

T. Fukuda is with the Department of Micro-Nano Systems Engineering,Nagoya University, Nagoya 464-8603, Japan (e-mail: [email protected]).

T. Matsuno is with the Department of Natural Science and Technol-ogy, Okayama University, Okayama 700-8530, Japan (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TMECH.2013.2280147

NOMENCLATURE

in , jn ,kn Unit vectors in right hand coordinatesystem n.

On Original point of coordinate systemn.

‖ • ‖n Coordinate values in system n.b, rlw, rrw, fw Subscripts indicating vehicle body,

rear left or right wheels, and frontwheel respectively.

G,Gb , Gfw , Grlw , Grrw Center of gravity (COG) of the wholevehicle, vehicle body, front wheel,and rear left or right wheels, respec-tively.

ddt

nx Derivative of vector x with respect to

frame n.rI , J rI Position of point I and position

of point I with respect to J res-pectively.

Rnm Rotation matrix, converts coordinates

in system n to coordinates in systemm.

Sγ , Cγ Abbreviation of sin(γ) and cos(γ).The same abbreviations are used forangle θ, ψ, ξ, etc.

θ, θ, θ Vehicle tilt angle, tilt velocity and tiltacceleration.

ψ, ψ, ψ Vehicle yaw angle, yaw velocity andyaw acceleration.

δ, δ, δ Front wheel steering angle, angularvelocity and acceleration.

β, β, β Vehicle roll angle, roll velocity androll acceleration in critical stablecases.

y, y, y Vehicle lateral displacement, velocityand acceleration.

δ′ Direction angle of front wheel.ξ Steering head angle.γ Rotation angle of system {1′} with

respect to system {1}.v Longitudinal vehicle velocity.θ∗, θ∗ Desired tilt angle and angular velocity

at the equilibrium when v = v∗, δ =δ∗ are given.

lf , lr Horizontal distances from G to frontand rear wheels (lf = 0.88 m, lr =0.53 m).

L Wheel base of vehicle (L = lf + lr).h Height of G (0.625 m).

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1310 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 19, NO. 4, AUGUST 2014

b Distance between two rear wheels(0.495 m).

m Mass of vehicle (230 kg).Ixxb Moments of inertia of NTV body

about x-direction (35 kg·m2)Iyyb Moments of inertia of NTV body

about y-direction (35 kg·m2)Izzb Moments of inertia of NTV body

about z-direction (35 kg·m2)Cf1 Front wheel cornering stiffness

(4047 N/rad).Cr1 Rear wheel cornering stiffness

(507 N/rad).Cf2 Front wheel camber stiffness

(4589 N/rad).Cr2 Rear wheel camber stiffness

(556 N/rad).rrw , rfw Radius of rear and front wheel.R Radius of curvature of turn.aG,HG Acceleration and angular momentum

of G.Fx , Nx Lateral and normal forces on the front

or rear wheel of vehicle, where x ∈{f, r} .

Fextx ,Fgrav

x External and gravitational force vec-tors on the x part of NTV, wherex ∈ {b, fw, rlw, rrw}.

rFfw , rF

rw Moment arms of tire external forceswith respect to the COG of front orrear wheels.

Mt ,Mext Input active tilt torque and externaltorque vector on NTV.

Mt ,ΔMt Two components of input active tilttorque.

π1 Support triangle when all wheels onthe ground.

π2 Support triangle when one rear wheellifts (not contacting).

P, P ′ Points in the support triangle directlybelow G (parallel to k3 direction) insystem 3.

P1 , P2 , P3 Projections of the COG, lateral accel-eration, and resultant acceleration ofNTV to ground.

la Longitudinal distance between P2and P3 .

J, J∗ Intermediate definitions for nonlinearmodel.

δv , e Auxiliary variables used in the stabil-ity analysis.

aSA , aSA , aSA Nondimensional measurement ofNTV lateral acceleration and its re-lated auxiliary variables.

ΔSA , ΔSA Estimation residual of aSA and itsauxiliary variable.

ΔSAm , θm Upper bound of ΔSA and tilt angle θ.

x, z State and output of the state-spacemodel.

A,B,C System matrices of the state-spacemodel.

A0 ,ΔA Nominal and uncertain item ofmatrix A.

H,F,E,Q Intermediate matrices in the stabilityproof.

k1 , k2 , k3 Constants in the active tilt controller.θdes Desired tilt angular velocity in the

controller.μ, α, ε Intermediate variables in the stability

proof.

I. INTRODUCTION

WORLDWIDE, traffic problems are widespread andprevalent. First, the gas emissions caused by the com-

bustion of fossil fuel in vehicles are the largest source of green-house gases. Second, the traffic congestion is growing in urbanarea due to the rapidly increasing number of automobiles [1].Finally, it is hard to find a parking place in a metropolis for thesame reason. To solve the first problem, a promising solutionmight be to develop electric vehicles. Using electric power is theclear and green choice instead of gasoline. To alleviate the othertwo problems, an efficient approach is to increase the flow rateof a particular traffic artery. A converse approach to increasingthe flow rate is to make the vehicle smaller and narrower [2].The narrow body of these vehicles can share a single lane ofthe road; it occupies less space at the parking lot easing parkingcongestion. Before we build up a traffic system using narrowvehicles, we have to solve the problem of how to develop a safe,stable, and easily maneuverable platform.

Narrow vehicles have been studied for many years. In the caseof two-wheeled vehicles, Sharp built a model of two-wheeledmotorcycle, based on which stable motion control method can bedesigned [3]–[7]. Mobile Wheeled Inverted Pendulum is anothernotable model of two-wheeled vehicle which is used in theJOE [8], the Segway [9], the B2 [10], the UW-Car [11], etc.

The electric bicycle is a cost-efficient alternative to expensivecars or crowded public transport systems. It is estimated thatonly in China does the number of electric bicycles surpasses100 million. Although there is a huge market for this type ofmotorized transport systems, it is not easy for a novice to quicklylearn how to ride a bicycle or two-wheeled motorcycle. More-over, a lot of fall-related accidents occur due to undercorner-ing or overbraking, while the electric bicycle is in motion. Toovercome these deficiencies, as well as cater to a potential fu-ture market, low-cost, easily maneuverable, and stable electricbicycle-like vehicles should be developed. Three-wheeled ve-hicles (TWV) with adaptive tilt mechanisms might have theminimum configuration to meet these demands. The commonfeature of a TWV is that its shape is narrow and all three wheelscan tilt. Generally, we call the TWV a narrow tilt vehicle (NTV).Several projects of NTV have been proposed on the industrialside, including Ford’s Gyron, GM’s three-wheeled Lean Ma-chine, and B.S.A. group’s tricycle.

FURUICHI et al.: SWITCHING DYNAMIC MODELING AND DRIVING STABILITY ANALYSIS OF THREE-WHEELED NARROW TILTING VEHICLE 1311

There have been extensive modeling and control studieson many aspects of vehicles, including steering system con-trol [12], [13], power train scaling [14], tire sensing [15]–[17],clutch control [18], battery control [19], etc. However, so far,there are not many related academic results of TWV using ac-tive tilt technology. Rajamani et al. presented a new conceptTWV called Narrow Commuter Vehicle (NCV). This NCVcan tilt all three wheels so that it can corner at high speed[20]–[22]. Chiou and Chen proposed an intelligent personalmobility vehicle that features low weight and a vehicular tiltingmotion [23].

In this paper, we focus on an NTV which has one front wheeland two rear wheels. Driving safety and stability are concen-trated on in this study. Our goal is to enhance the stabilityof the NTV in all driving situations, including straightforwardrunning, turning around, etc. During normal driving, all threewheels of NTV should contact the ground. There is a “criti-cal stable” state between normal running and falling down, inwhich one of the rear wheels is not on the ground. This situ-ation often happens when the NTV turns around suddenly ata high speed and should be avoided. To our knowledge, mostof the current research works on NTV modeling only discussthe normal driving situation. Thus, this paper aims to proposea new switching dynamical model considering both normal andabnormal running states of NTV.

In several previous work of NTV (see [20] and [22]), the ac-tive tilt controller was designed by calculating desired tilt angleθ∗ according to steering input δ. It should be noted that the lat-eral acceleration of NTV is zero when it runs at an equilibrium.This lateral acceleration can be easily measured by sensors ina real NTV system. In this study, the desired lateral acceler-ation (zero) is chosen as the reference input of tilt controllerinstead of a desired tilt angle θ∗. In this way, the controllerdesign procedure is easier to realize from the point of view ofengineering.

Narrow vehicles have greater stability issues than nonnarrowvehicles. Although active tilt control technology can improvedriving stability, it remains an open problem how to design effi-cient tilt control strategy. Therefore, the evaluation of tilt controlstrategy is necessary in the study of NTV. However, evaluationby experiments might be dangerous for both the driver and thevehicle itself since falling may incur injury, especially in a high-speed driving mode. Thus, we applied our simulation platformto evaluate the efficiency of tilt controller for the NTV at highspeed.

The rest of this paper is organized as follows. Section II givesthe mechanical structure of our NTV. In Section III, the modelformulations of our new switching dynamical model of NTVare discussed. By using this new dynamical model, we made asimulation platform in MATLAB/Simulink which can be usedto analyze typical dynamical behaviors of NTV. A new active tiltcontroller is proposed and the local stability of the closed-loopNTV system is proved in Section IV. The driving stability is alsoanalyzed and the stability condition is obtained by the peak-to-peak gain analysis method. Finally, we verify the effectivenessof proposed model, simulation platform, control method, andstability conditions through simulations and experiments.

Fig. 1. Tilting concept of NTV (rear view). The tilt angle is denoted by θ,whose positive direction is counterclockwise in the rear view. The tilt motion isdriven by torque Mt generated from a motor.

Fig. 2. Picture of the NTV.

Fig. 3. Tilt mechanism layout of the NTV.

II. MECHANICAL STRUCTURE OF NTV

In this study, we focus on the NTV which has one front wheeland two rear wheels. The two rear wheels may be actively tiltedby a motor fixed on the rear part. The steering angle and movingvelocity of NTV can also be measured as inputs to control thetilt angle. Fig. 1 shows the tilting concept of the NTV. All wheelswill tilt when it turns around. This increases the stability whenit turns around and makes it possible to curve at a high speed.Fig. 2 shows a picture of our NTV.

The detailed active tilt mechanism is shown in Fig. 3. TheNTV body is tilted by activation of the tilt motor when thedriver turns the vehicle. The purpose of tilt motion is to directthe resultant force exerted on the COG point to the inner areaof the support triangle. Fig. 4 shows the whole control blockdiagram. The tilt motor and driving motors are independentlycontrolled at the same time. The former is determined by thelateral acceleration, while the latter depend on the driver’s twist-grip control input.


Fig. 4. Control system schematics of the NTV.

Fig. 5. Normal running and critical stable states of NTV (rear view).

Fig. 6. Coordinate systems {0} and {1}. (a) Full view. (b) Top view.

III. MODEL FORMULATION OF NTV

We can categorize the running states of our NTV into thefollowing types:T1) temporary running with right rear wheel not on the ground

[see Fig. 5(a)];T2) normal running [see Fig. 5(b)];T3) temporary running with left rear wheel not on the ground

[see Fig. 5(c)];T4) totally fall down.

In existing work of three-wheeled NTV, e.g., in [22], re-searchers only paid attention to normal state T2, whereas statesT1 and T3 often occur when the NTV turns around suddenlyat a high speed. These states are intermediate states betweenT2 and T4 and are important in the stability analysis. Thus, inthis study, we aim at establishing a switching system model forNTV, which can describe these running states.

Fig. 7. Coordinates systems {1}and {2}. Full view of NTV with right wheellifted.

Fig. 8. Coordinates systems {2}, {3}, {3’}, {4}, and {5}. (a) Rear view.(b) Side and Top view in {3}.

A. Kinematic Description

Using the modeling procedure proposed in [22], similar dy-namical model of NTV in state T2 can be derived. On theother hand, dynamical model of T3 can be obtained similarly tocomputing the model of T1. To simplify the illustration, weonly give the modeling procedure of state T1 in this paper. Allcoordinate system definitions are given as follows.

System {0} is the inertial frame. The local coordinate system{1} is attached to the NTV and rotates with the yawing motionψ of the vehicle body (see Fig. 6). Point P is the projection ofCOG G onto the ground when there is no tilt motion of TWV.The original point O1 of {1} is coincident with point P . Thesupport triangle is denoted by π1 , whose vertices are the threecontact points between wheels and the ground. In the case of theNTV lifting up its right wheel from the ground, the supportingtriangle π1 will rotate angle β around the axis crossing twocontact points between the ground and the other two wheels.We denote the new triangle by π2 and define a new system {2}fixed to π2 corresponding to system {1} fixed to π1 . Note thatthe original point of {2} is O2 , which is coincident with point P ′

corresponding to P in system {1}. The next useful coordinatesystem is system {3} whose original point is coincident with theCOG G. This system is fixed to the vehicle body and is formedby a rotation of tilt angle θ around i2 . To conveniently introducethe other coordinate systems, we also define an intermediatecoordinate system {3′}, which is fixed to the center of frontwheel and has the same orientation as system {3} (see Fig. 7).


Fig. 9. Coordinates systems {1}, {1′}, {2′}, and {2}.

Coordinate frame {1} rotates with the yawing motion of thebody while frame {3} rotates with the tilt motion of the body.The front wheel has one degree of freedom with respect to thevehicle body. We define two more coordinate frames to dealwith this. The first is the steering axis of the front wheel whichis oriented at an angle ξ from the k′

3 . The last system {5} isdefined from the steering angle δ (see Fig. 8).

To calculate the transformation matrix from system {1} to{2}, we also define two more systems {1′} and {2′}, which aredepicted in Fig. 9. All rotation matrices can be calculated asfollows:

R10 =

⎡⎣

Cψ −Sψ 0Sψ Cψ 00 0 1

⎤⎦, R3

2 =

⎡⎣

1 0 00 Cθ Sθ

0 −Sθ Cθ

⎤⎦

R21 =

⎡⎢⎣

C2γ + S2

γ Cβ Sγ Cγ (Cβ − 1) Sγ Sβ

Sγ Cγ (Cβ − 1) S2γ + C2

γ Cβ Cγ Sβ

−Sγ Sβ −Cγ Sβ Cβ

⎤⎥⎦

R43 =

⎡⎣

Cξ 0 −Sξ

0 1 0Sξ 0 Cξ

⎤⎦, R5

4 =

⎡⎣

Cδ −Sδ 0Sδ Cδ 00 0 1

⎤⎦.

All the displacement vectors between consecutive coordinatesystems can be obtained as follows:

0r1 =

⎡⎣

dxdy0

⎤⎦

0

, 2r3 =

⎡⎣

00h

⎤⎦

2

, 3r4 =

⎡⎣

lf00

⎤⎦

3

1r2 =

⎡⎣

lr (1 − C2γ − S2

γ Cβ ) − b2 Sγ Cγ (1 − Cβ )

lrSγ Cγ (1 − Cβ ) − b2 (1 − S2

γ − C2γ Cβ )

−lrSγ Sβ + b2 Cγ Sβ

⎤⎦

2

.

B. Dynamical Model of State T1

Some assumptions to avoid the complexity of the model areassumed in this paper, which are listed as follows.

1) The steer axis is perpendicular to the ground, i.e., system{4} is coincident with system {3′}.

2) The weights of wheels are much less than the body.3) The vehicle is moving at a constant speed.4) The external forces on two rear wheels are the same in state

T2. Tire forces in state T2 is shown in Fig. 7. The lateralforces are denoted by Ffw , Frlw , and Frrw , while normalforces by Nfw , Nrlw , and Nrrw . Although the accuratelateral tire forces cannot be measured, these forces can be

approximated by the following tire model [22]:

Ffw = Ff = Cf1

(δ − y + lf ψ

v

)+ Cf2θ

Frrw = Frlw = Fr = Cr1

(− y − lrψ

v

)+ Cr2θ. (1)

5) The driver is supposed to be well buckled with a seat belt.It means that there are no influence of the shift of COGcaused by the driver.

Similar to the modeling procedure in [22], by applying thetransportation theorem [24], we can obtain the Total DynamicalModel of state T1 as follows:

maG = Fextb + Fext

rlw + Fextrrw + Fext

fw (2)

d

dt

0

HG = Mext + (B rG r lw + rFrlw ) × Fext

rlw

+ (B rG fw + rFfw ) × Fext

fw + B rGb × Fgravb (3)

where the tilt torque, the gravity force, and external forces ex-erted on each part of NTV satisfy:

Mext =

⎡⎣

Mt00

⎤⎦

1

, Fextb = Fgrav

b =

⎡⎣

00

−mg

⎤⎦

1

,

Fextfw =

⎡⎣

0Ff

Nfw

⎤⎦

1

, Fextrlw =

⎡⎣

0Fr

Nrw

⎤⎦

1

, Fextrrw =

⎡⎣

000

⎤⎦

1

.

The moment arms are given by

rFfw =

⎡⎣

00

−rfw

⎤⎦

3

, rFrw =

⎡⎣

00

−rrw

⎤⎦

3

.

From (2) and (3), the motion equations of lateral translation y,roll motion β, yaw motion ψ, and tilt motion θ can be obtainedas follows:

Cγ · y = (•)1 · Sγ Cβ + (•)2 · Cγ Cβ + (•)3 · Sβ

− β(nβ1 · Sγ Cβ + nβ2 · Cγ Cβ + nβ3 · Sβ )

− ψ(nψ1 · Sγ Cβ + nψ2 · Cγ Cβ + nψ3 · Sβ )

− θ(nθ1 · Sγ Cβ + nθ2 · Cγ Cβ + nθ3 · Sβ ) (4)

{−nβ1 · mhSγ Cθ − nβ2 · mhCγ Cθ

− nβ3 · m(lf Sγ − hCγ Sθ ) + eβ1}β= −(•)1 · mhSγ Cθ − (•)2 · mhCγ Cθ

− (•)3 · m(lf Sγ − hCγ Sθ ) + (•)4

− y{−ny1 · mhSγ Cθ − ny2 · mhCγ Cθ

− ny3 · m(lf Sγ − hCγ Sθ )}

− ψ{−nψ1 · mhSγ Cθ − nψ2 · mhCγ Cθ

− nψ3 · m(lf Sγ − hCγ Sθ ) + eψ1}

− θ{−nθ1 · mhSγ Cθ − nθ2 · mhCγ Cθ

− nθ3 · m(lf Sγ − hCγ Sθ ) + eθ1} (5)


Fig. 10. State transition diagram of NTV.

Fig. 11. Lateral acceleration analysis when NTV turns around. (a) Full view.(b) Top view.

(eψ2 · Sβ − eψ3 · Cβ ) · ψ

= (•)5 · Sβ − (•)6 · Cβ − β(eβ2 · Sβ − eβ3 · Cβ )

− θ(eθ2 · Sβ − eθ3 · Cβ ) (6){

(lf Sγ Cβ − hCγ Cβ Sθ − hSβ Cθ )nθ3 −1m

Cβ · eθ1

}θ

= (lf Sγ Cβ − hCγ Cβ Sθ − hSβ Cθ )(•)3

− 1m

Cβ (•)4 +1m

Cβ Cγ Mt

− (lf Sγ Cβ − hCγ Cβ Sθ − hSβ Cθ )ny3 y

+{

1m

Cβ · eβ1−(lf Sγ Cβ −hCγ Cβ Sθ−hSβ Cθ )nβ3

}β

+{

1m

Cβ · eψ1−(lf Sγ Cβ −hCγ Cβ Sθ−hSβ Cθ )nψ3

}ψ.

(7)

All items in (4)–(7) are given in [25].

C. Switching Conditions for Models of Different States

The state transition diagram of NTV is shown in Fig. 10. Toobtain the state switching condition, we consider a general caseshown in Fig. 11. The NTV is supposed to be turning aroundand the turning radius is R. The instant speed and accelerationof NTV are equal to v and a, respectively. The resultant lateralacceleration perpendicular to the body includes two componentscaused by the gravity g and the centrifugal acceleration v2/R,respectively. The measured lateral acceleration of NTV is de-noted by aSA · g, where aSA is a nondimensional quantity that

will be used in the stability analysis and controller design. Thisnondimensional quantity is given by

aSA =(

gSθ −v2Cθ

R

)/g. (8)

As illustrated in [26], the relationship among the steeringangle δ, direction angle δ′, roll angle θ, and front wheel casterangle ξ is given by

(tan δ′)Cθ = (tan δ)Cξ , tan δ′ =L

R. (9)

It follows from (8) and (9) that we have

aSA = Sθ −v2 tan δCξ

gL. (10)

Point P1 is the projection of NTV COG to the ground. P2 isthe projection of lateral acceleration. P3 is the projection of theresultant force of gravity, the centrifugal force, and the inertialforce. Obviously, a state transition from T2 to T3 will occur ifP3 goes right and outside the support triangle. If the accelerationis zero, point P2 is coincident with P3 . From (10) and geometryknowledge, the coordinate value of P3 in coordinate system {1}can be calculated as

P3 = [P3x P3y P3z ]T1 = [ la haSA 0 ]T1 (11)

where la satisfies

la =ahCθ

g.

Accordingly, all the switching conditions can be concludedas follows:

R1. IF − b · (lf + la)2(lf + lr)

≤ P3y ≤ b · (lf + la)2(lf + lr)

AND βr

≤ 0 AND βr ≤ 0 AND I(k − 1)=T1, THEN I(k)=T2.

R2. IF P3y ≥ b · (lf + la)2(lf + lr)

AND I(k − 1) = T2,

THEN I(k) = T1.

R3. IF − b · (lf + la)2(lf + lr)

≤ P3y ≤ b · (lf + la)2(lf + lr)

AND βl ≤ 0

ANDβl ≤ 0 AND I(k − 1) = T3, THEN I(k) = T2.

R4. IF P3y ≤ −b · (lf + la)2(lf + lr)

AND I(k − 1) = T2,

THEN I(k) = T3.

R5. IF θ − βr ≤ −THRES AND I(k − 1) = T1,

THEN I(k) = T4.

R6. IF θ + βl ≥ THRES AND I(k − 1) = T3,

THEN I(k) = T4.

Note that we use I(k) to denote current NTV state in theswitching rule set. βr and βl are used to denote different rollangle β of state T1 and T3.


Fig. 12. Control diagram of NTV switching dynamical model.

D. Total Control Diagram

To stabilize the system shown in Fig. 11, the desired positionof P3 is within the support triangle and on the axis i1 . That is,the desired aSA is zero. In our real NTV system, the resultantlateral acceleration is easily measured by sensors. Therefore,aSA is very suitable to be used as a feedback quantity to designan active tilt controller, which will be further discussed in thefollowing section. The total control diagram of NTV is thenillustrated in Fig. 12, and the corresponding simulation platformwas implemented in MATLAB/Simulink.

IV. TILT CONTROLLER DESIGN AND DRIVING STABILITY

A. Tilt Controller Design

Using similar modeling method presented in [22], the tiltdynamics of NTV in state T2 can be derived. The only differencebetween modeling states T2 and T1 is that the external force onright rear wheel Fext

rrw should be the same as Fextrlw . The final tilt

dynamics is given by

Jθ = Mt + mghSθ − h2m(SθCθ )θ2

−(SθCθ )ψ2(Iyyb − Izz

b ) − h(Cθ )(Ff + 2Fr) (12)

where J satisfies

J = Ixxb + h2mS2

θ .

Note that aSA = 0 corresponds to an equilibrium of a runningNTV system. At this equilibrium, the acceleration of θ shouldbe zero if there are no active torque Mt . To guarantee the drivingstability, the aim of tilt controller is regulating the tilt angle θ toa desired value θ∗ such that aSA = 0 holds. The approximatedlinearized tilt dynamic model of NTV around θ = θ∗ is

J∗θ = Mt − h(−mg + Cf2 + 2Cr2)θ (13)

where J∗ = Ixxb + h2mS2

θ∗ .Considering the motor regulating tilt angle normally works in

a speed control mode, we propose an active tilt control strategyshown in Fig. 13. In the control block in Fig. 13, θdes denotesthe desired angular velocity of motor and is fed into the speedcontrol module. The active torque Mt consists of two parts, Mt

Fig. 13. Proposed active tilt control strategy.

and ΔMt . Mt is proportional to the difference between actualmotor angular velocity θ and the desired value θdes . ΔMt isused to compensate the nonacceleration term, satisfying ΔMt =h(−mg + Cf2 + 2Cr2)θ. Thus, the final linearized model is

J∗θ = Mt . (14)

For the proposed control strategy, the following stability resultholds.

Theorem 1 (Stability of Active Tilt Control): For a given veloc-ity v∗ and steering angle δ∗, if tilt controller parameters k1 and k2are positive, then the closed-loop tilt dynamical model depictedin Fig. 13 is locally stable around equilibrium [θ, θ]T = [θ∗, 0]T .

Proof: From (10), (14), and Fig. 13, the closed-loop tilt dy-namical model of NTV is given by

(J∗

k2

)θ + θ + k1hSθ =

k1hv∗2 tan δ∗Cξ

gL. (15)

Note that (15) is a second-order nonlinear differential equationwith respect to tilt angle θ. Although there are infinite equi-libriums of this differential equation, it is easy to find that thefollowing equilibrium is locally stable by linearizing (15):

[ θ∗ θ∗ ]T =[sin−1

(v ∗2 tan δ ∗Cξ

gL

)0]T

. (16)

�It should be pointed out that the tilt dynamics (12) will not

change even if the speed v of NTV is time varying. This con-clusion is easy to reach if considering longitudinal driven/braketorques of rear wheels in the modeling procedure, whereas thedesired tilt angle θ∗ will change if v is not a constant. This leadsto an uncertain J∗ in the linearized model (14) and makes thedriving stability analysis more difficult, which is discussed inthe next section.

B. Driving Stability Analysis

From switching condition analysis in Section III-C, anotherimportant stability problem is the driving stability which canbe guaranteed provided that the projection point P3 is alwayslocated inside the support triangle π1 . This stability condition isquantitatively given by

−b · (lf + la)2(lf + lr)

≤ P3y ≤ b · (lf + la)2(lf + lr)

(17)


or

−b · (lf + la)2h(lf + lr)

≤ aSA ≤ b · (lf + la)2h(lf + lr)

. (18)

Given the steering angle δ∗ and moving velocity v∗ of NTV,aSA = 0 is satisfied if the NTV system runs at the equilibrium[θ, θ]T = [θ∗, 0]T , where θ∗ is determined by (16). Nevertheless,this equilibrium state is easily broken if the driver suddenlychanges the steering angle or moving velocity. From experience,a fast motorcycle is easily overturned when the driver suddenlyturns around. Therefore, an interesting question comes up: whatare the constraints of steering angle δ and moving velocity vthat guarantee the driving stability of NTV when an active tiltcontroller is in use? To answer this question, it is needed toanalyze the relationship between the steering angle δ and aSA .It should be pointed out that the equilibrium (16) is not fixed asv varies with time. On the other hand, different drivers lead todifferent mass and COG height h. Therefore, the uncertainty ofparameter J∗ should be taken into account when studying thedriving stability.

From (10), (14), and (15), the relationship between δ and aSAcan be described by

⎧⎪⎪⎨⎪⎪⎩

aSA = Sθ −v2 tan δCξ

gL(J∗

k2

)θ + θ + k1hSθ =

k1hv2 tan δCξ

gL.

(19)

It is difficult to analyze the effect of steering angle δ to aSA from(19) due to its strong nonlinearity. Therefore, we introduce anauxiliary variable aSA which is given by

aSA = θ − k3δv (20)

where δv = v2 tan δ, k3 = Cξ

gL .In the real NTV system, the tilt angle θ is confined in a bound

for safety consideration. The bound value is denoted by θm andwe have

−θm ≤ θ ≤ θm . (21)

From (10), (20), and (21), aSA can be represented by

aSA = aSA + ΔSA (22)

where the residual ΔSA satisfies

|ΔSA | = |Sθ − θ| ≤ ΔSAm = θm − Sθm . (23)

From (19), (20), and (22), the relationship between aSA andδv is described by

¨aSA +(

k2

J∗

)˙aSA +

k1k2h

J∗ aSA

= −k3 δv − k2k3

J∗ δv − k1k2h

J∗ ΔSA . (24)

The differential equation (24) can be represented by the trans-fer function block diagram shown in Fig. 14. As shown inFig. 14, if we take ΔSA as a bounded disturbance, the driv-ing stability problem is then reduced to finding the bounds ofinput δv such that aSA satisfies (18). Thus, the peak-to-peakgain analysis method finds its application in this case [27].

Fig. 14. Block diagram of the relationship between δv and aSA .

Let us define x =[

eJ ∗ e

]T, u = δv , z = aSA . According to

Fig. 14, the relationship between δv and aSA can be describedby the following state-space model:

{x = Ax + Buz = Cx

(25)

where

A =

⎡⎢⎣

01J∗

−k1k2h − k2

J∗

⎤⎥⎦ , B =

[01

]

C = [−k2k3 −k3 ] .

The uncertain system matrix A can be rewritten as

A = A0 + ΔA(t) (26)

where A0 is a constant matrix and we can assume that uncertainpart ΔA(t) satisfies

ΔA(t) = HF(t)E,FT(t)F(t) ≤ I. (27)

Lemma 1 (see [28]): Given matrices Q,H,E, and R of ap-propriate dimensions and with Q,R symmetrical and R > 0,then

Q + HFE + ET FT HT < 0

for all F satisfying FT F ≤ R , if and only if there exists someε > 0 such that

Q + εHHT + ε−1ET RE < 0.

Theorem 2: For system (25), if the following optimizationproblem

mins.t.

μ

⎡⎢⎣

XAT0 + A0X + εHHT + αX XET B

EX −εI 0BT 0 −αI

⎤⎥⎦ < 0

[X XCT

CX μ2I

]> 0 (28)

has a solution μ∗, then we have ‖z(t)‖ ≤ μ∗, when ‖u(t)‖ = 1,i.e. ‖•‖PP = μ∗.


Proof: Suppose there is a symmetric positive matrix P for aV = xT Px such that

V + α(V − uT u

)< 0, α > 0 (29)

P − 1μ2 CT C > 0. (30)

Since we have ‖u(t)‖ = 1, from (29), it followsthat V + α (V − 1) < 0. Obviously, it implies that Θ ={x|V = xT Px ≤ 1

}is an attractive invariant set. From (30),

it is easy to know that ‖z(t)‖ ≤ μ · xT Px ≤ μ,∀x ∈ Θ. It fol-lows from (29) that

V + α(V − uT u

)

= xT(AT

0 P + PA0 + ET FT HT P + PHFE + αP)x

− αuT u + uT BT Px + xT PBu < 0.

According to Lemma 1, this inequality is equivalent to

xT(AT

0 P + PA0 + ε−1ET E + εPHHT P + αP)x

−αuT u + uT BT Px + xT PBu < 0.

Considering the property of Schur complements, the aboveinequality implies the first matrix inequality of (28) withX = P−1 . On the other hand, it is obvious that (30) is equiv-alent to the second matrix inequality of (28). Therefore, if theoptimization problem (28) is solvable, the solution μ∗ is thepeak-to-peak gain from u to z. �

The optimization problem (28) can be solved by LMI toolboxin MATLAB with a given α. Normally, α is between zero andhalf of the real part of least eigenvalue of A. The optimal valueof α can be obtained by a simple two point search (dichotomoussearch) algorithm. Then, we can obtain the main result of drivingstability as following.

Theorem 3 (Driving Stability Condition): Suppose that theoptimization problem (28) has a solution μ∗. Parameters k1 andk2 satisfy

(k2

J∗

)2

>4k1k2h

J∗ , k1 > 0, k2 > 0. (31)

Then, the driving stability can be guaranteed if inequality∣∣∣δv

∣∣∣ =∣∣∣v2 δ

[1 + (tan δ)2

]+ 2vv tan δ

∣∣∣

≤(

b · (lf + la)2h(lf + lr)

− 2ΔSAm

)/μ∗ (32)

holds.Proof: It follows from Theorem 2 that we have

∥∥∥∥aSA

δv

∥∥∥∥PP

= μ∗. (33)

Therefore, if inequality (32) holds, we have

|aSA | ≤b · (lf + la)2h(lf + lr)

− 2ΔSAm . (34)

From (31), we know that the relationship from ΔSA to ΔSAis described by an overdamped second-order system with unit

Fig. 15. Comparison of experimental and simulated trajectories of NTV whenit runs in state T2 (v ≈ 8 km/h).

gain. Thus, for any continuous bounded signal ΔSA , we have

ΔSA ≤ ΔSA ≤ ΔSAm . (35)

If follows from (34) and (35) that

|aSA | = |aSA + ΔSA | ≤b · (lf + la)2h(lf + lr)

. (36)

This implies the driving stability of NTV. �Note that if δ is small and v is constant, inequality (32) can be

regarded as a constraint of the steering velocity during drivingthe NTV. The tolerant steering velocity becomes higher as thewidth b between two rear wheels increases or the height of COGh decreases. This agrees with our finding that the NTV becomesmore stable with increasing area of the support triangle π1 orlower the driver’s COG position.

V. SIMULATION AND EXPERIMENT STUDY

To verify the correctness of our proposed switching modeland stability results, some simulations and experiments wereperformed on the NTV. In all the simulations, the control dia-gram shown in Fig. 12 was implemented in MATLAB/Simulink.The active tilt controller was designed as shown in Fig. 13.

A. Simulation and Experiment of Model T2

During most of the driving time, the NTV runs in state T2.That is, all wheels are on the ground. We first verified our switch-ing model in this special case. Both experiment and simulationwere performed. In the experiment, a driver was asked to ridethe NTV at will in an almost constant velocity. Both low-speed(v ≈ 8 km/h) and high-speed (v ≈ 30 km/h) cases were inves-tigated. All real-time NTV state variables were recorded simul-taneously. In the simulation, the real steering angle δ and driv-ing velocity v were input to our switching model in Simulink.Corresponding motion state variables were obtained. The realsteering angle trajectory and comparison results of simulationand experiment are shown in Figs. 15 and 17. It turns out thatthe simulated tilt angle trajectory matches the real one well.The maximum simulation error of θ is less than 0.03 rad in the


Fig. 16. Simulation error and slip angle when NTV runs in state T2 (v ≈8 km/h).

Fig. 17. Comparison of experimental and simulated trajectories of NTV whenit runs in state T2 (v ≈ 30 km/h).

low-speed case and less than 0.05 rad in a high-speed case. Theslip angle trajectories were calculated and given in Figs. 16 and18. It turns out that the slip angle is very small (less than 4◦),which guarantees that the linear model (1) can sufficiently andaccurately describe tire lateral forces. The desired and real tiltangular velocity θdes , θ were also shown in Figs. 15 and 17. Ourfindings show that the desired θdes can be tracked satisfactorilyby controlling the tilt motor.

B. Simulation and Experiment of Two-State Switching

To further verify our model, we also investigated the two-stateswitching situation. In the experiment, the NTV runs straightlyat a constant speed (≈ 2.7 m/s), then turns on a dime to the left.The recorded real steering angle δ and velocity v were input tothe model in Simulink.

The comparison results of simulation and experiment areshown in Fig. 19. From the simulated trajectory of roll angleβl , the NTV starts raising its left rear wheel approximately atinstant t = 1 [s]. This coincides with the real measured roll an-

Fig. 18. Simulation error and slip angle when NTV runs in state T2 (v ≈30 km/h).

Fig. 19. Comparison of experimental and simulated trajectories of NTV in aT2 → T1 switching case.

gle β. At the end of this experiment, the NTV fell down to theright, which agrees with the results from simulation. The slipangle is also given in Fig. 19. It should be noted that the sim-ulation accuracy deteriorates due to the significantly increasingslip angle as the roll angle builds up. On the other hand, themode switching is only verified in the low-speed case. The ver-ification of a high-speed case is dangerous because it is difficultto ensure sufficient safety measures in a fall test of high-speeddriving. The presented model is sufficient to be used in stabilityanalysis and tilt controller design as our aim is to control thesystem running in state T2 all the time.

C. Driving Stability Analysis

Theorem 3 gives the constraints of steering angle δ for NTVdriving stability. The effectiveness of this theoretical result isverified by simulations and experiments in the following.

The driving stability condition proposed in Section IV is veri-fied in the following. Both the low-speed driving mode and high-speed driving mode will be discussed. In all the experiments


Fig. 20. Experiment results of NTV normal running in a low-speed drivingcase (v ≈ 2.2 m/s). (a) Sampling points of steering angle and angular velocity(δ(t), δ(t)) (v ≈ 2.2 m/s). (b) Trajectory of aSA (v ≈ 2.2 m/s). The boundsof aSA are calculated from (18). (c) Trajectory of NTV velocity v(t) of theexperiment.

and simulations, the control parameters were chosen as k1 =2.54, k2 = 3900. The uncertain parameter J∗ is assumed to bewithin interval [0.9J0 , 1.1J0 ], where J0 is a nominal value.

In the case of low-speed driving mode, experiment data inSections V-A and V-B were used to validate the correctness of

Fig. 21. Experiment results of NTV falling down in a low-speed drivingcase (v ≈ 2.7 m/s). (a) Sampling points of steering angle and angular velocity(δ(t), δ(t)) (v ≈ 2.7 m/s). (b) Trajectory of aSA (v ≈ 2.7 m/s). The boundsof aSA are calculated from (18). (c) Trajectory of NTV velocity v(t) of theexperiment.

Theorem 2. The experiment data in Section V-A comes froma period of normal driving. During the driving, the velocity ofNTV was almost equal to 2.2 m/s ≈ 8 km/h [see Fig. 20(c)].With the almost constant v, the bounds of allowable steeringangle δ and angular velocity δ were calculated according to(32) and depicted by blue solid lines in Fig. 20(a). The samplingpoints of real steering angle and angular velocity sequence (δ, δ)are represented by red circles in the phase plane. All the pointsare inside the calculated bounds, which satisfies the drivingstability condition. Correspondingly, the NTV is in a normalstate all the time. The trajectory of aSA is also given. In theexperiment discussed in Section V-B, the NTV fell down aftera sudden turn. During this experiment, the driving velocity of


Fig. 22. Experiment results of NTV accelerating from standstill to 30 km/h.(a) Sampling points of steering angle, angular velocity and vehicle velocity(δ(t), δ(t), v(t)) (a ≈ 2 m/s2 ). (b) Trajectory of aSA (a ≈ 2 m/s2 ). (c) Tra-jectory of vehicle velocity (a ≈ 2 m/s2 ).

NTV is approximately equal to 2.7 m/s. In the same way asabove, sampling points of sequence (δ, δ) are plotted in thephase plane together with the calculated bounds. Note that thesudden change of steering angle generated some outliers of(δ, δ). These outliers are located outside the stability bounds,which results in a final falling down of NTV. Correspondingly,the trajectory of aSA exceeds the safety bounds [see Fig. 21(b)].

Fig. 23. Simulation results of NTV falling down in a high-speed case. (a) Sam-pling points of steering angle and angular velocity (δ(t), δ(t)) (v = 30 km/h).(b) Trajectory of aSA (v = 30 km/h). (c) Trajectory of all state responses(v = 30 km/h).

An acceleration experiment was also performed to further val-idate the stability condition (32). In this experiment, the vehiclewas accelerated to 8 m/s(≈ 30 km/h) within 3 s and then run at aconstant speed. The acceleration is approximately 2 m/s2 duringthe whole process. The stability bounds are then calculated from(32) and plotted in a 3-D space (with three components: v, δ,and δ) [see Fig. 22(a)]. Data series (v, δ, δ) of the NTV wererecorded and shown in this figure (red point dot). All data areinside the stability bounds so that the system is stable accordingto Theorem 3, which agrees with our experiment results.

In the high-speed driving mode, conducting intentional fallingexperiment of NTV is dangerous both to the driver and vehi-cle itself. Because the proposed simulation platform based onswitching dynamical NTV models was proved to be sufficiently


accurate, driving stability analysis of high-speed mode was per-formed by simulations. The velocity of NTV is assumed to be30 km/h (8.33 m/s). A falling down simulation was carried out.A dataset was generated by combining a sequence of allowable(δ(t), δ(t)) and a data sequence (δ(t), δ(t)) of sudden turns.The generated data sequence is fed into the simulation platformto calculate the state responses. All the simulation results aredepicted in Fig. 23. The NTV fell down at around t = 10.7 [s],which is reflected by the nonzero βl and abnormal aSA .

All the experimental and simulation results coincide with thetheoretical conclusion very well. It should be pointed out thatthe driving stability condition is also useful to design suitablesteer-by-wire controllers.

VI. CONCLUSION

The main contribution of this study lies in the following.1) A novel switching dynamical model of NTV was proposed

to describe multiple running states including normal running,temporary running with left or right rear wheel not on ground,and totally falling down. Based on this model, a simulationplatform was established to test dangerous driving behaviorsthat cannot be performed in a real vehicle.

2) A new active tilt controller was designed based on alinearized model about a state without lateral acceleration(aSA = 0). Using measured lateral acceleration to design tiltcontroller avoids both the inaccuracy of approximation andcomputation complexity. A potential risk of this method is over-reliance on the acceleration sensor’s single point of failure, ashortcoming that can be mitigated by redundant sensors.

3) The driving stability of NTV was analyzed and correspond-ing stability condition was obtained. This study might be the firstattempt to investigate the driving stability problem of NTV viapeak-to-peak gain analysis. Promising theoretical results wereobtained, which may be beneficial for the performance analysisand controller design of NTV.

ACKNOWLEDGMENT

The authors would like to thank the Equos Research Com-pany, Ltd., for their valuable experimental data of a real vehicle.

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Hiroki Furuichi received the B.S. and Master of En-gineering degrees from Nagoya University, Nagoya,Japan, in 2011 and 2013, respectively.

He is currently with Nissan Motor Company, Ltd.,Yokohama, Japan. His research interests include nar-row vehicles.


Jian Huang (M’07) received the Graduate, Master ofEngineering, and Ph.D. degrees from Huazhong Uni-versity of Science and Technology (HUST), Wuhan,China, in 1997, 2000, and 2005, respectively.

From 2006 to 2008, he was a Postdoc-toral Researcher in the Department of Micro-Nano System Engineering and Department ofMechano-Informatics and Systems, Nagoya Univer-sity, Nagoya, Japan. He is currently an AssociateProfessor in the School of Automation, HUST. Hismain research interests include rehabilitation robots,

robotic assembly, networked control systems, and bioinformatics.

Toshio Fukuda (M’83–SM’93–F’95) received theB.S. degree from Waseda University, Tokyo, Japan,in 1971, and the M.S. and Dr.Eng. degrees fromThe University of Tokyo, Tokyo, in 1973 and 1977,respectively.

From 1977 to 1982, he was with the National Me-chanical Engineering Laboratory, Tsukuba, Japan.From 1982 to 1989, he was with the Science Univer-sity of Tokyo, Tokyo. Since 1989, he has been withNagoya University, Nagoya, Japan, where he is cur-rently a Professor in the Department of Micro-Nano

Systems Engineering, mainly involved in the research fields of intelligent roboticsystems, cellular robotic systems, mechatronics, and micro/nanorobotics.

Dr. Fukuda was the President of the IEEE Robotics and Automation Society(1998–1999), the Director of IEEE Division X, Systems and Control (2001–2001), the Editor-in-Chief of the IEEE/ASME TRANSACTIONS ON MECHA-TRONICS (2000–2002), and the President of the IEEE Nanotechnology Council(2002–2005). He is the AdCom President of the IEEE Nanotechnology Counciland was the Director of IEEE Region 10 (2011–2012).

Takayuki Matsuno received the Graduate, Master ofEngineering, and Ph.D. degrees from Nagoya Uni-versity, Nagoya, Japan in 1998, 2000, and 2005,respectively.

He was a Research Associate in the Departmentof Micro-Nano Systems Engineering and Departmentof Mechano-Informatics and Systems, Nagoya Uni-versity, until March 2006. He was a Research As-sociate and then a Lecturer in Intelligent SystemsDesign Engineering, Toyama Prefectural University,until September 2011. He is currently a Lecturer in

the Graduate School of Natural Science and Technology, Okayama University,Okayama, Japan. He is mainly involved in the research fields of flexible objectmanipulation, automation of assembly tasks in factories, and dynamical systemcontrol.

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Narrow tilting dynamic