1470 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 20, NO. 3, JUNE 2015 …mercury.hau.ac.kr/sjkwon/Lecture/Capstone/2015-06 TMECH... · 2016-08-30 · 1470 IEEE/ASME TRANSACTIONS

1470 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 20, NO. 3, JUNE 2015

Tilting-Type Balancing Mobile Robot Platformfor Enhancing Lateral Stability

SangJoo Kwon, Sangtae Kim, and Jaerim Yu

Abstract—In this paper, a tilting-type balancing mobile robotplatform is investigated for enhancing lateral stability. In addi-tion to pitch, yaw, and straight motion by the conventional two-wheeled inverted pendulum mechanism, it can generate roll andvertical motion by an additional tilting mechanism. The static forceanalysis shows that body separation tilting is more advantageousin power consumption than single body tilting, specifically whenthe payload to body weight is relatively small. Some design con-siderations are given for the determination of body structure andactuator powers. For the dynamic modeling, the titling balancingplatform is assumed as a three-dimensional inverted pendulumwith moving base and the nonlinear equation of motion is derivedin terms of Kane’s method. Then, a velocity/posture control loop isconstructed, where the tilt angle reference is naturally generatedaccording to the centrifugal force variation in following a circu-lar path. Experimental results are given to validate the proposedmobile platform with the tilting control strategy.

Index Terms—Inverted pendulum robot, personal transporter,self-balancing, tilting vehicle, two-wheeled mobile robot.

I. INTRODUCTION

R ECENTLY, a variety of personal vehicles useful in termsof mobility and accessibility in congested urban environ-

ments and narrow tracks are being developed [1]–[13]. Many ofthem are closely concerned with a balancing mobile platformwhich utilizes just two wheels for steering and velocity control.In fact, the two-wheeled balancing mobile robot (2WBMR)has been an issue for a long time both in industry and in theacademy and a lot of design and control problems have been in-vestigated [1]–[7]. As some typical examples, Segway [1] wasproven to have stable driving performance as a personal trans-porter, and such as iBOT [2] and Genny [3] demonstrated thatthese new balancing robot wheelchairs for the disabled are goodalternatives to conventional electric wheelchairs. As another, aself-balancing electric vehicle EN-V [4] can save much roadspace due to its compact design and it looks good enough forcommuters. These commercial products indicate that personalvehicles based on the self-balancing technology will becomeeveryday items in the near future.

Thus far, however, the 2WBMRs have been much limited inturning speed because of the need to stop rollovers. They lack

Manuscript received May 30, 2014; revised August 24, 2014 and October 11,2014; accepted October 16, 2014. Date of publication October 15, 2014; dateof current version May 18, 2015. Recommended by Technical Editor G. Liu.This research was supported by Basic Science Research Program through theNational Research Foundation of Korea (NRF-2012R1A1B3003886). (Corre-sponding author: SangJoo Kwon.)

The authors are with the School of Aerospace and Mechanical Engi-neering, Korea Aerospace University, Goyang-City 412-791, Korea (e-mail:[email protected]; [email protected]; [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.2014.2364204

Fig. 1. Lateral stability increase in high speed turns by using tilting motion.(a) Nontilting. (b) Tilting.

the roll motion capability to compensate for centrifugal forcesarising in rapid turns. Hence, currently existing control schemesfor the 2WBMRs basically assume that the centrifugal force inthe lateral direction does not have a great effect on the posturalstability [5]. Actually, the rollover problem is an important issuein vehicle dynamics and control and diverse research works existon the rollover estimation and roll prevention systems [14].

A basic requirement for the 2WBMRs is to achieve highlyreliable pitch balancing performance, which can be generallyimplemented in terms of inverted pendulum control schemes[6], [7]. However, in applying the 2WBMR platforms to per-sonal transporters, robot wheelchairs, and other diverse formsof electric vehicles, lateral stability also should be seriously con-sidered for comfortable riding and coping with rollovers due toabrupt turns.

Here, if the 2WBMRs are equipped with a tilting mechanismin the lateral direction, it can generate roll motions around theforward direction vector. First of all, the tilting function is help-ful to prevent the mobile robot from falling down, as illustratedin Fig. 1, when the robot is moving fast following a circularpath and a large moment is exerted due to the centrifugal force.Actually, the tilting trains were developed in many countries toincrease the cornering speed and the riding quality on a fixed cur-vature of railroad [15], [16]. The tilting technology also enablesthe balancing robots to keep an upright posture on any inclinedterrain for any moving direction in terms of three-dimensional(3-D) posture control using roll, pitch, and yaw motion together.

The notion of lateral tilting has already been adopted fre-quently in the design of compact electric vehicles for individ-ual transportation, which includes three wheelers [8]–[11] andfour wheelers [12], [13], mostly in the convergence form ofcar and motorcycle. They require active roll control to make

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KWON et al.: TILTING-TYPE BALANCING MOBILE ROBOT PLATFORM FOR ENHANCING LATERAL STABILITY 1471

stable banked turns in cornering [8], [9]. Unlike tilting trains,these narrow tilting vehicles make good use of the phase dif-ference between the left and right two wheels to generate theroll motion. As well, the man-powered tilting bikes [17] ex-plain the effectiveness of the tilting technique for fast and stablecornering.

Contrary to the active developments of tilting passenger cars,examples of tilting technology being applied to the 2WBMRsare rarely found. Just the necessity of the centrifugal force com-pensation for the 2WBMR was raised in [18] and an initial formof the tilting 2WBMR of this paper can be found in [19]. On theone hand, a portable personal transporter recently announced in[20] has a passive tilt function to overcome the deficiency ofSegway type balancing vehicles.

We have announced basic principles for designing a tiltingrobot driven by rack and pinion in [21] and applied to a robotwheelchair in [22]. This paper presents the final form of thetilting-type 2WBMR with ball screw driven tilting mechanism,where the dynamic modeling in [21] and the force analysis in[22] are corrected and fully extended. In addition, an effec-tive tilting control strategy to reduce the wheel slippage on theground is given with new experiments. It is expected that theproposed tilting-balancing platform can be applied to personaltransporters or unmanned navigation vehicles.

As is shown in Section II, the proposed platform is symmet-rically separated into two parts and the tilting motion is realizedby making them slide reversely. Static analysis in Section IIIconfirms that body separation tilting is more advantageous inthe respect of power consumption compared with single bodytilting. In Section IV, a nonlinear equation of motion is derivedby assuming it as a 3D inverted pendulum with moving base.Section V discusses how to operate the tilting mechanism inarbitrary circular paths through the control design, and experi-mental results are shown in Section VI. Finally, the conclusionis drawn in Section VII.

II. STRUCTURAL DESIGN

A. Body-Separated Tilting Mechanism

In designing a tilting 2WBMR, roughly we can consider twokinds of tilting methods. The first one can be defined as the singlebody tilting shown in Fig. 2(a), where the bodyshell is rotatedwith the load or rider upon the fixed base of the chassis (bogie)by the operation of connecting rods in the tilting bolster [15].This type has been widely accepted in tilting trains carryingheavy weights [16]. In the second one, which can be calledchassis tilting, the chassis base and wheels are leaned togetheras in Fig. 2(b), where the tilting motion is implemented usuallyby using parallelogram mechanisms. Most of the narrow tiltingvehicles in [8]–[13] follow this type, because it is compatiblewith light weight vehicles.

As another version of chassis tilting, this paper suggests thebody separation tilting shown in Fig 3(a). The two body blocksin Fig 3(b) are equipped with wheel motor, ball screw, and linearguides for smooth sliding, respectively. The ball screws in theleft and the right sides transmit torques from the tilting actuatorsmounted in the top platform to both blocks. If they are moving

Fig. 2. Comparison of two tilting methods. (a) Single body tilting. (b) Chassistilting.

Fig. 3. Ball screw driven active tilting mechanism. (a) Body separation tilting.(b) Sliding motion of two body blocks.

in the opposite direction, the separated bodies are sliding againstone another and the lateral tilt is created by the phase differencebetween the two wheels. On the other hand, if they move in thesame direction, the upper platform follows the vertical motionof rise and fall, which is favorable when it is used as a balancingwheelchair platform.

In determining the dimensions of the body in Fig. 3, muchdepends on what the mobile platform is used for. When theperformance specifications are given, it is desirable to choosethe major design parameters considering mobility, structuralstrength, and also controllability through an extra process. Forexample, a large wheel diameter increases the top velocity butreduces the hill climbing ability. As the distance between thetwo wheels (called tread) gets smaller, the effect of the centrifu-gal force compensation in turning will become larger for thesame tilt angle but it may degrade roll stability in case of havingexternal disturbances. For a better pitch balancing performance,the total center of mass should be exactly placed above the wheelaxis, where the vertical offset from the wheel axis, that is, thelength of the inverted pendulum, determines its dynamic char-acteristic. Interestingly, the balancing robot is uncontrollablewhen the offset is zero [23].

The body separation design is disadvantageous to keeping thestiffness of the whole body. The middle plate denoted in Fig 3(b)


Fig. 4. Configuration and mobility of the tilting-type 2WBMR platform.Dimension: 593 × 450 × 522 mm (width × depth × height), weight: 45 kg,wheel diameter: 16 in, payload: 100 kg, maximum tilt angle: 15°, top speed:10 km/h, gradeability: 15°.

Fig. 5. Skew-symmetric bottom with belt–pulley connections.

crosses the left and right body and allows only vertical slidingbetween the two bodies by preventing them from deviating inthe lateral direction. As indicated in Fig. 4, the lower endsof the ball screws are connected to the middle plate and theupper ends to the top platform. In the first place, the prototypeof the tilting 2WBMR was intended to be used as the mobilebase of a balancing wheelchair. For the compactness of thesingle rider vehicle, it has the skew-symmetric bottom in Fig. 5,where the speed reduction ratio of the belt–pulley connectionwas determined to satisfy the requirements for the top velocityin plains and the gradeability on slopes.

B. 5-DOF Nonlinear Mobility

As denoted in Fig. 4, the tilting 2WBMR of this paper has themobility of five degrees of freedom, where the roll and verticalmotion are generated by the ball screw movement. By adding theroll motion to the conventional two-wheeled balancing mecha-nism with pitch and yaw motion, a 3-D pose of the top platformcan be determined. As well as being helpful for comfortableriding, it would be useful when the balancing platform is ap-plied to mobile manipulations. In Fig. 4, simultaneous rollingand vertical motion is possible only when the two ball screwsare independently moving. If just one of the two is connectedto a single actuator, only the roll motion is possible with halfspeed.

In spite of the inherent instability of the 2WBMR due to its in-verted pendulum structure, it has drawn much attention becauseit is mechanically simple, convenient for navigating in narrowand small areas, able to steer on a spot and maintain uprightposture on an inclined surface. Furthermore, it is an interestingnonlinear underactuation system in that the three states of pitch,yaw, and forward motion are controlled by the two inputs ofboth wheel motors. Hence, it has become a challenging topic inapplying advanced control schemes [6], [7]. The increased de-grees of freedom created by the titling mechanism will make thedynamic coupling among the states more complicated, specif-ically when it is designed to make the bandwidth of the rollmotion to be higher.

III. STATIC ANALYSIS

A. Required Tilting Forces

To compare the mechanical efficiency of the body separationtilting method with the conventional one, we consider the freebody diagrams in Fig. 6, where the reaction forces and momentsbetween the rigid bodies are properly defined. The dimensionsand weights of the chassis, bodyshell, and load are the samewith each other.

First, in the single body tilting in Fig. 6(a), it is assumed thatthe vehicle can be divided into two parts: chassis and bodyshellplus load. At the two supports, the reaction moments are ne-glected and the vertical reactions are equivalent to the effectiveforces generated by the tilting actuators. At a designated tiltingpose, the effective forces directly support the body plus load. Bytaking the force equilibrium and the moment equilibrium intoconsideration with respect to the points P and Q, we have

PV =hB WB + hD WD

dsinφ +

12WD cos φ + ΔF1 (1)

QV = −hB WB + hD WD

dsinφ +


where ΔF1 = (WB /2) cos φ.Second, in the body separation tilting of Fig. 6(b), the vehicle

has three moving parts: left body, right body, and the top plat-form plus load, where each of the left and right body includeshalf of the bodyshell and half of the chassis. For the convenienceof analysis, we assume that the reaction forces and moments be-tween the bodies are the sum of distributed loads along the ball


Fig. 6. Free body diagrams for static force analysis. WB : bodyshell weight,WC : chassis weight, WD : load plus platform weight, r: wheel radius, d: dis-tance between the wheels, and hb , hd : center of gravity of body and load,respectively, above the wheel axis. (a) Single body tilting. (b) Body separationtilting.

screw and the other contact points and they are exerted at thecenter of each body. As is shown, when the three bodies aremerged into one, the reaction forces and moments disappear asinternal forces. Then, we can readily apply the force equilibriumand the moment equilibrium for the three parts, respectively.

By arranging the equilibrium equations and removing all theother variables, finally we have the vertical reactions at the twoball screws, which correspond to the effective forces requiredfor keeping the tilting pose:

AV =hB WB + hD WD

dsin φ +


BV = −hB WB + hD WD

dsin φ +

12WD cos φ − ΔF2 (4)

where ΔF2 = (r/d)(WC + WB + WD ) sin φ.These results indicate that unlike the single body tilting, the

chassis weight and even the wheel radius contribute to the ef-fective forces in the body separation tilting. Specifically, whenthe tilting angle is zero, (1) and (2) of the single body tilting say

Fig. 7. Variation of the tilting force ratio (r/d = 0.3, WC /WB = 1).

that each actuator of the body tilting is going to support half ofthe load plus bodyshell weight. However, in the body-separatedtilting, (3) and (4) indicate that only half weight of the load willbe charged to each actuator.

In (1) to (4), the difference between the tilting forces for thetwo methods occurs in the additional terms ΔF1 and ΔF2 . Now,the force ratio (FR) between them can be defined as follows:

FR =ΔF2

ΔF1= 2

(WC + WB + WD

WB

)r

dtan φ. (5)

If the force ratio is greater than one, it means that the bodyseparation tilting requires greater actuator powers than the singlebody tilting.

Assuming that the chassis weight is the same as that ofbodyshell (WC = WB ) and the wheel radius is about one-thirdof the wheel distance, i.e., r/d = 0.3, we have Fig. 7. It showshow the force ratio in (5) is varied according to the changesof tilting angle and payload. That is, in case where the vehiclecarries much heavier loads and undergoes large tilting angles,the single body tilting will be more advantageous in powerconsumption. However, in general, mobile robots and personalvehicles are operated in the ranges where the force ratio is un-der one. Furthermore, if the tilting angle is the same, the bodyseparation mechanism induces a larger amount of transfer ofthe center of mass in the lateral direction than the single bodytitling.

The aforementioned results assume that the ratio of centrifu-gal force to body weight is very small and the centrifugal forceeffect is negligible in the static force analysis. In dynamic sit-uations when the robot is turning fast, a large centrifugal forcemay happen in the lateral direction in Fig. 6. Even in that case,it does not harm the general conclusion that the body separationtilting is mechanically more efficient for personal vehicles.

B. Required Actuator Powers

The suggested tilting-type balancing mobile platform isequipped with two wheel motors for straight driving and


Fig. 8. Applied forces to the robot body. (a) Longitudinal. (b) Lateral.

steering and also two tilting actuators for roll and vertical mo-tion. First, the wheel motors torque is determined to fulfill boththe top velocity and the gradeability which conflict with one an-other. When the robot is climbing a hill with constant velocityin Fig. 8, the effective driving force is balanced with the groundfriction and the gravity component as

T = f + W sin α = μW cos α + W sin α. (6)

Then, the input torque at the wheel axis is equal to

τin = r × T = r(μ cos α + sin α)W (7)

where r is the wheel radius, α the slope angle, W the total weight,and μ the coefficient of ground friction.

Also, the additional torque needed for constant accelerationcan be approximated as

τa = J · π

30· Δn

Δta(8)

where J is the total mass moment of inertia of the robot withrespect to the wheel axis and Δn is the RPM change of thewheel during the acceleration time (duration) Δta [24].

Then, the rated torque of each wheel motor required for con-tinuous driving is given by

τwheel =(

τin + τa

2

)· 1η

(9)

where η denotes the efficiency of power transmission. In theaforementioned, the wheel torques correspond to the value afterapplying the speed reduction ratio at the transmission gear andthe belt–pulley connection.

In the second place, the titling actuator power identified inFig. 6(b) should be enough to lift the static loads exerted tothe ball screws. Considering the screw mechanics in Fig. 9, theexternal load Q is transferred to the screw through the ball nutand the effective torque as much as τ = rsx P is applied to thescrew from the titling actuator, while the ball nut will be movingupward when the sum of the force components along the screwsurface overcomes the static friction. That is,∑

Fx > μ∑

Fy

→ P cos λ − Q sin λ > μ(Q cos λ + P sin λ). (10)

Fig. 9. Applied forces at the ball screw [25].

Then, we have

P > Q

(p + 2μπrs

2πrs − μp

). (11)

Consequently, after torque amplification in speed reductiongear and belt–pulley connection, each tilting actuator is requiredto have the rated torque of

τtilt = (rs × P ) · 1η

= (rs × Q)(

p + 2μπrs

2πrs − μp

)· 1η

(12)

where the variables are the lead p, the screw radius rs , and thepower transmission efficiency η and the maximum load Q canbe produced in terms of the relationship in (3) and (4).

As the screw radius is smaller, the torque requirement will bedecreased just as much. However, if the screw is too narrow, itmay cause flexible modes in high speed movements. The leadof screw is equivalent to the linear displacement of the ballnut for one revolution of the screw and actually it determinesthe linear speed of the screw for a specific RPM of the tiltingactuator. Hence, it must be also carefully determined to satisfythe specification of tilting speed (i.e., the roll rate).

C. Finite Element Analysis

Owing to the separated body structure and the asymmetricconfiguration, the tilting-balancing platform may be disadvan-tageous in maintaining the overall stiffness and strength. In thiscase, it is helpful to depend on the finite element analysis to findstructurally weak points and confirm the compatibility of themechanical elements.

For the given dimensions and material properties, we assumethe worst case that the maximum load of 100 kgf is applied tothe top while the robot is leaned with the maximum tilt angle.The stress distribution in Fig. 10 indicates that the stress is con-centrated in the vicinity of the wheel axis. Then, it is necessaryto keep the allowable stress of the material high enough overthe maximum stress which could happen in the body.

IV. DYNAMIC MODELING

To understand the dynamic characteristics of the tilting-typebalancing platform and reflect them in the mechanical designand posture control, the equation of motion must be prepared.Basically, wheeled mobile robot is a nonholonomic system withnonintegrable constraints and the 2WBMR case belongs to un-deractuated systems which generate three degrees of freedommotions with only two inputs. Formerly, a dynamic equation


Fig. 10. Stress distribution in the robot body.

Fig. 11. 4-DOF motion of the tilting-type 2WBMR prototype.

of motion for the 2WBMR was obtained in terms of Lagrangemechanics [26] and also by using Kane’s method [27]. Notably,if the robot has geometric and mechanical symmetries and it isunder pure rectilinear motion or spinning at a point, it becomesso-called quasi-holonomic and the Coriolis force term in thedynamic model disappears [23].

A. Kinematics

The configuration of the developed balancing mobile platformis shown in Fig. 11 and its main parameters are listed in Table I.As described in Fig. 4, it has a 5-DOF mobility. However, ifall the five axes are involved in the dynamic modeling, theresulting equations will be extremely complicated comparedwith the conventional 2WBMRs with 3-DOF motion. Hence, weexclude the vertical axis motion by assuming that it is relativelyslow. If necessary, it is enough to consider the varied length ofthe inverted pendulum due to the vertical motion in the finalequations.

As in the schematic shown in Fig. 12, the tilting-balancingrobot can be approximated as a 3-D inverted pendulum withthree rigid bodies: main body (B), conducting pitch motion, left

TABLE INOMENCLATURE OF MODEL PARAMETERS

Parameter Definition

D Distance between two wheelsL Height of the body mass center from the wheel axismB Mass of main body (except wheels)mw Mass of single wheelR Radius of wheelTL , TR , TT Wheel torques (left, right), tilting actuator torqueI1 , I2 , I3 Mass moment of inertia of the main body w.r.t. BJ, K Mass moment of inertia of the wheel w.r.t. R or Lφ , θ , ψ Roll rate, pitch rate, and yaw ratevO Robot speed at the origin O, vO = x

Fig. 12. 3-D inverted pendulum motion of the tilting-balancing platform.

(L), and right (R) wheels with peripherals, where the pitch axisis free to rotate to the bottom but the tilting axis rotation isstructurally limited to its maximum angle.

First, the angular velocities of the rotating frames are

ω{E } = ψn3 , ω{C } = ω{E } + φc1 , ω

{B } = ω{C } + θb2 .

(13)

Then, the angular velocities of the left wheel (L), right wheel(R), and the main body (B) with respect to {N} are

ωL = ω{C } + γL c2 = ψn3 + φc1 + γL c2

ωR = ω{C } + γR c2 = ψn3 + φc1 + γR c2

ωB = ω{B } = ψn3 + φc1 + θb2 (14)

where (γL , γR ) are the angular velocities of the left and rightwheel about the wheel axis.

From now on, we use the abbreviations: cos θ = cθ, sin θ =s θ. The unit vectors in (14) can be transformed into the same co-ordinate systems by using the transformation matrices between


the frames:

N RE =

⎡⎣ c ψ − s ψ 0

s ψ c ψ 00 0 1

⎤⎦ , E RC =

⎡⎣1 0 0

0 c φ − s φ0 s φ c φ

⎤⎦ ,

C RB =

⎡⎣ c θ 0 s θ

0 1 0− s θ 0 c θ

⎤⎦ . (15)

Consequently, we have

ωL = φc1 + (γL + ψ s φ)c2 + ψ c φc3

ωR = φc1 + (γR + ψ s φ)c2 + ψ c φc3

ωB = (φ c θ − ψ c φ s θ)b1 + (ψ s φ + θ)b2

+ (ψ c φ c θ + φ s θ)b3 (16)

Given the RPMs of the two wheels (γL , γR ), the velocity ofthe driving frame {E} is constrained by

vO =12r(γL + γR )e1 . (17)

Then, the linear velocity of each body at its center of masscan be written as

vL = vO + vL/O = vO + ω{E } × (d/2)e2

vR = vO + vR/O = vO − ω{E } × (d/2)e2

vB = vO + vB/O = vO + ω{B } × lb3 . (18)

Finally, we have

vL =(vO − (d/2)ψ

)e1

vR =(vO + (d/2)ψ

)e1

vB = vO e1 + (lθ + lψ s φ)b1 + (lψ c φ s θ − lφ c θ)b2 .

(19)

B. Nonlinear Equations of Motion

In deriving the equation of motion for a multi-body system,Kane’s method [28] provides rather a straightforward way to getthe final results by means of systematic vector operations. Ratherthan differentiating energy functions in Lagrangian formula-tion or examining interactive forces and torques in Newtonianmechanics, it is based on finding generalized active forces andgeneralized inertia forces as the functions of generalized coor-dinate variables.

Following the procedure in [28] first we define the generalizedcoordinates for the roll, pitch, yaw, and straight motion:

q =[φ θ ψ x

]T(20)

and the generalized velocities:

u1 = φ, u2 = θ, u3 = ψ, u4 = x. (21)

In Fig. 12, the three rigid bodies are subjected to the appliedforces from gravity and torques from actuators. Also, the wheel

movements are affected by frictional resistance as much as

τLf = −cw γL c2 , τRf = −cw γR c2 (22)

where cw is the damping coefficient considering ground effect,joints and bearings and the wheel angular velocities can beexpressed as

γL =1r

(x − d

2ψ

), γR =

1r

(x +

d

2ψ

). (23)

Then, the applied forces and torques for the left and rightwheel are given by

RL = RR = −mw gn3 ,

TL = (TL − cw γL ) c2 =(TL − cw (x − dψ/2)/r

)c2 ,

TR = (TR − cw γR ) c2 =(TR − cw (x + dψ/2)/r

)c2

(24)

For the main body, as well as gravity and actuator torques,it is greatly affected by Coulomb friction from the ball screws.Then, the applied forces and torques are

RB = −mB gn3 ,

TB = (TT − cT φ − QT )c1 − (TL + TR )

= (TT − cT φ − QT )c1 − (TL + TR − cw (2x/r)) c2

(25)

with cT the damping coefficient and QT the Coulomb friction.By differentiating (16) and (19) respectively, we have the

angular acceleration and the linear acceleration components foreach body. Then, the three bodies will be subjected to the inertiaforces and torques:

R∗P = −mP aP , T ∗

P = −IP · αP − ωP × IP · ωP (26)

where P = (B,L,R), and the inertia diadic of each body withrespect to its center of mass is described as

IB = I1 b1 · b1 + I2 b2 · b2 + I3 b3 · b3

IL = IR = Kc1 · c1 + Jc2 · c2 + Kc3 · c3 (27)

Here, the angular velocities (16) and the linear velocities (19)for the three bodies can be expressed as

ωP =4∑

r=1

(ωP )rur , vP =4∑

r=1

(vP )rur (P = B,L,R)

(28)

where the partial angular velocities and partial linear velocitiesfor the rth general coordinate are defined as

(ωP )r =∂ωP

∂ur, (vP )r =

∂vP

∂ur(r = 1, . . . , 4, P = B,L,R).

(29)

Now, the rth generalized active force is produced by summingdot products between the partial velocities (29) and the applied


forces (24) and torques (25) as follows:

Fr =∑P

(vP )r · RP +∑P

(ωP )r · TP (P = B,L,R) (30)

for r = 1 ∼ 4, which results in

F1 = mB lg s φ c θ + TT − cT φ − QT ,

F2 = mB lg c φ s θ − (TL + TR ) + 2cw x/r,

F3 = d(TR − TL )/2r − d(cw ψ)/r,

F4 = (TL + TR )/r − 2cw x/r2 . (31)

Also, the rth generalized inertia force is determined by takingdot products between the partial velocities (29) and the inertiaforces and torques (26) as

F ∗r =

∑P

(vP )r · R∗P +

∑P

(ωP )r · T ∗P (P = B,L,R) (32)

for r = 1 ∼ 4, which is too lengthy to display all the componentshere.

Finally, the equations of motion for the generalized coordi-nates are formulated by taking the sum:

Fr + F ∗r = 0 (r = 1, . . . , 4). (33)

As a consequence, the nonlinear dynamical equations of mo-tion can be arranged into

M(q)q + C(q, q)q + G(q) + D(q) = τ(t) − Q (34)

where M denotes the inertia matrix, C the Coriolis and cen-trifugal force, G the gravitational force, D the damping force,τ the input torque, and the last Q the Coulomb friction. Thedimensions of those matrices and vectors take the followingform, while the details about the elements can be found in theAppendix:

M =

⎡⎢⎢⎣

a11 0 a13 00 a22 a23 a24

a31 a32 a33 a340 a42 a43 a44

⎤⎥⎥⎦ , C =

⎡⎢⎢⎣

c11 c12 c13 c14c21 0 c23 c24c31 c32 c33 c34c41 c42 c43 0

⎤⎥⎥⎦

G =[g1(q) g2(q) 0 0

]T

D =[cT φ −2cw x/r cw ψd/r 2cw x/r2

]T

τ =[TT −(TL + TR ) (TR − TL )d/2r (TL + TR )/r

]T

Q =[QT 0 0 0

]T. (35)

In terms of the property of Euler–Lagrange systems, it isconfirmed that the inertia matrix M is symmetric and M − 2Cis skew-symmetric and also M = C + CT . If the roll motionby the tilting mechanism is neglected and the friction effectsare not considered, the aforementioned nonlinear equations ofmotion will be reduced to the results in [26] or [27].

C. Characteristics of Ball Screw Mechanism

In general, the ball screw system is characterized by the largeCoulomb friction and very low back drivability. It means that

the roll motion through the ball screw driven tilting mechanismwill not be as much affected by the state changes of the otheraxes as it can affect them through the dynamic coupling terms.Hence, as long as the coupling force from the other axes is notlarger than the Coulomb friction occurring in the ball screws,the roll dynamics can be decoupled from the other axes.

In the developed prototype, the Coulomb friction torque isroughly equivalent to the gravitational moment when the bodyis tilted about 35°. Even when the other state variables are at theirmaximum values, the coupling force to the roll axis is rarely overthe Coulomb friction level. In this case, the dynamic equationsin (34) and (35) can be written in the following reduced form:(

(I1 + mB l2) c 2θ + I3 s 2θ + 2K)φ + cT φ = TT (t) − QT

Mr (q)qr + Cr (q, q)qr + Gr (q) + Dr (q) = τr (t) (36)

with

qr =

⎡⎣ θ

ψx

⎤⎦ ,Mr =

⎡⎣a22 a23 a24

a32 a33 a34a42 a43 a44

⎤⎦ , Cr =

⎡⎣ 0 c23 c24

c32 c33 c34c42 c43 0

⎤⎦ ,

Gr =

⎡⎣ g2

00

⎤⎦ ,Dr =

⎡⎣−2cw x/r

cw ψd/r2cw x/r2

⎤⎦ , τr =

⎡⎣ −(TL + TR )

(TR − TL )d/2r(TL + TR )/r

⎤⎦ .

(37)

In the aforementioned, we have to note that the rolling equa-tion is free from the motion of the other axes, but the parametersof pitch, yaw, and straight motion are still dependent upon theroll angle and roll rate.

V. CONTROL SYSTEM DESIGN

A. Tilting Motion Control Strategy

The tilting-type balancing robot was motivated to increaselateral stability when the robot is following a circular path byusing the roll motion capability. Then, how to determine the tiltangle needed to compensate for the centrifugal forces arisingin the lateral direction? Actually, all the driving paths are thesuccession of arcs with different radii of curvature. Comingback to Fig. 8(b), in order to prevent rollovers, the gravitationalmoment about the right wheel corner must be always larger thanthe moment due to the centrifugal force.

In fact, the presence of the centrifugal force in turning motionchanges the normal reactions of the wheels to the ground. Bytaking the force equilibrium and moment equilibrium conditionwith arbitrary tilt angle in Fig. 8(b), we can readily derive

NL =(

l + r

dc φ s φ +

12

)W −

(l + r

d

)FC c 2φ

NR =(− l + r

dc φ s φ +

12

)W +

(l + r

d

)FC c 2φ

(38)

where the centrifugal force is described as

FC = mv2O /ρ = mψvO (39)

with vO the robot speed, ψ the yaw rate, and ρ the radius ofcurvature of the circular path.


Fig. 13. Velocity/posture control loop.

The difference of normal reactions will make the groundfriction exerted on both wheels inconsistent. It generates yawmoment to the robot body as a disturbance and the slippageof wheels may occur. As the centrifugal force increases athigh speed turns, the frictional disturbance from the bottomgets larger and the postural stability will deteriorate as much.Hence, it is a good idea to make the normal reactions at bothwheels as equal as possible by the controlled motion of the tiltingmechanism.

By letting NL = NR in (38), we have W sinφ = FC cosfindicating that the centrifugal force component along they-axis in the body frame is balanced with the gravitational forcecomponent. Then, we have the relationship of

tan φ =FC

W=

mψvO

mg→ φref = tan−1

(ψvO

g

).(40)

Hence, the roll angle reference can be determined in realtime by using the sensory information of yaw rate and forwardvelocity. In the case that the yaw rate for steering and velocitycommand are given, also we have

φref = tan−1(ψrefvref/g). (41)

In the body separation tilting mechanism, as denoted inFig. 3(a), the desired roll angle can be achieved by the rela-tive motion of the ball screws in the left and right body. Therequired displacements of the two screws are determined by

Δs = sL − sR = d tan φ

→ sL = (d/2) tan φref , sR = −(d/2) tan φref . (42)

By activating the tilting mechanism to follow the referencecommand (40) or (41), we can expect that the slippage occur-rence at the wheels is greatly reduced and the passenger’s ridingquality will be much improved.

B. Velocity/Posture Control Loop

The motion control system of the developed tilting-type2WBMR has four loops as shown in Fig. 13. Fundamentally, thebalancing robot is to keep the pitch angle near zero all the time

Fig. 14. Consecutive turning with lateral tilting motion.

while tracking a desired velocity. Because the pitch pendulummotion is rotated about the single pivot of the wheel axis, thefailure of pitch balancing control means that the robot falls tothe ground. Hence, the pitch control loop must depend on aninertial sensor to measure the absolute pitch angle.

In contrast, the roll motion of the suggested body separationtilting mechanism is like a twin inverted pendulum with twopivots (i.e., the ground contact points of both wheels) and struc-turally it is limited to the maximum tilting angle. Thus, a directroll angle feedback from an inertial sensor is not a prerequi-site for stabilizing the lateral motion. Moreover, when a mobilerobot is affected by large centrifugal accelerations, common gy-roscopic sensors show severe drifts in the roll angle outputs.However, adopting a high spec inertial sensor without such driftphenomena is not cost effective. As indicated in Fig. 13, thetilting control is realized through the position control of the ballscrew driven actuators in terms of the relationship in (42) andthe feedback of motor encoder signals.

VI. EXPERIMENT

The control strategy for the tilting balancing mobile platformsuggested in the former section is validated through experimen-tal study. As shown in Fig. 14, the robot is turning with nearconstant speed and yaw rate on the floor. The inertial sensoroutputs are compared in Fig. 15 for the tiling and non-tiltingcase. When the tilting mechanism is activated, about 10° of rollangle is produced to make the normal reactions of both wheelsthe same. Noting the accelerations along the y-axis in the bodyframe (i.e., {B} in Fig. 12), the non-tiling case is biased as muchas the centrifugal acceleration. But, the tilting case stays nearzero because the centrifugal acceleration component is balancedwith the gravitational acceleration along the y-axis. The biasedpitch angles indicate the eccentricity of the center of mass fromthe wheel axis. The quite different behaviors of the yaw anglewere generated by accident because it is not directly controlledalthough the yaw rate is regulated to the reference value.

Second, Fig. 16 compares the circular trajectories for the tilingand non-tilting case, respectively, after consecutive fast turns,where the ‘AHRS’ trajectories have been produced by usinginertial sensor measurements and the ‘encoder’ trajectories by


Fig. 15. Effects of the tilting action in comparison with nontilting case duringthe first 10 turns. (Inertial sensor: roll/pitch accuracy 0.5°, yaw accuracy 1.0°,and bias of accelerometer 40 μg).

the wheel odometers (motor encoders). The circular turningswere conducted in succession, but in plotting the trajectories, theinitial point after each turn was reset to zero for the convenienceof evaluations.

Although not exact, the differences in the trajectories depend-ing on the sensor outputs represent the quantity of slippage inthe lateral direction, because the slippage of wheels cannot goundetected by odometers unlike inertial sensors. As is shown,the tilting action much reduces the differences. For more accu-rate evaluation of slippage, a 3-D sensor system is required totrack the robot position with respect to a fixed frame. Related tothis, a visual odometry such as Kinect [29] can be convenientlyused for indoor robots because it can be loaded in the robotunlike other fixed 3-D sensing systems.

In order to quantitatively evaluate the tilting effect, we definea sort of slip function:

slip(t) = |ρAHRS(t) − ρencoder(t)| (43)

which corresponds to the difference of the turning radius ofcurvature generated by AHRS (inertial sensor) from the oneby wheel odometers. From (38), the radius of curvature of the

Fig. 16. Comparison of circular turning trajectories (32 turns). Encoder: wheelodometers. AHRS: attitude and heading reference system (inertial sensor).(a) Nontilting. (b) Tilting.

circular path can be determined every instant by

ρ(t) = vO (t)/ψ(t). (44)

In case of inertial sensors, the yaw rate is produced directlyand the robot speed can be obtained by integrating forwardaccelerations. In contrast, wheel odometers determine the speedand yaw rate in terms of the encoder outputs of both wheels asfollows:

vO =vR + vL

2, ψ =

2(vR − vL )d

. (45)

Then, the slip quantity (43) can be determined every time andalso the average for each turn. Now, the averages during the 32turns for the tilting and non-tilting case are compared in Fig. 17.The differences are observed more clearly as the turning speedgets higher.

Finally, the balancing robot is tracing S-shape curves inFig. 18, where the target velocity and yaw rate are 2 m/s and60°/s, respectively. Since the position control is not applied, thetitling and non-tilting driving show great difference in trajectoryas the time progresses. The slip quantities were generated every


Fig. 17. Comparison of the slip quantity with 1-σ bound.

Fig. 18. Tracing S-shape curves under velocity control.

0.6 s. The lateral instability of vehicles and rollovers are mainlycaused by the difference in tire force (i.e., normal reaction onthe ground) among the wheels [14]. The tilting action accordingto (40) keeps the difference between the two wheels minimaland enables less slippage.

VII. CONCLUSION

In this paper, we have discussed the design, analysis, model-ing, and control for a tilting-type two-wheeled balancing mobileplatform. It was shown that the body separation tilting is advan-tageous in saving actuation powers and the dynamic model-ing in terms of Kane’s method is intuitive and straightforward.Through the control design and experimental results, it wasclaimed that the titling action helps to prevent the wheels fromslipping and fundamentally increases the stability margin in thelateral direction.

Two-wheeled balancing mobile platforms are expected tobe widely applied in diverse fields in a near future, includ-ing personal transportation, social service, rehabilitation for thedisabled, and even military use. As the demand for the high

maneuverability of the mobile platform increases, the lateralstability issue will be of great importance. In this respect, bodyseparation tilting in terms of a ball screw driven mechanism canbe considered as a practical solution and can contribute to otherpossible designs.

APPENDIX

Elements of Matrices and Vectors in (35) and (37)

(Note) s θΔ= sin θ, c θ

Δ= cos θ, s 2θΔ= sin2 θ, c 2θ

Δ= cos2 θ

a11 = (I1 + mB l2) c 2θ + I3 s 2θ + 2K

a13 = a31 = (I3 − I1 − mB l2) c φ s θ c θ

a22 = I2 + mB l2

a23 = a32 = (I2 + mB l2) s φ

a24 = a42 = mB l c θ

a33 = mB l2( s 2φ + c 2φ c 2θ) + I1 c 2φ s 2θ

+ I2 s 2φ + I3 c 2φ c 2θ

+ 2mw d2 + 2J(d2/r2) + 2J s 2φ + 2K c 2φ

a34 = a43 = mB l s φ c θ + 2(J/K) s φ

a44 = mB + 2mw + 2(J/r2)

c11 = −(mB l2 + I1 − I3)θ s θ c θ

c12 = −(mB l2 + I1 − I3)φ s θ c θ

−{mB l2 + I2 + (mB l2 + I1 − I3) c 2θ

}(ψ/2) c φ

c13 = −((mB l2 − I3) c 2θ + I2 − I1 s 2θ + 2J − 2K

)×ψ s φ c φ

−{mB l2 + I2 + (mB l2 + I1 − I3) c 2θ

}(θ/2) c φ

− (J/r) x c φ

c14 = (mB l c θ + J/r) ψ c φ

c21 = −c12

c22 = 0

c23 = −(mB l2 + I1 − I3)ψ c 2φ s θ c θ

+{mB l2 + I2 + (mB l2 + I1 − I3) c 2θ

}(φ/2) c φ

c24 = mB lψ s φ s θ

c31 = (J/r) x c φ + (mB l2 + I1 − I3)φ s φ s θ c θ

−{(mB l2 + I1 − I3) c 2θ − (mB l2 + I2)

}(θ/2) c φ

+{(mB l2 − I3) c 2θ + I2 − I1 s 2θ + 2J − 2K

}×ψ s φ c φ

c32 = −{(mB l2 + I1 − I3) c 2θ−(mB l2 + I2)

}(φ/2) c φ

+(mB l2 + I1 − I3)ψ c 2φ s θ c θ

c33 =((mB l2 − I3) c 2θ−I1 s 2θ+I2 + 2J−2K

)φ s φ c φ

+(mB l2 + I1 − I3)θ c 2φ s θ c θ


c34 = (J/r) φ c φ + mB lψ s θ

c41 = −c14

c42 = −mB lθ s θ − mB lψ s φ s θ

c43 = {mB l c θ + (J/r)} φ c φ − mB lθ s φ s θ − mB lψ s θ

c44 = 0

g1(q) = −mB lg s φ c θ

g2(q) = −mB lg c φ s θ

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SangJoo Kwon received the B.S. degree in navalarchitecture and ocean engineering from Seoul Na-tional University, Seoul, Korea, in 1989, and theM.S. and Ph.D. degrees in mechanical engineeringfrom Pohang University of Science and Technol-ogy (POSTECH), Pohang, Korea, in 1991 and 2002,respectively.

He was with the Agency for Defense Developmentof Korea from 1991 to 1997 as a Research Scientistand the Korea Institute of Science and Technology,in 2003 and the Korea Institute of Industrial Technol-

ogy, in 2004 as a Senior Researcher. Currently, he is an Associate Professor inSchool of Aerospace and Mechanical Engineering, Korea Aerospace University,Goyang, Korea. His current research interests include mobile robot design andcontrol, and optimal planning, and filtering.

Sangtae Kim received the B.S. and M.S. degrees inaerospace and mechanical engineering from KoreaAerospace University, Goyang, Korea, in 2008 and2010, respectively. He is currently working towardthe Ph.D. degree in School of Aerospace and Me-chanical Engineering, Korea Aerospace University,Goyang, Korea.

His research interests include the design and anal-ysis of two-wheeled balancing mobile robot and non-linear optimal control.

Jaerim Yu received the B.S. degree in aerospaceand mechanical engineering from Korea AerospaceUniversity, Goyang, Korea, in 2013. He is cur-rently working toward the M.S. degree in Schoolof Aerospace and Mechanical Engineering, KoreaAerospace University.

His research interests include the design and con-trol of two-wheeled balancing robot wheelchairs.

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