Development of an Inverted Pendulum Cart with a Sliding Mechanismfor Posture Control

–Design and Manufacture of a Small Mobility and Experiment–

Yuki Otsuki1, Takayuki Matsuno1, Akira Yanou1, Mamoru Minami1

1Graduate school of Natural Science and Technology, Okayama University, Okayama, Japan(Tel: +81-86-251-8924; E-mail: otsukilab@s.okayama-u.ac.jp, {matsuno, yanou-a, minami-m}@cc.okayama-u.ac.jp)

Abstract: A lot of researches on inverted pendulum cart are conducted in recent years. And it is attracted as a personalvehicle which realizes energy saving for many practical applications, such as the Segway and the P.U.M.A. The personalmobility has the advantage of energy efficiency for transportation because the cart is small and light. However, since mostconventional personal vehicles require a certain level of physical ability from the driver, it is not suitable for elderly anddisabled people to drive. Therefore, an inverted pendulum cart with a sliding mechanism for posture control is developedas a personal mobility available for anyone. And we aim for realization of acceleration and deceleration while keeping theangle of the cart perpendicular. In this research, proposed inverted pendulum cart with a sliding mechanism for posturecontrol was designed and manufactured, and the posture control performance was verified through experiment.

Keywords: Inverted pendulum, Posture control, Personal mobility

1. INTRODUCTIONAn inverted pendulum is a mechanical system which

performs the same operation as the play which a childstands an umbrella or a stick upside down on his/her palmby automatic control. Because of no actuators in pendu-lum mechanism part structurally, this is also a kind of asystem with fewer actuators than degrees of freedom, so-called under actuated system [1]. The inverted pendulumcart is what adopted this inverted pendulum mechanismand controlled by using acceleration caused by intention-ally imbalance for movement. An inverted pendulum cartwas proposed by Yamafuji et al. [2] as example of appli-cation of modern control theory. And the realization offunction due to a difference in structure has been pro-posed by a number of researchers until today. In addi-tion, a lot of researches on inverted pendulum cart areconducted in recent years. It has been applied to a per-sonal vehicle and a mobile robot [3], and a fast passingover stairs robot [4]. It is attracted as a personal vehi-cle which realizes energy saving for many practical ap-plications, such as the Segway [5] and the P.U.M.A [6]and the MOBIRO [7]. Especially, Segway secured a nopractical problem level of stability, with only two paral-lel wheels, it has demonstrated the reliability of controlof wheel type inverted pendulum. These personal mobil-ities are small and light and have the advantage of energyefficiency for transportation per person [8]. This is a bigmerit of personal mobility. However, since most conven-tional personal vehicles require a certain level of physicalability because the driver controls posture in person, it isnot suitable for elderly and disabled people.

Therefore, an inverted pendulum cart with a slidingmechanism for posture control is developed as a personalmobility available for anyone to realize the energy sav-ings as a society. And we aim for realization of accel-eration and deceleration while keeping the angle of theinverted pendulum cart perpendicular. Because the pro-

Top plate

Bottom plate

Base plate

Cover plate

Front

Rear

Z

X

Y

Fig. 1 External view of the inverted pendulum cart

posed inverted pendulum cart always keep the angle ofthe cart perpendicular, the driver does not need makemoved the center of gravity for oneself when it accel-erates, decelerates and curves. In this research, proposedinverted pendulum cart with a sliding mechanism was de-signed and manufactured, and construct electrical circuitssystem including a microcomputer. Finally, the posturecontrol performance was verified through experiment.

2. CONTROL SYSTEMCONFIGURATION

2.1 StructureThe appearance of proposed inverted pendulum cart is

shown in Fig.1. The length of the cart is 420 [mm], thewidth is 385 [mm], and the height is 280 [mm]. The partsare defined as base plate, bottom plate, top plate, coverplate. A rack gear, rails, and stoppers are mounted onbase plate. Motor drivers are mounted on bottom plate

(a) Upside (b) Downside

Fig. 2 Structure of the inverted pendulum cart

Z

X

Y

Fig. 3 Movement of the slider

and a microcomputer is mounted on top plate. Castersare mounted on base plate in order to decrease the loadon the motor if the cart falls down. The casters will nottouch the ground during feedback control, because thecart can incline until about 25 degrees.

An upper part of the body, which is named as slider, isshown in Fig.2(a) and a lower part is shown in Fig.2(b).A proposed sliding mechanism enables to control posi-tion of the center of gravity and keep the angle of thecart perpendicular, because entire cart is not only movedbut also the slider shown in Fig.2(a) is moved to back andforth as Fig.3. The weight of the upper part is 3.5 [kg] andthe weight of the lower part is 5.6 [kg], and total weightof the cart is 9.1 [kg]. And, range of motion of the slider,or length of the rack gear is about 280 [mm]. It is enoughlength for posture control by movement of the slider.

There are ball screw mechanism, rack and pinionmechanism, and belt pulley mechanism as power trans-mission mechanism [9]. Rack and pinion mechanismhas feature that backlash is small, structure is simple, en-ergy loss is small and torque is easily transmitted becauserigidity is strong. This is the reason why rack and pinionmechanism is adopted as movement of the slider.

Two-stage speed reducer configured by combiningspur gear shown in Fig.4 is manufactured. And enoughreduction gear ratio not to lose the weight of the slider isset.

The proposed inverted pendulum cart needs to acquire

Motor plate

Base plate

Motor

Two-stage

speed

reducer

Fig. 4 Two-stage speed reducer

TIOC3C

TIOC3A

TCLKA

TCLKB

TCLKC

PWM ①

PWM ②

Phase A

Phase B

Phase A

WheelWheel

Slider

PE1-7D/A

PE12IN1 ①

PE13IN2 ①

PE14IN1 ②

PE15IN2 ②

PB1 D/A

PB2 Brake

PB3 Direction

AN2

AN3

AN5

AN6

TCLKC

TCLKD

Out X

Out Y

Out Z

G2

Phase A

Phase B

Accelerometer

gyroscope microcomputer

PE15IN2 ②

AN0CS ①

AN1CS ②

RxD1 USB

TxD1 USB

Current sensor

Fig. 5 Wiring diagram of microcomputer and peripheraldevices

Wheel

FRONT

1 2

Wheel

Fig. 6 Wheel number

the value of the accelerometer, the gyroscope and the en-coder in order to obtain state variables for feedback con-trol. Therefore, STK-7125 which has SH7125 micropro-cessor equipped with the number of ports required forA/D conversion is used. Wiring diagram of such as sen-sors and assignment of the number to wheels are shownin Fig.5 and Fig.6. And, D/A converter is required in or-der to transmit voltage to motor drivers. However, thereis not D/A converter on microcomputer. Therefore, D/Aconverter is made by combining digital output parts.

Two wheel motors have an encoder respectively, andalso a motor for the slider connect with encoder via gears.These encoders are counted by quad edge evaluation.

Fs

ï2

í1

íw

rw úw

l1

l2

z

xmw, Iw

m1, I1

m2, I2

Fig. 7 Model of the inverted pendulum cart with a slidingmechanism

Table 1 Parameters

SYMBOLS MEANINGSm1 [kg] Weight of cartm2 [kg] Weight of slidermw [kg] Weight of wheelr [m] Radius of wheelI1 [kgm2] Inertia moment of cartI2 [kgm2] Inertia moment of sliderIw [kgm2] Inertia moment of wheell1 [m] Height of COG of cart from center of

wheell2 [m] Height of COG of slider from center of

wheelD1 [Nms] Coefficient of viscous friction for

wheel axisD2 [Ns/m] Coefficient of viscous friction for ve-

locity of sliderDw [Nms] Coefficient of viscous friction or angu-

lar velocity of wheel angleg [m/s2] Gravitational acceleration

2.2 ModelingA model of dynamics of inverted pendulum cart is

shown in Fig.7. Here, the state variables for control arethe angle of the inverted pendulum cart, θ1 [rad], the ro-tation angle of the tire wheel, θw [rad], and the displace-ment of the slider, λ2 [m]. State variables are includedderivative of above mentioned parameters. The controlinputs are the torque of the motor wheel, τw [Nm], andthe actuator force to move the seat of the cart, Fs [N].The model parameters are described in Table 1.

Next, system dynamics of the inverted pendulum cartis derived. The kinetic energy T and the potential energyV is considered, and Lagrange L is obtained by Fig.7.

L = T − V. (1)

T =12mw(rw

˙θw)2 +12Iw

˙θw2

+12m1(rw

˙θw + l1θ̇1 cos θ1)2

+12m1(l1θ̇1 sin θ1)2

+12m2{rw

˙θw + (l2 cos θ1 − l2 sin θ1)θ̇1

+λ̇2 cos θ1}2

+12m2{(−l2 sin θ1

−λ2 cos θ1)θ̇1 − λ̇2 sin θ1}2

+12(I1 + I2)θ̇1

2. (2)

V = m1gl1 cos θ1

+m2g(l2 cos θ1 − λ2 sin θ1). (3)

Therefore,

L =12I∗w

˙θw2

+12(m1l

21 + m2l

22 + m2λ

22 + I1 + I2)θ̇1

2

+12m2λ̇2

2

+{m1l1 cos θ1 + m2(l2 cos θ1

−λ2 sin θ1)}rwθ̇1θ̇1

+m2rw cos θ1˙θwλ̇2

+m2l2θ̇1λ̇2

−m1gl1 cos θ1

−m2g(l2 cos θ1 − λ2 sin θ1). (4)

Here,

I∗w = Iw + (m1 + m2 + mw)r2w. (5)

By lagrange equation

d

dt

(∂L

∂ ˙θw

)− ∂L

∂θw+Dw

˙θw+D1( ˙θw−θ̇1) = τw,(6)

the equation is obtained as follows:

I∗wθ̈w + rw{(m1l1 + m2l2) cos θ1

−m2λ2 sin θ1}θ̈1

+m2rw cos θ1 λ̈2

= τw − (Dw + D1)θ̇w + D1θ̇1

+rw{(m1l1 + m2l2) sin θ1

+m2λ2 cos θ1} θ̇21

−2m2rw sin θ1θ̇1λ̇2. (7)

And by lagrange equation

d

dt

(∂L

∂θ̇1

)− ∂L

∂θ1+ Dθ1 θ̇1 = −τw, (8)

the equation is obtained as follows:

rw{(m1l1 + m2l2) cos θ1

−m2λ2 sin θ1} θ̈w

+(m1l21 + m2l

22 + m2λ

22 + I1 + I2) θ̈1

+m2l2 λ̈2

= −τw − D1θ̇1 + D1˙θw − 2m2λ2θ̇1λ̇2

+(m1l1 + m2l2)g sin θ1

+m2gλ2 cos θ1. (9)

And by lagrange equation

d

dt

(∂L

∂λ̇2

)− ∂L

∂λ2+ D2λ̇2 = Fs, (10)

the equation is obtained as follows:

m2rw cos θ1 θ̈w + m2l2 θ̈1 + m2 λ̈2

= Fs − D2λ̇2 + m2λ2θ̇12

+ m2g sin θ1. (11)

Sliding mechanism provides possibility of quick accel-eration with keeping the angle of the cart perpendicular.However, if motion of the slider and acceleration of cartare not synchronized, the angle of the cart may be dis-turbed for under actuated system. Such a situation desta-bilizes cart system. It is known that it is impossible tocontinue acceleration while keeping pendulum cart per-pendicularly by state feedback control. This problem iscaused due to linear projections in coordinate orthogonal-ized variables of state. Therefore, it is difficult to alwayskeep only particular variables of state at zero.

Matsuno et al. proposed the control method to realizequick acceleration with keeping the angle of the cart per-pendicular for inverted pendulum cart through this modelin a previous study [10]. And, the effectiveness of pro-posed methods to keep the angle of the cart perpendicularis confirmed through simulations.

2.3 Control systemControl system of inverted pendulum cart is shown in

Fig.8. Values of accelerometer and gyroscope are ac-quired by A/D conversion and the angle of the cart iscalculated in microcomputer. Equally, values of wheelencoder and slider encoder are acquired by A/D conver-sion and amount of movement is calculated. Each wheelis controlled by applying the appropriate voltage based oncalculated values to wheel motor driver and slider motordriver. Two direction arrow between slider motor driverand slider motor show using a moror with hall sensors.Data sampling period and control cycle are set as 10 [ms].

3. VERIFICATION OF POSTURECONTROL PERFORMANCE

In order to verify posture control performance of theinverted pendulum cart with a sliding mechanism, exper-iments are conducted with the manufactured cart.

Slider

Microcomputer

Accelerometer Encoder

Motor driver Motor

WheelGyroscope Encoder

MotorMotor driver

Fig. 8 Configuration of control system

Cart+

-

+

-

x0É xÉ e u

K

RE

x

Fig. 9 Block diagram of state feedback

3.1 Control algorithmExperiments are carried out by state feedback in this

paper. The wheel torque, the slider torque and the sliderdisplacement are drawn up by using the angle of the cartθ1 obtained from accelerometer. Block diagram of statefeedback is shown in Fig.9.

Here,

x = [θ1 λ2 λ̇2]T , (12)x∗ = [θ∗1 λ∗

2 λ̇∗2]

T , (13)x∗

0 = [0 0 0]T , (14)

u = [τw Fs]T , (15)e = x∗ − x, (16)

E =

0 0 0ε 0 00 0 0

, (17)

K =

[Kp 0 00 Ks Kd

]. (18)

And, controller is shown as follows:

[τw

Fs

]=

[Kp 0 00 Ks Kd

]

θ1

λ2

λ̇2

(19)

Each gain is decided heuristically as follows:

Ks = 0.8 (20)Kd = 0.03 (21)

ε = 7.0 × 10−5 (22)

Since the threshold values by which wheels begin tomove are different from right and left, The wheel gain is

(a) 5.65 [s] (b) 5.75 [s]

(c) 5.8 [s] (d) 5.85 [s]

(e) 6.0 [s] (f) 6.35 [s]

(g) 6.4 [s] (h) 6.65 [s]

Fig. 10 First experiment for posture control

set as follows:

KpL= 1.4 (23)

KpR= 2.0 (24)

Here, the left gain is KpL, the right gain is KpR

.

3.2 Experimental resultThe situation of experiments are shown in Fig.10. The

angle of the inverted pendulum cart, θ1, the torque of themotor wheel, τw, and the rotation angle of the tire wheel,θw on first experiment are shown in Fig.11(a)-(c). And,the angle of the inverted pendulum cart, θ1, the actuator

force to move the seat of the cart, Fs, and the displace-ment of the slider, λ2 on second experiment are shown inFig.11(d)-(f).

3.3 DiscussionAs shown in Fig.11, intentional movement can be real-

ized until fall in any experiment. Therefore, although theoperation which tends toward a stable state is seen, this isnot maintainable.

When attention is paid to Fig.11(d) and Fig.11(f), Fs

also changes in accordance with λ2 at 12 [s] approxi-mately. Because motor gear separates from rack gearon the both edges, torque is cut off in this moment. Af-ter that, the slider hit and rebound from the stopper, andgears engage and return to the initial value. This meansthat the moderating ratio of the slider is large enough.

Moreover, the fact that the angle of the inverted pen-dulum cart calculated only one accelerometer may affectserror of θ1. Since accelerometer also detect accelerationby motion of the cart [11] as above-mentioned. By usinga gyroscope together, it appears that this detection errordecreases.

4. CONCLUSIONIn this report, proposed inverted pendulum cart with a

sliding mechanism for posture control is introduced as apersonal mobility available for anyone. And a small mo-bility was designed and manufactured. Rack and pinionmechanism is adopted to this mobility and rotational mo-tion of the motor is converted to linear motion. Thus, itis possible for not only the movement of the cart but alsoposture control by seesaw of the slider. Finally, posturecontrol performance was verified through experiment. Itis a future work that it keeps standing a little longer time.

The following are mentioned as improvements: stabi-lization of a posture, the angle of the inverted pendulumcart which used the accelerometer and the gyroscope to-gether. Moreover, wireless in order not to affect tensionby a power cable or a communication cable, digitizationof each sensor, and so on are needed. We wrestle theseassignments and aim to realize acceleration and deceler-ation with keeping the angle of the cart perpendicular inthe future.

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-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

5.0 6.0 7.0 8.0 9.0

Th

e an

gle

of

the

cart

[rad

]

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(a) θ1 (1st)

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5.0 6.0 7.0 8.0 9.0

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e to

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eel

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-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0

Th

e a

ctu

ato

r fo

rce t

o

move t

he s

lid

er

[N]

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(e) Fs (2nd)

-1.5

-1

-0.5

0

0.5

1

1.5

5.0 6.0 7.0 8.0 9.0

Th

e ro

tati

on a

ng

le o

f

the

wh

eel

[rad

]

Time [s]

(c) θw (1st)

-80

-60

-40

-20

0

20

40

60

7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0

Th

e d

isp

lace

men

t o

f

the

slid

er[m

m]

Time [s]

(f) λ2 (2nd)

Fig. 11 Experimental results

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