Cascade Sliding Mode Control for Bicycle Robot

Xiuli Yu，Shimin Wei and Lei Guo Department of Automation

Beijing University of Posts and Telecommunications Beijing, China

alicexiuli@163.com

Abstract—The nonlinear characteristics in the bicycle

robot dynamic system make the bicycle robot control much more difficult. Aiming at this problem, the cascade sliding-mode control method was applied to the bicycle robot nonlinear system. A kind of bicycle robot nonlinear dynamic model presented based on Lagrange method was analyzed and the cascade sliding-mode controller of multi-sliding surface was designed with the method of variable structure control theory. This method divided system states into two sub-systems. Firstly, two state variables of subsystem are chosen to construct the first-layer sliding surface, on the basis of the first-layer sliding function, the first layer sliding surface and one of the residual state variables are used to construct the second-layer sliding surface ,at last, all of the system states are included in final sliding surface. The ultimate control is obtained by using Lyapunov stability theorem. since each sub-sliding surface contains the equivalent components of sliding surface, which makes the system states move along the each sub-sliding surface. The cascade sliding-mode controller has the bicycle robot controlled effectively, which has been verified in the simulation experiment.

Keywords- Cascade sliding mode control; under-actuated systems;bicycle robot; stability

I. INTRODUCTION In recent years, more and more attention has been

devoted to the study of bicycle robot. Much of the effort has been spent on both bicycle dynamic model and devising control algorithms.

Some researchers proposed a kind of bicycle dynamic model at a high-speed running state in [1], and analyzed it by using the classical control theory because of basing on the linear feature of the bicycle moving with high speed. Karl J. Åström described a dynamic model of bicycle robot, and designed corresponding controller in [2]. A kind of bicycle dynamic model was proposed based on the equilibrium of moment inertia. Then the dynamic model described with a nonlinear differential equation was disposed with approximate linearization method. And a controller was designed with linear control theory in [3]. When bicycle robot was disposed as linear model, the systematic errors are too large. A bicycle robot system shows dynamic characteristics of nonlinear, time-varying, delay and inertia in actual motion. However, the nonlinear characteristics in the bicycle robot dynamic system make the bicycle robot control much more difficult. Aiming at this problem, a nonlinear controller of the robot is designed based on fuzzy sliding-mode control theory in [4]. The fuzzy sliding-mode controller also has better performance than the DFL

nonlinear controller presented in [5]. At present, the non-linear control theory has been an unprecedented attention. Sliding mode control (SMC) theory was developed and studied as a control method [6]. Scholar Gao Weibing, and others [7,8] studied variable structure control problem of general nonlinear systems, turned nonlinear system into a controllable canonical type system by using nonlinear transform, and proposed a "reaching law" concept. In addition, the variable structure control theory has been applied more successfully in robot control, flight control and other areas.

In this paper, A kind of bicycle robot nonlinear dynamic model presented based on Lagrange method was analyzed and the cascade sliding-mode controller of multi-sliding surface was designed to stable the bicycle robot system with the method of variable structure control theory [9] . Since each sub-sliding surface contains the equivalent components of sliding surface, which makes the system states move along the each sub-sliding surface. The cascade sliding controller can realize the stabilization of bicycle robot system.

II. THE DYNAMICS MODEL OF BICYCLE ROBOT SYSTEM

A. Ideal Assumption and Variables Denotation To present the bicycle dynamic model based on Lagrange

theorem, we take the reasonable analysis and assumption as follows. When the bicycle goes forward vertically with the time invariable velocity 0V , any slight disturbance will make the robot move away from the balance point and fall down (the initial position of the robot is an unstable equilibrium point). To simplify the analysis, the triangle frame of the bicycle is taken as a rigid body; the front wheel and rear wheel are transfigured as rigid body. In order to keep the robot vertical to the ground, the front wheel of the bicycle should be rotated a relevant angle (steering angleα )[4].

Let qm denote the mass of front wheel. Let hm denote the mass of rear wheel. om means the mass of the triangle frame of the bicycle excluding battery. jm means the mass of the triangle frame of the bicycle including battery. dm means the mass of the battery. Let qr be the radius of the front wheel. Let hr be the radius of the rear wheel. Let s be the distance between the COG of front wheel and the rotation axis (line AB) of the front fork. Let β denote the roll angle of the COG of the bicycle with respect to the

This work is partially supported by National Natural Science Foundation (50775012、50875027) ,National 973 Project(2004CB318000),National 863 Program thematic topics(2007AA04Z211) and the Beijing Natural Science Foundation (3092015).

2010 International Conference on Artificial Intelligence and Computational Intelligence

978-0-7695-4225-6/10 $26.00 © 2010 IEEE

DOI 10.1109/AICI.2010.20

62

2010 International Conference on Artificial Intelligence and Computational Intelligence

978-0-7695-4225-6/10 $26.00 © 2010 IEEE

DOI 10.1109/AICI.2010.20

62

vertical direction. Let α denote the steering angle of the rotation of front fork in order to keep the robot vertical. Let

1h be the height of COG of the bicycle. Let 2h be the height of COG of the battery, as Fig. 1, Fig.2 and Fig.3 show below. In order to maintain a balance torque M must be applied, torque M can keep the bicycle robot balance by changing the angle of handlebar.

Figure 1. Rear view of the triangle frame

Figure 2. Platform of bicycle robot

Figure 3. Side elevation of the bicycle robot

Figure 4. Motion analysis of front wheel

Figure 5. Motion analysis of rear wheel

B.Motion Analysis of The Bicycle Robot Wherever Times is specified, Times Roman or Times

New Roman may be used. If neither is available on your word processor, please use the font closest in appearance to Times. Avoid using bit-mapped fonts if possible. True-Type 1 or Open Type fonts are preferred. Please embed symbol fonts, as well, for math, etc.

Front wheel: qV means the velocity of front wheel. Front wheel rotates around the axis of the front fork with the

velocity sα•

. 1J denotes its moment of inertia. And front

wheel rotates around the axis oA with the qr β•

.Let

2J denote its moment of inertia. What's more, Front wheel rotates around the axis O1 with the velocity 0V .We denote

3J its moment of inertia. The angle between the direction of

velocity sα•

and the direction of velocity qr β•

is β (as shown in Fig.1). We can get the expression of qV presented based on vector synthesizing theory as follows:

2 2 2 20 [( ) ( ) 2 cos ]q q qV V s r s rα β α β β

• • • •= + + + .

The kinetic energy of front wheel is represented as follows.

2 22 20

1 2 31 1 1 1 ( )2 2 2 2q q q

q

VT m V J J J

rα β• •

= + + + .

In which, 112

2q qJ m r= ， 2

12

2q qJ m r= ， 3

2q qJ m r=

Rear wheel: hV means the velocity of the rear wheel. Rear

wheel rotates around the axis oC with the velocity hr β•

.Let

4J denote its moment of inertia. Front wheel rotates around the axis O1 with the velocity 0V . 5J denotes its moment of inertia. And the direction of velocity 0V is vertical to the

direction of the velocity hr β•

(as shown in Fig.5 ). Then

hV can be represented as follows. 2 2 2

0 ( )h hV V r β•

= + .

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The kinetic energy of rear wheel is represented as follows.

22 20

4 51 1 1 ( )2 2 2h h h

h

VT m V J J

rβ•

= + + .

In which, 412

2h hJ m r= , 5

2h hJ m r=

Triangle frame of bicycle: Let jV denote the velocity of the triangle frame of bicycle. It moves forward with the

0V velocity. And it rotates with the velocity of 1h β•

around the axis o(as shown in Fig. 1 ) , 6J denotes its moment of inertia. Then jV can be represented as follows.

2 2 2 20 1 1( ) ( ) 2 cosjV V h s p h s pβ α β α β

• • • •= + + + .

The kinetic energy of triangle frame is represented as follows.

22

61 12 2j j jT m V J β

•= + .

In which, 6 12

jJ m h= . The kinetic energy of bicycle is represented as follows.

q h jT T T T= + + . The Lagrange function can be represented as follows. L = T. The generalized forces of the system are presented as

follows. Q M uα = = ， 1 2( ) sinq q h h o dQ m r m r m h m h gβ β= + + + . The dynamic equations of system can be deduced based

on Lagrange theorem as follows.

1 2

( )

( ) ( ) sinq q h h o d

d T T Mdt

d T T m r m r m h m h gdt

αα

βββ

•

•

∂ ∂⎧ − =⎪ ∂∂⎪⎨ ∂ ∂⎪ − = + + +⎪ ∂∂⎩

（1）

Equations (1) can be identically transformed as follows.

(Equations (2))

1 2 2 2 2( + ) ( s+ ) cos12

( s+ ) sin13 32 2 2( 2 ) ( ) cos1 12 2( ) sin1 2

m r m s m s p m r m h sp βq q q j q q j2m r m h sp β Mq q j

m r m r m h m sr m h spq q h h j q q jm r m r m h m h gq q h h o d

α β

β

β α β

β

⎧⎪⎪ + +⎪⎪⎪⎪⎪⎪− =⎪⎪⎨⎪⎪⎪ + + + +⎪⎪⎪⎪⎪= + + +⎪⎪⎩

（2）

Let α be input, Let β be output, so state variables of

system are{ , , , }α α β β• •

.

Let 1 2 3 4, , ,x x x xα α β β• •

= = = = ，Equations (2) can be identically transformed as follows. (Equations (3))

1 2

12 4 3

2 2 2 2

1 24 3

2 2 2 2 2 2 2 2

3 4

14 2 32 2 2

1

1 22

cos12

1sin1 12 2

2( )cos

3 3 4

2( )3 3

q q j

q q q j

q q j

q q q j q q q j

q q j

q q h h j

q q h h o d

q q h

x xmrs mhsp

x x xmr ms ms p

mrs mhspx x M

mr ms ms p mr ms ms p

x xmsr mhsp

x x xmr mr mh

mr mr mh mh gmr m

•

• •

•

• •

=+

=−+ +

++ +

+ + + +

=+

=−+ +

+ + ++

+ 32 2 sin4h j

xr mh

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪

+⎪⎩

（3）

The output equation is represented as follows.

3y xβ= = Let

11 2

2 2 2 212

q q j

q q q j

m r s m h spa a

m r m s m s p

+= − = −

+ +.

32 2 2 2

112 q q q j

am r m s m s p

=+ +

.

11 2 2 2

1

2( )3 3 4

q q j

q q h h j

m sr m h spb

m r m r m h+

= −+ + .

1 22 2 2 2

2( )3 3 4

q q h h o d

q q h h j

m r m r m h m h gb

m r m r m h+ + +

=+ +

.

Equations (3) can be identically transformed as

follows.(Equations (4))

1 2

21 2 3 2 4 3 3

2 21 1 3

3 4

22 1 4 3 2 3 3 1 3

4 21 1 3

1 sin 2 sin2

1 cos

1 sin 2 sin cos2

1 cos

x x

a b x a x x a ux

a b x

x x

a b x x b x a b x ux

a b x

•

•

•

•

⎧ =⎪⎪

+ +⎪⎪ =

−⎪⎨⎪ =⎪⎪ + +⎪

=⎪ −⎩

（4）

The output equation is represented as follows. 3y xβ= =

Let 3

2113 xcos1)C(x ba−= ,Given the specific value of the

coefficients 1 2 3 1 2( , , , , )a a a b b ，it can be verified that for any

3x R∈ there is always that: 3C(x ) 0≠

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Their expression is represented as follows. T

4321 xxxx ][=x .

22

2 1 3 2 3 42

1 1 3

42

1 2 3 4 2 32

1 1 3

x0.5 sin(2x ) sinx x

1 cos x( )

x0.5 sin(2x )x sinx

1 cos x

b a aba

x

ba bba

f

⎡ ⎤⎢ ⎥+⎢ ⎥⎢ ⎥−

= ⎢ ⎥⎢ ⎥⎢ ⎥+⎢ ⎥⎢ ⎥−⎣ ⎦

32

1 1 3

1 3 32

1 1 3

0

1 cos x( )

0cosx

1 cos x

ab a

x

b ab a

g

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥−

= ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

−⎢ ⎥⎣ ⎦

In which , 2

2 1 3 2 3 41 2

1 1 3

0.5 sin(2 x ) sinx x( )

1 cos xb a a

f Xb a

+=

− .

21 2 3 4 2 3

2 21 1 3

0.5 sin(2 x ) x sinx( )

1 cos xb a b

f Xb a

+=

−.

31 2

1 1 3

( )1 cos x

ag X

b a=

− . 1 3 3

2 21 1 3

cosx( )

1 cos xb a

g Xb a

=−

.

Equations (4) can be identically transformed as follows. (Equations (5))

1 2

2 1 1

3 4

4 2 2

( ) ( )

( ) ( )

x x

x f X g X u

x x

x f X g X u

•

•

•

•

⎧ =⎪⎪

= +⎪⎨⎪ =⎪⎪ = +⎩

（5）

3y x= .

III. CASCADE SLIDING MODE CONTROLLER DESIGN Consider Equations (5) expressed in the above form: Where 1 2 3 4( , , , )TX x x x x= is state variables

vector, 1( )f X , 2 ( )f X , 1 ( )g X and 2 ( )g X are the nominal nonlinear functions, and u is the control input. Cascade sliding mode controller is designed as follows, firstly, states 1 2( , )x x and 3 4( , )x x can be treated as the states of two subsystems with canonical form. State 1 2( , )x x variables of subsystem are chosen to construct the first-layer sliding surface, on the basis of the first-layer sliding function, the first layer sliding surface and one of the residual state variables are used to construct the second-layer sliding surface ,at last, all of the system states are included in final sliding surface.

For the state variables we construct suitable sliding surfaces as the following level:

1 1 1 2s c x x= + ， 2 2 3 1s c x s= + ， 3 3 4 2s c x s= + (6) Where 1c , 2c and 3c are positive constants. A set of Lyapunov function is as following:

21 1

12

V s= 22 2

12

V s= 23 3

12

V s= . (7)

If its derivative value 1V , 2V ,and 3V are negative, then the system is stable and its system trajectory will approach the sliding surface on till converging toward the origin. This is a well-known sliding-mode condition

1 1 1 0V s s= ≤ , 2 2 2 0V s s= ≤ , 3 3 3 0V s s= ≤ (8)

1c , 2c and 3c are coefficient, which are defined as

1 1 2 0c x x > , 2 1 3 0c s x > , 3 2 4 0c s x > We can get the following results

1 2 3V V V≤ ≤ Therefore, sliding coefficients are defined as follows: 1 1 1 2( )c C sign x x= , 2 2 1 3( )c C sign s x= ,

3 3 2 4( )c C sign s x=

In which, ic（ 1,2,3i = ）is positive.

Using the equivalent control method the equivalent control law of the subsystems can be obtained as

1 1 21

1

( )( )eq

f X c xu

g X+

= − . （9）

1 2 2 4 12

1

( )( )eq

c x c x f Xu

g X+ +

= − . （10）

1 2 2 4 1 3 2 23

3 2 2 1

( ) ( ) ( )( ) ( ) ( )eq

c x c x f X c f X sign su

c g X sign s g X+ + +

= −+

. （11）

For an under-actuated bicycle robot system it is difficult to control several outputs with fewer actuators. Therefore to ensure that each subsystem follows its own sliding surface, the total control law must include some portion of the equivalent control law of each subsystem. We define the total control as

3eq swu u u= + (12) In which,

3 3 3 3 2 2 4 1 2 1 3 2 1[ ( ) ( ) ( ( ) ( ))( )]eq swV s s s c f x c x c x f x c g x g x u u= = + + + + + +1

3 2 1 3 3( ( ) ( )) [ ( ) ]swu c g x g x sign s ksη−= − + + . （13） 1

3 3 2 1 3 2 2 4 1 2 1( ( ) ( )) [ ( ) ( )]equ c g x g x c f x c x c x f x−= − + + + + . (14） In which,η and k are positive, so

2 23 3 3 3 3( ) 0V s sign s ks s ksη η= − − = − − ≤ . (15)

where 3equ is called the equivalent control which is used when the system states are in the sliding mode and swu is called the switching control which drives the system states toward the sliding mode. The switching control includes a sign function and an exponential law which assures the system states reach the sliding mode faster; 3s is called the

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switching function because the control action switches its sign on the two sides of the switching surface 3s =0.

When 3s is converged to the origin, its various states will also be equal to zero, that is 4 2 0x s= = , so that cascade sliding-mode controller will be degraded into two sliding surfaces. The equivalent control 3equ will also be degraded into the equivalent control 2equ , the system will be 2 0s = ; Similarly, when the second-level sliding plane is converged to zero, that is 3 1 0x s= = .At the same time, the sliding cascade sliding-mode controller will degenerate into a plane, equivalent control will also be degraded into a sliding surface. Therefore the sliding surface ensures the stability of the whole system.

IV. COMPUTER SIMULATION Simulation results: The overhead bicycle robot

parameters as shown in Table 1 . The initial conditions of the overhead bicycle robot

system are 0.8α = ° and 0.3β = ° .Cascade sliding surface is as follow.

1 1 1 2s c x x= + , 2 2 3 1s c x s= + , 3 3 4 2s c x s= + . In which, 1c =8， 2c =0.1， 3c =0.1, k =10,η =6. Figure6,7 show output curves of controlled variables.

Simulation results also show this method gives a viable solution for the second-order under-actuated bicycle robot systems.

0 5 10 15 20 25 30

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

time(s)

fron

tbar

Ang

le o

f bi

cycl

e(de

g)

Figure 6. Response of front bar angle of bicycle

0 5 10 15 20 25 30-3

-2

-1

0

1

2

3

4

time(s)

rolli

ng A

ngle

of

bicy

cle(

deg)

Figure 7. Response of rolling angle of bicycle

V. CONCLUSION The nonlinear under-actuated bicycle robot system with

the cascade sliding mode control method can control effectively the front bar angle α , while the rolling angle β of bicycle robot vibrates at equal amplitude and is not divergent. As a result of the sliding mode control, when the system states enter into the sliding surface, the whole system can resist interference. So it has better performance, it also shows that the cascade sliding mode controller is more robust. Computer simulation testifies that the algorithm presented can keep the robot stable.

REFERENCES [1] Shashikanth Suryanrayanan, Masayoshi Tomizula and Matt Weaver.

"System dynamics and control of bicycles at high speeds". Proceedings of the American Control Conference. Anchorage, AK:IEEE, 2002:845-850.

[2] Karl J. Åström, Richard E. Klein, Anders Lennartsson. "Bicycle dynamics and control". IEEE Control Systems Magazine, 2005, 25(4).pp.26-47

[3] Yasuhito Tanaka, Toshiyuki Murakami. "Self sustaining bicycle robot with steering controller". The 8th IEEE International Workshop on Advanced Motion Control. Kawasaki, Japan:IEEE, 2004.pp.193-197

[4] Lei Guo, Qizheng Liao, Shiming Wei. "Design of Fuzzy Sliding-mode Controller for Bicycle Robot Nonlinear System". Proceedings of the 2006 IEEE International Conference on Robotics and Biomimetics, Kunming, China, December, 2006.pp.176-180

[5] Guo Lei, Liao Qizheng. "Design of Motion Controller for Bicycle Robot Based on DFL Nonlinear Control Method ", Proceedings of the 6th World Congress on Intelligent Control and Automation (WCICA 06), Vol 11,DaLian, China, June, 2006. pp. 9047-9011

[6] Wang Hongbin, Chen Weidong, Wang Hongrui, et al. "Design for a Kind of Nonlinear Variable Structure Controller" . Journal of Yanshan University .2001, 25(2):150-155

[7] Sun Debin,Cheng Mian,Gao Weibing. "The Synthesis of Nonlinear Variable Structure Control Systems". 1989,15(3).pp.248-253

[8] Cheng Daizhan, et al. "A Surview and Prespect in Research of Nonlinear Systems. Control and Decision". 1995, 6(5).pp.394-400

[9] Yinxing Hao, Jianqiang Yi, Dongbin Zhao et al. "Proposal incremental sliding mode control". Journal of Yantai University (Natural Science and Engineering). 2006,7(19) Special.pp.53-58

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