1. Stabilization of Inertia Wheel
Pendulum using Multiple Sliding
Surface Control Technique
A paper from Multtopic Conference, 2006. INMIC’ 06. IEEE
By Nadeem Qaiser, Naeem Iqbal, and Naeem Qaiser
Dept. of Electrical Engineering, Dept. of Computer Science
and Information technology, PIEAS Islamabad, Pakistan
CONTROL OF ROBOT ANDMoonmangmee
Speaker: Ittidej VIBRATION LABORATORY
No.6 Student ID: 5317500117
January 17, 2012

2. 2/18
Key references for this presentation
Proceedings:
[1] M.W. Spong, P. Corke, and R. Lozano, “Nonlinear Control
of the Inertia Wheel Pendulum”, Automatica, 2000.
PhD Thesis:
[2] Reza Olfati-Saber, Nonlinear Control of Undeactuated
Mechanical Systems with Application to Robotics and
Aerospace Vehicles, MIT, PhD Thesis 2001.
Textbook:
[3] Bongsob Song and J. Karl Hedrick, Dynamic
Surface Control of Uncertain Nonlinear
Systems: An LMI Approach, Springer,
New York, 2011.

3. 3/18
Classifications & Styles of Control Paper
Control Application (or Control Engineering)
 Dynamic model
 Controller and/or observer design Engineers
 Experimental setup
 Simulations vs. experimental results
Control Theory (or Mathematical
Type 2:
Control Theory or Control Sciences)
 Problem formulation & Assumptions
Type 1:  Mathematical proofs (rigorously):
 Dynamic model (a benchmark) definition, lemma, proposition,
 Controller and/or observer theorem, corollary, etc.
design  No experiments
 Computer simulation via  Illustrated examples
compare with other methods  Sometimes has no simulations
Engineers &
Mathematicians
Mathematicians
are in a majority

4. 4/18
Control Sciences
Stabilization of Inertia Wheel
Pendulum using Multiple Sliding
Surface Control Technique
Control Theory (or Mathematical Control Theory or
Control Sciences)

5. 5/18
Outline
 Underactuated Mechanical Systems
 Overview of Control System Design
 Dynamic Model
 Controller Design
 Stability Analysis
 Simulation Results
 Concluding Remarks

6. 6/18
Underactuated Mechanical Systems
q2 Fully actuated: #Control I/P = #DOF.
q2 Underactuated: #Control I/P < #DOF.
q1
S
q1
E
Pendubot Inverted Pendulum
L
P
q2
q2
M
q1
q1
Rotational Inverted q3
A
Acrobot Pendulum
q1
q2
X
q2 q2
E
q1 q1 q4
Rotary Prismatic Perpendicular Rotational “Fish Robot”
System Inverted Pendulum [Mason and Burdick, 2000]

7. 7/18
Underactuated Mechanical Systems

8. 8/18
The Inertia-Wheel Pendulum
I2 q2
S
q1
E
I1 , L 1
g
L
P
M
 Single-Input-Single-Output (SISO)
 Nonlinear time-invariant
A
 Underactuated mechanical system
X
 Simple mechanical system
[Spong et al, 2000] Euler-Lagrange (EL) equations
E
of motion

9. 9/18
Control System Architecture
Step 2:
Step 1:
Step 3:

10. 10/18
Dynamic Model
é
m 11 m 12 ùé& ù
q& é ( m l + m L )g sin (q ) ù
- é0 ù
Step 1: ê úê 1 ú+ ê 1 1 2 1 1 ú ê út
I2 q2 ê úê& ú
q& ê ú= ê1 ú
ê 21 m 22 úê 2 ú
m
ë 42 44443 û
ûë ê
ë
0 ú
û ê ú
ë û
1444 144444444442 44444444443 {
M (q ) g (q ) Q (q )
q1
2 2
I1 , L 1 w h e r e m 11 = m 1l 1 + m 2 L 1 + I 1 + I 2
g
a n d m 12 = m 21 = I 2 a r e con st a n t s
í m q& + m q& - ( m l + m L )g sin (q ) = 0
ï & &
ï 11 1 12 2 1 1 2 1 1
W: ì
ï m 21q& + m 22q& = t
& &
ï
î 1 2

& &
L e t x 1 = q1, x 2 = q1, x 3 = q 2, a n d x 4 = q 2
í
ï x& = x 2
ï 1
ï
ï x& = ..... + t
ï
ï 2
S t a t e e qu a t io n : ì
ï x& = x 4
ï 3
ï
ï x& = ..... + t
ï
ï
î 4

11. 11/18
Collocated Partial Feedback Linearization
General Form of the EL Equations of Motion for
an Underactuated Mechanical System: [Spong et al, 2000]
Proposition There exists a global
é (q ) invertible change of control in the form
m m 1 2 ( q ) ù é& ù
q& é (q , q ) ù
h & é0 ù
ê 11
W: ê ú ê 1 ú+ ê1 ú= ê ú
m 2 2 (q ) ú ê& ú ê (q , q ) ú êt ú &
t = a (q )u + b (q , q )
ê 2 1 (q )
m q&
úê 2 ú h
ê2
&
ú ê ú
ë ûë û ë û ë û
where
C on figu r a t ion v ect or : - 1
C on t r ol v ect or : a ( q ) = m 2 2 ( q ) - m 2 1 ( q )m 1 1 ( q )m 1 2 ( q )
T (n - m ) m
q = [q 1 , q 2 ] Î ¡ ´ ¡ ,t Î ¡ m
( m ) con t r ols & & - 1
&
b ( q , q ) = h 2 ( q , q ) - m 2 1 ( q )m 1 1 ( q )h 1 ( q , q )
( n - m ) u n d er a ct u a t ed coor d in a t es
such that the dynamics of transformed
( m ) a ct u a t ed coor d in a t es to the partially linearized system.
Remarks:
 fully linearized system (using a change of control)  Impossible
 partially linearized system (q2 transform into a double integrator)  Possible
 after that, the new control u appears in the both (q1, p1) & (q2, p2) subsystems
 this procedure is called collocated partial linearization

12. 12/18
Collocated Partial Feedback Linearization
í
ï &
q1 = p1 ü
ï
ï ï
ï ý ( q 1 , p 1 ) n o n lin e a r s u bs y s t e m
ï ï
ï &
p 1 = f 0 ( q , p ) + g 0 ( q )u ï
Wn ew : ï
ì þ
ï &
q2 = p2 ü
ï
ï ï
ï ý ( q 2 , p 2 ) lin e a r s u bs y s t e m
ï ï
ï &
p2 = u ï
ï
î þ
- 1
where  is an m m positive definite symmetric matrix and g 0 (q ) = - m 11 (1)m 12 (q )
(q)
Step 2:
Transform to the Partial Feedback Linearization form
í a (q , q ) = ( m m - m 2 ) / m
ï &
ï 11 22 21 11
t = a (q , q )u + b (q ) w h er e ì
&
ï b (q ) = ( m 21 / m 11 ) ( m 1l1 + m 2 L 1 )g sin ( q 1 )
ï
ï
î
Define new state variables New state equation in the Strict Feedback Form
í &
ï z = (m l + m L ) g sin ( z )
íz = m q + m q
ï & &
ï 1
ï 1 1 2 1 2 } N o n lin a e r (C o r e o r R e d u c e d )
ï 1 ï
ï 11 1 12 2
ï m 12 ü
ï
ï z = q ( p en d u lu m a n gle) ï 1 ï
ì 2 ì z& = z1 - z3 ï
ï L in e a r (o r O u t e r )
ï 1
ï 2 m 11 m 11 ý
ï z = q ( w h eel v elocit y ) ï
ï 3 & ï ï
ï
î 2 ï z& = u ï
ï 3 ï
ï
ï
î þ

13. 13/18
Controller Design
Step 3: Outer Subsystem Controller Design
Goal: Stabilizes z 2 ® 0, z 3 ® 0 Second define the sliding surface
ì S 1 = z 2 - z 2d
ï
ï
ï
z& =
1 (m 1l 1 + m 2L 1 ) g sin ( z 2 ) } C ore 1 m 12
ï
ï ü
ï
S& = z& - z&d = z1 - z 3 - z&d
ï 1 m
ï
1 2 2
m 11 m 11
2
í z& = z1 - 12
z3 ï
ï O uter
ï 2
m m ý
ï
ï 11 11
ï To achieve this condition, we choose
ï z& = u ï
ï
ï
î 3 ï
ï
þ
m 11 æ
çK S + 1 z - z& ÷
ö
z 3d = ç 1 1 ÷
ç 2d ÷
Core Subsystem Controller Design m 12 ç
è m 11
1
÷
ø
First Design the synthetic inputs z2d for the
core subsystem achieves the Lyapunov stability Third Design again the synthetic I/P, z3d
2
V ( z i ) = 1 z i > 0 ( p osit iv e d efin it e) S 2 = z 3 - z 3d
2
Þ V& z i ) = z i z& < 0 ( n ega t iv e d efin it e)
( S& = z& - z&d = u - z&d
2 3 3 3
i
To achieve this condition, we choose Finally, the control law chosen to drive
- 1
S20
z 2d = - a t a n (cz 1 ) , 0 < a £ p
2
,c > 0 u = z&d - K 2S 2
3

14. 14/18
Stability Results
Theorem 1: Theorem 3:
í S& = - K S - ( m / m )S
ï 1
SN : { z& = sin ( z 2 d + S 1 )
1
SL
ï
:ì &
1 1 12 11 2
í S& = - K S - ( m / m )S
ï 1 ï S 2 = - K 2S 2
ï 1 1 12 11 2
ï
î
SL : ì
ï S& = - K 2S 2  is global asymptotic stability
ï 2
î L

( ,  ) is global asymptotic stability
N L  is global exponential stability
L
SN
S1= 0
: { z& =
1
sin (a t a n
- 1
(cz 1 ) )
Proposition 1:  |S1 = 0 is globally Lipschitz.
N
Theorem 2:  |S1 = 0 if 0 < a ≤  and c > 0
N /2
then z1 = 0 is global asymptotic stability.
Remark: we left out all of the proofs from the presentation

15. 15/18
Simulation Results
I2
Initial state:
q2(T) = 0
(q1(0), q2(0)) = ( 0)
, Plant parameters:
m11 = 4.83 10-3
m12 = m21 = m22 I1 , L 1
g = 32 10-6 g q1(T) = 0
w = 379.26 10-3
q1(0) =  Controller parameters:
I1 , L 1 a =  c = 9,
/2,
Final state:
K1 = 4, K2 = 6,
and  = 0.001
T (q1(T), q2(T)) = (0, 0)
where the plant parameters are setted
I2 as same as in Olfati-Saber (2001) and
Spong (2000).

16. 16/18
Simulation Results
Pendulum angle, velocity Pendulum angle, velocity
MSS Controller
[Olfati-Saber, 2001]
2.2 sec 3.6 sec
time (second) time (second)
Wheel velocity VS Wheel velocity
3 sec 3.7 sec
time (second) time (second)

17. 17/18
Simulation Results
Control effort (Nm) Control effort (Nm)
0.43 Nm 0.33 Nm
VS
time (second) time (second)
MSS Controller [Olfati-Saber, 2001]

18. 18/18
Concluding Remarks
 The collocated partial feedback linearization was presented
for transform a nonlinear underactuated mechanical system
into the strict feedback form
 A Multiple Sliding Surface controller is designed to achieves
global asymptotically stable of the pendulum angle and the
wheel velocity (neglect the wheel angle)
 The MSS has advantages that the two controllers, i.e.
 no supervisory switching required as in Spong’s design
(2000) (more simple structure)
 the response is faster than the designs by Olfati-Saber
(2001) (more better performance)
 However, more control effort required for MSS

19. Thank you
Please comments and suggests!
CONTROL OF ROBOT AND VIBRATION LABORATORY