Stabilization of Inertia Wheel Pendulum using Multiple Sliding Surface Control Technique
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This paper presents a control technique for stabilizing an inertia wheel pendulum system using multiple sliding surface control. The inertia wheel pendulum is an underactuated mechanical system with one control input and two degrees of freedom. The paper first develops a dynamic model using Euler-Lagrange equations of motion. It then designs a controller based on multiple sliding surface control and analyzes the stability of the closed-loop system. Simulation results demonstrate the effectiveness of the proposed control method.
![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.](https://image.slidesharecdn.com/seminar2012mss-130118001238-phpapp02/85/Stabilization-of-Inertia-Wheel-Pendulum-using-Multiple-Sliding-Surface-Control-Technique-2-320.jpg)
![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]](https://image.slidesharecdn.com/seminar2012mss-130118001238-phpapp02/85/Stabilization-of-Inertia-Wheel-Pendulum-using-Multiple-Sliding-Surface-Control-Technique-6-320.jpg)
![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](https://image.slidesharecdn.com/seminar2012mss-130118001238-phpapp02/85/Stabilization-of-Inertia-Wheel-Pendulum-using-Multiple-Sliding-Surface-Control-Technique-8-320.jpg)
![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](https://image.slidesharecdn.com/seminar2012mss-130118001238-phpapp02/85/Stabilization-of-Inertia-Wheel-Pendulum-using-Multiple-Sliding-Surface-Control-Technique-11-320.jpg)
![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)](https://image.slidesharecdn.com/seminar2012mss-130118001238-phpapp02/85/Stabilization-of-Inertia-Wheel-Pendulum-using-Multiple-Sliding-Surface-Control-Technique-16-320.jpg)
![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]](https://image.slidesharecdn.com/seminar2012mss-130118001238-phpapp02/85/Stabilization-of-Inertia-Wheel-Pendulum-using-Multiple-Sliding-Surface-Control-Technique-17-320.jpg)
Stabilization of Inertia Wheel Pendulum using Multiple Sliding Surface Control Technique
- 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]
- 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 S20 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