CRV CONTROL OF ROBOT AND VIBRATION LABORATORY Stabilization of Inertia Wheel Pendulum using Multiple Sliding Surface Control Technique January 17, 2012 Speaker: Ittidej Moonmangmee No.6 Student ID: 5317500117 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
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Stabilization of Inertia Wheel Pendulum using Multiple Sliding Surface Control Technique
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CRVCONTROL OF ROBOT AND VIBRATION LABORATORY
Stabilization of Inertia Wheel Pendulum using Multiple Sliding Surface Control
Single-Input-Single-Output (SISO) Nonlinear time-invariant Underactuated mechanical system Simple mechanical system
Euler-Lagrange (EL) equations of motion
Control System Architecture
Step 1:
Step 2:
Step 3:
9/18
{
11 12 1 1 1 2 1 1
21 22 2
( )( ) ( )
2 211 1 1 2 1 1 2
12 21 2
11 1 12
( ) sin( ) 0
0 1
are constants
:
Q qM q g q
m m q ml mL g q
m m q
where m ml mL I I
and m m I
m q m
té ùé ù é ù éù- +ê úê ú ê ú êú+ =ê úê ú ê ú êúê úê ú ê ú êúë ûë û ë û ëû
= + + +
= =
+W
&&
&&14444244443 14444444444244444444443
&& &2 1 1 2 1 1
21 1 22 2
( ) sin( ) 0q ml mL g q
m q m q t
ìï - + =ïíï + =ïî
&
&& &&
q1
q2I2
I1, L1g
Dynamic Model
Step 1:
10/18
1 1 2 1 3 2 4 2
1 2
2
3 4
4
, , ,
.....:
.....
Let x q x q x q and x q
x x
xStateequation
x x
x
t
t
= = = =ìï =ïïï = +ïïíï =ïïï = +ïïî
& &
&
&
&
&
Collocated Partial Feedback Linearization
11 12 1 1
21 22 2 2
( ) ( ) ( , ) 0:
( ) ( ) ( , )
m q m q q h q q
m q m q q h q q t
é ùé ù é ù é ùê úê ú ê ú ê úW + =ê úê ú ê ú ê úê úê ú ê ú ê úë ûë û ë û ë û
&& &
&& &
General Form of the EL Equations of Motion for an Underactuated Mechanical System: [Spong et al, 2000]
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
Proposition There exists a global invertible change of control in the form
where
such that the dynamics of transformedto the partially linearized system.