Approach for balancing a ball at any desired position on a hinged beam free to move according to gravity of ball.

1. Ball Balance on a
Plank

2. AIM
To balance a ball on an
inclined plane at some
random position irrespective of the
angle of inclination of the plank
α

3. Approach
 Balance the ball on a straight plank
for α=0 at central position.
 Balance it for some position other
than the central position.
 Balance it on the plank at some
inclination angle α

4. The Open Loop Control
G(s)
R(s)ϴ(s)

5. Mathematical Modelling
Assumptions:-
 We will assume the plank to be
frictionless(only then we can apply
lagrangian method: valid only for
conservative forces).
 The angle change while balancing
the ball to be very negligible.

6. Lagrangian method
beamballball JwJwvmenrgyKinetic 222
2
1
2
1
2
1

)sin(
2
1
)sin(  MgLmgrenergyPotential 
   
      
UKL
JrrmJK
rrvv
rrvry
rrvrx
vvv
ballbeam
yx
y
x
yxball






22222
22222
222
2
1
2
1
)(
2
1
)cos()sin()sin(
)sin()cos()cos(





7.  
 
0)()sin(
)sin(
2
2
2
2
2
22































mrmgrm
R
J
mgmr
dr
dL
rm
R
J
rd
dL
r
m
R
J
R
r
Jrm
rd
dL
ball
ball
ball
ball

8.  To linearize the system assuming θ’=0
 Transfer function of system can be
given as
  


L
d
mgrm
R
J
L
d
LLd
ball









2
)sin(
 
  2
2
1
s
m
R
J
L
mgd
s
sR
ball











9.  State space model for the system




































m
R
J
L
mgd
r
r
r
r
ball
2
0
00
10