Ball Balance on a Plank

AIMTo balance a ball on an inclined plane at some random position irrespective of the angle of inclination of the plank

α

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

The Open Loop Control

G(s)R(s)ϴ(s)

Mathematical ModellingAssumptions:- 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.

Lagrangian method

beamballball JwJwvmenrgyKinetic 222

21

21

21

)sin(21)sin( MgLmgrenergyPotential

UKL

JrrmJK

rrvv

rrvry

rrvrx

vvv

ballbeam

yx

y

x

yxball

22222

22222

222

21

21)(

21

)cos()sin()sin(

)sin()cos()cos(

0)()sin(

)sin(

22

2

2

222

mrmgrmRJ

mgmrdrdL

rmRJ

rddL

rm

RJ

RrJrm

rddL

ball

ball

ballball

To linearize the system assuming θ’=0

Transfer function of system can be given as

Ldmgrm

RJLd

LLd

ball

2

)sin(

2

2

1sm

RJL

mgdssR

ball

State space model for the system

mRJL

mgd

rr

rr

ball2

0

0010