Pole balancing robot and some control strategies

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Pole balancing robot and some control strategies
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Introduction

• A sketch of PBR (Pole Balancing Robot)

Figure1 Pole balancing Robot
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• Robotic vehicle would operate on the top of the
  table provided (refer to Fig.2a).

Fig.2.a. Pole Balancing Robot Table

• Tabletop will have a slight gradient at the start
  (region A) and the end (region C) ltshown in
  Fig.2agt

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• A metallic wedge of cross section shown in
  Fig.2b.(not to scale) will be used as an
  obstacle.
• The length of the wedge will match the width of
  the table.
• The wedge will be painted to match the table
  surface.
• A retro-reflective tape will be stuck to it at
  the middle to match the one on the table.
• The judge will place the wedge in region B any
  where between the inner edges of the two
  innermost tapes so that the wedge is
  perpendicular to the path.
• It will not be moved thereafter.

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• Vehicle will be placed within the region A (See
  Fig. 2b).

Fig.2.b Wedge Section (Enlarged View)

• The operator may move the inverted pendulum to an
  upright position and release it upon receiving
  the signal from the judges.
• The vehicle must balance the pole in the upright
  position for a minimum of 20 seconds without the
  vertical pole crossing the line X-X'.

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• Upon completion of the task above,
• vehicle should move across the line X-X' once
• move through the region B, until the pole clears
  the line Y-Y', without losing balance during
  transit
• ? not hitting any part of the table or its own
  chassis?

• Upon completion of the task above,
• vehicle must retrace the path, cross the line
  X-X' again and get back to region A.
• ?? This will complete one cycle. ??
• This time, during the retrace, the vehicle need
  not stay any length of time at region B or A,
  before the start of the second cycle.

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• When an electronic sensing system is used for
  detecting the pole crossing Y-Y and X-X lines,
  the pole sensors at both sides will be placed
  such that the line of sight of the sensors will
  be 20 cm above the lines marked on the platform.
• This may warrant that the robot moves further for
  the pole to intercept the line of sight of the
  sensors.
• This is important since many robots have their
  poles inclined inwards towards the centre of
  platform at these points of turning back.

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• Furthermore, no part of the robot other than the
  pole should be above 15 cm so that no other part
  of the robot (except the pole) would trigger the
  sensor.
• The vehicle should repeat these cycles.
• To count these cycles as successful cycles they
  must be followed by at least 20 seconds of static
  balancing at region A.
• The robot may continue on (untouched) for more
  cycles, and complete them with 20 seconds of
  static balancing at the end, which if successful
  will be counted cumulatively.
• If a robot is touched by the handler during the
  trial, it must be restarted for the next attempt.

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Pole Balancing Robot Dynamics

• The line diagram of a pole balancing robot is
  shown in Fig.1.
• The following equations can be written to
  describe the dynamics of the robot movement and
  the pole angle,?.

(Mm)s2 Bs X(s) (ml)s2 (b/l)s ?(s)
F(s)
ms2 X(s) mls2 (b/l)s - mg ? (s) 0
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• One Possible Approach

where, M mass of the vehicle B linear
equivalent friction of the vehicle m mass of
the pole b rotational friction of the pole g
9.81 m/s2 l half length of the
pole x distance ? angle in radians
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V(s) s.X(s)

• But,

• We can move a S from RHS to denominator. Hence,

• The above Eqn.5 can be represented by Fig.3.

Figure.3. Angle versus velocity
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• Getting back to the problem at hand, for any
  vehicle with

• a mass, M and
• friction coefficient, B,
• we can draw,
• for a given force f,

Fig 4.Force acting on a vehicle

• But torque is written as,

• Also torque can be written as,

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• Where r? diameter of the driving wheel

It has been assumed that there is only one
motor

• However, defining back emf as Eb,

Eb Kb.?m
Where, ?m is the motors rotational velocity.
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• v r?.?w

But,
Where, ?w is the driving wheels rotational
velocity
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• Connecting the above equations, a block diagram
  can be drawn

Figure.5.Block diagram relating applied voltage
to velocity
This can be simplified as
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• With the above system as the core plant , one can
  produce a velocity control block diagram.

Figure 6.Velocity Control of robotic vehicle
Where
Tpwm PWM Period half period del ON
fraction G Numerical Gain
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• Block diagram can be drawn.

Figure 7.Complete Block Diagrams

• It all looks very complicated.
• Note that it is still first order dynamics.

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Figure 8.Simplified Dynamics

• Once you get the numbers, it is not as
  complicated as it appears.

Numerical Example Let G 1000 Tpwm 1000 Kt
0.033 Ng 8.0 (gear ratio) r? 3 cm M
2.5 Kg Ra 6 ? B 2 Kb Kt Vs 28
Volts Then the parameters can be easily
computed.
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• To get the overall picture, let us combine Fig 3
  and Fig.8

Figure 9. Angle Versus Velocity Dynamics of a
Pole balancing robot
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Control Options

• Common Strategy
• Since there two outputs but only one manipulated
  variable.
• In all our design we use two loop system.
• The position reference is in the outer loop and
  the error generated is used as the angle
  reference to the inner loop to keep the pole
  vertical.
• Actually many variations are possible.

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• Implementation can be done using one of the
  following techniques
• Polynomial based controllers In this
  controllers, one can describe the transfer
  function and form z-domain system and use a pole
  placement or LQC algorithms to derive a
  controllers. At times LQC controller may go
  unstable.
• State space controllers One can take a state
  apace model with x, v, angle, angular velocity
  as states. Again pole placement controllers can
  be implemented.
• PD Controller Simple proportional and derivative
  controllers also would work. But such system is
  only conditionally stable.

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Typical PBR

• In our design,
• use eZdsp mother board
• robot uses potentiometer, and one drop encoder.
• drivers are H-bridge drivers controlling two
  motors.

Fig.9. Typical PBR
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CONCLUSION

• During this brief talk,
• The basic competition event was described
• The model of a pole balancing robot as a single
  input, multi-output system was derived
• Possibility of a two loop controller structure
  was discussed
• A few controller design options were suggested.