Balancing and Vision G

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Balancing and VisionGábor StépánDepartment of
Applied MechanicsBudapest University of
Technology and Economics
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Contents

• Introduction 1 2 (sports local nonlinearity)
• Robotic balancing (sampling and round-off)
• Micro-chaos (stable unstable impressionists)
• Human balancing
• Critical delay and threshold
• The labyrinth
• Vision a mechanical view
• Conclusions

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Chaos is amusing

• Unpredictable games strong nonlinearitiesthrow
  dice, play cards/chess, computer games ball
  games (football, soccer, basketball impact)plus
  nonlinear rules (tennis 6/4,0/6,6/4,
  snooker)balancing (skiing, skating, kayak,
  surfing,)
• Ice-hockey (one of the most unpredictable
  games)- impacts between club/puck/wall- impacts
  between players/wall - self-balancing of players
  on ice (non-holonomic)- continuous and fast
  exchanging of players

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Small-scale nonlinearities

• Engineering in small-scale,
  everything is linear
• not everything is linear in
  small-scale
• everything is non-linear in
  small-scale?
• For example, digital effects- quantization in
  time sampling linear - quantization in space
  round-off errors at ADA converters
  non-linear

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Position control

• 1 DoF models ? x
• Blue trajectoriesQ 0
• Pink trajectories Q Px Dx

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Force control

• Desired contact forceFd kyd
• Sensed force Fs ky
• Control force Q P(Fd Fs) DFs Fs or d

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Stabilization (balancing)

• Control forceQ Px Dx
• Special case of force control with k lt 0

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Alices Adventures in Wonderland

• Lewis Carroll (1899)

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Balancing inverted pendulum

• Higdon, Cannon (1962) 10-20 papers / year
• n 2 DoF ? ?, x x cyclic coordinate
• linearization at ?
  0

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Digital balancing

• 1) Q 0 - no control
• ? ? 0 is unstable
• 2) Q(t) P?(t) D?(t) (PD control)
• ? 0 is exponentially stable ? D gt 0, P gt
  mg
• 3) Q(t) P? (tj ?) D? (tj ?) (with
  sampling ? )

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Sampling delay of digital control
delay
ZOH
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Digitally controlled pendulum

• ,
• 
  ,

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Stability of digital control sampling

• 
  Hopf
• 
  pitchfork

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ABB

• Sampling frequency of industrial robots 30 Hz
  for the years 1990 2005 above 100 Hz recently
• Force control (EU 6FP RehaRob project),and
  balancing (stabilization-)tasks

Balancing
RehaRob
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Vertical direction?
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Random oscillations of robotic balancing

• 
  sampling time
  and
• quantization (round-off)

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Stability of digital control round-off

• h one digit converted to control force
• det(?I
  B) 0 ?
• ?1 e?
  gt1, ?2 e?, ?3 0

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1D cartoon the micro-chaos map

• Drop 2 dimensions, rescale x with h ? a ? e?,
  
  b ? P
• A pure math approach ( p gt 0 , p lt q )
• solution with xj y(j) leads to ?-chaos map,
• a ep, b q(ep 1)/p ? a gt 1, (0 lt) a b
  lt 1
• small scale xj1 a xj , large scale xj1 (a
  b) xj

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Digital stabilization of stick-slip

• normal form
• digital effects round-off
  sampling

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Stick-slip discrete map

• Solution for one sampling interval
• By we obtain the micro-chaos
  map

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Micro-chaos map

• large scale
• small scale
• Typical in digitallycontrolled machines

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Butterfly effect

• Prop. 1 The map has sensitive dependence on
  initial conditions
• Horseshoe (Smale) invariant Cantor set on
  which the map is topologically conjugate to a
  Bernoulli shift on 2 symbols.

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Attractive set

• Prop. 2 A is a positivelyinvariantattractive
  set

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Symbolic dynamics
Transition matrix A
All elements ofAK-1 are non-zero
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Characterization of ?-chaos (a5/2, b2)

• Fractal dim.

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2D micro-chaos map

• ZOH delay, and round-off for 1st order process
• 
  (p gt 0, p lt q)
• Solution and Poincare lead to
• 
  (a gt1, a b lt 1)
• Linearization at fixed points leads to
  eigenvalues
• So in 1 step the solution settles at an attractor
  that has a graph similar to the 1D micro-chaos map

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Csernak,Stepan (subm. 07)
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3D micro-chaos

Enikov,Stepan (J Vib Cont, 98)
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Transient micro-chaos

• PID control of machines in the presence of
  Coulomb friction
• Switch of robots from position control to force
  control, transient impacts with an elastic
  environment
• Stabilization of an unstable equilibrium or an
  unstable periodic motion of a machine (e.g.
  balancing, control of chaos, )

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Stability problems of polishing tools
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Transient micro-chaos map

• a1.5
• b1.2
• S2/15
• I00.1

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Trivial micro-chaos map
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Kick-out number

• Fibonacci series fn fn-1 fn-2 f5 3
• Length of intervals 1/2n 2
• 
  1
• 
  1
• 
  0

Csernak,Stepan (J Nonl Sci 05)
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Non-trivial transient micro-chaos

• a1.4
• b1.2
• I00.2

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Non-trivial mean kick-out numbers

• M
• a1.4
• b1.2
• I0gt

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Robotic balancing

• Even if vibration problems are all settled, there
  are still serious drawbacks
• Balancing is possible on any inclination, without
  knowing the exact vertical direction
• Balancing works in space (and not in plane only)
• Balancing should incorporate gyroscopic effects
• Study human balancing in more details!elderly
  people, sportsmen
  delay, threshold, stochasticity

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Human balancing

• Analogous or digital?Winking, eye-motion
  self-samplingplus neurons firing still, not
  digital
• 1) Q(t) P?(t) D?(t) (PD control)
• ? 0 is exponentially stable ? D gt 0, P gt
  mg
• 2) Q(t) P?(t ?) D?(t ?) (with reflex
  delay ? )

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Stability chart critical delay

• 
  ? instability

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Stability chart critical reflex delay

• 
  ? instability

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Experimental observations

• Kawazoe (1992)untrained manual control
• (Dagger, sweep, pub)
• Self-balancingBetzke (1994)target shooting0.3
  0.7 Hz

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Stability is the art of keeping the balance
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Labyrinth human balancing organ
Dynamic receptor
Static receptor
Both angle and angular velocity signals are
needed!
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Artificial labyrinth

• 
  Stepan, Kollar (Math Comp Model, 2000)

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Vision and balancing

• Vision can help balancing even when labyrinth
  does not function properly (e.g., dry ear
  effect)
• The visual system also provides the necessary
  angle and angular velocity signals!
• But the vertical direction is needed (buildings,
  trees), otherwise it fails
• Delay in vision and thinking (Ilona Kovacs)

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brain

• Colliculus superior

t gt 0.6 s
Medial Temporal Loop
MTL
t 0.1s
arm
eyes
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Human balancing is also chaotic

• We could reduce the delay below critical value
  through the MTL (Medial Temporal Loop)
• But we cannot reduce much the thresholds of our
  sensory system (glasses...)
• Both delay and threshold increases with age see
  increasing number of fall-overs in elderly homes
• Reduce gains, add stochastic perturbation to
  signal to decrease threshold at a 3rd sensory
  system our feet (Moss,Milton, Nature, 2003)

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Balancing the self-balanced

• Warning only a father has the right to do this

kid