Lecture%209:%20Behavior%20Languages

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Lecture 9Behavior Languages

• CS 344R Robotics
• Benjamin Kuipers

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Alternative Approaches To Sequencers

• Roger Brockett, MDL
• Hristu-Varsakelis Andersson, MDLe.
• Jim Firby, RAPS
• there are others
• The right answer is not completely clear.

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Motion Description Languages

• Problem Describe continuous motion in a complex
  environment as a finite set of symbolic elements.
• Applicability sequencing
• Termination condition or time-out.
• Roger Brockett defined MDL.
• Extended to MDLe by Manikonda, Krishnaprasad, and
  Hendler.

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This is an instance ofour framework for control
laws

• A local control law is a triple ?A, Hi, ??
• Applicability predicate A(y)
• Control policy u Hi(y)
• Termination predicate ?(y)

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The Kinetic State Machine

• The MDLe state evolution model is
• This is an instance of our general model
• There is also
• a set of timers Ti
• a set of boolean features ?i(y)
• U(t, x) is a general control law which can be
  suspended by the timer Ti or the interrupt ?i(y)

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The Kinetic State Machine
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Q What is the role of G(x)?

• In the state evolution model
• x is in Rn. Motor vector U(t,x) is in Rk.
• G is an n?k matrix whose columns gi are vector
  fields in Rn.
• Each column represents the effect on x of one
  component of the motor vector.

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MDL Programs

• The simplest MDL program is an atom
• To run an atom,
• apply U to the kinetic state machine model,
• until the interrupt function ?(y) goes false, or
• until T units of time elapse.

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Compose Atoms to Behaviors

• Given atoms
• Define the behavior
• Which means to do the atoms sequentially until
  the interrupt ?b or time-out Tb occurs.
• Behaviors nest recursively to make plans.

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Example Interrupts

• (bumper)
• (wait T)
• (atIsection b)
• b specifies 4 bits whether obstacle is required
  (front, left, back, right).
• Interrupt occurs when a location of that
  structure is detected.

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Example Atoms

• (Atom interrupt_condition control_law)
• (Atom (wait ?) (rotate ?))
• (Atom (bumper OR atIsection(b)) (go v, ?))
• (Atom (wait T) (goAvoid ?, kf, kt))
• (Atom (ri(t)rj(t)) (align ri rj))
• Select ideas from here for your controllers.

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Environment Model

• A graph of local maps.
• We will study local metrical maps later.
• Likewise topological maps.
• Edges in the graph represent behaviors.
• Compact and effective
• Local metrical maps are reliable.
• Describe geometry only where necessary.

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Experiment

• They built a model of three places in their
  laboratory.
• They demonstrated MDLe plans for travel between
  pairs of places.

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Limitations

• Simple sequential FSM model.
• No parallelism or combination of control laws.
• No success/failure exits from control laws.
• Much can pack into the interrupt conditions.
• Limited evaluation
• No exploration or learning.
• No test of reliability.

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Next Observers

• Probabilistic estimates of the true state, given
  the observations.
• Basic concepts
• Probability distribution Gaussian model
• Expectations

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Estimates and Uncertainty

• Conditional probability density function

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Gaussian (Normal) Distribution

• Completely described by N(?,?)
• Mean ?
• Standard deviation ?, variance ? 2

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The Central Limit Theorem

• The sum of many random variables
• with the same mean, but
• with arbitrary conditional density functions,
• converges to a Gaussian density function.
• If a model omits many small unmodeled effects,
  then the resulting error should converge to a
  Gaussian density function.

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Expectations

• Let x be a random variable.
• The expected value Ex is the mean
• The probability-weighted mean of all possible
  values. The sample mean approaches it.
• Expected value of a vector x is by component.

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Variance and Covariance

• The variance is E (x-Ex)2
• Covariance matrix is E (x-Ex)(x-Ex)T

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Covariance Matrix

• Along the diagonal, Cii are variances.
• Off-diagonal Cij are essentially correlations.

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Independent Variation

• x and y are Gaussian random variables (N100)
• Generated with ?x1 ?y3
• Covariance matrix

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Dependent Variation

• c and d are random variables.
• Generated with cxy dx-y
• Covariance matrix