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Lecture%209:%20Behavior%20Languages

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Title: Lecture%209:%20Behavior%20Languages


1
Lecture 9Behavior Languages
  • CS 344R Robotics
  • Benjamin Kuipers

2
Alternative Approaches To Sequencers
  • Roger Brockett, MDL
  • Hristu-Varsakelis Andersson, MDLe.
  • Jim Firby, RAPS
  • there are others
  • The right answer is not completely clear.

3
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.

4
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)

5
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)

6
The Kinetic State Machine
7
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.

8
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.

9
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.

10
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.

11
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.

12
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.

13
Experiment
  • They built a model of three places in their
    laboratory.
  • They demonstrated MDLe plans for travel between
    pairs of places.

14
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.

15
Next Observers
  • Probabilistic estimates of the true state, given
    the observations.
  • Basic concepts
  • Probability distribution Gaussian model
  • Expectations

16
Estimates and Uncertainty
  • Conditional probability density function

17
Gaussian (Normal) Distribution
  • Completely described by N(?,?)
  • Mean ?
  • Standard deviation ?, variance ? 2

18
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.

19
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.

20
Variance and Covariance
  • The variance is E (x-Ex)2
  • Covariance matrix is E (x-Ex)(x-Ex)T

21
Covariance Matrix
  • Along the diagonal, Cii are variances.
  • Off-diagonal Cij are essentially correlations.

22
Independent Variation
  • x and y are Gaussian random variables (N100)
  • Generated with ?x1 ?y3
  • Covariance matrix

23
Dependent Variation
  • c and d are random variables.
  • Generated with cxy dx-y
  • Covariance matrix
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