1. Lecture 9: Behavior 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 of our 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.
goAvoid moves in direction \psi, with gains k_f and k_t on the forward and turn controllers, responding to distances to obstacles.

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. Limited evaluation:
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 E[x] 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-E[x])2 ]
Covariance matrix is E[ (x-E[x])(x-E[x])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 (N=100)
Generated with x=1 y=3
Covariance matrix:

23. Dependent Variation c and d are random variables.
Generated with c=x+y d=x-y
Covariance matrix: