Lecture 9: Behavior Languages CS 344R: Robotics Benjamin Kuipers

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

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.

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)

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)

The Kinetic State Machine

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.

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.

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.

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.

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.

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.

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

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.

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

Estimates and Uncertainty Conditional probability density function

Gaussian (Normal) Distribution Completely described by N(,) Mean  Standard deviation , variance  2

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.

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.

Variance and Covariance The variance is E[ (x-E[x])2 ] Covariance matrix is E[ (x-E[x])(x-E[x])T ]

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

Independent Variation x and y are Gaussian random variables (N=100) Generated with x=1 y=3 Covariance matrix:

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