Lecture 9: Behavior LanguagesAlternative Approaches To SequencersMotion Description LanguagesThis is an instance of our framework for control lawsThe Kinetic State MachineSlide 6Q: What is the role of G(x)?MDL ProgramsCompose Atoms to BehaviorsExample InterruptsExample AtomsEnvironment ModelExperimentLimitationsNext: ObserversEstimates and UncertaintyGaussian (Normal) DistributionThe Central Limit TheoremExpectationsVariance and CovarianceCovariance MatrixIndependent VariationDependent VariationLecture 9:Behavior LanguagesCS 344R: RoboticsBenjamin KuipersAlternative 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 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)€ Α(y)€ u=H(y)€ Ω(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)€ ˙ x =f (x)+G(x)U(t,x) y=h(x)€ ˙ x = F(x,u)The Kinetic State MachineQ: 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.€ ˙ x =f (x)+G(x)U(t,x) y=h(x)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.€ σ = U,ξ,TCompose 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.€ σ1= U1,ξ1,T1σ2= U2,ξ2,T2€ b= (σ1,σ2),ξb,TbExample 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.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.–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–ExpectationsEstimates and Uncertainty•Conditional probability density functionGaussian (Normal) Distribution•Completely described by N(,) –Mean  –Standard deviation , variance  2€ 1σ 2πe−(x−μ)2/2σ2The 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.€ E[x] = x p(x) dx∫≈ x =1Nxi1N∑ € E[x]=x =[x 1,L x n]TVariance and Covariance•The variance is E[ (x-E[x])2 ]•Covariance matrix is E[ (x-E[x])(x-E[x])T ]€ σ2=E[(x−x )2] =1N(xi−x )21N∑€ Cij=1N(xik−x i)(xjk−x j)k=1N∑Covariance Matrix•Along the diagonal, Cii are variances.•Off-diagonal Cij are essentially correlations. € C1,1= σ12C1,2C1,NC2,1C2,2= σ22O MCN ,1L CN ,N= σN2 ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥Independent Variation•x and y are Gaussian random variables (N=100)•Generated with x=1 y=3•Covariance matrix:€ Cxy=0.90 0.440.44 8.82 ⎡ ⎣ ⎢ ⎤ ⎦ ⎥Dependent Variation•c and d are random variables.•Generated with c=x+y d=x-y•Covariance matrix:€ Ccd=10.62 −7.93−7.93 8.84 ⎡ ⎣ ⎢ ⎤ ⎦