Dipartimento di Elettronica Informazione e Bioingegneria Cognitive Robotics

Transcription

1 Dipartimento di Elettronica Informaione e Bioingegneria Cognitive Robotics robabilistic self localiation and 205

2 Implementations of rational agent 2 Today implementations of rational agents are hybrid declarative knowledge for high level planning declarative/probabilistic /neural knowledge for motion planning qualitative physics or behavior for monitoring and recovery

3 Why? The urge to ask why and the capacity to find causal eplanations came early in human development. When God asks: Did you eat from that tree? This is what Adam replies: The woman whom you gave to be with me She handed me the fruit from the tree; and I ate. Eve is just as skillful: The serpent deceived me and I ate. The thing to notice about this story is that God did not ask for eplanation only for the facts it was Adam who felt the need to eplain. The message is clear: causal eplanation is a man-made concept. J earl

4 causality - human reasoning We humans are fundamentally storytellers. We like to organie events into chains of causes and effects that eplain the consequences of our actions. It is tempting to believe that our stories of causes and effects are how the world works. Actually they are just a framework that we use to manipulate the world and to construct eplanations. Did Newton F Ma eplain how a force causes a mass to accelerate? The cause-effect paradigm works well in engineering to arrange the world for our convenience. The computer is a perfect eample. The inputs affect the outputs but not vice versa. The components used to construct the computer are constructed to be atomic building blocks of cause-and-effect.

5 limits The notion of cause-and-effect breaks down when the parts that we would like to think of as outputs affect the parts that we would prefer to think of as inputs. The paradoes of quantum mechanics are a perfect eample. Our mere observation of a particle can "cause" a distant particle to be in a different state. It also falls apart when we try to use causation to eplain comple dynamical systems. A gene does not "cause" the trait like height or a disease like cancer. Science will need more powerful eplanatory tools and we will learn to accept the limits of our old methods of storytelling. We will come to appreciate that causes and effects do not eist in nature that they are just convenient creations of our own minds.» W. Daniel Hillis

6 Galileo - how two maims: description first eplanation second that is the how precedes the why description is carried out in the language of mathematics; namely equations. Ask not whether an object falls because it is pulled from below or pushed from above. Ask how well you can predict the time it takes; that time will vary from object to object and as the angle of the track changes. Moreover do not attempt to answer such questions in human language; say it in the form of mathematical equations. description first eplanation second was taken very seriously by the scientists and changed the character of science from speculative to empirical. 638

7 Hume - how David Hume: the why is not merely second to the how the why is totally superfluous as it is subsumed by the how. Thus causal connections are the product of observations. Causation is a learnable habit of the mind almost as fictional as optical illusions and as transitory as avlov s conditioning. How do people ever acquire knowledge of causation? Spuriuos correlations the rooster crow stands in constant conjunction to the sunrise yet it does not cause the sun to rise.

8 Causality in a pure logic system

9 Computing Causality causal relations can be reduced to other concepts mechanistic theory reduces causal relations to physical processes. probabilistic account reduces causal relations to probabilistic relations; causal relations induce probabilistic dependencies counterfactual account reduces causal relations to counterfactual conditionals. C is a direct cause of E if» If C where to occur than E would occur» If C where not to occur than E would not occur agent-oriented account reduces causal relations to the ability of agents to achieve goals by manipulating their causes. C causes E if and only if reaching C would be an effective way for an agent to reach E.

10 Discovering causal relationships 2 general strategies: Hypothetico deductive opper Make hypothesis of a causal relationship Deduce predictions from hypothesis Compare predictions and true values A causal eplanation of an event effect is natural laws +initial conditionscause Inductive Bacon Make a large number of observations Compile a table of positive instances negative and partial instances Induce causal relationship from data in steps Today: causal Bayesian networks J. earl

11 Causality or probability In most of causal connections causality is a matter of degree. robabilities can be used "by default" to model graduation but they do not represent the core of a causal process. They are only a measure of the eternal manifestations of the inner mechanism of change. Marianne Belis The causal roots of probability - in "Causality and probability in the sciences" editors: F.Russo and J.Williamson Volume 5 pp College ublication King's College London 2007

12 Non-determinism in Science 2 In his h. D. Thesis Niels Bohr 90 introduced a demonstration of diamagnetism based on statistics Heisemberg introduced the uncertainty principle in physics 927 In 997 Ilya rigogine contended that determinism is no longer a viable scientific belief

13 Schrodinger s cat parado A cat a flask of poison a radioactive source in a sealed bo. When eactly quantum superposition ends and reality collapses into one possibility or the other? When there is an eternal observation. If an internal monitor detects radioactivity a single atom decaying the flask is shattered releasing the poison that kills the cat. After a while the cat is simultaneously alive and dead. Yet when one looks in the bo one sees the cat either alive or dead. Eplanations consistent with microscopic quantum mechanics require that macroscopic objects such as cats do not always have unique classical descriptions. Our intuition says that no observer can be in a miture of states.

14 probabilistic approach in AI 4 LANNING IN UNCERTAIN DOMAINS Choosing the best action requires thinking about more than just the immediate effects of your actions. There is a lot of uncertainty about the future. Models developed in Dynamic rogramming and Operation Research have been adopted in AI Markov decision process

15 probabilistic robotics Classical Robotics mid-70 s eact models no perception necessary Reactive aradigm mid-80 s no models relies heavily on good perception Hybrids since 90 s model-based at higher levels reactive at lower levels robabilistic Robotics since mid-90 s seamless integration of models and perception inaccurate models inaccurate sensors

16 robabilistic Robotics S. Thrum Key ideas: Eplicit representation of uncertainty using the calculus of probability theory. applications independent from the sensors used. robability both for perception from action

17 museum robot RHINO 996 AAAI - 97 Bonn Navigation Environment crowded unpredictable Environment unmodified Invisible haards Walking speed or faster High failure costs Interaction Individuals and crowds Museum visitors first encounter Age 2 through 99 Spend less than 5 minutes

18 Localiation as an estimate The localiation problem can be described as a Bayesian estimation problem: We want to estimate the location of a robot given noisy measurements the robot has a belief about where it is. at any time it does not consider one location but the whole space of possible locations. based on all available information the robot can believe to be at a certain location to a certain degree. The localiation problem consists of estimating the probability density function over the space of all locations.

19 SLAM SLAM Simultaneous Localiation And Mapping Figure out where the robot is and what the world looks like at the same time Localiation Where am I? osition error accumulates with movement Mapping What does the environment look like? Sensor error not independent of position error

20 Random Variables Discrete X can take on a finite number of values in { 2 n} Xi or i is the probability that the random variable X takes value i Continuous X takes on values in the continuum. px or p is a probability density function. r [ a b] p d b a p

21 Theorem of Total robability Discrete case Continuous case p d independent y y y y y p p y dy p p y p y dy

22 Conditional robability X and Yy y If X and Y are independent then y y conditional probability we know that the Y value is y we would like to know the probability that the X value is conditioned on that fact y probability of given y y y / y if y>0 y y y If X and Y are independent then y y / y y/y

23 Bayes rule evidence prior likelihood y y y y y y y We want to connect y to its inverse y

24 Meaning of Bayes rule is a quantity that we want to infer from y y is data is prior probability distribution knowledge we have prior to incorporate data y y is the posterior probability distribution Bayes rule is a way to compute a posterior probability using the conditional and prior probabilities. y describes how the state variable causes sensor measurements y

25 Normaliation y y y y y y η η y y y y y au : au au : η η The denominator y of Bayes rule does not depend on y - will be the same for any so call it η normalier calculus

26 Meaning in robotics pd η pd p pd is the probability of the position being true given the sensor measurement d pd is the probability of the sensor measurement being d given an object at p is the prior probability of the map

27 Eample - State Estimation Suppose a robot with a sensor in front of a door It makes a measurement What is open?

28 Causal vs. Diagnostic Reasoning open is diagnostic. open is causal. Often causal knowledge is easier to obtain. Bayes rule allows to transform causal into diagnostic knowledge: count frequencies open open open

29 Eample open 0.6 open 0.3 open open 0.5 open open open open p open + open p open open raises the probability that the door is open

30 Combining Evidence Suppose our robot obtains another observation 2. How can we integrate this new information? More generally how can we estimate... n? We want to estimate the probability of being at state considering the history of sensors measures

31 Conditioning Conditioning on random variable Z gives y y y

32 Recursive Bayesian Updating n n n n n n Markov assumption: n is independent of... n- if we know n i i n n n n n n n n η η

33 Eample: Second Measurement 2 open open 0.6 open 2/ open open open open open open open 2 lowers the probability that the door is open

34 Robot actions the world is dynamic actions carried out by the robot actions carried out by other agents or just the time passing by changes the world. How can we incorporate such actions? Actions are never carried out with absolute certainty. In contrast to measurements actions generally increase the uncertainty.

35 Modeling Actions To incorporate the outcome of an action u into the current belief we use u This term specifies the probability density function that eecuting u changes the state from to.

36 Eample: Closing the door

37 State Transitions for action u u for u close door : open closed 0 If the door is open the action close door succeeds in 90% of all cases if it is closed in 00%.

38 Integrating the Outcome of Actions Continuous case: u u ' ' d' Discrete case: u u ' '

39 Eample: The resulting Belief ' ' ' ' u closed closed closed u open open open u open u open u open closed closed u closed open open u closed u closed u closed

40 Bayes Filters Given: Stream of observations and action data u Sensor model. Action model u. rior probability of the system state. Wanted: Estimate of the state X The posterior of the state is called Belief: Bel t t u 2 ut t

41 Bayes Filters 2 2 t t t t t u u u u η Bayes observation u action state 2 t t t t u u Bel Markov assumption 2 t t t t u u η t t t t t t t d Bel u η Markov 2 t t t t t t t t d u u u η 2 2 t t t t t t t t d u u u u η Total prob. redicts the state given action and belief Corrects the estimate using the perceptual model

42 Bayes filter Algorithm Bayes_filter Beld : η0 if d is a perceptual data item then for all do for all do Bel ' η η + Bel' Bel' η Bel Bel' else if d is an action data item u then for all do return Bel Bel' u ' Bel ' d'

43 Bayes filters Represent the state at time t by random variables i At each point in time a probability distribution over i Bel i represents the uncertainty Sequentially estimate the belief over the state space conditioned on all sensor information to make the computation tractable assume the Markov hypothesis Recursive Bayes filter updating reduces the uncertainty of being at a location state s

44 Eample A person or robot with a door-sensing sensor moves in a corridor osition and current belief in black robability to make door observation in red

45 Bayes Filters implementations different implementations article filters Kalman Filters and EKF Etended KF Dynamic Bayes networks Hidden Markov models artially Observable Markov Decision rocesses OMDs

46 State Representations for Localiation Discrete Representations Grid Based approaches Continuous Representations Kalman Tracking article Filters topologicall

47 State representation Grid based approaches Maintain a grid of discrete positions in memory and upgrade them Topological approaches Maintain a graph representation of the environment article filters represent beliefs of present position by set of samples particles Belt St {< i t w i t> i..n} each i t is a state w i t are non negative weight factors that sum up to n is the number of particles At each iteration samples constitue an approimation of the posterior probability

48 article filters: the idea The vertical black lines are particles. Each particle represents a possible location at random along the hallway. article filters use a sampling procedure At each state different samples of the variable to estimate are taken and the probabilities computed; only particles with probabilities higher than a threshold are used and the weights arranged to sum up to.

49 article filter method article filters represent beliefs posterior density function by sets of samples or particles a set of random samples with associated weights w that sum up to. At each iteration samples constitute an approimation of the posterior probability new samples obtained by augmenting each of the present samples with the new state and derive the weights A common problem is the degeneracy after iterations all but one particle will have negligible weights. resampling is a way to eliminate particles that have small weights

50 The particle filter algorithm

51 Eample of partile filter a door sensor says if we are in front of a door The vertical black lines are particles; Each particle represents a possible Location; at start chosen at random along the hallway.

52 p Bel Bel p w Bel p Bel α α α Sensor Information the door sensor currently tells in front of a door.

53 Robot Motion When the robot drives forward particles move the same amount Bel p u ' Bel ' d '

54 Sensor Information Bel w α p Bel α p Bel Bel α p

55 Robot Motion Bel p u ' Bel ' d ' The net time we move we can rule out most of the particles and be more confident about the real location

56 Eample - ultrasounds

57 Markov Decision rocess MD 57 At each discrete time step an agent must choose an action MD is defined by SATR S finite set of states - A finite set of actions T State transition function - from SA to probability distributions over S. Tsas` is the probability of being state s` when agent was in state s and has chosen action a. Actions have nondeterministic effect. R Reward Function - from SA to real numbers. The making agent starts in some state and chooses an action according to its policy. This determines a reward and causes a stochastic transition to the net state Optimiation -> find the policy that leads to the highest total reward over T finite horion Because of the Markov property the policy does not have to remember previous states

58 Generaliations to MD 58 Two dimensions of generaliation: artial observability Decentraliation

59 OMD 59 Real agents cannot directly observe the state.state is estimated SE The agent must maintain a probability distribution over the set of possible states the belief state based on a set of observations and observation probabilities Optimally solving a generic discrete OMD is computationally epensive observation World SE belief state b π action Agent

60 Successful Applications Industrial outdoor navigation [Durrant-Whyte 95] Underwater vehicles [Leonard et al 98] Coal Mining [Singh 98] Missile Guidance Indoor navigation [Simmons et al 97] RoboCup [Lenser et al 2000] Museum Tour-Guides [Burgard et al 98; Thrun 99] + many others

61 Real application: eploring a mine On 30 May 2003 CMU robot Groundhog successfully eplored and mapped a main corridor 308 meters of the abandoned Mathies mine ennsylvania. Groundhog is designed to autonomously eplore and acquire 3-D maps. is built out of the front halves of two all terrain vehicles ATVs with identical steering mechanisms on either end. is equipped with tiltable laser range finders on either end.

62 The core of the Groundhog navigation system is a software package SLAM that acquires 2-D maps. Eample: A 2-D map obtained from a dataset lacking any odometry information. A 3D reconstruction. -

63 Advantages and pitfalls of SLAM Can accommodate inaccurate models Can accommodate imperfect sensors Robust in real-world applications No need for perfect world model Computationally demanding Consider entire probability densities False assumptions Approimate Represents continuous probability distribution

64 workshops Affordances EmbodiedLanguage MachineConsciuosness rostheticrobotics RoboticsEperiments SwarmRobotics