7630 Autonomous Robotics Probabilities for Robotics

Transcription

1 7630 Autonomous Robotics Probabilities for Robotics Basics of probability theory The Bayes-Filter Introduction to localization problems Monte-Carlo-Localization Based on material from R. Triebel, R. Kästner

2 Recommended Reading PhD Thesis: Probabilistic Robotics Textbook: Thrun, Burgard, Fox: Probabilistic Robotics SLAM Summer School: Stockolm 02

3 BASICS OF PROBABILITY THEORY

4 Basics of Probability Theory Definition 1.1: A sample space outcomes of a given experiment. Examples: Coin toss experiment: Distance measurement: Definition 1.2: A random variable is a function that assigns a real number to each element of. Example: Coin toss experiment: is a set of Values of random variables are denoted with small letters, e.g.:

5 Discrete and Continuous If is countable then is a discrete random variable, else it is a continuous random variable. The probability that takes on a certain value is a real number between and. It holds: Discrete case Continuous case More detailed: see, e.g. P. Brémaud: An Introduction to Probabilistic Modeling

6 A Discrete Random Variable Suppose a robot knows that it is in a room, but it does not know in which room. There are 4 possibilities: Kitchen, Office, Bathroom, Living room Then the random variable Room is discrete, because it can take on one of four values. The probabilities are, for example:

7 A Continuous Random Variable Suppose a robot travels 5 meters forward from a given start point. Its position is a continuous random variable with a Normal distribution: Shorthand:

8 Joint and Conditional Probability The joint probability of two random variables and is the probability that the events and occur at the same time: Shorthand: The conditional probability of given is defined as:

9 Independency Two random variables and are independent iff: For independent random variables and we have:

10 Law of Total Probability Marginalisation Theorem 1.1: For two random variables and it holds: Discrete case Continuous case The process of obtaining from by summing or integrating over all values of is called marginalisation.

11 Bayes Formula Theorem 1.2: For two random variables and, it holds: Bayes Rule Proof: I. (definition) II. (definition) III. (from II.)

12 Bayes Rule: Background Knowledge For it holds: Background knowledge Shorthand: Normalizer

13 Computing the Normalizer Bayes rule Total probability can be computed without knowing

14 Conditional Independence Definition 1.5: Two random variables and are conditional independent given a third random variable iff: This is equivalent to: and

15 Expectation and Covariance Definition 1.6: The expectation of a random variable defined as: is (discrete case) (continuous case) Definition 1.7: The covariance of a random variable defined as: is

16 Localization and Mapping Why a probabilistic approach for mobile robot localization? Because the data coming from the robot sensors are affected by measurement errors, we can only compute the probability that the robot is in a given configuration. The key idea in probabilistic robotics is to represent uncertainty using probability theory: instead of giving a single best estimate of the current robot configuration, probabilistic robotics represents the robot configuration as a probability distribution over the all possible robot poses. This probability is called belief. R. Siegwart, D. Scaramuzza ETH Zurich - ASL

17 5 17 Example probability distributions for representing robot configurations In the following we consider a robot which is constrained to move along a straight rail. The problem is one dimensional. The robot configuration is the position x along the rail. 0 x R. Siegwart, D. Scaramuzza ETH Zurich - ASL

18 R. Siegwart, D. Scaramuzza ETH Zurich - ASL 5 18 Example 1: Uniform distribution 5 - Localization and Mapping This is used when the robot does not know where it is. It means that the information about the robot configuration is null. 1 N N x Remember that in order to be a probability distribution a function p(x) must always satisfy the following constraint: p ( x ) dx 1

19 R. Siegwart, D. Scaramuzza ETH Zurich - ASL 5 19 Example 2: Multimodal distribution 5 - Localization and Mapping The robot is in the region around a or b. x

20 R. Siegwart, D. Scaramuzza ETH Zurich - ASL 5 20 Example 3: Dirac distribution 5 - Localization and Mapping The robot has a probability of 1.0 (i.e. 100%) to be at position a. p ( x ) ( x a ) Note, the Dirac distribution is usually represented with an arrow. The Dirac function ( x ) is defined as: x if x 0 ( x ) with ( x ) dx 1 0 otherwise

21 5 21 Example 4: Gaussian distribution 5 - Localization and Mapping The Gaussian distribution has the shape of a bell and is defined by the following formula: p ( x ) 1 2 e ( x ) where μ is the mean value and σ is the standard deviation. The Gaussian distribution is also called normal distribution and is usually abbreviated with N(μ,σ ). x R. Siegwart, D. Scaramuzza ETH Zurich - ASL

22 Introduction to Localization Problems Example: Mobile Robot Indoor environment The Robot has to fulfill a given task, for example: Cleaning

23 Introduction First Idea: Start cleaning at current position Move straight until an obstacle is encountered Choose a new direction of motion Go back to step 2. This is called a Reactive Agent

24 Introduction Problems with Reactive Agents: When are we ready? What is a good decision for a new direction? Can we guarantee that the whole room will be cleaned? What if the robot needs to refill or recharge?

25 Introduction Second approach: Obtain a map of the environment Plan a path using the given map Move along the obtained path While moving: find position in map End when the whole map is covered This is called a Model-based Agent

26 Introduction Advantages of Model-based Agents: Ready when the path is traversed Easier to decide what to do next We can guarantee that the room will be cleaned at some time The robot can include the refilling station into its path

27 Introduction Examples: Roomba vacuum cleaner Reactive Electrolux Trilobite vacuum cleaner Model-based

28 Introduction Two major requirements for Model-based Agents: Mapping: We need an accurate map of the environment. The map is created with sensor measurements. We need to know where the robot sensed the env. Localization: We need to know where the robot is with respect to our map

29 Introduction The chicken-and-egg problem: mapping Known positions Known map localization

30 Introduction Solution to the chicken-and-egg problem: Robot positions and map are computed simultaneously This is usually called: Simultaneous Localisation and Mapping (SLAM)

31 Introduction SLAM techniques are probabilistic. This means that they Model the uncertainty of the robot s sensors using probability theory Compute the most likely solution for the robot positions and the map Textbook: Thrun, Burgard, Fox: Probabilistic Robotics

32 BAYES DATA FUSION

33 Bayesianism vs. Frequentism Bayesianism (Bayes) Frequentism (Fisher, Neyman, Pearson) The state of nature can be modeled as a random variable Probabilities can be assigned a-priori to arbitrary statements Subjective (through opinion) or objective (through data) Popular in decision theory Obviously lacks objectivity The frequency of occurrence of an event determines its measure of probability Probabilities only result from well-defined random experiments Conservative but guaranteed to be objective Empirical, sampling theory Built-in subjectivity

34 Simple Example of State Estimation Suppose a robot needs to know the state of a door. It obtains a measurement from its sensor. What is??

35 Causal vs. Diagnostic Reasoning Searching for is called diagnostic reasoning. Searching for is called causal reasoning. Often causal knowledge is easier to obtain. Bayes rule allows us to use causal knowledge:

36 Example with Numbers Assume we have: Then: raises the probability that the door is open

37 Combining Evidence Suppose our robot obtains another observation. Question: How can we integrate this new information? Formally, we want to estimate. Using Bayes formula with background knowledge:??

38 Markov Assumption If we know the state of the door at time the measurement does not give any further information about.. Formally: and are conditional independent given. This means: then

39 Example with Numbers Assume we have: Then: (from above) lowers the probability that the door is open

40 General Form Measurements: Markov assumption: and are conditionally independent given the state.

41 APPLICATION: COLOUR SEGMENTATION

42 Colour Segmentation Learning: Use colour patch as example Build a probabilistic model Exploitation: Use probabilistic model to estimate how likely a given pixel is to be similar to the patch. Example: Ground segmentation

43 Solution Learning: build histogram of P R, G, B Sample set P R, G, B Sample as uniform. Exploitation: for each pixel (r,g,b), compute P Sample r, g, b = P(Sample,r,g,b) P(r,g,b) This is why we need the negative distribution = P Sample P(r,g,b Sample) P Sample P r,g,b Sample +P Sample P(r,g,b Sample)

44 APPLICATION: THE HOUGH TRANSFORM

45 Model estimation Input: data points Xi = (xi,1, xi,n) Model: f(xi, P) = 0 Objective: find a distribution over the parameter P based on the data. Example: Hough Transform Any fitting algorithm

46 Solution P π, X 1, X n = P π P X 1 π P X 2 X 1, π P X n X 1,, X n 1, π = P π P X i π Conditional independence P π X 1,, X n = P π P(X i π) P(X 1,,X n ) = ηp π P(X i π) = P π P(X i π) X1,,Xn P(X 1,,X n ) Algorithm: For each value v of π compute, P π = v P(X i π = v) Compute η = π P π P(X i π) Return 1 P π P(X η i π)

47 APPLICATION: LOCALIZATION AND DATA ASSOCIATION

48 Localization Input: observation of landmarks L 1, L n Objective: estimate of state X Specialization: Full-pose landmarks (AR-Tags) Bearing only (image features, real beacons) Range only (radio signal) For each case, express the sensor model and the fusion process.

49 Bearing-only localization

50 Data association Known Map Observation

51 Data association Core problem in robotics Make localization a combinatorial problem Several approaches: Features or discriminative observations Nearest neighbour Consistency enforcement: Probabilistic Approach (Ex) Clique JCDA / JCBB

52 Probabilistic approach Add a match variable M i [0, p] for each measurement. M i = 0 means that Z i is an outlier. P Z 1,, Z n, M 1,, M n, L 1,, L p, X = P X P(L 1:p ) P(M i X) P(Z i X, M i, L 1:p ) i P(M i X) can be uniform or include some prior based on current state P(Z i X, M i 0, L 1:p ) follows a Gaussian model P(Z i X, M i = 0, L 1:p ) could be uniform. i

53 Probabilistic approach Localisation P X Z 1,, Z n, L 1,, L p, X = η P X P(L 1:p ) P(M i = j X) p i j=0 P(Z i X, M i, L 1:p )

54 Range measurement

55 Range with outliers

56 2 Ranges

57 2 Ranges with outliers

58 2 Ranges with 2 Landmarks

59 MORE ON DATA ASSOCIATION

60 Consistency enforcement GCDA/JCBB See external document: Mingo, Cliques See external document: Bailey, c - Perception - Vision 4c 60

61 Clique (1/n)

62 Clique (2/n)

63 Clique (3/n)

64 Clique (4/n)

65 Clique (5/n)

66 Clique (6/n)

67 Clique (7/n)

68 Clique (n/n) Finding the largest clique is NP- Hard in general. Heuristic exists to make it quasi-linear in sparse matching graphs.

69 Matching Tree A1 A2 A3 A4 A5 B1 B2 B3 B4 B5 C1 C2 C3 C4 C5 D1 D2 D3 D4 D5 Depth: 2

70 Matching Tree A1 A2 A3 A4 A5 B1 B2 B3 B4 B5 C1 C2 C3 C4 C5 D1 D2 D3 D4 D5 Depth: 2

71 Matching Tree A1 A2 A3 A4 A5 B1 B2 B3 B4 B5 C1 C2 C3 C4 C5 D1 D2 D3 D4 D5

72 Matching Tree A1 A2 A3 A4 A5 B1 B2 B3 B4 B5 C1 C2 C3 C4 C5 D1 D2 D3 D4 D5 Depth: 4

73 Matching Tree Result Deepest path: 4 mathes, (A2, B5, C3, D4) Branch & Bound Pruning helps reducing the combinatorial complexity Possibility to integrate robot pose distribution or sensing uncertainty into the compatibility check.

74 R. Siegwart and D. Scaramuzza, ETH Zurich - ASL 4c 74 Example: Build a Panorama 4c - Perception - Vision This panorama was generated using AUTOSTITCH (freeware), available at

75 R. Siegwart and D. Scaramuzza, ETH Zurich - ASL 4c 75 How do we build panorama? 4c - Perception - Vision We need to match (align) images

76 R. Siegwart and D. Scaramuzza, ETH Zurich - ASL 4c 76 Matching with Features 4c - Perception - Vision Detect feature points in both images

77 4c 77 Matching with Features 4c - Perception - Vision Detect feature points in both images Find corresponding pairs R. Siegwart and D. Scaramuzza, ETH Zurich - ASL

78 4c 78 Matching with Features 4c - Perception - Vision Detect feature points in both images Find corresponding pairs Use these pairs to align images R. Siegwart and D. Scaramuzza, ETH Zurich - ASL

79 4c 79 Matching with Features 4c - Perception - Vision Problem 1: Detect the same point independently in both images no chance to match! We need a repeatable detector R. Siegwart and D. Scaramuzza, ETH Zurich - ASL

80 4c 80 Matching with Features 4c - Perception - Vision Problem 2: For each point correctly recognize the corresponding one? We need a reliable and distinctive descriptor R. Siegwart and D. Scaramuzza, ETH Zurich - ASL

81 R. Siegwart and D. Scaramuzza, ETH Zurich - ASL 4c 81 Most Famous Feature Detectors 4c - Perception - Vision Overview Harris Detector (1988) SIFT Detector (2004)

82 BAYES FILTERING

83 Example: Sensing and Acting Now the robot senses the door state and acts (it opens or closes the door).

84 Action Model An action is modeled as a random variable in our case means close the door. State transition example: where o p e n c lo s e d 1 0 If the door is open, the action close door succeeds in 90% of all cases.

85 The Outcome of Actions Prediction For a given action we want to know the probability: We do this by integrating over all possible previous states. If the state space is discrete: If the state space is continuous:

86 Back to the Example Example

87 Sensor Update and Action Update So far, we learned two different ways to update the system state: Sensor update: Action update: Now we want to combine both: Definition 2.1: Let be a sequence of sensor measurements and actions until time. Then the belief of the current state is defined as

88 Dynamic Bayes Network We can describe the overall process using a Dynamic Bayes Network: This incorporates the following Markov assumptions: (measurement) (state)

89 The Bayes Filter (Bayes) (Markov) (Tot. prob.) (Markov) (Markov)

90 The Bayes Filter Algorithm Bayes_filter : 1. if is a sensor measurement then for all do for all do 7. else if is an action then 8. for all do 9. return

91 Bayes Filter Variants The Bayes filter principle is used in Kalman filters Particle filters (Monte-Carlo-Localization) Hidden Markov models Dynamic Bayesian networks Partially Observable Markov Decision Processes (POMDPs)

92 R. Siegwart, D. Scaramuzza ETH Zurich - ASL 5 92 The localisation problem 5 - Localization and Mapping Consider a mobile robot moving in a known environment. As it starts to move, say from a precisely known location, it might keep track of its location using odometry. However, after a certain movement the robot will get very uncertain about its position => update using an observation of its environment Observation leads also to an estimate of the robot position which can then be fused with the odometric estimation to get the best possible update of the robots actual position.?

93 R. Siegwart, D. Scaramuzza ETH Zurich - ASL 5 93 Illustration of probabilistic bap based localization: Improving belief state by moving 5 - Localization and Mapping The robot is placed somewhere in the environment but it is not told its location x The robot queries its sensors and finds it is next to a pillar The robot moves one meter forward. To account for inherent noise in robot motion the new belief is smoother The robot queries its sensors and again it finds itself next to a pillar Finally, it updates its belief by combining this information with its previous belief

94 R. Siegwart, D. Scaramuzza ETH Zurich - ASL 5 94 Description of the probabilistic localization problem 5 - Localization and Mapping Consider a mobile robot moving in a known environment.

95 5 95 Description of the probabilistic localization problem 5 - Localization and Mapping Consider a mobile robot moving in a known environment. As it starts to move, say from a precisely known location, it can keep track of its motion using odometry. R. Siegwart, D. Scaramuzza ETH Zurich - ASL

96 5 96 Description of the probabilistic localization problem 5 - Localization and Mapping Consider a mobile robot moving in a known environment. As it starts to move, say from a precisely known location, it can keep track of its motion using odometry. Due to odometry uncertainty, after some movement the robot will become very uncertain about its position. R. Siegwart, D. Scaramuzza ETH Zurich - ASL

97 5 97 Description of the probabilistic localization problem 5 - Localization and Mapping Consider a mobile robot moving in a known environment. As it starts to move, say from a precisely known location, it can keep track of its motion using odometry. Due to odometry uncertainty, after some movement the robot will become very uncertain about its position. To keep position uncertainty from growing unbounded, the robot must localize itself in relation to its environment map. To localize, the robot might use its on-board exteroceptive sensors (e.g. ultrasonic, laser, vision sensors) to make observations of its environment Sensor reference frame R. Siegwart, D. Scaramuzza ETH Zurich - ASL

98 5 98 Description of the probabilistic localization problem 5 - Localization and Mapping Consider a mobile robot moving in a known environment. As it starts to move, say from a precisely known location, it can keep track of its motion using odometry. Due to odometry uncertainty, after some movement the robot will become very uncertain about its position. To keep position uncertainty from growing unbounded, the robot must localize itself in relation to its environment map. To localize, the robot might use its on-board exteroceptive sensors (e.g. ultrasonic, laser, vision sensors) to make observations of its environment The robot updates its position based on the observation. Its uncertainty shrinks. Sensor reference frame R. Siegwart, D. Scaramuzza ETH Zurich - ASL

99 5 99 Action and perception updates 5 - Localization and Mapping In robot localization, we distinguish two update steps: 1. ACTION update: the robot moves and estimates its position through its proprioceptive sensors. During this step, the robot uncertainty grows. 2. PERCEPTION update: the robot makes an observation using its exteroceptive sensors and correct its position by opportunely combining its belief before the observation with the probability of making exactly that observation. During this step, the robot uncertainty shrinks. Probability of making this observation Robot belief before the observation R. Siegwart, D. Scaramuzza ETH Zurich - ASL

100 5 100 The solution to the probabilistic localization problem 5 - Localization and Mapping Solving the probabilistic localization problem consists in solving separately the Action and Perception updates R. Siegwart, D. Scaramuzza ETH Zurich - ASL

101 5 101 The solution to the probabilistic localization problem 5 - Localization and Mapping A probabilistic approach to the mobile robot localization problem is a method able to compute the probability distribution of the robot configuration during each Action and Perception step. The ingredients are: 1. The initial probability distribution p ( x ) t 0 2. The statistical error model of the proprioceptive sensors (e.g. wheel encoders) 3. The statistical error model of the exteroceptive sensors (e.g. laser, sonar, camera) 4. Map of the environment (If the map is not known a priori then the robots needs to build a map of the environment and then localizing in it. This is called SLAM, Simultaneous Localization And Mapping) R. Siegwart, D. Scaramuzza ETH Zurich - ASL

102 R. Siegwart, D. Scaramuzza ETH Zurich - ASL The solution to the probabilistic localization problem 5 - Localization and Mapping How do we solve the Action and Perception updates? Action update uses the Theorem of Total probability (or convolution) Perception update uses the Bayes rule

103 R. Siegwart, D. Scaramuzza ETH Zurich - ASL Action update 5 - Localization and Mapping In this phase the robot estimates its current position based on the knowledge of the previous position and the odometric input Using the Theorem of Total probability, we compute the robot s belief after the motion as which can be written as a convolution

104 R. Siegwart, D. Scaramuzza ETH Zurich - ASL Perception update 5 - Localization and Mapping In this phase the robot corrects its previous position (i.e. its former belief) by opportunely combining it with the information from its exteroceptive sensors p ( x t z t ) p ( z t x t p ( z ) t p ) ( x t ) bel ( x ) p ( z x ) bel ( x t t t t ) where η is a normalization factor which makes the probability integrate to 1.

105 R. Siegwart, D. Scaramuzza ETH Zurich - ASL Perception update 5 - Localization and Mapping Explanation of bel ( x ) p ( z x ) bel ( x t t t t ) This is the probability of observing z given that the robot is at position x, that is: p z x ) ( t t p ( x t ) p ( x t z t ) p ( z t x t p ( z ) p ( x t ) t ) p ( x t )

106 R. Siegwart, D. Scaramuzza ETH Zurich - ASL Illustration of probabilistic bap based localization 5 - Localization and Mapping Initial probability distribution p ( x ) t 0 x p ( z x ) t t Perception update bel ( x ) p ( z x ) bel ( x ) t t t t Action update p ( z x ) t t Perception update bel ( x ) p ( z x ) bel ( x ) t t t t

107 R. Siegwart, D. Scaramuzza ETH Zurich - ASL Illustration of probabilistic bap based localization 5 - Localization and Mapping Initial probability distribution p ( x ) t 0 x p ( z x ) t t Perception update bel ( x ) p ( z x ) bel ( x ) t t t t Action update p ( z x ) t t Perception update bel ( x ) p ( z x ) bel ( x ) t t t t

108 R. Siegwart, D. Scaramuzza ETH Zurich - ASL Illustration of probabilistic bap based localization 5 - Localization and Mapping Initial probability distribution p ( x ) t 0 x p ( z x ) t t Perception update bel ( x ) p ( z x ) bel ( x ) t t t t Action update p ( z x ) t t Perception update bel ( x ) p ( z x ) bel ( x ) t t t t

109 R. Siegwart, D. Scaramuzza ETH Zurich - ASL Illustration of probabilistic bap based localization 5 - Localization and Mapping Initial probability distribution p ( x ) t 0 x p ( z x ) t t Perception update bel ( x ) p ( z x ) bel ( x ) t t t t Action update p ( z x ) t t Perception update bel ( x ) p ( z x ) bel ( x ) t t t t

110 MARKOV LOCALISATION

111 5 111 Markov localization 5 - Localization and Mapping Markov localization uses a grid space representation of the robot configuration. For sake of simplicity let us consider a robot moving in a one dimensional environment (e.g. moving on a rail). Therefore the robot configuration space can be represented through the sole variable x. Let us discretize the configuration space into 10 cells R. Siegwart, D. Scaramuzza ETH Zurich - ASL

112 R. Siegwart, D. Scaramuzza ETH Zurich - ASL Markov localization 5 - Localization and Mapping Let us discretize the configuration space into 10 cells Suppose that the robot s initial belief is a uniform distribution from 0 to 3. Observe that all the elements were normalized so that their sum is 1.

113 5 113 Markov localization 5 - Localization and Mapping Initial belief distribution Action phase: Let us assume that the robot moves forward with the following statistical model This means that we have 50% probability that the robot moved 2 or 3 cells forward. Considering what the probability was before moving, what will the probability be after the motion? R. Siegwart, D. Scaramuzza ETH Zurich - ASL

114 R. Siegwart, D. Scaramuzza ETH Zurich - ASL Markov localization: Action update 5 - Localization and Mapping The solution is given by the convolution of the two distributions *

115 R. Siegwart, D. Scaramuzza ETH Zurich - ASL Markov localization: Action update - Explanation 5 - Localization and Mapping Here, we would like to explain where this formula actually comes from. In order to compute the probability of position x 1 in the new belief state, one must sum over all the possible ways in which the robot may reach x 1 according to the potential positions expressed in the former belief state x 0 and the potential input expressed by u 1. Observe that because is limited between 0 and 3, the robot can only reach the states between 2 to 6, therefore You can now verify that equations that these equations are computing nothing but the convolution of the Action update:

116 R. Siegwart, D. Scaramuzza ETH Zurich - ASL Markov localization: Perception update 5 - Localization and Mapping Let us now assume that the robot uses its onboard range finder and measures the distance from the origin. Assume that the statistical error model of the sensors is This plot tells us that the distance of the robot from the origin can be equally 5 or 6 units. What will the final robot belief be after this measurement? The answer is again given by the Bayes rule: bel x ) p ( z x ) bel ( x ) ( t t t t

117 R. Siegwart, D. Scaramuzza ETH Zurich - ASL Markov Localization: Case Study 2 Grid Map (5) 5 - Localization and Mapping Example 2: Museum Laser scan 1 Courtesy of W. Burgard

118 R. Siegwart, D. Scaramuzza ETH Zurich - ASL Markov Localization: Case Study 2 Grid Map (6) 5 - Localization and Mapping Example 2: Museum Laser scan 2 Courtesy of W. Burgard

119 R. Siegwart, D. Scaramuzza ETH Zurich - ASL Markov Localization: Case Study 2 Grid Map (7) 5 - Localization and Mapping Example 2: Museum Laser scan 3 Courtesy of W. Burgard

120 R. Siegwart, D. Scaramuzza ETH Zurich - ASL Markov Localization: Case Study 2 Grid Map (8) 5 - Localization and Mapping Example 2: Museum Laser scan 13 Courtesy of W. Burgard

121 R. Siegwart, D. Scaramuzza ETH Zurich - ASL Markov Localization: Case Study 2 Grid Map (9) 5 - Localization and Mapping Example 2: Museum Laser scan 21 Courtesy of W. Burgard

122 5 122 Drawbacks of Markov localization 5 - Localization and Mapping In the more general planar motion case the grid-map is a three-dimensional array where each cell contains the probability of the robot to be in that cell. In this case, the cell size must be chosen carefully. During each prediction and measurement steps, all the cells are updated. If the number of cells in the map is too large, the computation can become too heavy for real-time operations. The convolution in a 3D space is clearly the computationally most expensive step. As an example, consider a 30x30 m environment and a cell size of 0.1 m x 0.1 m x 1 deg. In this case, the number of cells which need to be updated at each step would be 30x30x100x360=32.4 million cells! Fine fixed decomposition grids result in a huge state space Very important processing power needed Large memory requirement R. Siegwart, D. Scaramuzza ETH Zurich - ASL

123 5 123 Drawbacks of Markov localization 5 - Localization and Mapping Reducing complexity Various approached have been proposed for reducing complexity One possible solution would be to increase the cell size at the expense of localization accuracy. Another solution is to use an adaptive cell decomposition instead of a fixed cell decomposition. Randomized Sampling / Particle Filter The main goal is to reduce the number of states that are updated in each step Approximated belief state by representing only a representative subset of all states (possible locations) E.g update only 10% of all possible locations The sampling process is typically weighted, e.g. put more samples around the local peaks in the probability density function However, you have to ensure some less likely locations are still tracked, otherwise the robot might get lost R. Siegwart, D. Scaramuzza ETH Zurich - ASL

124 Summary Model-based agents maintain an internal representation of their environment. They are capable of localizing themselves in the environment by means of filtering techniques, such as the Bayes filtering. The Bayes-Filter applies Bayesian probability theory and the Markov assumption in order to recursively estimate state (e.g. pose) from observations (e.g. range sensor readings) and actions (e.g. robot motion). The Bayes-Filter is a statistical filter, it estimates distributions over random state, observation, and action variables.

125 Summary 2 The belief is a key concept of Bayesian state estimation. It distinguishes the robot s true state from its internal belief with regard to the that state. Various methods exist to represent the belief distribution. Some of these methods apply approximation techniques to overcome complexity issues (see Monte-Carlo methods).