Recursive Bayes Filtering

Transcription

1 Recursive Bayes Filtering CS485 Autonomous Robotics Amarda Shehu Fall 2013 Notes modified from Wolfram Burgard, University of Freiburg

2 Physical Agents are Inherently Uncertain Uncertainty arises from four major factors: Environment stochastic, unpredictable Robot stochastic Sensor limited, noisy Models inaccurate

3 Example: Museum Tour-Guide Robots Rhino, 1997 Minerva, 1998

4 Technical Challenges Navigation Environment crowded, unpredictable Environment unmodified Invisible hazards Walking speed or faster High failure costs Interaction Individuals and crowds Museum visitors first encounter Age 2 through 99 Spend less than 15 minutes

5 Nature of Sensor Data Odometry Data Range Data

6 Nature of Sensor Data Odometry Data Range Data

7 Probabilistic Techniques for Physical Agents Key idea: Explicit representation of uncertainty using the calculus of probability theory Perception = state estimation Action = utility optimization

8 Advantages of Probabilistic Paradigm Can accommodate inaccurate models Can accommodate imperfect sensors Robust in real-world applications Best known approach to many hard robotics problems

9 Pitfalls Computationally demanding False assumptions Approximate

10 Outline Introduction Probabilistic State Estimation Robot Localization Probabilistic Decision Making Planning Between MDPs and POMDPs Exploration Conclusions

11 Axioms of Probability Theory Pr(A) denotes probability that proposition A is true. 0 Pr( A) 1 Pr(True) = 1 Pr( A B ) = Pr( A) + Pr( B ) Pr( A B ) Pr( False) = 0

12 A Closer Look at Axiom 3 Pr( A B ) = Pr( A) + Pr( B ) Pr( A B ) True A A B B B

13 Using the Axioms Pr( A A) = Pr( A) + Pr( A) Pr( A A) Pr(True) 1 Pr( A) = = = Pr( A) + Pr( A) Pr( False) Pr( A) + Pr( A) 0 1 Pr( A)

14 Discrete Random Variables X denotes a random variable. X can take on a finite number of values in {x1, x2,, xn}. P(X=xi), or P(xi), is the probability that the random variable X takes on value xi.

15 Continuous Random Variables X takes on values in the continuum. p(x=x), or p(x), is a probability density function. b Pr( x [a, b]) = p ( x)dx a p(x) E.g. x

16 Joint and Conditional Probability P(X=x and Y=y) = P(x,y) If X and Y are independent then P(x,y) = P(x) P(y) P(x y) is the probability of x given y P(x y) = P(x,y) / P(y) P(x,y) = P(x y) P(y) If X and Y are independent then P(x y) = P(x)

17 Law of Total Probability, Marginals Discrete case Continuous case P( x) = 1 p( x) dx = 1 P ( x ) = P ( x, y ) p ( x) = p ( x, y ) dy P( x) = P( x y ) P( y ) p ( x) = p ( x y ) p ( y ) dy x y y

18 Bayes Formula P ( x, y ) = P ( x y ) P ( y ) = P ( y x ) P ( x ) P ( y x) P ( x) likelihood prior P( x y ) = = P( y ) evidence

19 Normalization P( y x) P( x) P( x y ) = = η P ( y x) P( x) P( y ) 1 1 η = P( y) = P( y x)p( x) x Algorithm: x : aux x y = P( y x) P( x) 1 η= aux x y x x : P ( x y ) = η aux x y

20 Conditioning Total probability: P ( x y ) = P ( x y, z ) P ( z y ) dz Bayes rule and background knowledge: P ( y x, z ) P ( x z ) P( x y, z ) = P( y z )

21 Simple Example of State Estimation Suppose a robot obtains measurement z What is P(open z)?

22 Causal vs. Diagnostic Reasoning P(open z) is diagnostic. P(z open) is causal. Often causal knowledge is easier to obtain. Bayes rule allows us to use causal knowledge: P ( z open) P (open) P (open z ) = P( z )

23 Causal vs. Diagnostic Reasoning P(open z) is diagnostic. P(z open) is causal. Often causal knowledge is easier to obtain. count frequencies! Bayes rule allows us to use causal knowledge: P ( z open) P (open) P (open z ) = P( z )

24 Example P(z open) = 0.6 P(z open) = 0.3 P(open) = P( open) = 0.5 P(open z) =?

25 Example P(z open) = 0.6 P(z open) = 0.3 P(open) = P( open) = 0.5 P( z open) P (open) P (open z ) = P ( z open) p (open) + P( z open) p ( open) P (open z ) = = = z raises the probability that the door is open.

26 Combining Evidence Suppose our robot obtains another observation z2. How can we integrate this new information? More generally, how can we estimate P(x z1...zn )?

27 Recursive Bayesian Updating P ( zn x, z1,, zn 1) P ( x z1,, zn 1) P ( x z1,, zn) = P ( zn z1,, zn 1)

28 Recursive Bayesian Updating P ( zn x, z1,, zn 1) P ( x z1,, zn 1) P ( x z1,, zn) = P ( zn z1,, zn 1) Markov assumption: zn is independent of z1,...,zn-1 if we know x.

29 Recursive Bayesian Updating P ( zn x, z1,, zn 1) P ( x z1,, zn 1) P ( x z1,, zn) = P ( zn z1,, zn 1) Markov assumption: zn is independent of z1,...,zn-1 if we know x. P ( zn x) P( x z1,, zn 1) P ( x z1,, zn) = P ( zn z1,, zn 1) = η P ( zn x) P ( x z1,, zn 1) = η1...n P( z x) P( x) i i =1...n

30 Example: Second Measurement P(z2 open) = 0.5 P(open z1)=2/3 P(open z2, z1) =? P(z2 open) = 0.6

31 Example: Second Measurement P(z2 open) = 0.5 P(open z1)=2/3 P(z2 open) = 0.6 P ( z 2 open) P(open z1 ) P(open z 2, z1 ) = P( z 2 open) P(open z1 ) + P( z 2 open) P( open z1 ) = = = z2 lowers the probability that the door is open.

32 Actions Often the world is dynamic since actions carried out by the robot, actions carried out by other agents, or just the time passing by change the world. How can we incorporate such actions?

33 Typical Actions The robot turns its wheels to move The robot uses its manipulator to grasp an object Plants grow over time Actions are never carried out with absolute certainty. In contrast to measurements, actions generally increase the uncertainty.

34 Modeling Actions To incorporate the outcome of an action u into the current belief, we use the conditional pdf P(x u,x ) This term specifies the pdf that executing u changes the state from x to x.

35 Example: Closing the door

36 State Transitions P(x u,x ) for u = close door : open closed 0 If the door is open, the action close door succeeds in 90% of all cases. 1

37 Integrating the Outcome of Actions Continuous case: P( x u)= P ( x u,x' ) P( x' )dx' Discrete case: P( x u )= P ( x u,x' )P ( x' )

38 Example: The Resulting Belief P( closed u )= P(closed u,x' )P ( x' ) =P(closed u,open )P (open ) +P(closed u,closed )P (closed ) = + = P( open u)= P (open u,x' ) P( x' ) =P(open u,open) P( open) +P(open u,closed )P( closed ) = + = =1 P (closed u )

39 Bayes Filters: Framework Given: Stream of observations z and action data u: d t = {u1, z 2, ut 1, zt } Sensor model P(z x). Action model P(x u,x ). Prior probability of the system state P(x). Wanted: Estimate of the state X of a dynamical system. The posterior of the state is also called Belief: Bel ( xt ) = P( xt u1, z 2, ut 1, zt )

40 Markov Assumption Zt-1 Xt-1 Zt+1 Zt ut-1 Xt ut Xt+1 p(d t, d t 1,..., d 0 xt, d t + 1, d t + 2,...) = p(d t, d t 1,..., d 0 xt ) p(d t, d t + 1,... xt, d1, d 2,..., d t 1 ) = p(d t, d t + 1,... xt ) p( x t ut 1,x t 1,d t 2,...,d 0 )=p ( x t u t 1,x t 1 ) Underlying Assumptions Static world Independent noise Perfect model, no approximation errors

41 Bayes Filters Bel ( xt ) = P( xt u1, z 2, ut 1, zt ) z = observation u = action x = state

42 Bayes Filters z = observation u = action x = state Bel ( xt ) = P( xt u1, z 2, ut 1, zt ) Bayes = η P ( zt xt, u1, z 2,, ut 1 ) P( xt u1, z 2,, ut 1 )

43 Bayes Filters z = observation u = action x = state Bel ( xt ) = P( xt u1, z 2, ut 1, zt ) Bayes = η P ( zt xt, u1, z 2,, ut 1 ) P( xt u1, z 2,, ut 1 ) Markov = η P( zt xt ) P ( xt u1, z 2,, ut 1 )

44 Bayes Filters z = observation u = action x = state Bel ( xt ) = P( xt u1, z 2, ut 1, zt ) Bayes = η P ( zt xt, u1, z 2,, ut 1 ) P( xt u1, z 2,, ut 1 ) Markov = η P( zt xt ) P ( xt u1, z 2,, ut 1 ) Total prob. = η P ( zt xt ) P ( xt u1, z 2,, ut 1, xt 1 ) P( xt 1 u1, z 2,, ut 1 ) dxt 1

45 Bayes Filters z = observation u = action x = state Bel ( xt ) = P( xt u1, z 2, ut 1, zt ) Bayes = η P ( zt xt, u1, z 2,, ut 1 ) P ( xt u1, z 2,, ut 1 ) Markov = η P ( zt xt ) P( xt u1, z 2,, ut 1 ) Total prob. Markov = η P ( zt xt ) P( xt u1, z 2,, ut 1, xt 1 ) P ( xt 1 u1, z 2,, ut 1 ) dxt 1 = η P ( zt xt ) P( xt ut 1, xt 1 ) P ( xt 1 u1, z 2,, ut 1 ) dxt 1

46 Bayes Filters z = observation u = action x = state Bel ( xt ) = P( xt u1, z 2, ut 1, zt ) Bayes = η P ( zt xt, u1, z 2,, ut 1 ) P ( xt u1, z 2,, ut 1 ) Markov = η P ( zt xt ) P( xt u1, z 2,, ut 1 ) Total prob. Markov = η P ( zt xt ) P( xt u1, z 2,, ut 1, xt 1 ) P ( xt 1 u1, z 2,, ut 1 ) dxt 1 = η P ( zt xt ) P( xt ut 1, xt 1 ) P ( xt 1 u1, z 2,, ut 1 ) dxt 1 = η P ( zt xt ) P( xt ut 1, xt 1 ) Bel ( xt 1 ) dxt 1

47 Bayes Filters z = observation u = action x = state Bel ( xt ) = P( xt u1, z 2, ut 1, zt ) Bayes = η P ( zt xt, u1, z 2,, ut 1 ) P ( xt u1, z 2,, ut 1 ) Markov = η P ( zt xt ) P( xt u1, z 2,, ut 1 ) Total prob. Markov = η P ( zt xt ) P( xt u1, z 2,, ut 1, xt 1 ) P ( xt 1 u1, z 2,, ut 1 ) dxt 1 = η P ( zt xt ) P( xt ut 1, xt 1 ) P ( xt 1 u1, z 2,, ut 1 ) dxt 1 = η P ( zt xt ) P( xt ut 1, xt 1 ) Bel ( xt 1 ) dxt 1

48 Bayes Filter Algorithm Algorithm Bayes_filter( Bel(x),d ): η=0 if d is a perceptual data item z then For all x do Bel ' ( x) = P ( z x) Bel ( x) η = η + Bel ' ( x) For all x do Bel ' ( x) = η 1 Bel ' ( x) else if d is an action data item u then For all x do Bel ' ( x) = P( x u, x' ) Bel ( x' ) dx' return Bel (x) Bel ( xt ) = η P ( zt xt ) P ( xt ut 1, xt 1 ) Bel ( xt 1 ) dxt 1

49 Bayes Filters are Familiar! Bel ( xt ) = η P ( zt xt ) P ( xt ut 1, xt 1 ) Bel ( xt 1 ) dxt 1 Kalman filters Particle filters Hidden Markov models Dynamic Bayes networks Partially Observable Markov Decision Processes (POMDPs)

50 Application to Door State Estimation Estimate the opening angle of a door and the state of other dynamic objects using a laser-range finder from a moving mobile robot and based on Bayes filters.

51 Result

52 Lessons Learned Bayes rule allows us to compute probabilities that are hard to assess otherwise. Under the Markov assumption, recursive Bayesian updating can be used to efficiently combine evidence. Bayes filters are a probabilistic tool for estimating the state of dynamic systems.

53 Tutorial Outline Introduction Probabilistic State Estimation Localization

54 The Localization Problem Using sensory information to locate the robot in its environment is the most fundamental problem to providing a mobile robot with autonomous capabilities. [Cox 91] Given Wanted Map of the environment. Sequence of sensor measurements. Estimate of the robot s position. Problem classes Position tracking Global localization Kidnapped robot problem (recovery)

55 Representations for Bayesian Robot Localization Kalman filters (late-80s?) Discrete approaches ( 95) Topological representation ( 95) uncertainty handling (POMDPs) occas. global localization, recovery Grid-based, metric representation ( 96) global localization, recovery Particle filters ( 99) sample-based representation global localization, recovery AI Gaussians approximately linear models position tracking Robotics Multi-hypothesis ( 00) multiple Kalman filters global localization, recovery

56 Localization with Bayes Filters bel ( xt ) = η p(st xt ) p( xt xt 1, at 1 ) bel ( xt 1 ) dxt 1 bel ( xt m) = η p( st xt, m) p( xt xt 1, at 1, m) bel ( xt 1 m) dxt 1

57 Localization with Bayes Filters bel ( xt ) = η p(st xt ) p( xt xt 1, at 1 ) bel ( xt 1 ) dxt 1 bel ( xt m) = η p( st xt, m) p( xt xt 1, at 1, m) bel ( xt 1 m) dxt 1 map m

58 Localization with Bayes Filters bel ( xt ) = η p(st xt ) p( xt xt 1, at 1 ) bel ( xt 1 ) dxt 1 bel ( xt m) = η p( st xt, m) p( xt xt 1, at 1, m) bel ( xt 1 m) dxt 1 p(s a,s,m) s a s a

59 Localization with Bayes Filters bel ( xt ) = η p(st xt ) p( xt xt 1, at 1 ) bel ( xt 1 ) dxt 1 bel ( xt m) = η p( st xt, m) p( xt xt 1, at 1, m) bel ( xt 1 m) dxt 1 p(s a,s,m) s a s a observation o laser data map m p(o s,m) p(o s,m)

60 Localization with Bayes Filters bel ( xt ) = η p(st xt ) p( xt xt 1, at 1 ) bel ( xt 1 ) dxt 1 bel ( xt m) = η p( st xt, m) p( xt xt 1, at 1, m) bel ( xt 1 m) dxt 1 laser data p(o s,m)

61 Localization with Bayes Filters bel ( xt ) = η p(st xt ) p( xt xt 1, at 1 ) bel ( xt 1 ) dxt 1 bel ( xt m) = η p( st xt, m) p( xt xt 1, at 1, m) bel ( xt 1 m) dxt 1 ( xt ) = p( x x Motion: bel Perception: bel ( xt ) = η p( st xt ) bel t t 1, at 1 ) bel ( xt 1 ) dxt 1 ( xt ) is optimal under the Markov assumption

62 What is the Right Representation? Kalman filters Multi-hypothesis tracking Grid-based representations Topological approaches Particle filters

63 Gaussians p ( x) ~ N ( µ, σ 2 ) : µ 1 ( x µ ) σ2 1 p ( x) = e 2 2π σ 2 Univariate σ -σ p (x) ~ Ν (μ,σ) : p ( x) = 1 (2π ) d /2 Σ 1/ 2 e Multivariate 1 ( x μ ) t Σ 1 ( x μ ) 2 µ

64 Kalman Filters Estimate the state of processes that are governed by the following linear stochastic difference equation. x t+1 =Ax t +But +v t z t =Cx t +w t The random variables vt and wt represent the process measurement noise and are assumed to be independent, white and with normal probability distributions.

65 Kalman Filters [Schiele et al. 94], [Weiß et al. 94], [Borenstein 96], [Gutmann et al. 96, 98], [Arras 98]

66 Kalman Filters [Schiele et al. 94], [Weiß et al. 94], [Borenstein 96], [Gutmann et al. 96, 98], [Arras 98]

67 Kalman Filters [Schiele et al. 94], [Weiß et al. 94], [Borenstein 96], [Gutmann et al. 96, 98], [Arras 98]

68 Kalman Filters [Schiele et al. 94], [Weiß et al. 94], [Borenstein 96], [Gutmann et al. 96, 98], [Arras 98]

69 Kalman Filter Algorithm Algorithm Kalman_filter( <µ,σ>, d ): If d is a perceptual data item z then K = ΣC ( CΣC + Σ obs ) µ = µ + K ( z Cµ ) Σ = ( I KC )Σ T T 1 Else if d is an action data item u then µ = Aµ + Bu Σ = AΣAT + Σ act Return <µ,σ>

70 Non-linear Systems Very strong assumptions: Linear state dynamics Observations linear in state What can we do if system is not linear? Linearize it: EKF Compute the Jacobians of the dynamics and observations at the current state. Extended Kalman filter works surprisingly well even for highly non-linear systems.

71 Kalman Filter-based Systems (1) [Gutmann et al. 96, 98]: Match LRF scans against map Highly successful in RoboCup mid-size league Courtesy of S. Gutmann

72 Kalman Filter-based Systems (2) Courtesy of S. Gutmann

73 Kalman Filter-based Systems (3) [Arras et al. 98]: Laser range-finder and vision High precision (<1cm accuracy) Courtesy of K. Arras

74 Localization Algorithms - Comparison Kalman filter Sensors Gaussian Posterior Gaussian Efficiency (memory) ++ Efficiency (time) ++ Implementation + Accuracy ++ Robustness - Global localization No

75 Multihypothesis Tracking [Cox 92], [Jensfelt, Kristensen 99]

76 Localization With MHT Belief is represented by multiple hypotheses Each hypothesis is tracked by a Kalman filter Additional problems: Data association: Which observation corresponds to which hypothesis? Hypothesis management: When to add / delete hypotheses? Huge body of literature on target tracking, motion correspondence etc. See e.g. [Cox 93]

77 MHT: Implemented System (1) [Jensfelt and Kristensen 99,01] Hypotheses are extracted from LRF scans Each hypothesis has probability of being the correct one: H i = { xˆ i, Σ i, P ( H i )} Hypothesis probability is computed using Bayes rule P( s H i ) P( H i ) P( H i s) = P( s) Hypotheses with low probability are deleted New candidates are extracted from LRF scans C j = {z j, R j }

78 MHT: Implemented System (2) Courtesy of P. Jensfelt and S. Kristensen

79 MHT: Implemented System (3) Example run # hypotheses P(Hbest) Map and trajectory Courtesy of P. Jensfelt and S. Kristensen Hypotheses vs. time

80 Localization Algorithms - Comparison Kalman filter Multi-hypot hesis tracking Sensors Gaussian Gaussian Posterior Gaussian Multi-modal Efficiency (memory) Efficiency (time) Implementation + o Robustness - + Global localization No Yes Accuracy

81 Piecewise Constant [Burgard et al. 96,98], [Fox et al. 99], [Konolige et al. 99]

82 Piecewise Constant Representation bel ( xt =< x, y,θ >)

83 Grid-based Localization

84 Tree-based Representations (1) Idea: Represent density using a variant of Octrees

85 Tree-based Representations (2) Efficient in space and time Multi-resolution

86 Localization Algorithms - Comparison Kalman filter Multi-hypot hesis tracking (fixed/variable) Sensors Gaussian Gaussian Non-Gaussian Posterior Gaussian Multi-modal Piecewise constant Efficiency (memory) /+ Efficiency (time) o/+ Implementation + o +/o /++ Robustness Global localization No Yes Yes Accuracy Grid-based

87 Localization Algorithms - Comparison Kalman filter Multi-hypot hesis tracking (fixed/variable) Topological maps Sensors Gaussian Gaussian Non-Gaussian Features Posterior Gaussian Multi-modal Piecewise constant Piecewise constant Efficiency (memory) /+ ++ Efficiency (time) o/+ ++ Implementation + o +/o +/o /++ - Robustness Global localization No Yes Yes Yes Accuracy Grid-based

88 Particle Filters Represent density by random samples Monte Carlo filter, Survival of the fittest, Condensation, Bootstrap filter, Particle filter Filtering: [Rubin, 88], [Gordon et al., 93], [Kitagawa 96] Estimation of non-gaussian, nonlinear processes Computer vision: [Isard and Blake 96, 98] Dynamic Bayesian Networks: [Kanazawa et al., 95]

89 Monte Carlo Localization (MCL) Represent Density Through Samples

90 Importance Sampling Weight samples: f w= g

91 MCL: Global Localization

92 MCL: Sensor Update Bel ( x) α p ( z x) Bel ( x) α p( z x) Bel ( x) w = α p( z x) Bel ( x)

93 MCL: Robot Motion Bel ( x) p( x u x' ) Bel ( x' ), d x'

94 MCL: Sensor Update Bel ( x) α p ( z x) Bel ( x) α p( z x) Bel ( x) w = α p( z x) Bel ( x)

95 MCL: Robot Motion Bel ( x) p( x u x' ) Bel ( x' ), d x'

96 Particle Filter Algorithm 1. Algorithm particle_filter( St-1, ut-1 zt): S t =, For η =0 Generate new samples i = 1 n Sample index j(i) from the discrete distribution given by wt-1 1. Sample xti from p ( xt xt 1, ut 1 ) using xtj (1i ) and 2. wti = p ( zt xti ) Compute importance weight 3. η = η + wti Update normalization factor 4. St = St {< xti, wti >} Insert 5. For 6. i = 1 n wti = wti / η Normalize weights ut 1

97 Particle Filter Algorithm Bel ( xt ) = η p( zt xt ) p( xt xt 1, ut 1 ) Bel ( xt 1 ) dxt 1 draw xit 1 from Bel(xt 1) draw xit from p(xt xit 1,ut 1) Importance factor for xit: target distribution w = proposal distribution η p ( zt xt ) p ( xt xt 1, ut 1 ) Bel ( xt 1 ) = p ( xt xt 1, ut 1 ) Bel ( xt 1 ) p( zt xt ) i t

98 Resampling Given: Set S of weighted samples. Wanted : Random sample, where the probability of drawing xi is given by wi. Typically done n times with replacement to generate new sample set S.

99 Resampling Algorithm 1. Algorithm systematic_resampling(s,n): 1. S ' =, c1 = w1 2. For i = 2 n Generate cdf ci = ci 1 + w u1 ~ U [0, n ], i = 1 Initialize threshold i For j = 1 n Draw samples u j = u1 + n 1 ( j 1) Advance threshold While ( u j > ci) Skip until next threshold reached i = i +1 { } S ' = S ' < x i, n 1 > Insert 1. Return S Also called stochastic universal sampling

100 Motion Model p(xt at-1, xt-1) Model odometry error as Gaussian noise on α, β, and δ

101 Motion Model p(xt at-1, xt-1) Start

102 Model for Proximity Sensors The sensor is reflected either by a known or by an unknown obstacle: Laser sensor Sonar sensor

103

104

105

106

107

108

109

110

111

112

113

114

115

116

117

118

119

120

121

122 Recovery from Failure Problem: Samples are highly concentrated during tracking True location is not covered by samples if position gets lost Solutions: Add uniformly distributed samples [Fox et al., 99] Draw samples according to observation density [Lenser et al.,00; Thrun et al., 00]

123 MCL: Recovery from Failure [Fox et al. 99]

124 The RoboCup Challenge Dynamic, adversarial environments Limited computational power Multi-robot collaboration Particle filters allow efficient localization [Lenser et al. 00]

125 Using Ceiling Maps for Localization [Dellaert et al. 99]

126 Vision-based Localization P(s x) s h(x)

127 Under a Light Measurement: Resulting density:

128 Next to a Light Measurement: Resulting density:

129 Elsewhere Measurement: Resulting density:

130 MCL: Global Localization Using Vision

131 Vision-based Localization Odometry only: Vision: Laser:

132 Localization for AIBO robots

133 Adaptive Sampling

134 MCL: Adaptive Sampling (Sonar)

135 MCL: Adaptive Sampling (Laser)

136 Performance Comparison Grid-based localization Monte Carlo localization

137 Particle Filters for Robot Localization (Summary) Approximate Bayes Estimation/Filtering Full posterior estimation Converges in O(1/ #samples) [Tanner 93] Robust: multiple hypotheses with degree of belief Efficient in low-dimensional spaces: focuses computation where needed Any-time: by varying number of samples Easy to implement

138 Localization Algorithms - Comparison Kalman filter Multi-hypot hesis tracking Topological maps Grid-based Particle filter (fixed/variable) Features Non-Gaussian Non-Gaus sian Sensors Gaussian Posterior Gaussian Multi-modal Piecewise constant Piecewise constant Samples Efficiency (memory) /+ +/++ Efficiency (time) o/+ +/++ Implementation + o + +/o /++ ++ Robustness /++ Global localization No Yes Yes Yes Accuracy

139 Bayes Filtering: Lessons Learned General algorithm for recursively estimating the state of dynamic systems. Variants: Hidden Markov Models (Extended) Kalman Filters Discrete Filters Particle Filters