nips nips2001 nips2001-102 knowledge-graph by maker-knowledge-mining

102 nips-2001-KLD-Sampling: Adaptive Particle Filters

Source: pdf

Author: Dieter Fox

Abstract: Over the last years, particle ﬁlters have been applied with great success to a variety of state estimation problems. We present a statistical approach to increasing the efﬁciency of particle ﬁlters by adapting the size of sample sets on-the-ﬂy. The key idea of the KLD-sampling method is to bound the approximation error introduced by the sample-based representation of the particle ﬁlter. The name KLD-sampling is due to the fact that we measure the approximation error by the Kullback-Leibler distance. Our adaptation approach chooses a small number of samples if the density is focused on a small part of the state space, and it chooses a large number of samples if the state uncertainty is high. Both the implementation and computation overhead of this approach are small. Extensive experiments using mobile robot localization as a test application show that our approach yields drastic improvements over particle ﬁlters with ﬁxed sample set sizes and over a previously introduced adaptation technique.

Reference: text

Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 edu Abstract Over the last years, particle ﬁlters have been applied with great success to a variety of state estimation problems. [sent-3, score-0.586]

2 We present a statistical approach to increasing the efﬁciency of particle ﬁlters by adapting the size of sample sets on-the-ﬂy. [sent-4, score-0.726]

3 The key idea of the KLD-sampling method is to bound the approximation error introduced by the sample-based representation of the particle ﬁlter. [sent-5, score-0.568]

4 Our adaptation approach chooses a small number of samples if the density is focused on a small part of the state space, and it chooses a large number of samples if the state uncertainty is high. [sent-7, score-0.847]

5 Extensive experiments using mobile robot localization as a test application show that our approach yields drastic improvements over particle ﬁlters with ﬁxed sample set sizes and over a previously introduced adaptation technique. [sent-9, score-1.867]

6 1 Introduction Estimating the state of a dynamic system based on noisy sensor measurements is extremely important in areas as different as speech recognition, target tracking, mobile robot navigation, and computer vision. [sent-10, score-0.823]

7 Over the last years, particle ﬁlters have been applied with great success to a variety of state estimation problems (see [3] for a recent overview). [sent-11, score-0.586]

8 It is due to this representation, that particle ﬁlters combine efﬁciency with the ability to represent a wide range of probability densities. [sent-14, score-0.485]

9 The efﬁciency of particle ﬁlters lies in the way they place computational resources. [sent-15, score-0.485]

10 By sampling in proportion to likelihood, particle ﬁlters focus the computational resources on regions with high likelihood, where things really matter. [sent-16, score-0.551]

11 So far, however, an important source for increasing the efﬁciency of particle ﬁlters has only rarely been studied: Adapting the number of samples over time. [sent-17, score-0.738]

12 While variable sample sizes have been discussed in the context of genetic algorithms [10] and interacting particle ﬁlters [2], most existing approaches to particle ﬁlters use a ﬁxed number of samples during the whole state estimation process. [sent-18, score-1.497]

13 An adaptive approach for particle ﬁlters has been applied by [8] and [5]. [sent-20, score-0.554]

14 This approach adjusts the number of samples based on the likelihood of observations, which has some important shortcomings, as we will show. [sent-21, score-0.345]

15 In this paper we introduce a novel approach to adapting the number of samples over time. [sent-22, score-0.342]

16 Our technique determines the number of samples based on statistical bounds on the samplebased approximation quality. [sent-23, score-0.277]

17 Extensive experiments using a mobile robot indicate that our approach yields signiﬁcant improvements over particle ﬁlters with ﬁxed sample set sizes and over a previously introduced adaptation technique. [sent-24, score-1.505]

18 The remainder of this paper is organized as follows: In the next section we will outline the basics of particle ﬁlters and their application to mobile robot localization. [sent-25, score-1.115]

19 In Section 3, we will introduce our novel technique to adaptive particle ﬁlters. [sent-26, score-0.514]

20 2 Particle ﬁlters for Bayesian ﬁltering and robot localization Particle ﬁlters address the problem of estimating the state of a dynamical system from sensor measurements. [sent-28, score-0.937]

21 The goal of particle ﬁlters is to estimate a posterior probability density over the state space conditioned on the data collected so far. [sent-29, score-0.67]

22 Particle ﬁlters represent this belief by a set of weighted samples distributed according to : ¢ ¥¤ ¢ © § ¢ ¨¦ £¡ ¢ § © ¦ ¢ 5 1) 20' A 9 7 53 ¢ % # ! [sent-33, score-0.33]

23 by the importance weight , the Importance sampling: Weight the sample likelihood of the sample given the measurement . [sent-41, score-0.386]

24 After iterations, the importance weights of the samples are normalized so that they sum up to 1. [sent-43, score-0.322]

25 ¢ © § ¨¦ Particle ﬁlters for mobile robot localization We use the problem of mobile robot localization to illustrate and test our approach to adaptive particle ﬁlters. [sent-45, score-2.452]

26 Robot localization is the problem of estimating a robot’s pose relative to a map of its environment. [sent-46, score-0.367]

27 The mobile robot localization problem comes in different ﬂavors. [sent-48, score-0.949]

28 Here the initial robot pose is known, and localization seeks to correct small, incremental errors in a robot’s odometry. [sent-50, score-0.812]

29 More challenging is the global localization problem, where a robot is not told its initial pose, but instead has to determine it from scratch. [sent-51, score-0.847]

30 Robot position (a) (b) Start Robot position Robot position (c) (d) Fig. [sent-52, score-0.303]

31 b)-d) Map of an ofﬁce environment along with a series of sample sets representing the robot’s belief during global localization using sonar sensors (samples are projected into 2D). [sent-54, score-0.651]

32 b) After moving 5m, the robot is still highly uncertain about its position and the samples are spread trough major parts of the free-space. [sent-56, score-0.868]

33 c) Even as the robot reaches the upper left corner of the map, its belief is still concentrated around four possible locations. [sent-57, score-0.566]

34 d) Finally, after moving approximately 55m, the ambiguity is resolved and the robot knows where it is. [sent-58, score-0.494]

35 ¢ In the context of robot localization, the state of the system is the robot’s position, which is typically represented in a two-dimensional Cartesian space and the robot’s heading direction. [sent-60, score-0.52]

36 The state transition probability describes how the position of the robot changes using information collected by the robot’s wheel encoders. [sent-61, score-0.649]

37 The perceptual model describes the likelihood of making the observation given that the robot is at location . [sent-62, score-0.519]

38 Figure 1 illustrates particle ﬁlters for mobile robot localization. [sent-64, score-1.115]

39 While such a high number of samples might be needed to accurately represent the belief during early stages of localization (cf. [sent-67, score-0.732]

40 1(a)), it is obvious that only a small fraction of this number sufﬁces to track the position of the robot once it knows where it is (cf. [sent-68, score-0.654]

41 QV¤ 3 Q¢ 9 ¢ U R P¢ R P ¢ ¡ ¢ ¤ ¢ ¢ 9 ¢ U ¡ ¢ 3 Adaptive particle ﬁlters with variable sample set sizes The localization example in the previous section illustrates that the efﬁciency of particle ﬁlters can be greatly increased by changing the number of samples over time. [sent-71, score-1.689]

42 Before we introduce our approach to adaptive particle ﬁlters, let us ﬁrst discuss an existing technique. [sent-72, score-0.554]

43 1 Likelihood-based adaptation We call this approach likelihood-based adaptation since it determines the number of samples such that the sum of non-normalized likelihoods (importance weights) exceeds a prespeciﬁed threshold. [sent-74, score-0.499]

44 This approach has been applied to dynamic Bayesian networks [8] and mobile robot localization [5]. [sent-75, score-0.989]

45 The intuition behind this approach can be illustrated in the robot localization context: If the sample set is well in tune with the sensor reading, each individual importance weight is large and the sample set remains small. [sent-76, score-1.233]

46 If, however, the sensor reading carries a lot of surprise, as is the case when the robot is globally uncertain or when it lost track of its position, the individual sample weights are small and the sample set becomes large. [sent-79, score-0.911]

47 The likelihood-based adaptation directly relates to the property that the variance of the importance sampler is a function of the mismatch between the proposal distribution and the distribution that is being approximated. [sent-80, score-0.262]

48 Consider, for example, the ambiguous belief state consisting of four distinctive sample clusters shown in Fig. [sent-82, score-0.272]

49 Due to the symmetry of the environment, the average likelihood of a sensor measurement observed in this situation is approximately the same as if the robot knew its position unambiguously (cf. [sent-84, score-0.772]

50 Likelihood-based adaptation would therefore use the same number of samples in both situations. [sent-86, score-0.345]

51 1(b) requires a multiple of the samples needed to represent the belief in Fig. [sent-88, score-0.366]

52 2 KLD-sampling The key idea of our approach is to bound the error introduced by the sample-based representation of the particle ﬁlter. [sent-91, score-0.584]

53 For such a representation we can determine the number of samples so that the distance between the maximum likelihood estimate (MLE) based on the samples and the true posterior does not exceed a pre-speciﬁed threshold . [sent-93, score-0.775]

54 In what follows, we will ﬁrst derive the equation for determining the number of samples needed to approximate a discrete probability distribution (see also [12, 7]). [sent-95, score-0.346]

55 Then we will show how to modify the basic particle ﬁlter algorithm so that it realizes our adaptation approach. [sent-96, score-0.577]

56 ¡ To see, suppose that samples are drawn from a discrete distribution with different bins. [sent-97, score-0.288]

57 Now we can determine the probability that this distance is smaller than , given that samples are drawn from the true distribution: § 7 C & A (3) RP £2F0 8 U 3 §U @ 7 XC A 7 9 & B U 3 §U @ & E8 D! [sent-112, score-0.35]

58 From (7) we see that the required number of samples is proportional to the inverse of the bound, and to the ﬁrst order linear in the number of bins with support. [sent-124, score-0.347]

59 ¡ ¡ It remains to be shown how to apply this result to particle ﬁlters. [sent-127, score-0.485]

60 The problem is that we do not know the true posterior distribution (the estimation of this posterior is the main goal of the particle ﬁlter). [sent-128, score-0.709]

61 To be more speciﬁc, we estimate for the proposal distribution resulting from the ﬁrst two steps of the particle ﬁlter update. [sent-132, score-0.536]

62 Sampling is stopped as soon as the number of samples exceeds the threshold speciﬁed in (7). [sent-134, score-0.299]

63 An update step of the resulting KLD-sampling particle ﬁlter is given in Table 1. [sent-135, score-0.512]

64 ¡ ¡ ¡ Q¢ (Q¢ ¤ 3 (Q¢ 9 ¢ U R P R P R P © § 6¦ ¡ The implementation of this modiﬁed particle ﬁlter is straightforward. [sent-136, score-0.507]

65 1, and to particle ﬁlters with ﬁxed sample set sizes. [sent-143, score-0.605]

66 For the ﬁxed approach we varied the number of samples, for the likelihood-based approach we varied the threshold used to determine the number of samples, and for our approach we varied , the bound on the K-L distance. [sent-145, score-0.335]

67 ¡¢ © falls into empty bin ) then R P (Q¢ 1 1 © ' T( S' ; R P Q¢ S /* Generate samples from the discrete distribution given by the weights in using and ¢ ! [sent-162, score-0.39]

68 T ¢ £ 5 if /* Initialize */ control measurement do R P FQ¢ © § 6¦ Inputs: /* Compute importance weight */ /* Update normalization factor */ /* Insert sample into sample set */ /* Update number of bins with support */ non-empty I GR P PP R H£2 0 R 2 £ B ! [sent-165, score-0.428]

69 £2 '0) ¢ 1 ¢ ¥ return for /* Update number of generated samples */ /* until K-L bound is reached */ D D FED while Table 1: KLD-sampling algorithm. [sent-168, score-0.285]

70 Since the ground truth for these posteriors is not available, we compared the sample sets generated by the different approaches with reference sample sets. [sent-170, score-0.328]

71 These reference sets were generated using a particle ﬁlter with a ﬁxed number of 200,000 samples (far more than actually needed for position estimation). [sent-171, score-0.963]

72 2(a) plots the average K-L distance along with 95% conﬁdence intervals against the average number of samples for the different algorithms (for clarity, we omitted the large error bars for KL distances above 1. [sent-175, score-0.362]

73 Each data point represents the average of 16 global localization runs with different start positions of the robot (each run itself consists of approximately 150 sample set comparisons at the different points in time). [sent-177, score-0.959]

74 The curves also illustrate the superior performance of our approach: While the ﬁxed approach requires about 50,000 samples before it converges to a KL distance below 0. [sent-179, score-0.322]

75 25, our approach converges to the same level using only 3,000 samples on average. [sent-180, score-0.271]

76 This is also an improvement by a factor of 12 compared to the approximately 36,000 samples needed by the likelihood-based approach. [sent-181, score-0.267]

77 In essence, these experiments indicate that our approach, even though based on several approximations, is able to accurately track the true posterior using signiﬁcantly smaller sample sets on avarage than the other approaches. [sent-182, score-0.337]

78 To test the performance of our approach under realistic conditions, we performed multiple global localization experiments under real-time considerations using the timestamps in the data sets. [sent-184, score-0.44]

79 5 0 0 20000 40000 60000 Average number of samples 80000 100000 (a) 0 20000 40000 60000 Average number of samples 80000 (b) Fig. [sent-190, score-0.506]

80 a) The -axis plots the K-L distance between the reference densities and the sample sets generated by the different approaches (real-time constraints were not considered in this experiment). [sent-192, score-0.285]

81 b) The -axis represents the average localization error measured by the distance between estimated positions and reference positions. [sent-193, score-0.455]

82 The U-shape in b) is due to the fact that under real-time conditions, an increasing number of samples results in higher update times and therefore loss of sensor data. [sent-194, score-0.378]

83 As a natural measure of the performance of the different algorithms, we determined the distance between the estimated robot position and the corresponding reference position after each iteration. [sent-199, score-0.776]

84 The U-shape of all three graphs nicely illustrates the trade-off involved in choosing the number of samples under real-time constraints: Choosing not enough samples results in a poor approximation of the underlying posterior and the robot frequently fails to localize itself. [sent-202, score-1.046]

85 The smallest average localization error is 44cm in contrast to an average error of 79cm and 114cm for the likelihood-based and the ﬁxed approach, respectively. [sent-206, score-0.377]

86 This result is due to the fact that our approach is able to determine the best mix between more samples during early stages of localization and less samples during position tracking. [sent-207, score-0.959]

87 5 Conclusions and Future Research We presented a statistical approach to adapting the sample set size of particle ﬁlters onthe-ﬂy. [sent-209, score-0.694]

88 The key idea of the KLD-sampling approach is to bound the error introduced by the sample-based belief representation of the particle ﬁlter. [sent-210, score-0.683]

89 At each iteration, our approach generates samples until their number is large enough to guarantee that the K-L distance between the maximum likelihood estimate and the underlying posterior does not exceed a prespeciﬁed bound. [sent-211, score-0.492]

90 Thereby, our approach chooses a small number of samples if the density is focused on a small subspace of the state space, and chooses a large number of samples if the samples have to cover a major part of the state space. [sent-212, score-0.986]

91 Extensive experiments using mobile robot localization as a test application show that our approach yields drastic improvements over particle ﬁlters with ﬁxed sample sets and over a previously introduced adaptation approach [8, 5]. [sent-214, score-1.912]

92 ter approximations using only 6% of the samples required by the ﬁxed approach, and using less than 9% of the samples required by the likelihood adaptation approach. [sent-216, score-0.606]

93 So far, KLDsampling has been tested using robot localization only. [sent-217, score-0.786]

94 We conjecture, however, that many other applications of particle ﬁlters can beneﬁt from this method. [sent-218, score-0.485]

95 As a result, more samples are used in “important” parts of the state space, while less samples are used in other parts. [sent-224, score-0.537]

96 Another area of future research is the thorough investigation of particle ﬁlters under real-time conditions. [sent-225, score-0.485]

97 In many applications the rate of incoming sensor data is higher than the update rate of the particle ﬁlter. [sent-226, score-0.61]

98 This introduces a trade-off between the number of samples and the amount of sensor data that can be processed (cf. [sent-227, score-0.351]

99 Branching and interacting particle systems approximations of feynamkac formulae with applications to non linear ﬁltering. [sent-243, score-0.507]

100 Monte Carlo Localization: Efﬁcient position estimation for mobile robots. [sent-264, score-0.289]

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