nips nips2002 nips2002-169 knowledge-graph by maker-knowledge-mining
Source: pdf
Author: Cody Kwok, Dieter Fox, Marina Meila
Abstract: Particle filters estimate the state of dynamical systems from sensor information. In many real time applications of particle filters, however, sensor information arrives at a significantly higher rate than the update rate of the filter. The prevalent approach to dealing with such situations is to update the particle filter as often as possible and to discard sensor information that cannot be processed in time. In this paper we present real-time particle filters, which make use of all sensor information even when the filter update rate is below the update rate of the sensors. This is achieved by representing posteriors as mixtures of sample sets, where each mixture component integrates one observation arriving during a filter update. The weights of the mixture components are set so as to minimize the approximation error introduced by the mixture representation. Thereby, our approach focuses computational resources (samples) on valuable sensor information. Experiments using data collected with a mobile robot show that our approach yields strong improvements over other approaches.
Reference: text
sentIndex sentText sentNum sentScore
1 edu £ Abstract Particle filters estimate the state of dynamical systems from sensor information. [sent-7, score-0.305]
2 In many real time applications of particle filters, however, sensor information arrives at a significantly higher rate than the update rate of the filter. [sent-8, score-0.888]
3 The prevalent approach to dealing with such situations is to update the particle filter as often as possible and to discard sensor information that cannot be processed in time. [sent-9, score-0.901]
4 In this paper we present real-time particle filters, which make use of all sensor information even when the filter update rate is below the update rate of the sensors. [sent-10, score-0.871]
5 This is achieved by representing posteriors as mixtures of sample sets, where each mixture component integrates one observation arriving during a filter update. [sent-11, score-0.387]
6 The weights of the mixture components are set so as to minimize the approximation error introduced by the mixture representation. [sent-12, score-0.365]
7 Thereby, our approach focuses computational resources (samples) on valuable sensor information. [sent-13, score-0.439]
8 Experiments using data collected with a mobile robot show that our approach yields strong improvements over other approaches. [sent-14, score-0.277]
9 1 Introduction Due to their sample-based representation, particle filters are well suited to estimate the state of non-linear dynamic systems. [sent-15, score-0.492]
10 Over the last years, particle filters have been applied with great success to a variety of state estimation problems including visual tracking, speech recognition, and mobile robotics [1]. [sent-16, score-0.619]
11 The increased representational power of particle filters, however, comes at the cost of higher computational complexity. [sent-17, score-0.512]
12 The application of particle filters to online, real-time estimation raises new research questions. [sent-18, score-0.528]
13 The key question in this context is: How can we deal with situations in which the rate of incoming sensor data is higher than the update rate of the particle filter? [sent-19, score-0.831]
14 The prevalent approach in real time applications is to update the filter as often as possible and to discard sensor information that arrives during the update process. [sent-21, score-0.512]
15 Obviously, this approach is prone to losing valuable sensor information. [sent-22, score-0.348]
16 At first sight, the sample based representation of particle filters suggests an alternative approach similar to an any-time implementation: Whenever a new observation arrives, sampling is interrupted and the next observation is processed. [sent-23, score-0.704]
17 In this paper we introduce real-time particle filters (RTPF) to deal with constraints imposed by limited computational resources. [sent-25, score-0.511]
18 (b) Aggregate observations within a window and integrate them in one step. [sent-36, score-0.395]
19 samples among the different observations arriving during a filter update. [sent-38, score-0.286]
20 The weights of the mixture components are computed so as to minimize the approximation error introduced by the mixture representation. [sent-40, score-0.365]
21 The resuling approach naturally focuses computational resources (samples) on valuable sensor information. [sent-41, score-0.439]
22 The remainder of this paper is organized as follows: In the next section we outline the basics of particle filters in the context of real-time constraints. [sent-42, score-0.454]
23 Then, in Section 3, we introduce our novel technique to real-time particle filters. [sent-43, score-0.454]
24 ¦ §¥ £¡ ¤¢ (1) ¨£¡ ¦ ¥©¤¢ Here is a sensor measurement and is control information measuring the dynamics of the system. [sent-46, score-0.275]
25 The basic form of the particle filter realizes the recursive Bayes filter according to a sampling procedure, often referred to as sequential importance sampling with resampling (SISR): 1. [sent-49, score-0.61]
26 Under realtime conditions, however, it is possible that the update cannot be completed before the next sensor measurement arrives. [sent-59, score-0.309]
27 This can be the case for computationally complex sensor models or whenever the underlying posterior requires large sample sets [2]. [sent-60, score-0.408]
28 The majority of filtering approaches deals with this problem by skipping sensor information that arrives during the update of the filter. [sent-61, score-0.421]
29 While this approach works reasonably well in many situations, it is prone to miss valuable sensor information. [sent-62, score-0.348]
30 zt 1 St1 zt 3 zt 2 St2 α1 St3 α2 zt+11 St+11 zt+12 St+12 St+13 α2 α1 ’ ’ α3 z t+13 α3 ’ Estimation window t+1 Estmation window t Figure 2: Real time particle filters. [sent-63, score-1.181]
31 The samples are distributed among the observations within one estimation interval (window size three in this example). [sent-64, score-0.399]
32 The belief is a mixture of the individual sample sets. [sent-65, score-0.391]
33 Let be the number of samples required by the particle filter. [sent-70, score-0.58]
34 Assume that the resulting update cycle of the particle filter takes and is called the estimation interval or estimation window. [sent-71, score-0.709]
35 The -th observation and state within window are denoted and , respectively. [sent-76, score-0.343]
36 1 illustrates different approaches to dealing with window sizes larger than one. [sent-78, score-0.346]
37 Here, observations arriving during the update of the sample set are discarded, which has the obvious disadvantage that valuable sensor information might get lost. [sent-81, score-0.645]
38 For example, it assumes that observations can be aggregated optimally, and that the integration of an aggregated observation can be performed as efficiently as the integration of individual observations, which is often not the case. [sent-85, score-0.297]
39 1(c), simply stops generating new samples whenever an observation is made (hence each sample set contains only samples). [sent-87, score-0.302]
40 While this approach takes advantage of the any-time capabilities of particle filters, it is susceptible to filter divergence due to an insufficent number of samples [2, 1]. [sent-88, score-0.626]
41 & ) 10R 3 Real time particle filters In this paper we propose real time particle filters (RTPFs), a novel approach to dealing with limited computational resources. [sent-89, score-1.083]
42 The key idea of RTPFs is to consider all sensor measurements by distributing the samples among the observations within an update window. [sent-90, score-0.571]
43 Additionally, by weighting the different sample sets within a window, our approach focuses the computational resources (samples) on the most valuable observations. [sent-91, score-0.388]
44 If needed, however, the complete belief can be generated by considering the dynamics between the individual mixture components. [sent-97, score-0.335]
45 The belief state that is propagated by RTPF to the next estimation interval is a mixture distribution where each mixture component is represented by one of the sample sets, all generated independently from the previous window. [sent-106, score-0.658]
46 Thus, the belief state propagation is simulated by sample trajectories, that for computational convenience are represented at the points in time where the observations are integrated. [sent-107, score-0.441]
47 The key idea is to choose the weights that minimize the KL-divergence between the mixture belief and the optimal belief. [sent-110, score-0.342]
48 The optimal belief is the belief we would get if there was enough time to compute the full posterior within the update window. [sent-111, score-0.417]
49 The optimal belief at the end of an estimation window results from iterative application of the Bayes filter update on each obseration [3]: & (2) ¤ ¡ 2 ¦ ¥ 4 16 5 ¦ ¥ 4 ¦ "¥©¤I §©) ¦ ¥ ¦ "¥¨ §© ¦ ¥ # ¦ ¨ 6 6 ¦ 0( ¨£¡ 6 ¤ © ¡ 6 16 ¦ "¥¨ 3 7I 6 ¦ £¡ ! [sent-114, score-0.504]
50 In essence, (2) computes the belief by integrating over all trajectories through the estimation interval, where the start position of the trajectories is drawn from the previous belief . [sent-117, score-0.552]
51 Now let denote the belief resulting from integrating only the observation within the estimation window. [sent-119, score-0.295]
52 In contrast to (2), however, each trajectory selectively integrates only one of the observations within the estimation interval1. [sent-127, score-0.272]
53 2 Optimizing the mixture weights We will now turn to the problem of finding the weights of the mixture. [sent-129, score-0.267]
54 Optimizing the weights of mixture approximations can be done using EM [6] or (constrained) gradient descent [7]. [sent-139, score-0.295]
55 3 Monte Carlo gradient estimation The exact computation of the gradients in (6) requires the computation of the different beliefs, each in turn requiring several particle filter updates (see (2), (3)), and integreation over all states . [sent-152, score-0.577]
56 The approach is based on the observation that the beliefs in (6) share the same trajectories through space and differ only in the observations they integrate. [sent-155, score-0.345]
57 Therefore, we first generate sample trajectories through the estimation window without considering the observations, and then use importance sampling to generate the beliefs needed for the gradient from a sample estimation. [sent-156, score-0.787]
58 Trajectory generation is done as follows: we draw a sample is given by the set of the previous mixture belief, where the probability of chosing a set mixture weights . [sent-157, score-0.432]
59 This sample is then moved forward in time by consecutively drawing samples from the distributions at each time step . [sent-158, score-0.324]
60 The number of independent samples needed to represent the belief, the update rate of incoming sensor data, and the available processing power determine the size of the estimation window and hence the number of mixture components. [sent-175, score-0.943]
61 RTPF computes the optimal weights of the mixture distribution at the end of each estimation window. [sent-176, score-0.297]
62 The resulting weights are used to generate samples for the individual sample sets of the next estimation window. [sent-178, score-0.408]
63 The task of the robot was to determine its position using data collected by two distance measuring devices, one pointing to its left, the other pointing to its right. [sent-183, score-0.292]
64 4 Experiments In this section we evaluate the effectiveness of RTPF against the alternatives, using data collected from a mobile robot in a real-world environment. [sent-184, score-0.249]
65 The task of the robot was to determine its position within the map, using data collected by two laser-beams, one pointing to its left, the other pointing to its right. [sent-186, score-0.313]
66 Between each observation the robot moved approximately 50cm (see [3] for details on robot localization and sensor models). [sent-188, score-0.889]
67 Localization performance was measured by the average distance between the samples and the reference robot positions, which were computed offline. [sent-190, score-0.317]
68 In the experiments, our real-time algorithm, RTPF, is compared to particle filters with skipping observations, called “Skip data” (Figure 1a), and particle filters with insufficient samples, called “Naive” (Figure 1c). [sent-191, score-0.964]
69 First, we fix the sample set size which is sufficient for the robot to localize itself. [sent-198, score-0.281]
70 In our experiment is set empirically to 20,000 (the particle filters may fail at lower , see also [2]). [sent-199, score-0.454]
71 We then vary the computational resources, resulting in different window sizes . [sent-200, score-0.316]
72 Larger window size means lower computational power, and the ). [sent-201, score-0.277]
73 number of samples that can be generated for each observation decreases to ( R & 0R ) R & R Figure 4 shows the evolutions of average localization errors over time, using different window sizes. [sent-202, score-0.613]
74 Furthermore, RTPF shows the least degradation with limited computational power (larger window sizes). [sent-207, score-0.308]
75 The key advantage of RTPF over “Uniform” lies in the mixture weighting, which allows our approach to focus computational resources on valuable sensor information, for example when the robot passes an informative feature in one of the hallways. [sent-208, score-0.763]
76 4(a)), this advantage is not very strong since in this environment, most features can be detected in several consecutive sensor measurements. [sent-210, score-0.288]
77 4(a)-(c): Performance of the different algorithms for window sizes of 4, 8, and 12 respectively. [sent-221, score-0.286]
78 The -axis plots the localization error measured in average distance from the reference position. [sent-223, score-0.25]
79 4(d) represents the localization speedup of RTPF over “Skip data” for various window sizes. [sent-227, score-0.501]
80 To see this, consider one estimation window of length . [sent-233, score-0.297]
81 In such a situation, “Uniform” considers this landmark every time the robot passes it. [sent-235, score-0.287]
82 In contrast to both approaches, RTPF detects all landmarks and generates more samples for the landmark detections, thereby gaining the best of both worlds, and Figures 4(a)–(c) show this is indeed the case. [sent-240, score-0.304]
83 & & R & 0R ) & In Figure 4(d) we summarize the performance gain of RTPF over “Skip data” for different window sizes in terms of localization time. [sent-241, score-0.51]
84 We considered the robot to be localized if the average localization error remains below 200 cm over a period of 10 seconds. [sent-242, score-0.47]
85 The -axis represents the window size and the -axis the localization speedup. [sent-244, score-0.45]
86 For each window size speedups were determined using -tests on the localization times for the 30 pairs of data runs. [sent-245, score-0.45]
87 At small window sizes the speedup is 20-50%, but it goes up to 2. [sent-250, score-0.361]
88 7 times for larger windows, demonstrating the benefits of the RTPF approach over traditional particle filters. [sent-251, score-0.454]
89 Ultimately, for very large window sizes, the speedup decreases again, which is due to the fact that none of the approaches is able to reduce the error below 200cm within the run time of an experiment. [sent-252, score-0.371]
90 ¡ ( ¥ 5 Conclusions In this paper we tackled the problem of particle filtering under the constraint of limited computing resources. [sent-253, score-0.481]
91 Our approach makes near-optimal use of sensor information by dividing sample sets between all available observations and then representing the state as a mixture of sample sets. [sent-254, score-0.744]
92 We showed that RTPF produces significant performance improvements in a robot localization task. [sent-257, score-0.398]
93 7 times faster than the original particle filter approach, which skips sensor data. [sent-260, score-0.695]
94 Based on these results, we expect our method to be highly valuable in a wide range of real-time applications of particle filters. [sent-261, score-0.54]
95 RTPF yields maximal performance gain for data streams containing highly valuable sensor data occuring at unpredictable time points. [sent-262, score-0.356]
96 So far RTPF uses fixed sample sizes and fixed window sizes. [sent-266, score-0.376]
97 For example, by the method of [2] we can change the sample size on-the-fly, which in turn allows us to change the window size. [sent-268, score-0.337]
98 Ongoing experiments suggest that this combination yields further performance improvements: When the state uncertainty is high, many samples are used and these samples are spread out over multiple observations. [sent-269, score-0.313]
99 On the other hand, when the uncertainty is low, the number of samples is very small and RTPF becomes identical to the vanilla particle filter with one update (sample set) per observation. [sent-270, score-0.671]
100 Branching and interacting particle systems approximations of feynamkac formulae with applications to non linear filtering. [sent-293, score-0.454]
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