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Efficient Histogram-Based Localization in Probabilistic Robotics

This document presents a comprehensive approach to histogram localization, a fundamental method in probabilistic robotics, as outlined by Sebastian Thrun and Alex Teichman. The paper discusses the discrete Bayesian Filter Algorithm for belief update during sensory input and actions, emphasizing the use of bounded Gaussian models to optimize computational efficiency. Key considerations include monitoring delocalization and utilizing grid-based and tree-based representations for efficient state space management. This algorithm effectively combines probabilistic modeling with practical implementation challenges in robotic navigation.

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Efficient Histogram-Based Localization in Probabilistic Robotics

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  1. Probabilistic Robotics: Historgam Localization Sebastian Thrun & Alex Teichman Stanford Artificial Intelligence Lab Slide credits: Wolfram Burgard, Dieter Fox, Cyrill Stachniss, Giorgio Grisetti, Maren Bennewitz, Christian Plagemann, Dirk Haehnel, Mike Montemerlo, Nick Roy, Kai Arras, Patrick Pfaff and others

  2. Bayes Filters in Localization

  3. Histogram =Piecewise Constant

  4. Piecewise Constant Representation

  5. Discrete Bayes Filter Algorithm • Algorithm Discrete_Bayes_filter( Bel(x),d ): • h=0 • Ifd is a perceptual data item z then • For all x do • For all x do • Else ifd is an action data item uthen • For all x do • ReturnBel’(x)

  6. Implementation (1) • To update the belief upon sensory input and to carry out the normalization one has to iterate over all cells of the grid. • Especially when the belief is peaked (which is generally the case during position tracking), one wants to avoid updating irrelevant aspects of the state space. • One approach is not to update entire sub-spaces of the state space. • This, however, requires to monitor whether the robot is de-localized or not. • To achieve this, one can consider the likelihood of the observations given the active components of the state space.

  7. + 1/16 1/16 1/4 1/8 1/4 1/2 1/4 1/8 1/8 1/2 1/4 1/16 1/16 1/4 1/8 Implementation (2) • To efficiently update the belief upon robot motions, one typically assumes a bounded Gaussian model for the motion uncertainty. • This reduces the update cost from O(n2) to O(n), where n is the number of states. • The update can also be realized by shifting the data in the grid according to the measured motion. • In a second step, the grid is then convolved using a separable Gaussian Kernel. • Two-dimensional example: • Fewer arithmetic operations • Easier to implement

  8. Markov Localizationin Grid Map

  9. Grid-based Localization

  10. Sonars and Occupancy Grid Map

  11. Tree-based Representation Idea: Represent density using a variant of octrees

  12. Tree-based Representations • Efficient in space and time • Multi-resolution

  13. Xavier: Localization in a Topological Map

  14. Summary:Discrete Markov Localization Most basic probabilistic localization algortihm Handles global uncertainty (e.g., global localization) Handles local uncertainty (e.g., tracking) Scales exponentially with number of dimensions; requires further algorithmic work for efficiency

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