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Efficient Implementation of Discrete Bayesian Filters for Probabilistic Robotics

This document outlines the Discrete Bayesian Filter algorithm for probabilistic robotics, focusing on piecewise constant representations and their efficient implementation. It details the process for updating beliefs based on perceptual and action data, emphasizing the importance of modular state space updates during localization tasks. The use of bounded Gaussian models for motion uncertainty leads to significant computation reductions. Techniques for grid-based localization and tree-based representations are discussed, providing insights into multi-resolution approaches for efficient state representation.

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Efficient Implementation of Discrete Bayesian Filters for Probabilistic Robotics

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  1. Probabilistic Robotics Bayes Filter Implementations Discrete filters

  2. Piecewise Constant

  3. 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)

  4. Piecewise Constant Representation

  5. 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.

  6. + 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

  7. Grid-based Localization

  8. Sonars and Occupancy Grid Map

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

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

  11. Xavier: Localization in a Topological Map [Courtesy of Reid Simmons]

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