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Paper Discussion: “Simultaneous Localization and Environmental Mapping with a Sensor Network”, Marinakis et. al. ICRA 2011
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Outline
• Problem:– Simultaneous mapping and localisation in static,
continuous and smooth field
• Solution– Expectation Maximisation (EM)
• Implementation – Grid-based representation of all PDFs– In simulation and practical
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Contributions
• Claim: Use of smoothly varying parameters in the environment to aid localization
• Simultaneous mapping of continuous field (with uncertainty) and localisation of sensors.
• Interesting idea, but implementation does not fully take advantage of continuous field
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Background: Expectation Maximisation
• Maximum likelihood estimator
• Two steps in each iteration– Expectation – compute likelihood of observations with
current model
– Maximisation – using likelihood of observations, maximise likelihood of model parameters
• Also used as Maximum a Posteriori estimator– How this paper uses EM
– Maximisation step uses MAP rather than ML
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Background: Expectation Maximisation
• Example: fitting Gaussian mixture models– Problem
• Inputs: set of data points, number of Gaussians in mixture• Outputs: weights, means and covariances of each Gaussian• Weights must sum to 1.0
– Expectation• Compute likelihood of each point being in
each Gaussian
– Maximisation• Update weights, means and covariances
based on likelihoods using “frequentist”definition
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Notation
• = sensor pose(s) – Grid representation of domain– Probability of occupancy represented as grid– = prior
• = model parameters– Grid representation of domain– Environmental parameter(s) represented by
(multivariate) Gaussian at each cell – = estimate of model parameters
• = observations of environmental parameters– Vector of measurements of environmental
parameter(s)
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Discussion
• Considers static sensors– A motion model can be incorporated in Expectation
step
• Grid representation of world– Continuous representation of world– Continuous representation of sensor network cost
• Communications cost