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Not sure I can add much to the slides you included in your question, but maybe I can simplify and rephrase. Here goes:

GMapping solves the Simultaneous Localization and Mapping (SLAM) problem. Unlike, say Karto, it employs a Particle Filter (PF), which is a technique for model-based estimation. In SLAM, we are estimating two things: the map and the robot's pose within this map. You can kind of think of each particle in the PF as a candidate solution to the problem. Together, the set of particles approximates the true probability distribution (see Importance Sampling). To clarify, I'm talking about the probability of the map and the robot's pose given the control inputs (e.g. motor encoder counts) and sensor readings (e.g. LiDAR). In addition, there's a motion model and a sensor model involved in the calculation of the probability distribution.

GMapping is much more sophisticated than the outline I gave above. If you are interested, check out one of the research papers describing the approach: TRO 2007 I also highly recommend the chapter on SLAM in the book Probabilistic Robotics.