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I've read the VFH paper and now reading the VFH+. Odometry does not seem to be necessary, but I'm kind of confused about the grid and how to track how much distance the robot has traveled between samples.

Should I read sensor readings and calculate distances to obstacles without taking into account the distance traveled?

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No. VFH is a method to identify the next heading direction, which, from the current information available seem to be the best choice.

It is independent of how the current position is determined and how the information about the surroundings of the robot is required. As far as I know, both VFH and VFH+ are independent of distance travelled. The only depend on the current surroundings and target position, so, to answer your question regarding sensor readings: yes, the Histogram for VHF is build up using obstacle distances relative to the current position of the robot, so there is no need to calculate the distance travelled.

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