Squirrels have fuzzy tails.
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I don't know your time frame but you can do visual odometry with the web cam...Do you have an IMU (or are you on a flat floor where you can make assumptions that the roll and pitch are constant, and that heading is found by looking at past GPS points), if so you can just align each sonar scan to overlap the previous scan based from the distance traveled and your heading.This way you can close the loop by using GPS and if your ambitious you can do some scan matching on your map to confirm that you actually closed the loop.
HiSo an IMU will give you your RPY (roll,pitch,yaw) that you will need to acuratly patch togethor the scans. However they can be pricey. I f you can ignore the effects of the rocks you might just want to use a digital compass for your heading.
You hinted at the $$$ for IMU? Couldn't one just buy a couple of cheap accelerometers and a gyro and mock up the IMU? How much are they exactly $$$-$$$$?