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MIT2_017JF09_p12

# MIT2_017JF09_p12 - 12 RANGING MEASUREMENTS IN THREE-SPACE...

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12 RANGING MEASUREMENTS IN THREE-SPACE 24 12 Ranging Measurements in Three-Space The global positioning system (GPS) and some acoustic instruments provide long-baseline navigation - wherein a number of very long range measurements can be used to triangulate. We call the item that we want to track the Target , and the nodes in the navigation system the Satellites . The locations of the satellites are assumed to be well known, and what we measure during tracking are ranges from the satellites to the target. In a plane, you can appreciate that two satellites would provide two range measurements, and the target could then be located on one of two points that form the intersection of two circles. 1. For a planar setting, how many satellites are required to uniquely locate a target at an arbitrary location, and how these should be laid out? Include sketches as needed to explain your reasoning. Three satellites give three ranges, corresponding with three circles. The intersection of circles is - in the best case - a single point, the unique localization result. There are several notable conditions where you’ll get bad results with three satellites. If they are colinear, you get no information about location in the direction perpendicular to the line. When noise is included (as below), we don’t want to be even close to colinear: we want low aspect-ratio (”not too long and thin”) triangles. More fundamentally, if the target is located on or near the line connecting any two satellites, then it is as if we have lost one satellite - the second range measurement does us very little good. 2. Consider a problem now in three-space. Two satellites are given, with locations [X,Y,Z] of [500 , 500 , 1000] m and [ 500 , 500 , 1000] m . The target ranges are measured at 724 m and 768 . 2 m respectively,

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MIT2_017JF09_p12 - 12 RANGING MEASUREMENTS IN THREE-SPACE...

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