# MIT 2 017J - Ranging Measurements in Three-Space (5 pages)

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**View the full content.**## Ranging Measurements in Three-Space

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## Ranging Measurements in Three-Space

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- Pages:
- 5
- School:
- Massachusetts Institute of Technology
- Course:
- 2 017j - Design of Electromechanical Robotic Systems

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12 RANGING MEASUREMENTS IN THREE SPACE 12 24 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 724m and 768 2m respectively and suppose we know also that the target is at an altitude z of less than 1000m Can you make an estimate for the target location x y z in three space If so give it No you cannot make an estimate because you only have two constraints Think of each range

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