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

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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
Design of Electromechanical Robotic Systems Documents
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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 measurement as a sphere in three space the intersection of two spheres is a circle Without more information you can t say what is the x z location although the y location is easy to get 3 Augment these two satellites with third having position 500 500 1000 m the cor responding range measurement is 649 8m Can you make an estimate for the target location in three space If so give it Yes we have enough information now to do a full localization See the attached code for a numerical approach Note that at least one student gured out the answer through algebraic manipulation despite the fact that the solution satis es three nonlinear equa tions The location of the target is 33 51 950 m There is another solution that will satisfy the three range equations 33 51 1050 m it is the mirror through the plane of the satellites 4 Almost all sensors have noise and such range based navigation systems are no ex ception What is the sensitivity of your best computed target location above to a 12 RANGING MEASUREMENTS IN THREE SPACE 25 one meter error in each of the three range measurements Perturb the range measurements separately Give an explanation for what you see happening using sketches Adding one meter onto each of the three ranges in turn gives solutions of 32 3 51 7 950 2 m 33 8 51 0 943 3 m and 33 0 50 3 943 3 m respectively The worst error here is about 6 7m pretty bad for a one meter range error It occurs because the target is fairly close to the plane of the satellites as alluded to in the planar case these measurements provide only poor quality information about the direction normal to the plane and it does not stand up to noise 5 Find a location for a fourth satellite that will bring down the sensitivity of the localization to noisy range measurements Demonstrate that the whole system is now better behaved with noise and explain why using sketches if necessary Since the problem is that the target is close to the plane of satellites a logical solution is to put a satellite far from the plane Indeed if we put one at 0 0 0 m we create a very nice tetrahedron like shape and the errors from perturbations in all four channels go to 32 3 51 7 950 0 m 33 4 51 4 949 9 m 32 7 50 7 950 0 m 33 0 51 1 951 0 m The positioning error now is on a par with the range error and this system is considerably more robust against noise In the algorithm you notice that we are here solving four equations in three unknowns As written this is an over determined problem and we are performing a least squares fit the sum of squares of the range equation errors is minimized I use the fact that the given ranges are Euclidean distances The MATLAB command fminsearch synonymous with fmins or fsolve in some versions does multi variable function minimization well enough for this problem For this use as your error function to be minimized the sum of the squared errors of all the range equations These questions are in the flavor of the geometric dilution of precision GDOP problem in GPS navigation the performance we get from the system is very strongly related to the physical arrangement of the satellites and target Try the Wikipedia page on GDOP Three dimensional Ranging MIT 2 017 FSH Sept 2009 clear all global X Y Z R X 500 500 500 0 Satellite locations in Cartesian space Y 500 500 500 0 Z 1000 1000 1000 0 12 RANGING MEASUREMENTS IN THREE SPACE x 33 y 51 z 950 true target location for i 1 length X calculate the ranges R i sqrt X i x 2 Y i y 2 Z i z 2 end plot the layout of sensors figure 1 clf hold off for i 1 length X plot3 X i X i Y i Y i 0 Z i hold on plot3 X i Y i Z i o LineWidth 2 end plot3 x x y y 0 z r plot3 x y z rs LineWidth 2 grid xlabel x m ylabel y m zlabel z m apply per channel range errors to see what happens to the estimates R 4 R 4 1 posCalc fminsearch rangeError 0 0 0 search for the minimum error solution err rangeError posCalc disp sprintf Solution Consistency Error get the consistency error for this solution g err here s the error with respect to the true solution disp sprintf Solution Error wrt Actual g norm posCalc x y z 2 add the points to the plot xCalc posCalc 1 yCalc posCalc 2 zCalc posCalc 3 plot3 xCalc xCalc yCalc yCalc 0 zCalc m plot3 xCalc yCalc zCalc m LineWidth 2 text xCalc 60 yCalc zCalc Target Estimate function err rangeError pos …


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