diff --git a/src/python/mlat-test.py b/src/python/mlat-test.py new file mode 100755 index 0000000..3bb5e73 --- /dev/null +++ b/src/python/mlat-test.py @@ -0,0 +1,7 @@ +#!/usr/bin/python +import mlat +import numpy + +ans = mlat.mlat(mlat.teststations, mlat.teststamps, mlat.testalt) +error = numpy.linalg.norm(numpy.array(mlat.llh2ecef(ans))-numpy.array(mlat.testplane)) +print error diff --git a/src/python/mlat.py b/src/python/mlat.py new file mode 100755 index 0000000..1ce063c --- /dev/null +++ b/src/python/mlat.py @@ -0,0 +1,218 @@ +#!/usr/bin/python +import math +import numpy +from scipy.ndimage import map_coordinates + +#functions for multilateration. + +#NB: because of the way this solver works, at least 3 stations and timestamps +#are required. this function will not return hyperbolae for underconstrained systems. +#TODO: get HDOP out of this so we can draw circles of likely position and indicate constraint + +##########################NOTES######################################### +#you should test your solver with a reference dataset that you've calculated. let's put one together. +#let's say we have you here in SF, ER in MV, and some place in Fremont. the plane can be at SFO at 24000'. +#your pos: 37.76225, -122.44254, 300' +#ettus: 37.409044, -122.077748, 300' +#fremont: 37.585085, -121.986395, 300' +#airplane: 37.617175,-122.380843, 24000' + +#calculated using a third-party tool and verified against the algorithms in this file: +#you: -2708399, -4260759, 3884677 +#ettus: -2693916, -4298177, 3853611 +#fremont: -2680759, -4292379, 3869113 +#airplane: -2712433, -4277271, 3876757 + +#here's how I did it in Octave... +#prange_est = ((stations(:, 1) - xguess(1)).^2 + (stations(:, 2) - xguess(2)).^2 + (stations(:,3) - xguess(3)).^2).^0.5; +#dphat = prange_obs - prange_est; +#H = [[-(stations(:,1)-xguess(1)) ./ prange_est, -(stations(:,2)-xguess(2)) ./ prange_est, -(stations(:,3)-xguess(3)) ./ prange_est]]; +#xerr = (H'*H\H'*dphat)'; +#xguess = xguess + xerr +#remember the last line of stations is the 0 vector. remember the last line of prange_obs is the height of the a/c above geoid. + +#use one of the station positions as a starting guess. calculate alt-above-geoid once as though it were located directly above the starting guess; 300 miles +#is like 4 degrees of latitude and i don't think it'll introduce more than a few meters of error. + +#well, there are some places where 4 degrees of latitude would count for 20 meters of error. but since our accuracy is really +/- 250m anyway it's probably not +#a big deal. saves a lot of CPU to only do the lookup once. + +#so, the above solver works for pseudorange, but what if you only have time-of-flight info? +#let's modify the prange eq to deal with time difference of arrival instead of pseudorange +#knowns: time of arrival at each station, positions of each station, altitude of a/c +#from this, you can say "it arrived x ns sooner at each of the other stations" +#thus you can say "it's this much closer/farther to station y than to me" + +#the stations vector is RELATIVE TO YOU -- so it's [ettus-me; fremont-me; [0,0,0]-me] +#prange_obs is a vector of TDOAs; the first value (earliest) is truncated since the difference is zero (just like stations) +#this implies the TDOAs should arrive sorted, which is probably a good idea, or at the very least the closest station should be first + +#prange_est = [norm(stations(1,:)-xguess); +# norm(stations(2,:)-xguess); +# norm(stations(3,:)-xguess)]; #only valid for the three-station case we're testing Octave with +#dphat = prange_obs - prange_est; +#H = [[-(stations(:,1)-xguess(1)) ./ prange_est, -(stations(:,2)-xguess(2)) ./ prange_est, -(stations(:,3)-xguess(3)) ./ prange_est]]; +#xguess += (H'*H\H'*dphat)'; +#err=norm(airplane-(xguess+me)) #just for calculating convergence + +#it converges for 500km position error in the initial guess, so it seems pretty good. +#seems to converge quickly in the terminal phase. 250m timing offset gives 450m position error +#250m time offset in the local receiver (830ns) gives 325m position error + +#the last question is how to use the altitude data in prange_obs; calculate height above geoid for YOUR position at a/c alt and use that to get height above [0,0,0] +#this will be close enough. you could iterate this along with the rest of it but it won't change more than the variation in geoid height between you and a/c. +#maybe after convergence you can iterate a few more times with the a/c geoid height if you're worried about it. +#there's probably a way to use YOU as the alt. station and just use height above YOU instead. but this works. + +########################END NOTES####################################### + + +#this is a 10x10-degree WGS84 geoid datum, in meters relative to the WGS84 reference ellipsoid. given the maximum slope, you should probably interpolate. +#NIMA suggests a 2x2 interpolation using four neighbors. we'll go cubic spline JUST BECAUSE WE CAN +wgs84_geoid = numpy.array([[13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13], #90N + [3,1,-2,-3,-3,-3,-1,3,1,5,9,11,19,27,31,34,33,34,33,34,28,23,17,13,9,4,4,1,-2,-2,0,2,3,2,1,1], #80N + [2,2,1,-1,-3,-7,-14,-24,-27,-25,-19,3,24,37,47,60,61,58,51,43,29,20,12,5,-2,-10,-14,-12,-10,-14,-12,-6,-2,3,6,4], #70N + [2,9,17,10,13,1,-14,-30,-39,-46,-42,-21,6,29,49,65,60,57,47,41,21,18,14,7,-3,-22,-29,-32,-32,-26,-15,-2,13,17,19,6], #60N + [-8,8,8,1,-11,-19,-16,-18,-22,-35,-40,-26,-12,24,45,63,62,59,47,48,42,28,12,-10,-19,-33,-43,-42,-43,-29,-2,17,23,22,6,2], #50N + [-12,-10,-13,-20,-31,-34,-21,-16,-26,-34,-33,-35,-26,2,33,59,52,51,52,48,35,40,33,-9,-28,-39,-48,-59,-50,-28,3,23,37,18,-1,-11], #40N + [-7,-5,-8,-15,-28,-40,-42,-29,-22,-26,-32,-51,-40,-17,17,31,34,44,36,28,29,17,12,-20,-15,-40,-33,-34,-34,-28,7,29,43,20,4,-6], #30N + [5,10,7,-7,-23,-39,-47,-34,-9,-10,-20,-45,-48,-32,-9,17,25,31,31,26,15,6,1,-29,-44,-61,-67,-59,-36,-11,21,39,49,39,22,10], #20N + [13,12,11,2,-11,-28,-38,-29,-10,3,1,-11,-41,-42,-16,3,17,33,22,23,2,-3,-7,-36,-59,-90,-95,-63,-24,12,53,60,58,46,36,26], #10N + [22,16,17,13,1,-12,-23,-20,-14,-3,14,10,-15,-27,-18,3,12,20,18,12,-13,-9,-28,-49,-62,-89,-102,-63,-9,33,58,73,74,63,50,32], #0 + [36,22,11,6,-1,-8,-10,-8,-11,-9,1,32,4,-18,-13,-9,4,14,12,13,-2,-14,-25,-32,-38,-60,-75,-63,-26,0,35,52,68,76,64,52], #10S + [51,27,10,0,-9,-11,-5,-2,-3,-1,9,35,20,-5,-6,-5,0,13,17,23,21,8,-9,-10,-11,-20,-40,-47,-45,-25,5,23,45,58,57,63], #20S + [46,22,5,-2,-8,-13,-10,-7,-4,1,9,32,16,4,-8,4,12,15,22,27,34,29,14,15,15,7,-9,-25,-37,-39,-23,-14,15,33,34,45], #30S + [21,6,1,-7,-12,-12,-12,-10,-7,-1,8,23,15,-2,-6,6,21,24,18,26,31,33,39,41,30,24,13,-2,-20,-32,-33,-27,-14,-2,5,20], #40S + [-15,-18,-18,-16,-17,-15,-10,-10,-8,-2,6,14,13,3,3,10,20,27,25,26,34,39,45,45,38,39,28,13,-1,-15,-22,-22,-18,-15,-14,-10], #50S + [-45,-43,-37,-32,-30,-26,-23,-22,-16,-10,-2,10,20,20,21,24,22,17,16,19,25,30,35,35,33,30,27,10,-2,-14,-23,-30,-33,-29,-35,-43], #60S + [-61,-60,-61,-55,-49,-44,-38,-31,-25,-16,-6,1,4,5,4,2,6,12,16,16,17,21,20,26,26,22,16,10,-1,-16,-29,-36,-46,-55,-54,-59], #70S + [-53,-54,-55,-52,-48,-42,-38,-38,-29,-26,-26,-24,-23,-21,-19,-16,-12,-8,-4,-1,1,4,4,6,5,4,2,-6,-15,-24,-33,-40,-48,-50,-53,-52], #80S + [-30,-30,-30,-30,-30,-30,-30,-30,-30,-30,-30,-30,-30,-30,-30,-30,-30,-30,-30,-30,-30,-30,-30,-30,-30,-30,-30,-30,-30,-30,-30,-30,-30,-30,-30,-30]], #90S + dtype=numpy.float) + +#ok this calculates the geoid offset from the reference ellipsoid +#combined with LLH->ECEF this gets you XYZ for a ground-referenced point +def wgs84_height(lat, lon): + yi = numpy.array([9-lat/10.0]) + xi = numpy.array([18+lon/10.0]) + return float(map_coordinates(wgs84_geoid, [yi, xi])) + +#WGS84 reference ellipsoid constants +wgs84_a = 6378137.0 +wgs84_b = 6356752.314245 +wgs84_e2 = 0.0066943799901975848 +wgs84_a2 = wgs84_a**2 #to speed things up a bit +wgs84_b2 = wgs84_b**2 + +#convert ECEF to lat/lon/alt without geoid correction +#returns alt in meters +def ecef2llh((x,y,z)): + ep = math.sqrt((wgs84_a2 - wgs84_b2) / wgs84_b2) + p = math.sqrt(x**2+y**2) + th = math.atan2(wgs84_a*z, wgs84_b*p) + lon = math.atan2(y, x) + lat = math.atan2(z+ep**2*wgs84_b*math.sin(th)**3, p-wgs84_e2*wgs84_a*math.cos(th)**3) + N = wgs84_a / math.sqrt(1-wgs84_e2*math.sin(lat)**2) + alt = p / math.cos(lat) - N + + lon *= (180. / math.pi) + lat *= (180. / math.pi) + + return [lat, lon, alt] + +#convert lat/lon/alt coords to ECEF without geoid correction, WGS84 model +#remember that alt is in meters +def llh2ecef((lat, lon, alt)): + lat *= (math.pi / 180.0) + lon *= (math.pi / 180.0) + + n = lambda x: wgs84_a / math.sqrt(1 - wgs84_e2*(math.sin(x)**2)) + + x = (n(lat) + alt)*math.cos(lat)*math.cos(lon) + y = (n(lat) + alt)*math.cos(lat)*math.sin(lon) + z = (n(lat)*(1-wgs84_e2)+alt)*math.sin(lat) + + return [x,y,z] + +#do both of the above to get a geoid-corrected x,y,z position +def llh2geoid((lat, lon, alt)): + (x,y,z) = llh2ecef((lat, lon, alt + wgs84_height(lat, lon))) + return [x,y,z] + + +c = 299792458 / 1.0003 #modified for refractive index of air, why not + +#here's some test data to validate the algorithm +teststations = [[37.76225, -122.44254, 100], [37.409044, -122.077748, 100], [37.585085, -121.986395, 100]] +testalt = 8000 +testplane = numpy.array(llh2ecef([37.617175,-122.380843, testalt])) +testme = llh2geoid(teststations[0]) +teststamps = [10, + 10 + numpy.linalg.norm(testplane-numpy.array(llh2geoid(teststations[1]))) / c, + 10 + numpy.linalg.norm(testplane-numpy.array(llh2geoid(teststations[2]))) / c, + ] + +#this function is the iterative solver core of the mlat function below +def mlat_iter(rel_stations, prange_obs, xguess = [0,0,0], numrounds = 10): + for i in range(0,numrounds): + prange_est = [] + for station in rel_stations: + prange_est.append([numpy.linalg.norm(station - xguess)]) + dphat = prange_obs - prange_est + H = [] + for row in range(0,len(rel_stations)): + H.append((numpy.array(-rel_stations[row,:])-xguess) / prange_est[row]) + H = numpy.array(H) + #now we have H, the Jacobian, and can solve for residual error + xerr = numpy.dot(numpy.linalg.solve(numpy.dot(H.T,H), H.T), dphat).flatten() + xguess += xerr + return xguess + +#func mlat: +#uses a modified GPS pseudorange solver to locate aircraft by multilateration. +#stations is a list of listening station positions in X,Y,Z ECEF format, geoid corrected +#timestamps is a list of times at which the correlated squitters were heard +#altitude is the barometric altitude of the aircraft as returned by the aircraft +#returns the estimated position of the aircraft in (lat, lon, alt) geoid-corrected WGS84. +def mlat(stations, timestamps, altitude): + if len(timestamps) != len(stations): + raise Exception("Must have x timestamps for x stations reporting!") + + me_llh = stations[0] + me = llh2geoid(stations[0]) + + rel_stations = [] #list of stations in XYZ relative to me + for station in stations[1:]: + rel_stations.append(numpy.array(llh2geoid(station)) - numpy.array(me)) + rel_stations.append([0,0,0]-numpy.array(me)) #arne saknussemm, reporting in + rel_stations = numpy.array(rel_stations) #convert list of arrays to 2d array + + tdoa = [] + for stamp in timestamps[1:]: + tdoa.append(stamp - timestamps[0]) + + prange_obs = [] + for stamp in tdoa: + prange_obs.append([c * stamp]) + + #so here we calc the estimated pseudorange to the center of the earth, using station[0] as a reference point for the geoid + #this is a necessary approximation since we don't know the location of the aircraft yet + #if the dang earth were actually round this wouldn't be an issue + prange_obs.append( [numpy.linalg.norm(llh2ecef((me_llh[0], me_llh[1], altitude)))] ) #use ECEF not geoid since alt is MSL not GPS + #prange_obs.append( [numpy.linalg.norm(testplane)]) #test for error + prange_obs = numpy.array(prange_obs) + + xyzpos = mlat_iter(rel_stations, prange_obs) + llhpos = ecef2llh(xyzpos+me) + + #now, we could return llhpos right now and be done with it. + #but the assumption we made above, namely that the aircraft is directly above the + #nearest station, results in significant error due to the oblateness of the Earth's geometry. + #so now we solve AGAIN, but this time with the corrected pseudorange of the aircraft altitude + #this might not be really useful in practice but the sim shows >50m errors without it + prange_obs[-1] = [numpy.linalg.norm(llh2ecef((llhpos[0], llhpos[1], altitude)))] + + xyzpos_corr = mlat_iter(rel_stations, prange_obs, xyzpos) #start off with a really close guess + llhpos = ecef2llh(xyzpos_corr+me) + + return llhpos