200 lines
11 KiB
Python
Executable File
200 lines
11 KiB
Python
Executable File
#!/usr/bin/python
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import math
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import numpy
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from scipy.ndimage import map_coordinates
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#functions for multilateration.
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#this library is more or less based around the so-called "GPS equation", the canonical
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#iterative method for getting position from GPS satellite time difference of arrival data.
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#here, instead of multiple orbiting satellites with known time reference and position,
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#we have multiple fixed stations with known time references (GPSDO, hopefully) and known
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#locations (again, GPSDO).
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#NB: because of the way this solver works, at least 3 stations and timestamps
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#are required. this function will not return hyperbolae for underconstrained systems.
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#TODO: get HDOP out of this so we can draw circles of likely position and indicate constraint
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########################END NOTES#######################################
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#this is a 10x10-degree WGS84 geoid datum, in meters relative to the WGS84 reference ellipsoid. given the maximum slope, you should probably interpolate.
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#NIMA suggests a 2x2 interpolation using four neighbors. we'll go cubic spline JUST BECAUSE WE CAN
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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
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[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
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[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
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[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
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[-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
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[-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
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[-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
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[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
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[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
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[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
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[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
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[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
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[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
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[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
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[-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
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[-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
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[-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
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[-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
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[-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
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dtype=numpy.float)
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#ok this calculates the geoid offset from the reference ellipsoid
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#combined with LLH->ECEF this gets you XYZ for a ground-referenced point
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def wgs84_height(lat, lon):
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yi = numpy.array([9-lat/10.0])
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xi = numpy.array([18+lon/10.0])
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return float(map_coordinates(wgs84_geoid, [yi, xi]))
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#WGS84 reference ellipsoid constants
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wgs84_a = 6378137.0
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wgs84_b = 6356752.314245
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wgs84_e2 = 0.0066943799901975848
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wgs84_a2 = wgs84_a**2 #to speed things up a bit
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wgs84_b2 = wgs84_b**2
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#convert ECEF to lat/lon/alt without geoid correction
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#returns alt in meters
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def ecef2llh((x,y,z)):
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ep = math.sqrt((wgs84_a2 - wgs84_b2) / wgs84_b2)
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p = math.sqrt(x**2+y**2)
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th = math.atan2(wgs84_a*z, wgs84_b*p)
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lon = math.atan2(y, x)
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lat = math.atan2(z+ep**2*wgs84_b*math.sin(th)**3, p-wgs84_e2*wgs84_a*math.cos(th)**3)
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N = wgs84_a / math.sqrt(1-wgs84_e2*math.sin(lat)**2)
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alt = p / math.cos(lat) - N
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lon *= (180. / math.pi)
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lat *= (180. / math.pi)
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return [lat, lon, alt]
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#convert lat/lon/alt coords to ECEF without geoid correction, WGS84 model
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#remember that alt is in meters
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def llh2ecef((lat, lon, alt)):
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lat *= (math.pi / 180.0)
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lon *= (math.pi / 180.0)
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n = lambda x: wgs84_a / math.sqrt(1 - wgs84_e2*(math.sin(x)**2))
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x = (n(lat) + alt)*math.cos(lat)*math.cos(lon)
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y = (n(lat) + alt)*math.cos(lat)*math.sin(lon)
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z = (n(lat)*(1-wgs84_e2)+alt)*math.sin(lat)
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return [x,y,z]
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#do both of the above to get a geoid-corrected x,y,z position
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def llh2geoid((lat, lon, alt)):
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(x,y,z) = llh2ecef((lat, lon, alt + wgs84_height(lat, lon)))
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return [x,y,z]
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c = 299792458 / 1.0003 #modified for refractive index of air, why not
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#this function is the iterative solver core of the mlat function below
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#we use limit as a goal to stop solving when we get "close enough" (error magnitude in meters for that iteration)
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#basically 20 meters is way less than the anticipated error of the system so it doesn't make sense to continue
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#it's possible this could fail in situations where the solution converges slowly
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#TODO: this fails to converge for some seriously advantageous geometry
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def mlat_iter(rel_stations, prange_obs, xguess = [0,0,0], limit = 20, maxrounds = 100):
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xerr = [1e9, 1e9, 1e9]
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rounds = 0
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while numpy.linalg.norm(xerr) > limit:
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prange_est = [[numpy.linalg.norm(station - xguess)] for station in rel_stations]
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dphat = prange_obs - prange_est
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H = numpy.array([(numpy.array(-rel_stations[row,:])+xguess) / prange_est[row] for row in range(0,len(rel_stations))])
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#now we have H, the Jacobian, and can solve for residual error
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xerr = numpy.linalg.lstsq(H, dphat)[0].flatten()
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xguess += xerr
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#print xguess, xerr
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rounds += 1
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if rounds > maxrounds:
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raise Exception("Failed to converge!")
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break
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return xguess
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#func mlat:
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#uses a modified GPS pseudorange solver to locate aircraft by multilateration.
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#replies is a list of reports, in ([lat, lon, alt], timestamp) format
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#altitude is the barometric altitude of the aircraft as returned by the aircraft
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#returns the estimated position of the aircraft in (lat, lon, alt) geoid-corrected WGS84.
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#let's make it take a list of tuples so we can sort by them
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def mlat(replies, altitude):
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sorted_replies = sorted(replies, key=lambda time: time[1])
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stations = [sorted_reply[0] for sorted_reply in sorted_replies]
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timestamps = [sorted_reply[1] for sorted_reply in sorted_replies]
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me_llh = stations[0]
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me = llh2geoid(stations[0])
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#list of stations in XYZ relative to me
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rel_stations = [numpy.array(llh2geoid(station)) - numpy.array(me) for station in stations[1:]]
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rel_stations.append([0,0,0] - numpy.array(me))
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rel_stations = numpy.array(rel_stations) #convert list of arrays to 2d array
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#differentiate the timestamps to get TDOA, multiply by c to get pseudorange
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prange_obs = [[c*(stamp-timestamps[0])] for stamp in timestamps[1:]]
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#so here we calc the estimated pseudorange to the center of the earth, using station[0] as a reference point for the geoid
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#in other words, we say "if the aircraft were directly overhead of station[0], this is the prange to the center of the earth"
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#this is a necessary approximation since we don't know the location of the aircraft yet
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#if the dang earth were actually round this wouldn't be an issue
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prange_obs.append( [numpy.linalg.norm(llh2ecef((me_llh[0], me_llh[1], altitude)))] ) #use ECEF not geoid since alt is MSL not GPS
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prange_obs = numpy.array(prange_obs)
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#xguess = llh2ecef([37.617175,-122.400843, 8000])-numpy.array(me)
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#xguess = [0,0,0]
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#start our guess directly overhead, who cares
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xguess = numpy.array(llh2ecef([me_llh[0], me_llh[1], altitude])) - numpy.array(me)
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xyzpos = mlat_iter(rel_stations, prange_obs, xguess)
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llhpos = ecef2llh(xyzpos+me)
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#now, we could return llhpos right now and be done with it.
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#but the assumption we made above, namely that the aircraft is directly above the
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#nearest station, results in significant error due to the oblateness of the Earth's geometry.
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#so now we solve AGAIN, but this time with the corrected pseudorange of the aircraft altitude
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#this might not be really useful in practice but the sim shows >50m errors without it
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#and <4cm errors with it, not that we'll get that close in reality but hey let's do it right
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prange_obs[-1] = [numpy.linalg.norm(llh2ecef((llhpos[0], llhpos[1], altitude)))]
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xyzpos_corr = mlat_iter(rel_stations, prange_obs, xyzpos) #start off with a really close guess
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llhpos = ecef2llh(xyzpos_corr+me)
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#and now, what the hell, let's try to get dilution of precision data
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#avec is the unit vector of relative ranges to the aircraft from each of the stations
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# for i in range(len(avec)):
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# avec[i] = numpy.array(avec[i]) / numpy.linalg.norm(numpy.array(avec[i]))
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# numpy.append(avec, [[-1],[-1],[-1],[-1]], 1) #must be # of stations
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# doparray = numpy.linalg.inv(avec.T*avec)
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#the diagonal elements of doparray will be the x, y, z DOPs.
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return llhpos
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if __name__ == '__main__':
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#here's some test data to validate the algorithm
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teststations = [[37.76225, -122.44254, 100], [37.680016,-121.772461, 100], [37.385844,-122.083082, 100], [37.701207,-122.309418, 100]]
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testalt = 8000
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testplane = numpy.array(llh2ecef([37.617175,-122.400843, testalt]))
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testme = llh2geoid(teststations[0])
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teststamps = [10,
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10 + numpy.linalg.norm(testplane-numpy.array(llh2geoid(teststations[1]))) / c,
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10 + numpy.linalg.norm(testplane-numpy.array(llh2geoid(teststations[2]))) / c,
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10 + numpy.linalg.norm(testplane-numpy.array(llh2geoid(teststations[3]))) / c,
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]
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print teststamps
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replies = []
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for i in range(0, len(teststations)):
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replies.append((teststations[i], teststamps[i]))
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ans = mlat(replies, testalt)
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error = numpy.linalg.norm(numpy.array(llh2ecef(ans))-numpy.array(testplane))
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range = numpy.linalg.norm(llh2geoid(ans)-numpy.array(testme))
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print testplane-testme
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print ans
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print "Error: %.2fm" % (error)
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print "Range: %.2fkm (from first station in list)" % (range/1000)
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