Functional Basics PDF

Basic Principles of Positioning Fabio Dovis Dept. of Electronics and Telecommunications delta t_u is iunknow OUTLINE ▪ Introduction to radionavigation and some historical notes ▪ Classification of positioning systems ▪ Global Navigation Satellite Systems ▪ The PVT solution ▪ The Geometrical Dilution of Precision © Copyright NavSAS 2 HOW PEOPLE USE GNSS… ▪ New challenging applications require the user position and pose strict requirements to the performance double frequency uses more power LOW POWER CONSUMPTION INTEGRATION WITH COMMUNICATION SYSTEMS ACCURACY AND PRECISION © Copyright NavSAS 3 HOW PEOPLE SHOULD USE GNSS… AUTONOMOUS VEHICLES: CARS, TRUCKS, DRONES, …. VEHICULAR NETWORKS PRECISION FARMING EU Agency for Space Programmes - GNSS Market report v6 © Copyright NavSAS 4 WHAT PEOPLE BELIEVE ABOUT GPS… ▪ Pinching the tiny object between his fingers, Langdon pulled it out and stared in astonishment. It was a metallic, button- shaped disk, about the size of a watch battery. He had never seen it before. "What the…?“ ▪ "GPS tracking dot," Sophie said. "Continuously transmits its location to a Global Positioning System satellite that DCPJ can monitor. We use them to monitor people's locations. It's accurate within two feet anywhere on the globe. They have you on an electronic leash. Digita qui il testo © Copyright NavSAS 5 IF DAN BROWN WOULD TEACH GPS… ▪ You would learn that: The GPS tracks people A GPS receiver is a small metal disk that can transmit to a satellite 20000 Km far away GPS works anytime and anywhere, even in the basement or underground floors of the Louvre museum GPS never fails and you can always rely on it GPS is precise within 2 feet (60 cm) even inside a building Dan Brown would never pass an exam of Satellite Navigation Systems ! © Copyright NavSAS 6 WHAT IS A POSITION? © Copyright NavSAS 7 WHERE ARE WE? A position make sense only if it is associated to a reference system © Copyright NavSAS 8 WHO IS CALCULATING THE POSITION? ▪ In mass market terminals, such GNSS antenna in the smartphone as smartphones, the position is provided by a set of sensors ULTRATIGHT Satellite navigation Front end Odometer TIGHT Inertial systems TIGHT Gyroscopes processing Proprioceptive Information from the sensors Compasses communication network Sonar LOOSE … Positioning Unit LOOSE Infra-red using the signal and other sensor Position Exteroceptive (inertial sensor) Ultra-sound Velocity ▪ Quality of the position and sensors navigations also depends on Camera Time Data Data fusion algorithms Ultra-Wide Communication Interface Band “Tricks” applied at Differential processing level (e.g. use of Networks maps) Assistance Data © Copyright NavSAS 9 MAP-MATCHING the position is transferred to the closest road ▪ A 2D position expressed in an absolute reference frame (e.g., longitude and latitude when using a geodetic reference frame), is transferred to a local reference frame on a digital map. ▪ Coordinates are translated in parameters allowing the vehicle to be localized on a road map, such the node number, direction, and distance to the node. ▪ Lanes are generally not modelled, in which case it is impossible to know, from the map-matched position, where the vehicle is in the cross-lane direction. in tunnel propagation of the position, not real position © Copyright NavSAS 10 THE NAVIGATION PROBLEM ▪ The problem of knowing the position with respect to some reference frame or a map ▪ The early navigators and mapmakers relied on celestial observations to determine both time and position on Earth ▪ The science of timekeeping and the advent of clocks allowed for an improvement of navigation (especially at open sea) © Copyright NavSAS 11 LATITUDE AND LONGITUDE λ = 60 deg longitudine + difficile da determinare Observation of a celestial event: A difference of 4 hours = 1/6 of the rotation period= 60 degrees Sextant Warning: the Earth is not a sphere! Accurate measurement of time was needed! © Copyright NavSAS 12 SOME HISTORICAL NOTES ▪ The longitude estimations made comparing the observed map of the sky with the one at the origin harbour and measuring a time difference for the position of the stars. ▪ Time measurements made by the clock on board the ship In 1500 a good watch had an error of 10 minutes a day... ▪ Need to refer to clearly identifiable and predictable celestial events... Galileo (1600): observation of the moons of Jupiter Newton-Halley (1700): observation of the lunar orbits ▪ A discussion that lasted centuries between astronomers and watchmakers on who had the merits or the faults of the navigation problems... © Copyright NavSAS 13 THE LONGITUDE ACT John Harrison competition-> solution of longitude calculation H1 H4 showed an error of 5 seconds on a trip 3 months (1714) long from UK to Jamaica © Copyright NavSAS 14 RADIONAVIGATION PRINCIPLES GPS doesn't work without anatomic clock ▪ Determination of position and speed of a mobile by means of the estimation of parameters of an electromagnetic signal Propagation time Phase Received Signal Strength x2,y2 … ▪ Such parameters are converted into estimated distances with respect to reference points the position of which is known ▪ The position is obtained by the intersection of geometrical loci, x1,y1 x3,y3 ▪ Line of Positions (in 2D) ▪ Volume of Positions (in 3D) © Copyright NavSAS 15 WHO IS ESTIMATING THE POSITION? ▪ Two possible architectures ▪ the reference transceivers localise the user-> localisation ▪ the user self-estimates the position -> positioning x2,y2 Triangulation:based on measured angles Trilateration: based on measured ranges The user can be collaborative or not x1,y1 depending if there are signals exchanged or x3,y3 not between the system and the user The position of the reference beacons is known in a reference system © Copyright NavSAS 16 OUTLINE ▪ Introduction to radionavigation and some historical notes ▪ Classification of positioning systems ▪ GNSS functional principles ▪ The PVT solution ▪ The Geometrical Dilution of Precision © Copyright NavSAS 17 CONICAL SYSTEMS ▪ Sources placed at know locations in a reference system ▪ The position P is obtained by intersection of straight lines (2D) or cones (3D) ▪ The receiver estimates the Angle of Arrival (AOA) ▪ The estimated AOA can be obtained by means of: Antenna arrays Shaped radiation patterns of the antenna Doppler measurements © Copyright NavSAS 18 HYPERBOLIC SYSTEMS ▪ Sources placed at know locations in a reference system ▪ The position is obtained by intersection of hyperbola (2D) or hyperboloids (3D) ▪ The receiver typically measures the Time Difference of Arrival (TDOA) of the signals coming from two sources the phase difference of two carriers ▪ The LORAN system for maritime positioning is based on this principle © Copyright NavSAS 19 HYPERBOLIC SYSTEMS ▪ The position of the user is obtained solving the equation for xu and yu ⇢ p p R2 R1 = p(x2 xu )2 + (y2 yu ) 2 p(x1 xu )2 + (y1 yu ) 2 R3 R1 = (x3 xu )2 + (y3 yu ) 2 (x1 xu )2 + (y1 yu ) 2 ▪ In general, it is easier (and cost-effective) to measure the distance difference with respect to the distance absolute values © Copyright NavSAS 20 SPHERICAL SYSTEMS ▪ Sources placed at know locations in a reference system ▪ The position is obtained by intersection of circles (2D) or spheres (3D) ▪ The receiver evaluates a parameter of the signal incoming from the sources whose value is proportional to the distance Time of Arrival (TOA): the signals must be timestamped with the transmission time Received Signal Strength: the power and the propagation channel must be known ▪ The sources are the center of the spheres and the distance the radius ▪ The position must be inferred by the intersection of at least three spheres © Copyright NavSAS 21 SPHERICAL SYSTEMS BASED ON TOA TWO WAYS MEASUREMENT ▪ reference beacon The round-trip time is measured Transmitters can be not synchronized A precise time reference is not required, it has to be stable for the duration of the measurement “Privacy” problems, the user is seen by the system: they have to cooperate Radars Deep space ranging } based on signals UWB ranging Localisation in mobile networks } based on packet transmission * same principles apply if the system is localising the users © Copyright NavSAS 22 SPHERICAL SYSTEMS BASED ON TOA ONE WAY MEASUREMENT reference beacon The users measure the time using only a received signal Transmitter and user must be synchronised with high precision (within fractions of ns). A misalignment of the time scales of the two clocks generates large errors in distance GNSS systems use a one way method for the measurement of the propagation time How can we keep user and satellite synchronised? © Copyright NavSAS 23 OUTLINE ▪ Introduction to radionavigation and some historical notes ▪ Classification of positioning systems ▪ Global Navigation Satellite Systems ▪ The PVT solution ▪ The Geometrical Dilution of Precision © Copyright NavSAS delta t_u is sconosciuto 24 GLOBAL NAVIGATION SATELLITE SYSTEMS ▪ Global Navigation Satellite Systems provide signals from a constellation of satellites ▪ They aim at providing an almost global coverage on the Earth surface ▪ They are a spherical positioning systems in which all the transmitters (satellites) are synchronised, the user is not synchronised the user measures the time of arrival a signal © Copyright NavSAS 25 GLOBAL NAVIGATION SATELLITE SYSTEMS NAVSTAR Global Positioning System (GPS) Owned and operated by operated by the U.S. Air Force on behalf of the U.S. Government ГЛОНАСС ГЛОбальная НАвигационная Спутниковая Система Owned and managed by the Russian federation (Global'naja Navigacionnaja Sputnikovaja Sistema) Initiative of the European Union (EU) and the European Space Agency (ESA), in collaboration Galileo with European Industries BEIDOU Owned and operated by the Chinese government Other initiatives are ongoing: ๏a quasi GNSS system by Japan ๏Commercial initiatives based on LEO satellites, with possible integration in megaconstellations for communication (LEO-PNT) © Copyright NavSAS 26 GNSS SEGMENTS ▪ space satellites constellation (Low or Medium Earth Orbit) ▪ control tracking stations continuously monitoring the orbital data master station data processing, update orbits and time scale up-loading stations transmit updated data to satellites ▪ users receivers determining their own position, velocity and time © Copyright NavSAS 27 SPACE SEGMENT © Copyright NavSAS 28 SATELLITE CONSTELLATION FROM THE GROUND - SKYPLOT North HORIZON SATELLITE VEHICLE (SV) West East The SV position is provided as ZENITH Azimuth and Elevation in this reference system Different colours identify satellites belonging to different constellations © Copyright NavSAS South 29 EXAMPLES OF REAL SKY PLOTS OPEN SKY URBAN ENVIRONMENT VARIABLE OVER TIME MULTICONSTELLATION Number of visible satellites over Turin on 30/09/2020 Elevation of GPS visible satellites over Turin on 30/09/2020 BeiDou Galileo GLONASS GPS © Copyright NavSAS 30 VISIBILITY AND FIX ▪ The fact that a satellite is theoretically visible does not mean that it is actually used for obtaining the position (fix): presence of obstacles and low quality of the signal a satellite is visible if the receiver is able to “sense” and process the signal broadcasted by such a satellite low signal quality can be due to many effects (that will be described later on), and the receiver might not be able to successfully process the signal receiver software choices some satellite might be excluded by the calculation of the position because of computational limitations or because of a specific statistical assessment that discourage their use in order to not degrade the quality of the estimated position # Satellites used for the fix ≤ # Satellites in view © Copyright NavSAS 31 THE CONTROL SEGMENT ▪ A network of monitoring stations, distributed all around the Earth, checks the status of the satellites and of the signals ▪ A Master control station keeps the control of the time scale GPS Ground Segment ▪ Some ground stations are able to communicate to the satellite in order to Control the satellite Adjust the signal generation Upload data that are broadcast to the users © Copyright NavSAS 32 USER SEGMENT ▪ The user segment is made of a wide range of different receivers, with different performance levels ▪ The receiver estimates the position of the user on the basis of the signals transmitted by the satellites MASS-MARKET PROFESSIONAL © Copyright NavSAS 33 USER SEGMENT ▪ The core functionalities common to any kind of receiver can be summarized as Identification of the satellites in view Estimation of the distance user-satellite Trilateration © Copyright NavSAS 34 USER SEGMENT ▪ Additional functionalities aim at Easing and/or improving the position estimation (aidings and augmentations) Improve the user interface Provide added value services route calculation integration with communication systems … © Copyright NavSAS 35 GETTING FAMILIAR WITH GNSS ▪ Download on your smartphone an application for GNSS ▪ Check which constellations are processed by your smartphone ▪ Check how many satellites are visible (on average) at your place ▪ Visit gnssmissionplanning.com and check the visibility of satellites at - Your location ATES R DIN O - Cape Town - South Africa E CO H KT EC - Rovaniemi CH which constellation provides satellites at the highest elevation? If you include buildings or set a cutoff elevation, how many satellites do you loose from visibility? © Copyright NavSAS 36 OUTLINE ▪ Introduction to radionavigation and some historical notes ▪ Classification of positioning systems ▪ Global Navigation Satellite Systems ▪ The PVT solution ▪ The Geometrical Dilution of Precision © Copyright NavSAS 37 FUNCTIONAL BASICS ▪ Let assume that a satellites transmit a signal at time instant TTX ▪ Such a signal is received by the user at time TRX = TTX + τ ▪ The distance between the transmitter and the receiver can be estimated as: we need two measurements R = c ⋅ (TRX − TTX ) = c ⋅ τ where c is the speed of light ▪ If both the oscillators are synchronised the knowledge of TTX and the measure of TRX = TTX + τ allows to obtain the geometric distance R © Copyright NavSAS delta t_u is iunknown 38 FUNCTIONAL BASICS ▪ The satellites’ payloads host high grade atomic clocks that are very stable in time we assume that all the satellites measure the time respect with the same time reference and time scale synchronous clocks on board of the satellites ▪ It is not possible to have user clocks aligned with the satellite time scale at low cost and complexity delta t of the user ▪ The user clock has a bias δtu with respect to the satellite time ▪ The measurement of TRX is affected by the bias δtu of the user clock ▪ The measured distance is then different from the geometric range and it named as pseudorange ρ = c ⋅ τ + c ⋅ δtu [m] it isn't a geometrical distance, it hasn't physical meaning © Copyright NavSAS 39 FUNCTIONAL BASICS task of control segment-> control delta t_s ▪ The clock of the satellite might be slightly misaligned with respect to the ideal GNSS time scale of a satellite clock bias δt S ▪ δt S is small and stable in time, and anyway known since it is monitored by the control segment ▪ Later we will consider δt S = 0 since we will see how it can be corrected delta t_u cannot be controlled © Copyright NavSAS delta t_u is unknow 40 TIME SCALES δt S trasmission time on the satellite time scaling Satellite time t S = tGNSS + δt S S TTX tS τ GNSS time tGNSS reference TTX tGNSS Receiver time t u = tGNSS + δtu u TRX tu δtu Propagation time: τ u S Pseudo transmit time: TRX − TTX = τ + δtu − δt S but delta t_s can be compensated later Pseudorange: ρ = c ⋅ (τ + δtu − δt S ) © Copyright NavSAS delta t_u is un 41 FUNCTIONAL BASICS ▪ δtu cannot be corrected, so we keep it as an unknown of our positioning problem, and we combine the pseudoranges ▪ The user, by measuring 4 pseudoranges with respect to 4 satellites with known coordinates can determine 4 unknowns: (xu, yu, zu) User coordinates δtu Bias of the user clock with respect to the GNSS time scale © Copyright NavSAS 42 FUNCTIONAL BASICS sphere with the center in the position of the satellite ρ1 = (x1 − xu)2 + (y1 − yu)2 + (z1 − zu)2 + but ρ2 = (x2 − xu)2 + (y2 − yu)2 + (z2 − zu)2 + but TRILATERATION ρ3 = (x3 − xu)2 + (y3 − yu)2 + (z3 − zu)2 + but ρ4 = (x4 − xu)2 + (y4 − yu)2 + (z4 − zu)2 + but solving the problem we get: position and time (xj, yj, zj) Satellite position - center of the pseudo-sphere ρj Pseudorange - radius of the pseudo-sphere but = c ⋅ δtu Range bias due to the clock bias 4 unknowns © Copyright NavSAS we estimate position and time information that are related 43 THE PSEUDORANGE EQUATION we need the position of the satellite It is derived through Satellite position information sent by the core of PVT solution satellites and it is then KNOWN ρj = (xi − xu)2 + (yi − yu)2 + (zi − zu)2 + but Pseudorange - MEASURED - done by the receiver User position - UKNOWN - Bias of the user clock Due to the misalignment of signal-> trasmission time-> receiving time and I can compute the SV and user timescales UNKNOWN © Copyright NavSAS 44 REMARKS ▪ In order to estimate its position a receiver must have at least four satellites in view at least 4 signals ▪ The satellite must be in Line-of-sight the propagation time should be correct, so line of sight in the room the signal could arrive from reflection or others ▪ If a larger number of satellites is in view a better estimation is possible. number of channels ▪ Modern receivers use at least to 12 channels in order to perform the 12 signals at the same time, 12 pseudoranges position estimation Multiconstellation from GPS, Galileo,... Multifrequency © Copyright NavSAS 45 SATELLITES SIGNAL-IN-SPACE ▪ The propagation time is estimated processing a signal transmitted by each satellite the shape of each signal is different from a satellite to another +A ▪ As an example in the baseline civil GPS: -A each satellite transmits over the carrier L1 a binary code - a BPSK modulated sequence named Coarse/Acquisition (C/A) code all satellites use the same carrier frequencies each satellite is identified by a different code (CDMA scheme) © Copyright NavSAS 46 GPS RECEIVER MEASUREMENTS ▪ Code phase measurements: the propagation time between SV and user is estimated measuring the Δt between a local replica of the C/A code and the received SIS looking at the sequence we know which satellite sent that code questo è presente in ricevitori + economici Received satellite signal ρ = cΔt Locally generated signal Δt in this range there is the b_ut CODE PHASE MEASUREMENTS we compare the two signals, we are comparing the phase differences, disallignment ▪ The user clock bias is included in Δt since the clock measures the time on a local time scale that is not synchronised to the GNSS time © Copyright NavSAS delta t_u is 47 GNSS RECEIVER MEASUREMENTS ▪ Carrier phase measurements: the phase difference between a local carrier and the received one is evaluated; better essendo + complicato questo è implementato in ricevitori + costosi resolution if I want measurements of centimeter the number N of integer cycles (Integer Ambiguity) is estimated through proper techniques x(t)*cos(2pi fL1t+ phi)=z(t) trasmitted signal f L1 CARRIER PHASE MEASUREMENTS Nλ ρϕ = (N + ϕ)λ Nλ integer part we introduce another unknown very complex algorithm ϕλ fractional part © Copyright NavSAS 48 THE LINEARISATION PROCESS we consider a clock ▪ The solution of the of non-linear equations can be simplified considering the large distance of the satellites with respect to the users (R ≥ ∼ 20000Km) Reduced computational complexity ▪ The equation of the pseudo-sphere can be linearised, thus transforming the trilateration process in the intersection of the planes tangent to the pseudosphere in the user position approssimazione che prevede l'intersezione dei piani tangenti alla sfera 2 solutions © Copyright NavSAS 49 THE LINEARISATION PROCESS ▪ The solution of the of non-linear equations can be simplified considering the large distance of the satellites with respect to the users (R ≥ ∼ 20000Km) Reduced computational complexity ▪ The equation of the pseudo-sphere can be linearised, thus transforming the trilateration process in the intersection of the planes tangent to the pseudosphere in the user position © Copyright NavSAS 50 THE NAVIGATION SOLUTION ▪ The generic pseudorange f(x)=f(x0)+ [df(x)/dx|in x=x0]*(x-x0) x0 is 4 dimensions point ρj = (xi − xu)2 + (yi − yu)2 + (zi − zu)2 + but ▪ Can be approximated through the Taylor expansion around a known location/time with coordinates (xu0, yu0, zu0, but0 ) (xi − xu ) + (yi − yu ) + (zi − zu ) 0 2 0 2 0 2 ρj0 = + but0 * it is not a restriction to choose but0 = 0 © Copyright NavSAS 51 THE LINEARIZATION SCENARIO that one that we measure ⇢j ρj0 that one that we evaluate xu 0 xu APPROXIMATED TRUE POSITION KNOWN POSITION (1 DIMENSION) (1 DIMENSION) is unknown xu © Copyright NavSAS 52 LINEARISATION PROCESS ▪ We can then write the Taylor expansion truncated at the first order for the multidimensional equation as: ∂ρj ∂ρj ∂ρj ∂ρj ρj = ρj0 + (xu − xu0) + (yu − yu0) + (zu − zu0) + (but − but0 ) ∂xu ∂yu ∂zu ∂but xu=xu0 yu=yu0 zu=zu0 but=but0 © Copyright NavSAS 53 THE NAVIGATION SOLUTION ▪ Then, at a first order approximation, if we define Δρj = ρj0 − ρj and Δxu = xu − xu0 Δyu = yu − yu0 Δzu = zu − zu0 Δbut = but − but0 ▪ we obtain the linearised equation which is function of the displacement with respect to the approximation point Δρj = axjΔxu. + ayjΔyu + azjΔzu − Δbut © Copyright NavSAS 54 THE NAVIGATION SOLUTION ▪ The coefficients are: xj − xu0 yj − yu0 zj − zu0 axj = , ayj = , azj = rj0 rj0 rj0 ▪ and range between the satllite and the approximation point rj0 = (xj − xu0)2 + (yj − yu0)2 + (zj − zu0)2 ▪ is the geometrical distance between the linearization point and the satellite Homework: derive the linearised equation and prove the values of the coefficients above © Copyright NavSAS 55 THE LINEARIZATION SCENARIO centered in the approximation point aj = (axj, ayj, azj) unitary vectors steering from the approximation point towards the j-th satellite ⇢j ρj0 the equation depends on the position of the satellite the performance of the pVT solution depends on the relative position of satellite and receiver aj 0 xu xu APPROXIMATED TRUE POSITION KNOWN POSITION (1 DIMENSION) (1 DIMENSION) xu © Copyright NavSAS 56 THE NAVIGATION SOLUTION Δρ1 = ax1Δxu + ay1Δyu + az1Δzu − Δbut ATTENZIONE i satelliti devono Δρ2 = ax2Δxu + ay2Δyu + az2Δzu − Δbut essere lungo direzioni diverse Δρ3 = ax3Δxu + ay3Δyu + az3Δzu − Δbut Δρ4 = ax4Δxu + ay4Δyu + az4Δzu − Δbut delta_x=H^(-1)*delta_rho matrix must be squared, deterimant diverso da 0 le linee non sono linearmnete indipendenti quando i due satelliti sono lungo la stessa direzione Δρ1 ax1 ay1 az1 1 Δxu Δρ2 ax2 ay2 az2 1 Δyu Δρ = H= Δx = Δρ3 ax3 ay3 az3 1 Δzu Δρ4 ax4 ay4 az4 1 −Δbut matrix H geometrical matrix MEASUREMENTS it contains: the first row related to satellite 1, the second to 2 and so on STATE VECTOR Δρ = HΔx © Copyright NavSAS 57 THE NAVIGATION SOLUTION ▪ In case four satellite are used Δx = H−1Δρ dal momento che è difficile ottenere 4 satelliti lungo direzioni diverse prendo più misurazioni possibili ▪ If a larger number of satellite is used ax1 ay1 az1 1 ax2 ay2 az2 1 H= ⋮ ⋮ ⋮ ⋮ axn ayn azn 1 © Copyright NavSAS 58 THE NAVIGATION SOLUTION ▪ For n>4 a least square solution must be used ▪ The solution is given by the value of Δx that minimizes the square of the residual: 2 the minimum squared error RSE ( x) = (H x ⇢) ▪ The solution can be obtained differentiating with respect to Δx to obtain the gradient of RSE. T T T rRSE = 2 ( x) H H 2 ( ⇢) H © Copyright NavSAS 59 THE NAVIGATION SOLUTION ▪ The gradient is set to zero and solved for Δx to seek a minimum value ▪ Taking the transpose and setting it to zero: T T 2H H ( x) 2H ( ⇢) = 0 ▪ Provided that (HT H)−1 is non-singular, the solution is : T 1 T x= H H H ⇢ the approximation point is kind of irrelevant pseudoinverse ▪ A GNSS receiver solves the equation by recursive methods or a Kalman filter © Copyright NavSAS delta t_u is 60 GENERALISATION OF THE PROBLEM ▪ measurement vector ρ, a noisy version of the true ρ = r(x) + n ranging function vector r that depends on the state x and on specific ranging technique ▪ TOA, TDOA, FOA (Doppler), FDOA ▪ The problem is linearised around an approximation point x0 taking the Jacobian of r(x) the byas is one of our dimensions of the problems © Copyright NavSAS 61 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sum of the elements on the diagonal from 1 to huge numbers in our applications GDOP between 1 and 2, maximum 10 © Copyright NavSAS 73 THE GEOMETRICAL PROBLEM ▪ The impact of the pseudorange error on the final estimated position depends on the displacement of the satellites (reference points) the output of the estimation process is an area: the area in which i can have t he uncertainty with a given probability better geometrical distribution of the other, because the second one are 'more parallel' than the first one © Copyright NavSAS 74 DILUTION OF PRECISION ▪ Partial factors can be defined: Position Dilution of Precision p PDOP = g11 + g22 + g33 Time Dilution of Precision p TDOP = g44 Horizontal Dilution of Precision p HDOP = g11 + g22 © Copyright NavSAS 75 REMARKS ▪ Root Mean Square Error σx = σx2u + σy2u + σz2u + σb2ut = GDOP ⋅ σUERE ▪ The RMSE σx is the radius of a hypersphere in 4 dimensions ▪ The error is actually distributed in un-even way in the 4-d space and only the entire Cx̂ describes it behaviour ▪ The RMSE is independent of the specific reference system used ▪ this derives from the properties of the trace, which is invariant to translations © Copyright NavSAS 76 REMARKS ON THE GDOP ▪ The minimum value of DOP is 1 if you are looking at the best 4 satellites. ▪ However this may not necessarily be the case if you are considering all satellites in view. The nominal error in the calculated position is a function of the relative satellite geometry and the satellite pseudorange errors. If you include a large number of satellites in the GDOP calculation then the accuracy of the calculated position can improve correspondingly and it possible that the GDOP may be reduced to less than 1 in this case. Situations where this could be true would be: High elevation vehicles. Multiple constellations enabled. Multiple SV's in view. © Copyright NavSAS 77 REAL GDOP AND ERROR BEHAVIOUR The error grows proportionally to the value of GDOP in each time #Satellites instant GDOP grows when the number of satellites ti dice se una decreases H is bad conditioned posizione GDOP è affidabile There is no solution when the number of satellites is less than 4 Note: satellites may either Error [m] error is order of be not visible due to 5m obstructions or the receiver didn’t acquire them code based measurements with 3 satellites no solution GNSS to guide spacecraft, the problem is that the satellites are all in the same direction © Copyright NavSAS 78 LMS WITH NOISY MEASUREMENTS ▪ The derivation so far obtained is not general and works under specific assumptions ▪ In the real world such conditions are seldom satisfied ▪ The position estimation process can be applied to other positioning systems using different kind of measurements (e.g. Doppler) ▪ The error is characterised by the error covariance matrix σρ21 σρ21,ρ2 … σρ21,ρN T σρ22,ρ1 σρ22 … σρ22,ρN Cδρ = E{δρδρ } = ⋮ ⋮ ⋮ ⋮ ▪ σρ2N ,ρ1 … … σρ2N © Copyright NavSAS 79 LMS WITH NOISY MEASUREMENTS ̂ 2 ▪ The Least Square finds x̂ which minimises the residuals: R = ∥ỹ − y∥ ▪ note that the residual depends on the measured - the estimated and it has not to be ̂ 2 confused with the measurement error e = ∥ȳ − y∥ ▪ The receiver can obtain the residual after the position is estimated (with uncertainty) ▪ In the general case the navigation solution is the result of the estimate (Gauss Markov theorem)Δρ = HΔx + n where n is a N × 1 noise vector. Δx = (H Cδρ H) HT C−1 −1 T −1 ▪ the Best Linear Unbiased Estimator (BLUE) is: δρ Δρ ▪ and the covariance matrix of δx is Cδx = (H δρ H) −1 T C−1 ▪ the minimum variance of xi (an element of the vector of unknowns) is var(xi) = [(HT C−1 −1 δρ H) ]ii ▪ If the pseudorange measurements are truly Gaussian, then the BLUE is also the Minimum Variance Unbiased (MVU) estimator © Copyright NavSAS 80 WEIGHTED SOLUTION ▪ If the measurements are uncorrelated but have different uncertainties, the BLUE considers “weights” equal to the reciprocal of the variance of the measurement Δx = (HT C−1 δρ H) −1 HT C−1 δρ Δρ σ12 0 … … 1 0 … … σ12 WEIGHTS Cδρ = 0 σ22 0 … 1 ⋱ 0 0 … W= C−1 δρ = σ22 … … 0 σN2 ⋱ 1 … … 0 σN2 ▪ The best estimator give less relevance to the measurements that are more noisy © Copyright NavSAS 81 WEIGTHED GDOP ▪ The covariance matrix is Cδx = (HT WH) −1 ▪ In the literature we can find the definition of a weighted GDOP defined as trace {(H WH) } T −1 WGDOP = ▪ However, this is not a pure geometrical factor since it involves the variance of the measurements ▪ The WGDOP is equal to the RMSE of the solution (trace of Cδx) l'altezza è la coordinata + difficile da calcolare © Copyright NavSAS 82

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