Calibration of Magnetic Field Sensors with two mass-market GNSS receivers

Transcription

1 Calibration of Magnetic Field Senor with two ma-market GNSS receiver Patrick Henkel *,**, Philipp Berthold ** and Jane Jean Kiam ** * echniche Univerität München, Munich, Germany ** Advanced Navigation Solution - ANavS, Munich, Germany patrickhenkel@tumde, philippberthold@anavde, janekiam@anavde Abtract Global Navigation Satellite Sytem GNSS ignal, inertial meaurement angular rate, acceleration and magnetometer meaurement complement each other for poition determination: GNSS provide a precie and drift-free poition olution but i uceptible to ignal outage Inertial meaurement are continuouly available and of higher rate but uffer from integration drift Magnetic field meaurement provide an intantaneou orientation in tatic condition but are affected by both tatic and dynamic diturbance In thi paper, we provide a calibration method for magnetometer, which determine the biae and mialignment error of the magnetometer a well a the magnetic flux including tatic diturbance he method ue the iterative Gau-Newton method and precie attitude information heading, pitch obtained from two low-cot GPS receiver he attitude determination require a tree earch of the carrier phae integer ambiguitie uing a priori information on the ditance between both GPS receiver We alo verified the propoed method with kinematic meaurement from the CMPS1 enor We oberve an accuracy of a few degree for the unfiltered heading and a heading offet of le than 1 in 995% of all meaurement epoch Index erm Magnetic field enor, Attitude determination, Integer Ambiguity Reolution, Calibration I INRODUCION GPS receiver can track the carrier phae of GPS ignal with millimeter to centimeter-level accuracy Low-cot GPS receiver are becoming attractive in numerou application We focu on attitude determination for future driving aitance ytem A GPS ignal are weak and eaily blocked by hadowing, additional enor are required to provide a continuou olution Inertial enor are attractive to upport the GPS olution, ie they can be ued to coat the GPS olution during hort ignal outage eg paing below bridge or tree, to detect and correct cycle lip, to accelerate the ambiguity re-fixing after hort ignal outage, and to increae the data rate However, inertial enor are not beneficial for integer ambiguity reolution of tanding vehicle hi initial integer ambiguity reolution can lat a minute for low-cot GPS receiver a code multipath i often in the order of everal ten of metre Magnetometer can provide an initial orientation of the vehicle and, thereby, ubtantially horten the time of initial GPS carrier phae integer ambiguity reolution However, magnetometer need an initial calibration Craidi et al [1] and Piaki et al [2] ued extended and uncented Kalman filter to etimate the enor biae and caling factor In thi paper, we provide a calibration method which ue precie GPS-baed attitude information to determine mialignment error of the magnetometer, biae of the magnetic flux meaurement, and the magnetic flux including tatic diturbance We alo apply our method to three-dimenional magnetic flux meaurement of a low-cot CMPS1 enor and two lowcot u-blox LEA 6 GPS receiver, that were mounted on the roof of a vehicle during a tet drive We oberve an error of only a few degree for the calibrated magnetic heading, which i provided with a meaurement rate of 1 Hz II MEASUREMEN MODELS In thi ection, we introduce meaurement model for the magnetometer and for the double difference code and carrier phae meaurement of two low-cot GPS receiver A Meaurement model for magnetic field enor We ue the CMPS1 enor of Devantech, which provide three-dimenional magnetic flux meaurement a raw data We model thee magnetic flux meaurement in the enorfixed - frame centered in the enor and aligned with the principal axe of the enor at time t i a m xt i m yt i m zt i =ΛRnφt i +Δφ, θt i +Δθ, ψt i +Δψ m n t i +b + η m t i, 1 with the caling factor matrix Λ, the rotation matrix Rn from the navigation n- frame centered in the enor and aligned with the Eat, North and Up-axi into the -frame, the roll, pitch and heading angle {φ, θ, ψ} of the vehicle, the mialignment error {Δφ, Δθ, Δψ}, the tatic diturbance/ biae b of the magnetic field meaurement, and the noie η m including dynamic diturbance A the magnetic flux mainly point in Northern direction the magnetic declination i aigned to the mialignment error and a the caling factor i negligible for mot magnetic field enor, the general model of Eq 1 implifie Furthermore, the roll angle i negligible in automotive application uch that Eq 1 become m xt i = coθt i +Δθ inψt i +Δψ m N t i +b x + η mx t i m yt i = coψt i +Δψ m N t i +b y + η my t i 2

2 and m zt i = inθt i +Δθ inψt i +Δψ m N t i +b z + η mz t i, 3 which leave the attitude {ψ, θ}, the attitude mialignment error {Δψ, Δθ}, the biae {b x,b y,b z }, and the magnitude of the magnetic flux m N t i in northern direction a unknown B Meaurement model for GPS meaurement In thi ubection, we decribe our model for double difference DD meaurement of two low-cot GPS receiver baed on [3] and [4] he model need to take the particularitie of thee receiver into account, ie the lack of ynchronization, code multipath and λ/2 cycle lip [5] We model the double difference carrier phae meaurement for the receiver {1, 2} and atellite {k, l} a φ kl t i = e kl t i b t i +c kl t i + λn kl + λ/2δnt kl i +ε t φ kl i, 4 with the atellite-atellite ingle difference e kl = e k e l of the atellite-receiver line of ight vector, the unknown baeline vector b between two GPS receiver, the ynchronization correction c kl correcting for the receiver and atellite movement within the differential receiver clock offet, the wavelength λ, the unknown DD integer ambiguity N kl Z, the unknown DD cycle lip ΔN kl Z and the DD phae noie including multipath We ue a imilar model for the DD peudorange meaurement, ie ρ kl t i = e kl t i b t i +c kl t i +Δρ kl MP,t i +ε t ρ kl i, 5 with Δρ kl MP, being the unknown DD code multipath and ε ρ kl denoting the DD code noie A both GPS receiver are mounted on the roof of a vehicle, the length l = b can be aumed to be contant We aume that there i a baeline length meaurement available, which we model a with ε l l = b t i + ε l, 6 N,σ 2 l decribing the error of the baeline length meaurement III INEGER AMBIGUIY RESOLUION WIH LOW-COS GPS RECEIVERS In thi ection, we determine the DD carrier phae ambiguitie by integer leat-quare etimation uing baeline length a priori information A tatic baeline i conidered and DD phae and code meaurement from n = 8 epoch are ued to achieve ufficient redundancy he baeline vector b and ambiguitie N are determined by minimizing the weighted um of quared meaurement reidual and of quared baeline length reidual, ie min b,n H b AN 2 Σ 1 + b l 2 /σ 2 l, 7 with the DD carrier phae and peudorange meaurement = λφ,ρ of n epoch, ie the carrier phae λφ = λφ t 1,,λφ t n and ρ = ρ t 1,,ρ t n, the DD integer ambiguitie N from all available DD with a common reference atellite, and the mapping matrice H and A We apply a DD ynchronization correction to retore the integer property of DD ambiguitie, ie λφ t i = ρ t i = λφ t i c t i λφ 1K t i c 1K t i ρ t i c t i ρ 1K t i c 1K t i 8 he minimization of Eq 7 provide an optimized trade-off between minimizing the quared meaurement reidual and minimizing the quared baeline length reidual he firt term of Eq 7 can be further developed with eunien orthogonal error decompoition [6] [7] [8], ie min b,n P H H b AN 2 Σ 1 + P AP H AN 2 Σ 1 + b l 2 /σ 2 l + P A P H 2 Σ 1, 9 where P H i the projector on the pace of H, PH i the projector on it orthogonal complement, and A = PH Ahe firt and econd term can be alo expreed in term of the float ambiguity olution = A Σ 1 1 A A Σ 1, the fixed baeline etimate ˇ b =H Σ 1 H 1 H Σ 1 A ˇN, and the repective covariance matrice Σ and Σˇb A the minimization over N can not be olved in cloed form, a earch i required We earch for the bet candidate N inide a predefined earch pace volume χ 2, ie N 2 Σ 1 + P A P H 2 Σ 1 +min b ˇ b N b 2 Σ 1 ˇb + b l 2 /σ 2 l χ 2 1 We will now implify the notation by omitting the index for the reference atellite in all DD meaurement and ambiguitie eunien [6] ued the conditional ambiguity etimate = k 1 k σ k j 1,,j 1/σ 2 j 1,,j 1 k 1,,k 1 j=1 j 1,,j 1 [ to rewrite the firt term of Eq 1 a N 2 Σ 1 = K k=1 k 1,,k 1 j 1,,j 1 ], 11 N k 2 σ 2, k 1,,k 1

3 with σ 2 k 1,,k 1 being the variance of the conditional float ambiguity etimate Replacing the quared ambiguity reidual of Inequality 1 by and olving the quared inequality for N k yield a lower and an upper bound for the tree earch: N k N k with the multiplier k 1 κ k = χ 2 min b j=1 k 1,,k 1 κ k σ k 1,,k 1 k 1,,k 1 + κ k σ k 1,,k 1 j 1,,j 1 ˇN j σ 2 j 1,,j 1 2 P A P H 2 Σ 1 ˇ b N b 2 Σ 1 ˇb + b l 2 /σ 2 l IV HEADING DEERMINAION WIH LOW-COS GPS RECEIVERS In thi ection, we determine the heading of the vehicle baed on the fixed baeline etimate We aume that the baeline between the two GPS receiver i aligned with longitudinal axi of vehicle he heading i in northern direction and counted clock-wie In thi cae, the baeline vector i given by b t i =l inψt i coθt i coψt i coθt i inθt i, 15 with the baeline length l, the heading ψ and pitch θ of the vehicle hu, the heading and pitch angle etimate follow from the fixed baeline etimate by imple trigonometric propertie a ˆψt i = atan b x t i / b y t i 16 ˆθt i = atan b z t i / b 2 xt i + b 2 yt i 17 V CALIBRAION OF MAGNEIC FIELD SENSOR In thi ection, we provide a method for calibration of magnetic field enor with two low-cot GPS receiver, ie we take the heading ψ and pitch θ from the GPS carrier phae fixed attitude olution and conider it a true Let x be the vector of unknown, ie x =Δθ, Δψ, b x,b y,b z,m N t 1,,m N t n, 18 then we obtain a leat-quare etimate of x by minimizing the quared magnetic flux reidual, ie ˆx =argmin m m x 2, 19 x with m being the meaured magnetic flux and m being the computed magnetic flux meaurement of all epoch, ie m = m t 1 m t n, m = m t 1 m t n, 2 where m t i and m t i are given by m t i = m xt i,m yt i,m zt i m t i = coθt i+δθinψt i +Δψm N t i +b x coψt i +Δψm N t i +b y inθt i +Δθinψt i +Δψm N t i +b z 21 Obviouly, there i ome change in attitude required to eparate the mialignment error, magnetic flux and biae and, thu, to perform the calibration A Eq 19 repreent a non-linear leat-quare problem, we perform a linearization and olve it iteratively with the Gau-Newton method A aylor erie expanion i performed up to order 1 around the tate etimate ˆx j 1 of the previou iteration j 1 he linearized m at iteration j i modeled a m = m x=x j 1 + J x=x j 1 x x j 1 +η m, 22 with the Jacobian matrix J = J 1,J 2 It conit of two part, ie of one part including the partial derivative wrt the mialignment error and biae given by J 1 = Δθ Δψ b x b y b z, and of a econd part including the partial derivative wrt the magnetic flux for every epoch, ie J 2 = Nt 1 Nt n 23 he partial derivative can be eaily obtained from Eq 21, eg the entrie of the firt column and firt three row of J 1 are given by inθt t 1 1 +Δθinψt 1 +Δψm N t 1 = Δθ coθt 1 +Δθinψt 1 +Δψm N t 1 24 he leat-quare etimate of the tate parameter at the j-th iteration i then given by ˆx j =ˆx j 1 + J 1 x=ˆx J j 1 x=ˆx j 1 J x=ˆx m m j 1 25 Obviouly, the iterative Gau-Newton method need a every iterative method an initialization of the tate vector We aume only for the initialization that the biae b x, b y and b z are zero and that there are no mialignment error Δθ and Δψ In thi cae, m N t i can be derived directly from the meaurement a m N t i= m xt i 2 +m yt i 2 +m zt i 2 26 hu, the tate vector can be initialized with ˆx = 1 2+3,m N t 1,,m N n t 27

4 VI MEASUREMEN RESULS In thi ection, we verify the propoed calibration method with peudorange and carrier phae meaurement from two 5 Hz u-blox LEA 6 GPS receiver and three-dimenional magnetic flux meaurement from one CMPS1 enor he GPS receiver were connected to two L1 patch antenna that were mounted on the roof of the vehicle together with the CMPS1 enor he baeline length between both GPS receiver wa 14 m Fig 1 how the track of the vehicle during the calibration at Nymphenburg Catle, Munich, Germany Meaurement m y m y m y ime [epoch] 5 Hz Fig 3 Comparion of meaured and etimated magnetic flux in y-direction: he etimated magnetic flux follow again the track of the meaured magnetic flux, which indicate the correctne of the meaurement model Fig 1 rack of vehicle: he driving of numerou circle i ideally uited for calibration of magnetometer he tarting point of the track i marked by a red point A Calibration of magnetometer In thi ubection, we verify the propoed meaurement model for the CMPS1 magnetometer with real meaurement, apply the propoed GPS-baed calibration method of magnetometer, and how the achievable accuracy of the magneticfield baed heading Fig 2-4 how a comparion of the meaured and etimated magnetic flux meaurement baed on Eq 21 he etimated magnetic flux cloely follow the track of the meaured magnetic flux, which indicate the correctne of the meaurement model he z-component of the magnetic flux i contant for mot of the time, and only how ome ocillation during the harp turn Meaurement m z ime [epoch] 5 Hz Fig 4 Comparion of meaured and etimated magnetic flux in z-direction: he z-component of the magnetic flux i contant for mot of the time During the harp turn, the magnetic flux ocillate to a certain extent he noie level of the meaured magnetic flux i in the order of a few degree m z m z Meaurement m x m x m x ime [epoch] 5 Hz Fig 2 Comparion of meaured and etimated magnetic flux in x-direction: he etimated magnetic flux cloely follow the track of the meaured magnetic flux, which indicate the correctne of the meaurement model Fig 5 include a comparion of the GPS-baed heading and of the magnetic field-baed heading after calibration he noie level of the magnetic field baed heading i in the order of only a few degree, which make the magnetometer very attractive to upport GPS integer ambiguity reolution he noie level of the GPS baed heading i in the order of only 2 and, thu, can be ued to analyze the performance of the magnetometer Fig 6 how the difference between the GPS baed and the magnetic field baed heading etimate One can oberve an almot zero-mean noie of a few degree within the firt 8 epoch he offet increae around epoch to 15, which indicate ome local magnetic diturbance In total, the difference i le than 5 in 926% and le than 1 in 995% of all epoch B Heading determination with calibrated magnetometer In thi ection, we ue the calibration correction for teting the performance of the magnetic field baed heading with a longer data et 85 epoch Fig 7 how the obtained heading etimate in comparion to the GPS-baed heading he firt enlarged area how that the heading can deviate by

5 Heading [deg] Heading [deg] heading baed on 2 low cot u blox 2 LEA 6 GPS receiver heading baed on magnetic field enor ime [epoch] 5 Hz heading baed on 3 2 low cot u blox LEA 6 GPS receiver 35 heading baed on magnetic field enor ime [epoch] 5 Hz Fig 5 Comparion of GPS-baed and magnetic-flux baed heading etimate: he noie level of the magnetic-field baed heading i in the order of a few degree he GPS-baed heading how a noie level in the order of only 2 and, thu, erve a a reference Fig 7 Challenge of heading determination with calibrated magnetometer: he firt enlarged area how heading error of up to 4 during dynamic diturbance, while the econd enlarged area how that the noie level i ignificantly larger than in cae of GPS 2 5 Difference in heading etimate [deg] Difference in heading etimate [deg] calibration phae coating phae, ie application of calibration correction ime [epoch] 5 Hz ime [epoch] 5 Hz Fig 6 Difference in heading etimate between GPS and magnetometer: he difference i le than 5 in 926%and le than 1 in 995%of all epoch hu, the magnetometer can faten ambiguity reolution Fig 8 Difference in heading etimate baed on two u-blox LEA 6 GPS receiver and the heading baed on the calibrated CMPS1 magnetometer: he offet remain le than 1 in 954%of all epoch up to 4 in cae of dynamic diturbance eg croing below tram power cable, while the econd enlarged area how a comparion of both noie level: he noie level i in the order of 2 for GPS and in the order of a few degree for the magnetometer Fig 8 how the difference between both heading etimate: he offet remain le than 5 in 781% and le than 1 in 954% of all epoch VII CONCLUSION In thi paper, we preented a method to calibrate magnetometer with the help of two GPS receiver he method firt determine the attitude with GPS which implie a tree earch of the carrier phae integer ambiguitie Subequently, the mialignment error, meaurement biae and magnetic flux including tatic diturbance are computed with the iterative Gau-Newton method he propoed method wa teted with two low-cot GPS receiver and a magnetic field enor in a tet drive he calibrated heading differed by le than 1 in 995% of all epoch from the GPS-baed heading REFERENCES [1] J Craidi, K-L Lai and R Harman, Real-time attitude-independent three axi magnetometer calibration, J of Guidance, Control, and Dynamic, vol 28, no 1, pp 115-, 25 [2] M Piaki, F Martel and P Pal, hree-axi attitude determination via Kalman filtering of magnetometer data, Journal of Guidance, Control, and Dynamic, vol 13, no 3, pp , 199 [3] P Henkel and J Kiam, Maximum A Poteriori Probability Etimation of Integer Ambiguitie and Baeline, Proc of 55-th IEEE Symp ELMAR, pp , Zadar, Croatia, Sep 213 [4] P Henkel and C Günther, Attitude determination with low-cot GPS/ INS, Proc of the 26-th Intern echn Meeting of the Satellite Diviion of the Intitute of Navigation ION GNSS+, Nahville, N, USA, pp , Sep 213 [5] P Henkel and N Oku, Cycle lip detection and correction for heading determination with low-cot GPS/ INS receiver, IAG Proc of VIII Hotine Marui Sympoium, Rome, Italy, Jun 213 [6] P eunien, he leat-quare ambiguity decorrelation adjutment: a method for fat GPS ambiguity etimation, J of Geodey, vol 7, pp 65-82, 1995 [7] P eunien, he LAMBDA method for the GNSS compa, Artificial Satellite, vol 413, pp 89-13, 26 [8] P eunien, Integer leat-quare theory for the GNSS compa, J of Geodey, vol 84, pp , 21