Tightly Coupled Precise Point Positioning and Attitude Determination

Transcription

1 I. INTRODUCTION Tightly Coupled Pecise Point Positioning and Attitude Detemination PATRICK S. HENKEL Technische Univesität München Munich, Gemany A method is descibed fo joint pecise point positioning and attitude detemination with tight coupling of two single-fequency low-cost global navigation satellite system eceives and an inetial senso. The senso fusion is pefomed with an extended Kalman filte. The caie phase ambiguities ae detemined in a constained tee seach using soft a pioi infomation on the antenna distance. A code multipath paamete is detemined fo each satellite to impove the accuacy. Ionospheic coections ae estimated also by single-fequency eceives. Manuscipt eceived July 29, 204; evised Febuay, 205, June 27, 205; eleased fo publication June 28, 205. DOI. No. 0.09/TAES Refeeeing of this contibution was handled by A. Dempste. Autho s addess: Technische Univesität München, Electical Engineeing and Infomation Technology, Institute fo Communication and Navigation, Theesienstasse 90, Munich, Bavaia 80333, Gemany, patic.henel@tum.de) /5/$26.00 C 205 IEEE Both pecise point positioning PPP) [ 2] and attitude detemination have eceived a lot of attention ecently. Teunissen intoduced an aay-aided PPP in [3] to pefom PPP and attitude detemination jointly. Thus, the coelation intoduced by using global positioning system GPS) measuements fo both PPP and attitude detemination is exploited. A tight coupling of global navigation satellite system GNSS) and inetial navigation system INS) eceives fo position and attitude detemination has been analyzed in [4 6]. The coupling is attactive because both sensos ae complementay GNSS povides an unbiased absolute position and attitude infomation, while INS povides high-ate angula ate and acceleation measuements, which ae obust to the envionment. Howeve, cuent methods fo tight coupling do not focus on PPP and ae not suited fo low-cost mass-maet GNSS eceives. The lac of pecise synchonization, a code multipath of seveal tens of metes, and fequent half- and full-cycle slips that might affect multiple satellites simultaneously) have to be taen into account. The phase noise of mass-maet GNSS eceives is, howeve, in the ode of a few millimetes, such that a position accuacy compaable to geodetic eceives can still be obtained if the paticula eo souces of mass-maet GNSS eceives ae popely addessed. This pape povides a joint PPP and attitude detemination method with tight coupling of measuements fom two low-cost mass-maet GNSS eceives and an inetial measuement unit IMU). Recent wo on satellite bias detemination fo PPP used a two-step pocedue see Gabo and Neem [7], Ge et al. [8], and Lauichesse et al. [9]): Fist, factional wide-lane biases ae deived fom the geomety-fee, ionosphee-fee Melboune Wübbena combination to enable wide-lane intege ambiguity esolution. Subsequently, eceive and satellite cloc offsets, satellite phase biases, topospheic zenith delays, station coodinate coections, satellite obit coections, and caie phase intege ambiguities ae deived in a Kalman filte [0] using a netwo of GNSS eceives. As the ionosphee-fee combination and peviously esolved wide-lane ambiguities ae used at this step, naow-lane ambiguities have to be esolved. Lauichesse et al. analyzed the stability of the obtained satellite phase biases and obseved that these biases can be assumed constant on a daily basis but might vay by up to 2 naow-lane cycles ove a yea. Li et al. [ 2] and Ge et al. [3] included egional augmentation coections fo atmospheic eos to fasten undiffeenced intege ambiguity fixing. Zhang et al. [4] estimated esidual ionospheic eos, topospheic zenith delays, satellite cloc offsets, satellite phase biases lumped with absolute ambiguities of one efeence station), and double diffeence DD) intege ambiguities in a Kalman filte with a egional netwo of 382 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 5, NO. 4 OCTOBER 205

2 dual-fequency GNSS eceives. They obseved vey stable satellite phase bias estimates with a vaiation of less than 0. cycles ove 24 h. The obtained coections wee used fo PPP and eal-time inematic RTK) navigation. The absolute position was detemined with an uncetainty of.3 cm, 0.9 cm, and 5.0 cm in the noth, east and up diection, espectively. Odij et al. [5] consideed a egional netwo of geodetic dual-fequency GNSS eceives. They fist esolved the DD intege ambiguities and subsequently estimated the satellite cloc offsets, satellite phase biases, and intepolated ionospheic delays. A hypothesis testing was pefomed by analyzing the stability of the phase biases. The satellite phase bias time seies pe ac and the espective vaiances wee used to conclude that the biases wee stable. Howeve, the hypothesis testing elied on the coectness of the covaiance matix, which might not be the case. Odij et al. also veified thei coections, i.e., the absolute position, and ambiguities wee detemined fo a low-cost eceive using the peviously detemined satellite clocs, phase biases, and intepolated ionospheic coections. Odij distinguished between two types of PPP: a standad PPP using ionospheic coections fom global ionospheic maps, and an impoved PPP/RTK using a dense egional continuously opeating efeence station CORS) netwo to geneate moe pecise ionospheic coections with Kiging intepolation. Fo the latte one, a positioning accuacy of a few millimetes was obtained fo low-cost u-blox eceives. This accuacy was achieved unde ideal conditions; i.e., the code multipath was negligible, and the measuements wee taen in the sola minimum with low ionospheic vaiations Octobe 200). It shall also be noted that a low-cost GNSS eceive was only used fo PPP but not fo estimation of the satellite phase bias estimates. If a netwo of low-cost GNSS eceives wee used fo satellite bias detemination, an additional synchonization would be needed fo each measuement. Odij et al. futhe impoved the PPP accuacy in [6] by descibing the stochastic natue of the netwo coections by a covaiance matix and by using this covaiance matix fo modeling the uncetainty of the coected measuements. A closed-fom expession was povided fo the covaiance matix. The benefit of the impoved measuement covaiance matix was analyzed with low-cost u-blox eceives: The time fo ambiguity esolution was educed to 4 min, and a position accuacy of 0.25 m, 0.9 m, and 0.39 m was achieved fo PPP in east, noth, and up diection, espectively, and a position accuacy of 5.6 mm, 7. mm and 20 mm, espectively, was achieved fo PPP-RTK. It has to be noted that these accuacies wee again achieved unde ideal conditions, i.e., negligible code multipath, minimum sola activity, and no movement, which maes cycle slip detection easy. A senso fusion of GNSS and INS measuements is needed fo PPP in moe challenging envionments and conditions. Wen et al. additionally included a subset of code multipath delays in the state vecto to impove the PPP accuacy [7]. The method was veified with measuements fom a few Intenational GNSS Sevice IGS) stations. The position eo was educed by a facto of 2. Attitude detemination has also eceived a lot of attention. Teunissen deived the intege least-squaes estimation with a baseline length constaint in [8] and [9]. The othogonal decomposition of the sum of squaed eos was used as a stating point fo the deivation of the optimal estimato. Unfotunately, a pioi infomation solely on the baseline length is not sufficient to pefom single-epoch ambiguity esolution with low-cost GNSS eceives in multipath envionments because two out of thee eceive coodinates ae not sufficiently constained. Henel and Günthe suggested the use of a pioi infomation on both the baseline length and attitude in [20]. The a pioi infomation was included as soft constaints in the intege least-squaes estimation. Soft constaints have the advantage of enabling a moe eliable ambiguity fixing and being obust ove eos in the a pioi infomation, which is not the case fo had constaints [2]. II. OVERVIEW OF SYSTEM SETUP In this section, we povide an oveview of the system setup. It includes two components: The fist component consists of the estimation of ionospheic coections and satellite phase biases with a netwo of static eceives. The second component includes the joint estimation of the position and attitude of a moving object. A GPS/INS tightly coupled solution is detemined, which taes the ionospheic coections and satellite phase biases into account. Thee ae seveal new contibutions involved in both pocessing steps: In the fist step, a netwo of single-fequency low-cost GNSS eceives is consideed. This class of eceives enables dense netwos and, thus, the monitoing of local ionospheic gadients. As single-fequency eceives cannot exploit the dispesive behavio of the ionospheic delay, the estimation of ionospheic coections and satellite phase biases is ill-conditioned. We conside statistical a pioi infomation on ionospheic gadients and ates to enhance the sepaation of phase biases and ionospheic coections. In the second step, we estimate the absolute position, velocity, acceleation, attitude, angula ates, single and double diffeence ambiguities, code multipath, and acceleomete and gyoscope biases with an extended Kalman filte. A code multipath paamete is estimated fo each satellite to pevent a pojection of the multipath paamete into othe state paametes and to exploit time coelation of the multipath paamete. Statistical a pioi infomation on the baseline length is also included in the tee seach; i.e., the seach intevals ae substantially HENKEL: TIGHTLY COUPLED PRECISE POINT POSITIONING AND ATTITUDE DETERMINATION 383

3 Fig.. Estimation of satellite phase biases and ionospheic coections with netwo of GNSS eceives: Fist, absolute eceive positions and cloc offsets ae detemined by least-squaes estimation LS pos.) using pseudoange measuements only. Obtained eceive and satellite positions and cloc offsets ae then used to compute synchonization coection Sync) fo each satellite-satellite single diffeence SS-SD) measuement of each eceive except fist one, which seves as efeence). Subsequently, absolute and elative eceive positions, SD code multipath, DD ambiguities, vetical ionospheic delay coections, and satellite phase biases lumped with SD ambiguities of fist eceive) ae detemined with netwo of eceives in Kalman filte. Obtained float solution is impoved by subsequent ambiguity fixing and eadjustment of vetical ionospheic delay coections and satellite phase biases. educed, and the success ate of ambiguity esolution is significantly inceased. We obtain a centimete-level accuacy fo inematic positioning in a multipath envionment with two low-cost single-fequency eceives and inetial sensos, which has not been achieved so fa. A. Estimation of Ionospheic Coections and Satellite Phase Biases Fig. gives an oveview of the individual pocessing steps fo estimation of ionospheic coections and satellite-satellite single diffeence SD) phase biases. Fist, a synchonization coection c,l, is computed fo each DD measuement. It equies a ough estimate of the cloc offsets δt of eceives {, 2}, of the satellite positions x :, and of the line-of-sight vectos e :.The synchonization coection is needed fo low-cost GNSS eceives because the satellite movement within the time diffeence of two eceive cloc offsets can be on the ode of seveal metes. The DD ambiguities ae no longe intege valued if synchonization eos emain uncoected. The pepocessing also includes the fomation of satellite-satellite SD caie phase and pseudoange measuements {ϕ,l,,ρ,l, }, with a common efeence satellite indexed by l, the cycle slip coections CSC) denoted by N,l, [26], and the application of coections fo atmospheic delays, antenna phase cente offsets PCO) and vaiations PCV), eath tides, and phase windup. Subsequently, the absolute position x of one efeence eceive, the elative positions b,,whee {2,..., R}, the code multipath eos ρ MP,DD ambiguities N,l,, SD satellite phase biases β,l lumped with SD ambiguities of the fist eceive), vetical ionospheic delays, and thei ates ae detemined using a Kalman filte [0]. The obtained float solution is impoved by subsequent ambiguity fixing and eadjustment of satellite phase biases and vetical ionospheic delays. B. Kinematic PPP and Attitude Detemination The obtained satellite phase bias and vetical ionospheic delay estimates ae then povided to a mobile eceive fo joint PPP and attitude detemination. The tight coupling of measuements fom two low-cost GNSS eceives and an IMU equies a cetain initialization time to synchonize the GNSS and INS measuements. Once the synchonization is completed, a state vecto including the absolute position, velocity, acceleation, attitude angles, angula ates, SD ambiguities, code multipath eos, and gyoscope and acceleomete biases is detemined with a Kalman filte [0]. The caie phase, code phase, and Dopple measuements fom two GNSS eceives and the thee-dimensional 3D) angula ate and acceleation measuements fom an IMU ae used to update the state vecto as shown in Fig. 2. An altenating state update is pefomed as GNSS and INS measuements ae eceived at diffeent time instances and as the 384 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 5, NO. 4 OCTOBER 205

4 Fig. 2. Tightly coupled PPP and attitude detemination: State vecto is updated by both GPS and IMU measuements in altenating way. As inetial measuements have highe date ate than GPS measuements, subsequent IMU-based state updates exist additionally. pocessing is immediately pefomed afte signal eception. As inetial measuements have a highe date ate than GPS measuements, subsequent IMU-based state updates exist additionally. This pape is oganized as follows: In section II, models fo the caie phase, pseudoange, Dopple, acceleation, and angula ate measuements ae descibed. Section III includes a detailed desciption of the estimation of satellite phase biases and vetical ionospheic delays with a local netwo of single-fequency GNSS eceives. Section IV povides some measuement esults on the estimation of the vetical ionospheic delays. The tight coupling of two low-cost GNSS eceives and an inetial senso is descibed in detail in section V. It diffes fom conventional tight couplings by DD synchonization coections to estoe the intege popety of ambiguities, by including a code multipath offset fo each satellite in the state vecto to exploit the time coelation of the code multipath paamete and, theeby, to impove the absolute positioning accuacy, and by the fixing of undiffeenced ambiguities. Section VI includes an analysis of the measuement esults fom a test dive. Finally, section VII concludes this pape. III. MEASUREMENT MODELS In this section, measuement models fo joint PPP and attitude detemination with tight coupling ae intoduced. We tae the following paticulaities of low-cost GNSS eceives into account: ) lac of pecise synchonization, 2) code multipath of seveal tens of metes, 3) fequent half- and full-cycle slips, and 4) single-fequency eceives, i.e., no elimination of ionospheic delays by linea combination [23]. Low-cost GNSS eceives have no timing input and ae only synchonized with an accuacy of ms to GPS system time. Because satellites move with a speed of 3 m/s, the satellite position might change by seveal metes within the time of the eceive cloc offset. Fo elative positioning, the satellite movement within the time diffeence between both eceive cloc offsets affects the DD measuements. If this satellite movement is not coected, the DD ambiguities ae no longe intege valued, and the position accuacy is significantly educed. Theefoe, we tae the eceive cloc offset explicitly into account in each paamete of ou measuement models. The code multipath is also highe fo mass-maet GNSS eceives due to the smalle eceive bandwidth and smalle antenna size. Additionally, half-cycle slips occu much moe fequently than fo geodetic eceives. Satellite-satellite SD measuements with a common efeence satellite indexed by l) eliminate eceive-dependent biases and eep the absolute position infomation. We model the SD caie phase measuements fo satellites and l and eceive at time t n + δt t n ) whee t n is the eceived time as detemined by the eceive, and δt [t n ] is the eceive cloc offset at time t n )as λ ϕ,l t n + δt t n )) = λ ϕ t n + δt t n )) λ ϕ l t n + δt t n )) = e t n + δt t n )) x t n + δt t n )) + x ET t n ) x t n + δt t n )) x t n ) ) e l t n + δt t n )) x t n + δt t n )) + x ET t n ) x l t n + δt t n )) x l t n ) ) + cδt,l + λn,l + λ ϕ PW,l + λ/2 N,l with the following notations: λ t n δt e x x ET x x c δt I T N N β ϕ MP ϕ PW ϕ PCO ε t n ) I,l t n ) + T,l t n ) t n ) + λβ,l t n ) + λ ϕ MP,lt n ) t n ) + λ ϕ PCO,lt n ) + ε,l t n ), ) wavelength of L caie phase [0.9 m]; eceived time as detemined by eceive [s]; eceive cloc offset [s]; nomalized line-of-sight vecto pointing fom satellite to eceive ; position of eceive [m]; eceive position offset due to Eath tides [m]; position of satellite using pecise obits [m]; eo of satellite position [m]; speed of light in vacuum [m/s]; cloc offset of satellite [s]; slant ionospheic delay [m]; slant topospheic delay [m]; intege ambiguity [cycles]; cycle slip [half cycles]; satellite phase bias [cycles]; eceive phase multipath [cycles]; eceive phase windup [cycles]; antenna phase cente offset [cycles]; and phase noise [m]. In this pape, all tems ae denoted in the Eath-centeed, Eath-fixed ECEF) fame if not othewise stated. In ), we have consideed only the eceive and satellite positions explicitly at time t n + δt. All othe HENKEL: TIGHTLY COUPLED PRECISE POINT POSITIONING AND ATTITUDE DETERMINATION 385

5 paametes change less quicly, such that the change within δt is negligible. GNSS eceives also povide an unambiguous pseudoange measuement. We model the satellite-satellite SD of the pseudoange measuements simila to the SD caie phase measuement, i.e., ρ,l t n + δt t n )) = e t n + δt t n )) x t n + δt t n )) + x ET t n ) x t n + δt t n )) x l t n ) ) e l t n + δt t n )) x t n + δt t n )) + x ET t n ) x l t n + δt t n )) x l t n ) ) + cδt,l t n ) + I,l t n ) + T,l t n ) +b,l t n ) + ρ MP,lt n ) + η,l t n ), 2) with the satellite code bias b, the code multipath ρ MP, and the code noise η as additional paametes. GNSS eceives also tac the Dopple fequency fo each satellite and, theeby, povide infomation on the eceive velocity. We model the SD Dopple measuements accoding to Misa and Enge [24] as f,l D t n ) = f c e t n ) v t n ) v t n ))/c e l t n) v t n ) v l t n ))/c ) +f c δ τ,l t n ) + ε f,lt n ), 3) D with the caie fequency f c, the eceive velocity v,the satellite velocity v, the satellite cloc dift δ τ,andthe measuement noise ε f,l. D The SD caie phase measuements ae eaanged fo eceive ; i.e., all nown tems ae bought to the left side. We use the ultaapid half-pedicted) pecise obits and clocs fom IGS [2]. As the ionospheic coections ae not sufficiently accuate fo satellite phase bias estimation and ambiguity esolution, we detemine a vetical ionospheic delay fo each satellite at its espective ionospheic piece point. We conside the Chapman pofile of the ionosphee and use a multilaye model of the ionosphee as descibed by Hoque and Jaowsi in [25]. A slant delay I,i is detemined fo each laye, and the total slant delay is given by the sum of individual delays, I = i I,i, 4) whee the slant ionospheic delay of the ith laye is obtained fom pojective geomety and the Chapman distibution: I i v I,i = ) efh i+) efh i )), hi +R e )cose,i 2 ) h mipp +R e 5) with the height h i of the ith laye, the satellite elevation E,i at the ith laye, the pea ionization height h mipp,and the eo function ef ). The multilaye model taes the spatial gadient I v of the ionospheic delay into account; x i.e., the vetical delay at the ith laye is modeled by I i v = I v x mipp ) + I v x x IPP i x mipp ), 6) with x IPP i being the location of the ionospheic piece point of the ith laye. The obtained ionospheic slant delay is intepeted as the poduct of an impoved mapping function m I E )and a vetical ionospheic delay at the pea ionization height. The vetical ionospheic delay is modeled as the sum of a ough estimate Iv and a pecise coection Iv. Thus, the SD slant ionospheic delay is modeled as I,l = m I E ) I v + I v ) mi E l )I l v + I l v ) + η I l, 7) with the mapping function obtained fom the multilaye model. The topospheic delays ae decomposed into a dy component and a wet component, which ae detemined with the MOPS model RTCA DO-229D) of ESA. This leaves the absolute position, SD ambiguities and phase biases, vetical ionospheic delay eos, and SD phase multipath delays as unnowns: λϕ,l t n + δt t n )) : = λ ϕ,l t n + δt t n )) e t n + δt t n )) x ET t n ) x t n + δt t n )) x t n + δt t n )) ) + e l t n + δt t n )) x ET t n ) x l t n + δt t n )) x l t n + δt t n )) ) cδt,l t n) + m I E t n ))Iv t n) m I E l t n ))Iv l t n) T,l t n ) λ ϕ PW,lt n ) λ ϕ PCO,lt n ) λ/2 N,l t n) = e,l t n + δt t n )) x t n + δt t n )) + λn,l + β,l t n )) m I E t n)) Iv t n ) + m I E l t n)) Iv l t n ) + λ ϕ MP,lt n ) + ε,l t n). 8) The SD code measuements ae eaanged in the same manne. Fo the Dopple measuements, the satellite velocities and cloc difts ae a pioi nown [24], which enables us to ewite 3) as f,l D t n ):= f,l D t n ) f c e t n) v t n )/c + f c e l t n) v l t n )/c f c δ τ,l t n ) = f c e,l t n ) v t n )/c + ε f,lt n ). 9) D Inetial sensos povide high-ate acceleation and angula ate measuements, which ae not affected by GNSS signal eception conditions and enable a eliable detection and coection of cycle slips fo inematic eceives. The acceleation and angula ate ae sensed in the senso-fixed s-) fame, which is centeed at the senso s chip and aligned with the pincipal axes of the chip. We assume that the s-fame is aligned with the body-fixed b-) fame, which is centeed at the vehicle and aligned with the longitudinal and tansvesal axes of the vehicle. 386 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 5, NO. 4 OCTOBER 205

6 As GNSS and inetial measuements ae obtained in diffeent fames, a fame tansfomation is needed. We use the e-fame also ECEF fame) fo the senso fusion. It is centeed at the Eath s cente with the x-axis pointing in the equatoial plane towads the 0 meidian and the z-axis pointing towad the geogaphic noth pole. The navigation n-) fame is centeed at the vehicle and aligned with the east, noth, and up diections. The n-fame seves as a efeence fame fo the attitude of the vehicle. The acceleation measuement is povided in the b-fame and is modeled accoding to Jeeli [26]as a b t n ) = Rn b t n)re n t n)a e t n ) + ba b t n) sinθt n )) + g cosθt n )) sinϕt n )) + εa b t n), 0) cosθt n )) cosϕt n )) with the otation matices Re n and Rb n, the acceleation ae in the e-fame, the acceleation biases ba b of the senso in the b-fame, the gavitational acceleation g, the pitch angle θ, the oll angle ϕ, and the measuement noise εa b.the otation fom the e-fame into the n-fame depends on the latitude ϕ and longitude λ of eceive and is given by R n e t n) = R π/2 ϕ t n ))R 3 π/2 + λ t n )). ) The otation fom the n-fame into the b-fame depends on the heading ψ and pitch θ of the vehicle and is given by R b n t n) = R 2 θt n ))R 3 π/2 ψt n )). 2) The gyoscope senses the angula ates ωib b of the body-fixed b-) fame with espect to the inetial i-) fame in the b-fame. The angula ate measuements can be expessed as the sum of ωin b, ωb nb,abiasbb ω ib, and a noise ηω b ib ; i.e., ωib b t n) = Rn b t n)ωin n t n) + ωnb b t n) + bω b ib t n ) + ηω b ib t n ). 3) The angula ates ωnb b ae elated to the ates of the Eule angles accoding to Jeeli [26] as 0 0 ϕ ωnb b = R ϕ)r 2 θ) 0 + R ϕ) θ + 0 ψ sinθ) ϕ = 0 cosϕ) cosθ)sinϕ) θ, 4) 0 sinϕ) cosθ)cosϕ) ψ with R i α) being a otation aound the ith axis by an angle α. The otation ωin n of the navigation fame with espect to the inetial fame depends on the latitude ϕ, the ates { ϕ, λ } of latitude and longitude, and Eath s otation ate ω e, and it is given by Jeeli [24] as λ + ω e )cosϕ ) ωin n = ϕ. 5) λ + ω e )sinϕ ), IV. ESTIMATION OF SATELLITE PHASE BIASES AND VERTICAL IONOSPHERIC DELAYS In this section, the satellite phase biases ae detemined with a netwo of low-cost GNSS eceives. As the cloc offsets of low-cost GNSS eceives ae in the ode of ms, a synchonization coection is equied fo each SD measuement of eceive {2,..., R}. The fist eceive seves as a efeence eceive. The measuement model of 8) is used fo the efeence eceive. Fo all othe eceives, we bing all nown tems to the left side and conside the satellite and eceive position at time t n + δτ t n ) [instead oft n + δτ t n )] to obtain λϕ,l t n + δt t n )) : = λ ϕ,l t n + δt t n )) e t n) x ET t n ) x t n ) ) + e l t n) x ET t n ) x l t n ) ) + e t n + δt t n )) x t n + δt t n )) e l t n + δt t n )) x l t n + δt t n )) cδt,l t n ) T,l t n ) + m I E t n))iv t n ) m I E l t n))iv l t n ) λ ϕ PW,lt n ) λ ϕ PCO,lt n ) λ/2 N,l t n ), 6) which can also be expessed in tems of the unnown paametes as λϕ,l t n + δt t n )) = e,l t n + δt t n )) x t n + δt t n )) m I E t n)) Iv t n ) + m I E l t n)) Iv l t n ) +λn,l + β,l t n )) + c,l, t n) + ε,l t n ), 7) with the synchonization coection c,l,, which descibes the change of the eceive and satellite positions within δt t n ) δt t n ) and is given by c,l, t n) = e t n + δt t n )) x t n + δt t n )) x t n + δt t n ))) e l t n + δt t n )) x t n + δt t n )) x l t n + δt t n ))) e t n + δt t n )) x t n + δt t n )) x t n + δt t n ))) + e l t n + δt t n )) x t n + δt t n )) x l t n + δt t n ))). 8) The eceive cloc offsets δt t n ) can be obtained by a stand-alone standad least-squaes estimation of the eceive position and cloc offset. Cycle slips need to be coected as well. Since the eceives ae static fo satellite bias detemination, it is sufficient to use tiple diffeence TD) phase measuements fo the coection of half-cycle slips; i.e., Ň,l, t n) = [ λϕ,l,2 t n) λϕ,l,2 t n ) ) /λ/2) ]. 9) The position x, ={2,...,R}, can be expessed in tems of x and the baseline vecto b, between both eceives: x t n ) = x t n ) b, t n ). 20) HENKEL: TIGHTLY COUPLED PRECISE POINT POSITIONING AND ATTITUDE DETERMINATION 387

7 Similaly, we expess the SD ambiguities N,l of the th eceive in tems of N,l and of the DD intege ambiguities N,l, as N,l = N,l N,l,, 2) which enables us to ewite 6) as λϕ,l t n + δt t n )) := λ ϕ,l t n + δt t n )) e t n) x ET t n ) x t n ) ) + e l t n) x ET t n ) x l t n ) ) + e t n + δt t n )) x t n + δt t n )) e l t n + δt t n )) x l t n + δt t n )) cδt,l t n ) T,l t n ) + m I E t n))iv t n ) m I E l t n))i l v t n ) λ ϕ PW,l t n ) λ ϕ PCO,lt n ) λ/2 N,l t n ), 22) which can also be witten in tems of the emaining unnowns as λϕ,l t n + δt t n )) = e,l t n + δt t n )) x t n + δt t n )) b, t n + δt t n ))) m I E t n)) Iv t n ) + m I E l t n)) Iv l t n ) + λn,l N,l, + β,l t n )) + c,l, t n) + ε,l t n ). 23) It is sufficient to use only caie phase and pseudoange measuements fo satellite bias detemination with static eceives. We stac all measuements at time t n in a vecto to obtain z n = λϕ T t n),...,λϕ T R t n),ρ T t n),...,ρ T R t n) ) T, 24) whee SD phase and code measuements ae synchonized and half-cycle slips ae pecoected; i.e., λϕ T = λϕ,l c,l,l, λ/2 Ň,,..., ) T λϕ 32,l c 32,l, λ/2 Ň 32,l,, ) T. ρ T = ρ,l c,l,,...,ρ32,l c 32,l, 25) Additionally, all unnown paametes at time t n ae staced in the state vecto x n = x T t n), b,2 T t n),..., b,r T t n), with Iv T t n ),..., Iv T R t n ), Iv T t n ),..., Iv T R t n ), ρmp T t n ),..., ρmp T R t n ), N T t n),n T,2 t n),...,n T,R t n)) T, 26) I v = Iv,..., Iv 32 ) T R 32 ) T ρ MP = ρ MP,l,..., ρ MP 32,l R 32 N = N,l + β,l,...,n 32,l + β 32,l) T R 32 N, = ) T N,l,,...,N32,l, Z ) The modeling of the state paametes ove time shall also be consideed in the paamete estimation. Obviously, the absolute and elative positions and ambiguities ae constants. The vetical ionospheic delay coections Iv can be sepaated fom the absolute ambiguities lumped with satellite phase biases) only by a change of the ionospheic mapping function. Because an accuate sepaation is essential fo pecise satellite bias estimation, we conside the measuements of multiple epochs and intoduce a state space model. We model the tempoal behavio of the vetical ionospheic delay by a piecewise linea function ove time [27]. Iv t) = Iv t i ) + t t i ) I v t i ) + η I v t) fo t i <t<t i, 28) whee t i t i has to be chosen sufficiently lage such that it enables a sepaation of biases and ionospheic delays and sufficiently small such that the modeling eo is still small. The paamete estimation emains ill-conditioned because t i t i can be only in the ode of a few minutes. Theefoe, we include additional constaints on the ionospheic vetical delays [27] to impove the conditioning. The constaints ae added as statistical a pioi infomation, i.e., as soft instead of had constaints [28]. The a pioi infomation on the vetical ionospheic delay coection is modeled as I v = I v + η I v, η I v N ) 0,σ 2 I, 29) v with the Gaussian distibution N ), the tue vetical ionospheic delay coection Iv, and the vaiance σ 2 Iv of the a pioi infomation. Similaly, we constain the magnitude of the ate of the vetical ionospheic delay coection by modeling it as I v N ) 0,σ 2 I. 30) v The vetical ionospheic delays and thei ates ae also spatially coelated [28]. Theefoe, we also include some constaints on the diffeence between vetical ionospheic delays, i.e., ) Iv Iv l N 0,σ 2 Iv I : {, l}, 3) v l whee σ 2 Iv I depends on the distance between both v l ionospheic piece points. Simila a pioi infomation can be added on the diffeence in ates of the vetical ionospheic delay coections. The code multipath of static eceives is also coelated ove time. Theefoe, the SD code multipath is modeled as ρ MP,l t n ) = ρ MP,l t n ) + η ρmp,l t n ). 32) Obviously, the coelation intoduced by single diffeences with a common efeence satellite has to be taen into account. 388 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 5, NO. 4 OCTOBER 205

8 At any epoch, only a subset of satellites is visible. We denote the subset of available measuements by sn zz n). Similaly, the set sn x of indices is intoduced to descibe the subset of state paametes that can be updated at time t n. Both the measuements and the statistical a pioi infomation ae linea with espect to the unnown state paametes; i.e., s z ) n z n ) H z ) ) sn x x = n sn x n) x ηs z n) + n z n ), 33) with x n = H x n η s x n x n ) I v,..., Iv 32,..., Iv R,..., I v,..., v,..., I v R,..., I v I v l,..., Iv 32 I v l,..., I v R I v l R,..., I 32 I v l R, I 32 v R Iv 32 R, I 32 v R, I v I l v,..., v I l v,..., I 32 ) T I v R I l v R,..., v R I l v R, 34) I 32 and whee H n is implicitly defined by the measuement models of 8) and 23). We assume that the measuements and the a pioi infomation ae statistically independent and that thei noises ae Gaussian distibuted, η s z n z n ) N 0, s z n z n )) and η s x n x n ) N 0, s x n x n )). 35) The state tansition is also descibed by a linea model accoding to the pevious desciption. In matix-vecto notation, the state space model is witten as s x n x n) = s x n x n) + η s x n x n ), 36) with the state tansition matix and pocess noise η s x n x n ). The state vecto is estimated with a Kalman filte [0]. The a posteioi state estimate is given by s sn x z ) ) ˆx+ n ) = sx n ˆx n ) + K n z n ) n sn x x H n s xn n) ˆx n ), 37) with the pedicted state estimate sn x ˆx n ) = sx n ˆx+ n )and the Kalman gain. K n = s x n ˆx n )Hn T H n s x n ˆx n )Hn T + snz z n )) 38) The covaiance matix of the float solution of 37) is given by s x n ˆx + n ) = KH n ) s x n ˆx n ). 39) The accuacy of the state estimation can be significantly impoved if the intege popety of the DD ambiguities is taen into account. We denote by ˆN + the subset of state paametes descibing the DD float ambiguity estimates in 37). The ambiguities ae fixed by an intege least-squaes estimation [29], so Ň = ag min ˆN + N 2 N ˆN + = ag min N L ˆN + N) 2 D, 40) whee an LDL T decomposition was applied to the covaiance matix ˆN +, which was obtained fom ˆx +. The sum of squaed decoelated ambiguity esiduals can also be expessed in tems of conditional ambiguities. Let ˆN j,...,j, whee j {,...,K}, denote the jth conditional ambiguity, and let σ 2ˆN j,...,j = D j,j be its vaiance, then the intege estimate of N is given by Ň = ag min N,...,N K N j ˆN j,...,j ) 2 σ 2ˆN. 4) j,...,j j= The intege least-squaes estimation equies a seach of the best candidate inside a pedefined seach space volume χ 2 : ˆN + N 2 ˆN + χ 2. 42) Witing the sum of squaed ambiguity esiduals in tems of the conditional ambiguities [8], and eaanging, yields a quadatic inequality fo detemining N i : N i ˆN i,...,i ) 2 χ 2 N j ˆN j,...,j ) 2 σ 2ˆN i,...,i σ j i 2ˆN j,...,j i χ 2 N j ˆN j,...,j ) 2 σ 2ˆN. 43) j,...,j j= Solving 43) fo N i gives a lowe and an uppe bound: with N i ˆN i,...,i σ ˆN i,...,i αi N i ˆN i,...,i + σ ˆN i,...,i αi, 44) i α i = χ 2 N j ˆN j,...,j ) 2 σ 2ˆN. 45) j,...,j j= Once the tee seach is completed and the candidate of smallest sum of squaed eos is selected, all eal-valued paametes can be eadjusted. The obtained fixed solution has a fa highe accuacy than the float solution if the ambiguities ae fixed coectly. Finally, the vetical ionospheic delay coections and the eal-valued lumped sum of the SD satellite phase biases and intege ambiguities of the fist eceive seve as a coection fo inematic PPP. Fo the latte value, it is sufficient to povide the eal-valued pat of this sum, β,l + N,l [β,l N,l ] = β,l [β,l ], 46) as the inematic eceive is detemining an intege ambiguity fo each SD measuement. HENKEL: TIGHTLY COUPLED PRECISE POINT POSITIONING AND ATTITUDE DETERMINATION 389

9 TABLE I Measuement Noise Assumptions fo Bias Estimation Phase noise undiffeenced) Code noise undiffeenced) σϕ = {2 mm...4mm} depending on satellite elevation σ ρ = {0.5 m...m} depending on satellite elevation TABLE II A Pioi Infomation on Ionospheic Delays Vetical ionospheic delay coections Diffeence of vetical ionospheic delay coections Rates of vetical ionospheic delay Coections I v = 0 m {, } σ I v = 0 m {, } I v I v l = 0m σ I v I v l = 0.5m {, } I v = 0m {, } = 0. mm/s σ I v V. MEASUREMENT RESULTS FOR VERTICAL IONOSPHERIC DELAY ESTIMATION The poposed method fo vetical ionospheic delay and satellite phase bias estimation was veified with a local netwo of thee single-fequency low-cost u-blox LEA 6T GPS eceives and patch antennas. The eceives wee mounted on a field at λ = , ϕ = , λ 2 = , ϕ 2 = ,and λ 3 = , ϕ 3 = with open sy conditions on Febuay 26, 204. The intestation distance was 00 m between eceives and 2 and 60 m between eceives and 3. These shot distances esult in a high coelation of the ionospheic delays. This coelation is exploited by ou constaints on the ionospheic gadients and ates. Tables I and II include ou assumptions fo the measuement noise and a pioi infomation. The measuement noise statistics wee detemined in static conditions as follows: DD measuements wee pefomed and analyzed ove a shot) peiod of 30 s. As static DDs can be well appoximated by a linea model ove such shot time peiods, a least-squaes estimation of the coefficients of the linea model was pefomed. The standad deviations of the DD noises wee then obtained fom the oot mean squae of the esiduals of the least-squaes estimation. The standad deviations of the undiffeenced measuements wee obtained by scaling of the standad deviations of the DDs. In Table II, the a pioi infomation on the vetical ionospheic delay is athe loose to enable an ionospheic coection also in the absence of Euopean Geostationay Navigation Ovelay Sevice EGNOS) coections. Howeve, the ates of the vetical ionospheic delays wee assumed to be constant ove t i t i = 3 min and tightly constained to enable a sepaation of biases and ionospheic delays. The standad deviations of the pocess noise of the code multipath ae modeled by an elevation-dependent Fig. 3. Estimation of ionospheic delays with local netwo of low-cost single-fequency GNSS eceives: Slant ionospheic delays coespond to pojection of vetical ionospheic delay estimates into slant diection. Vaiation in time is mainly caused by changes in mapping functions and only to a small extent by changes in zenith delays. Noise level of ionospheic delay estimates is in ode of millimetes and, thus, much moe accuate than conventional code minus phase based ionosphee estimation. exponential function, σ ρmp E ) = c 0 e E /c, 47) whee the coefficients c 0 and c wee chosen such that σ ρmp {0.5 m,..., m} fo E {5,...,90 }. Fig. 3 shows the slant ionospheic delays obtained fom a pojection of the vetical ionospheic delay estimates. The vetical delays wee detemined jointly with the absolute and elative eceive positions, the SD ambiguities lumped with the SD satellite phase biases), the DD ambiguities, and the SD code multipath with a local netwo of low-cost GNSS eceives. The vaiation in time of the slant ionospheic delays is mainly caused by changes in the mapping functions and only to a small extent by changes in zenith delays. The noise level of the ionospheic delay estimates is in the ode of millimetes, as the estimation mainly elies on the pecise phase measuements. It is much moe accuate than a conventional code minus phase based ionospheic delay estimation, which is heavily affected by code multipath. Fig. 4 shows the caie phase esiduals fo the estimation of the absolute and elative eceive positions, SD ambiguities lumped with SD satellite phase biases), DD intege ambiguities, vetical ionospheic delays, and SD code multipath delays with a egional netwo of low-cost GNSS eceives. The esiduals ae unbiased and dift-fee, which indicate the coectness of the model. The noise level of the esiduals is in the ode of a few centimetes, which coesponds to some shot-tem ionospheic vaiations and SD phase multipath. 390 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 5, NO. 4 OCTOBER 205

10 i.e.,,l,l λϕt = λϕ,l c, λ/2 N, λβ,l [β,l ]) + mi E ) Iv mi El ) Ivl,... K,l K,l λ/2 N, λβ K,l [β K,l ]) λϕk,l c, T 50) + mi EK ) IvK mi El ) Ivl. The second subset includes the angula ate and acceleation measuements and is given by T T T b snz 2, zn ) = ωib tn )), a b tn )). Fig. 4. Caie phase esiduals of satellite phase bias and ionospheic delay estimation with local netwo of low-cost GNSS eceives: Residuals ae unbiased and dift-fee, which indicates coectness of model. Noise level of esiduals is in ode of a few centimetes, which coesponds to some shot-tem ionospheic vaiations and SD phase multipath. VI. KINEMATIC PPP AND ATTITUDE DETERMINATION: GPS/INS SENSOR FUSION WITH TIGHT COUPLING In this section, a joint PPP and attitude detemination method is descibed. The measuements fom two low-cost GNSS eceives, an acceleomete, and a gyoscope ae tightly coupled in an extended Kalman filte [0]. The vetical ionospheic delay and satellite phase bias estimates of 46) ae used to estoe the intege popety of the absolute SD ambiguities and, theeby, to impove the absolute position accuacy. As the gyoscope povides the ate of heading, pitch, and oll, we paameteize also the baseline vecto b,2 in tems of the attitude angles: sinψ) cosθ ) b,2 = x x 2 = l cosψ) cosθ ). 48) sinθ) The set of measuements is divided into two types γ n {,2} of subsets snz γn, zn ): A fist subset of GNSS measuements that is denoted by snz, zn ) and a second subset of INS measuements descibed by snz 2, zn ). The intoduction of the subset type γ n enables an elegant notation because the state estimation based on GNSS and INS measuements can be descibed with the same expessions. At each epoch, thee is only a subset of measuements subset of K satellites o gyoscope/acceleation measuements) available. The fist subset is given by snz, zn ) = λϕt tn ), λϕ2t tn ), ρt tn ), ρ2t tn ), T 49) fdt tn ), fdt2 tn ), whee the SD caie phases ae coected fo the ionospheic delays and SD satellite phase biases of 46) in addition to the synchonization and cycle-slip coections, 5) Similaly, we intoduce two subsets of state paametes: The set of state paametes updated by GNSS measuements [30] is witten as snx, xn ) = x T tn ), v T tn ), a T tn ), ψtn ), ψ tn ), θtn ), θ tn ), ϕtn ), ϕ tn ), T T T tn ), ρmp t ), ρmp t ), NT tn ), N,2 n 2 n T T T bωb ib tn )), bab tn )). 52) Note that the IMU biases ae also included in the state vecto, although they ae not diectly obsevable by GPS. Howeve, the Dopple measuements of the fist eceive povide unbiased infomation on the velocity. The change of the velocity ove time gives unbiased infomation on the acceleation and, thus, enables the coection of acceleation biases. Similaly, the Dopple measuements of the second eceive povide unbiased infomation on the angula ates and, theeby, can coect also the biases of the angula ates. These biases ae diectly obsevable by the gyoscope but cannot be sepaated fom the angula ates without GPS. The subset of state paametes updated by INS measuements is given by snx 2, xn ) = x T tn ), v T tn ), a T tn ), ψtn ), ψ tn ), θtn ), θ tn ), ϕtn ), ϕ tn ), T T T 53) bωb ib tn )), bab tn )). The measuements of both obsevation types ae used to update the state vecto in a tight coupling. As the measuement ate of most low-cost gyoscopes and acceleometes is 00 Hz and, thus, highe than the measuement ate of most low-cost GNSS eceives, the state vecto is updated moe often by IMU than by GNSS measuements. The measuements ae nonlinea functions of the state vecto due to the tigonometic elationship between the Eule angles and the baseline vecto; i.e., snz γn, zn ) = hn snx γn, xn )) + ηzn snz γn, zn )), 54) with hn ) being implicitly defined by the measuement models, and whee ηzn snz γn, zn )) N 0, snz γn,zn ) ) is the Gaussian measuement noise. The vehicle dynamics and tempoal vaiations of the code multipath eos, and gyoscope and acceleation biases ae descibed in the HENKEL: TIGHTLY COUPLED PRECISE POINT POSITIONING AND ATTITUDE DETERMINATION 39

11 state space model. We conside a linea state space model, s x n γ n,x n ) = s x n γ n,x n ) + η xn s x n γ n,x n )), 55) with the state tansition matix and the pocess noise η xn s x n γ )) N 0, x n s x n γ )) ). The state vecto is detemined with an extended Kalman filte [0]. It includes the state pediction sn x γ n, ˆx n ) = sx n γ n, ˆx + n ), 56) with the covaiance matix of the pedicted state s x n γ n, ˆx n ) = s x n γ n, ˆx + n ) T + s x n γ n,x n ), 57) and the state update s x n γ n, ˆx + n ) = sx n γ n, ˆx n ) + K n s z n γ n,z n ) h n s x n γ n, ˆx n ))). 58) The latte expession equies the evaluation of a nonlinea function. As the statistics of a nonlinea function cannot be detemined analytically, the baseline b,2 of 48) is lineaized as b,2 b,2 ) 0 + b,2 ψ b,2 θ θ=θ 0 θ=θ 0 ψ=ψ0 ψ ψ 0 ) + ψ=ψ0 θ θ 0 ). 59) The Dopple measuement of the second eceive depends on the velocity of the second eceive, which is given by v 2 = x 2 = x b,2 = x cosψ) cosθ) l sinψ) cosθ) ψ 0 + sinψ)sinθ) cosψ)sinθ) θ, 60) cosθ) and, thus, is also nonlinea in ψ and θ. Theefoe, the baseline ate is also lineaized as b,2 b,2 ) 0 + ψ ψ 0 θ θ 0 ψ ψ 0 θ θ 0 b,2 ψ, b,2 θ, b,2 ψ, b ),2 θ ψ=ψ0, ψ= ψ 0 θ=θ 0, θ= θ 0. 6) Once h n ) is lineaized, the covaiance matix of the state update can be easily detemined as s x n γ n, ˆx + n ) = K n H n ) s x n γ n, ˆx n ). 62) The accuacy of the state estimates impoves if the SD and DD intege ambiguities ae fixed coectly to integes. The initial ambiguity fixing can be pefomed aleady in static conditions, i.e., without inetial measuements. In this case, the state vecto of 52) simplifies to sn x γ n,x n ) = x T t n), b,2 T t n),n T t n),n,2 T t n), ρmp T t n ), ρmp T 2 t n ) ) T. 63) The eliability of the fixing of DD ambiguities inceases if the baseline length a pioi infomation is also taen into account. Theefoe, the unconstained intege least-squaes estimation of 40) is extended to [9, 8], whee N = ag min ˆN + N 2 N ˆN + ˇξN) ξ 2 ˇξN) + min ξ ) ) + Sξ l) 2 /σl 2, 64) with l and σ l being the a pioi infomation on the baseline length and its uncetainty, espectively, and whee S is a selection matix fo selecting the baseline coodinates in x n. The constained intege least-squaes estimation of 64) equies again a seach, which is witten as an inequality: ˆN + N 2 + min ξ ˆN ˇξN) + ξ 2 ˇξN) + Sξ l) 2 /σ 2 l ) χ 2. 65) Expessing the ambiguity esiduals in tems of the conditional ambiguity estimates and solving fo N i esult in a lowe and uppe bound fo each ambiguity accoding to 44) but with diffeent i α i = χ 2 N j ˆN j,...,j ) 2 min ξ j= σ 2ˆN j,...,j ˇξN) ξ 2 ˇξN) + Sξ l) 2 /σ 2 l ). 66) As the α i values of 66) ae smalle than those in 45), the seach intevals ae smalle fo the constained than fo the unconstained tee seach. Howeve, the computation of the lowe and uppe bounds becomes moe demanding. VII. MEASUREMENT RESULTS FOR KINEMATIC PPP AND ATTITUDE DETERMINATION In this section, the poposed inematic PPP and attitude detemination method is veified with eal measuements fom low-cost GNSS eceives and an inetial senso. Fist, the hadwae setup is descibed. Subsequently, the measuement and pocess noise assumptions ae povided, and the obtained measuement esults ae analyzed. The measuement test was pefomed with a vehicle on which the following hadwae was mounted: ) 2 u-blox LEA 6T GPS eceives with 5 Hz data ate; 2) 2 L patch antennas mounted on the oof of a vehicle along its longitudinal axis with a baseline length of l =.2 m and σ l = cm; 3) an MPU 950 inetial senso fom Invensense with a 3D gyoscope, a 3D acceleomete, and a 3D magnetomete mounted on the oof of the vehicle, poviding measuements with a ate of 00 Hz; and 392 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 5, NO. 4 OCTOBER 205

12 TABLE III. Measuement Noise Assumptions Dopple noise undiffeenced) σ fd ={Hz...0 Hz} depending on satellite elevation Angula ates σ ω b = 0.00 ad/s i {, 2, 3} ib,i Acceleation σ a b i = 0.m/s i {, 2, 3} TABLE IV. Pocess Noise Assumptions Acceleation σa n x = 2.5m/s 2 σa n y = 2.5m/s 2 σa n z = 0.25 m/s 2 Deivatives of angula ates σ ψ = 25 /s 2,σ θ = 25 /s 2 Single diffeence ambiguities σ N = 0. cycles Code multipath {2m,...,5m} depending on satellite elevation Gyoscope biases σ b b = ad/s, i {x,y,z} ωib,i Acceleomete biases σ b b ai = 0 9 m/s 2,i {x,y,z} Fig. 5. Nothen pat of vehicle s tac duing test dive as detemined by poposed PPP with tight coupling of two low-cost GNSS eceives, a gyoscope, and an acceleomete. Enlaged sections show stat and passing below a bidge. Each blue point efes to GPS-based state update. 4) a efeence system, consisting of a high-gade inetial senso and geodetic GNSS eceive tightly coupled). A. Measuement and Pocess Noise Assumptions The statistics of the phase and code measuement noises ae chosen accoding to Table I. Tables III and IV include ou assumptions fo the additional measuements and pocess noise statistics. Fo any measuement type, the noise statistics wee again obtained by ) taing the measuements in static conditions, 2) by intoducing a linea model fo the measuements ove a shot peiod of time, and 3) by deiving the noise standad deviations fom the esiduals of a linea least-squaes fitting. Let zt j ) be a measuement at time t j and {α 0, α } be the coefficients of a linea model, then the vaiance of the least-squaes fit is given by σ 2 z) = min α 0,α J zt j ) α 0 + α t j t 0 ))) 2. 67) j= It shall be noted that both the Dopple measuements and the diffeence of caie phase measuements fom two subsequent epochs) povide velocity infomation. Thus, thee is a cetain coelation between both measuements. Howeve, the caie phase measuements of one single epoch cannot povide any velocity infomation. The velocity obtained fom the phase diffeence efes to a diffeent time i.e., time between two Dopple measuements) than the Dopple measuement. The pocess noise assumptions wee chosen accoding to the dynamics of the vehicle, the tempoal vaiation of multipath eos, and the senso chaacteistics. The caie phase ambiguities ae in pinciple constant ove time. Because phase multipath, uncoected cycle slips, and Fig. 6. Southen pat of vehicle s tac. Enlaged section shows tac at highway cossing with two passages below bidge. Width of bidge is 45 m. Each mae efes to GPS-based state update. esidual ionospheic delays ae mapped to the ambiguities, we have chosen a small pocess noise of 0. cycles. B. Measuement Results Figs. 5 and 6 show the vehicle s tac as obtained fom the absolute position estimates ˆ x. The stating point is in the ight pat of Fig. 5. The initial heading was 98 ; i.e., the ca was oiented in a westen diection. The tac includes also passages below seveal bidges. These passages ae also enlaged with additional maes at evey epoch using a GPS-based state update. As INS-based state updates occu 20 times moe fequently than GPS-based state updates, no maes ae included. One can obseve a continuous path despite the GNSS signal inteuption and inceased multipath. As some satellites ae at low elevation and as signals ae also eflected, a GNSS-based state update can still be pefomed duing the fist metes HENKEL: TIGHTLY COUPLED PRECISE POINT POSITIONING AND ATTITUDE DETERMINATION 393

13 Fig. 7. Absolute positioning accuacy: Deviation between obtained position of low-cost GPS eceives deviates by only a few centimetes fom geodetic efeence solution. Fig. 8. Code multipath estimates fo eceive : Substantial coelation ove time indicates slow tempoal vaiation of multipath. Coelation between satellites aises fom common efeence satellite fo all single diffeences. of diving below the bidge. Once all tacing loops have lost loc, the eacquisition also taes some time, which explains the distance between the end of the bidge and the fist GPS-based state update afte the bidge. Fig. 7 shows the deviation between the absolute position obtained with the poposed tight coupling of two low-cost GPS eceives and the absolute position of a efeence system fo the fist pat of the tac. The diffeence is in the ode of a few centimetes, which indicates the accuacy of the peviously detemined ionospheic coections. The enlaged section shows the absolute position eo between two GPS-based state updates. It is a continuous cuve with 20 state updates based on the inetial senso. Fig. 8 shows the satellite-satellite SD code multipath estimates fo eceive. Each cuve epesents one SD with a common efeence satellite. The code multipath eflects the geomety of the envionment. The multipath offsets of up to 5 m exceed the code noise level and ae significantly coelated ove time and between satellites. 394 Fig. 9. Speed of vehicle in easten diection: Speed estimate of tightly coupled low-cost GPS/INS diffes by 0. m/s fom efeence system. Fig. 0. Heading estimation with tight coupling: Compaison of heading between low-cost GPS/INS hadwae and efeence solution. The coelation between satellites aises fom the single diffeencing with a common efeence satellite. The coelation ove time was not enfoced by ou model, as the pocess noise standad deviation of {2 m,..., 5 m} pe epoch allows much lage vaiations ove time. Thus, the coelation aises fom slow vaiations of the code multipath and/o fom othe eo souces that ae slowly vaying in time and ae pojected into the code multipath estimate. Clealy, the obseved substantial coelation indicates the stong benefit of estimating a code multipath paamete pe satellite. The estimated speed of the vehicle in the easten diection is shown in Fig. 9. The diffeence between the speed estimate using low-cost GPS and INS hadwae and the geodetic-gade GPS/INS efeence solution is in the ode of only 0. m/s. It is also dift-fee, which indicates coect modeling and estimation of the acceleation biases. Fig. 0 shows the heading of the vehicle. The heading estimate based on low-cost GPS and INS hadwae closely follows the efeence solution thoughout the complete measuement peiod. The enlaged section efes to a bidge undepass. The heading estimate is continuous, and IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 5, NO. 4 OCTOBER 205

14 Fig.. Heading estimation with tight coupling: Diffeence between heading based on low-cost GPS/INS hadwae and efeence solution. Fig. 3. Compaison of ate of heading estimate of low-cost GPS/INS system with efeence system. Accuacy is in ode of 0. /s. fimwae of cuently available low-cost GNSS eceives, the heading is deived fom the velocity, which is obtained solely fom the Dopple measuements of a single GNSS eceive. A dawbac of this GNSS appoach is that no meaningful heading infomation can be obtained in static conditions. Moeove, the Dopple fequency also cannot be taced below bidges. Theefoe, the availability of a pecise heading is significantly lowe than fo the poposed tight coupling with caie phase intege ambiguity esolution. Fig. 3 compaes the ate of heading estimate using low-cost GPS/INS hadwae with the ate of heading of the efeence system. The accuacy is in the ode of 0. /s. Fig. 2. Cumulative distibution of heading eo: Heading eo is less than 0.27 in 68.3% σ ) and less than 0.73 in 95.4% 2σ ) of all epochs. VIII. CONCLUSION the heading eo emains less than 0.2 despite the GNSS signal inteuption. This indicates ) an accuate coasting of the heading with the gyoscope measuements, 2) a pecise detemination of the gyoscope bias befoe the signal inteuption, 3) a eliable coection of all cycle slips which occu fequently at the edges of a bidge), and 4) a coect efixing of all DD intege ambiguities afte the bidge. Fig. shows the diffeence between the heading estimate obtained fom the low-cost GPS/INS hadwae and the efeence solution. The heading offset is less than 0.5 fo most of the time. As this eo coesponds to a elative position eo of only cm, the DD ambiguities wee most liely esolved coectly. Fig. 2 shows the cumulative distibution of the heading eo including all passages below bidges. The heading eo is less than 0.27 in 68.3% σ ) and less than 0.73 in 95.4% 2σ ) of all epochs. The figue also shows the statistics of the heading estimate fom a state-of-the-at technique: As the caie phase intege ambiguities ae not esolved in the eceive The highly automatized diving of vehicles equies a pecise and eliable position and attitude infomation with high ate. The use of low-cost hadwae is mandatoy fo this application. In this pape, we descibed a method fo satellite phase bias estimation and joint PPP and attitude detemination with low-cost GNSS eceives and an inetial senso. We too the paticulaities of these eceives into account and showed that the achievable accuacies ae compaable to the ones of geodetic eceives. Thee wee two pocessing steps: In the fist step, vetical ionospheic delays and satellite phase biases wee detemined by a netwo of static GNSS eceives. The absolute position of one GNSS eceive, the elative positions between this and all othe GNSS eceives, the SD code multipath delays, the DD ambiguities, the vetical ionospheic delays, thei ates, and the satellite phase biases lumped with SD ambiguities) wee estimated with a Kalman filte. In the second step, the vetical ionospheic delay and satellite phase bias estimates wee used fo inematic PPP and attitude detemination. A tight coupling of caie phase, pseudoange, Dopple, angula ate, and acceleation measuements was pefomed to estimate the HENKEL: TIGHTLY COUPLED PRECISE POINT POSITIONING AND ATTITUDE DETERMINATION 395

15 position, velocity, acceleation, attitude, angula ates, SD and DD ambiguities, code multipath offsets, and biases of the gyoscope and acceleomete. The eliability of ambiguity fixing was impoved by including a pioi infomation on the baseline length between both eceives into the tee seach. The poposed method was veified in a test dive with two single-fequency u-blox LEA 6T GPS eceives and the inetial senso MPU 950 of Invensense. The absolute position was found with centimete-level accuacy. A heading accuacy of 0.27 σ ) and of σ )was obseved fo a baseline length of.2 m. REFERENCES [] Zumbege, J. F., Heflin, M. B., Jeffeson, D. C., Watins, M. M., and Webb, F. H. Pecise point positioning fo the efficient and obust analysis of GPS data fom lage netwos. Jounal of Geophysical Reseach, 02, B3 Ma. 997), [2] Kouba, J., and Heoux, P. Pecise point positioning using IGS obit and cloc poducts. GPS Solutions, 5, 2 200), [3] Teunissen, P. 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16 In Poceedings of the 55th IEEE Symposium ELMAR, Zada, Coatia, Sep. 203, [29] Teunissen, P. The least-squaes ambiguity decoelation adjustment: A method fo fast GPS ambiguity estimation. Jounal of Geodesy, 70, 995), [30] Henel, P., and Iafancesco, M. Tightly coupled position and attitude detemination with two low-cost GNSS eceives. In Poceedings of the th IEEE Intenational Symposium on Wieless Communication Systems, Bacelona, Spain, Aug Patic Henel eceived his bachelo, maste, and Ph.D. with summa cum laude) degees fom the Technische Univesität München TUM), Gemany. He was a guest eseache at Delft Univesity of Technology TU Delft) in 2007, at Stanfod Univesity in 2008 and 200, and at Cutin Univesity in 203, and is cuently woing on his habilitation on pecise point positioning at TUM. D. Henel eceived the Piee Contensou Gold Medal in 2007, the fist pize in Bavaia at the Euopean Satellite Navigation Competition in 200, and the Dissetation Awad of the Vodafone Reseach Foundation in 20. D. Henel is also one of the foundes of ANavS Advanced Navigation Solutions, which develops ultapecise position and attitude detemination systems. HENKEL: TIGHTLY COUPLED PRECISE POINT POSITIONING AND ATTITUDE DETERMINATION 397