Integration of a strapdown gravimeter system in an Autonomous Underwater Vehicle

Transcription

1 Integration of a strapdown gravimeter system in an Autonomous Underwater Vehicle Clément ROUSSEL PhD - Student (L2G - Le Mans - FRANCE) April 17, 2015 Clément ROUSSEL ISPRS / CIPA Workshop April 17, / 24

2 Plan 1 Introduction Definition Interests & Applications Principle 2 Design & Equation Instrumentation & Carrier Equation of moving-base gravimetry 3 Performance assessment & Filtering Numerical simulations Filtering strategy 4 Further work Clément ROUSSEL ISPRS / CIPA Workshop April 17, / 24

3 Introduction What is gravimetry? scalar gravimetry: g vectorial gravimetry: g x g y g z Clément ROUSSEL ISPRS / CIPA Workshop April 17, / 24

4 Introduction Why do we measure gravity in the subsea domain? in geodesy: to improve the determination of the geoid in geophysics: to determine the distribution of masses in the ocean crust in navigation: to improve underwater navigation Clément ROUSSEL ISPRS / CIPA Workshop April 17, / 24

5 Introduction How do we measure gravity in the subsea domain? Gravity is usually measured in units of acceleration [1 m.s 2 = 10 5 mgal] An instrument used to measure gravity is known as a gravimeter One can regard gravimeters as special-purpose accelerometers Unmanned Underwater Vehicle (UUV) Clément ROUSSEL ISPRS / CIPA Workshop April 17, / 24

6 Design & Equation Instrumentation LiMo-g : Light Moving gravimetry system Geodesy and Geomatic Lab (L2G) & Geodesy Lab (LAREG) Doctoral Thesis of Bertrand de Saint-Jean (2008) GRAVIMOB : MOBile GRAVImetry system Oceanic Domains Lab (LDO) Marcia Maïa (Head scientist) & Jean-François d Eu (Instrumentation engineer) Clément ROUSSEL ISPRS / CIPA Workshop April 17, / 24

7 Design & Equation Instrumentation Strapdown sensor Six electrostatic accelerometers Two triads (α & β) installed in a waterproof sphere of about 40 cm diameter Clément ROUSSEL ISPRS / CIPA Workshop April 17, / 24

8 Design & Equation Carrier Autonomous Underwater Vehicule: AsterX IFREMER: French Research Institute for Exploitation of the Sea Navigation: INS + DVL + USBL able to dive down to 3, 000 m & travel up to 100 km total mass of 800 kg & scientific payload of 200 kg Clément ROUSSEL ISPRS / CIPA Workshop April 17, / 24

9 Design & Equation Equation of moving-base gravimetry Application of Newton s Second Law: X i α Ẍ i α = g i α + a i α (1) position vector of the proof mass M α Ẍ i α g i α a i α second-order derivative of X i α gravitationnal acceleration restoring force per unit of mass projected onto the i-frame Clément ROUSSEL ISPRS / CIPA Workshop April 17, / 24

10 Design & Equation Frames i-frame inertial-frame e-frame earth-frame (in rotation with respect to the i-frame) n-frame navigation-frame (defined by the geographic coordinates of the AUV: longitude λ P, latitude ϕ P and ellipsoidal height h P ) b-frame body-frame (defined by the attitude angles of the AUV: heading δ, pitch χ and roll η) Clément ROUSSEL ISPRS / CIPA Workshop April 17, / 24

11 Design & Equation Equation of moving-base gravimetry Only P position (λ P, ϕ P, h P ) is known: X i α = X i P + L i α = C i ex e P + C i bl b α (2) X P position vector of the vehicle reference point P L α C i e C i b lever arm PM α rotation matrix transforming e-frame into i-frame rotation matrix transforming b-frame into i-frame Clément ROUSSEL ISPRS / CIPA Workshop April 17, / 24

12 Design & Equation Equation of moving-base gravimetry (neglecting the rotation of the Earth) ( ) gα n = Ce n Ẍ P e + Cb n Ω b ebω b eb + Ω b eb L b α Cb n aα b (3) g n α gravitational vector at point M α C n e rotation matrix transforming e-frame into n-frame (λ P, ϕ P ) Ẍ e P second-order derivative of X e P (λ P, ϕ P ) C n b rotation matrix transforming b-frame into n-frame (δ P, χ P, η P ) Ω b eb skew symmetric matrix associated with the rotation of the b-frame with respect to the e-frame (λ P, ϕ P, δ P, χ P, η P ) L b α a b α lever arm PM b α restoring force per unit of mass Clément ROUSSEL ISPRS / CIPA Workshop April 17, / 24

13 Design & Equation Equation of moving-base gravimetry (neglecting the rotation of the Earth) ( gα n = Ce n Ẍ P e + Cb n Ω b ebω b eb + Ω ) b eb L b α Cb n aα b (4) g n β = C n e Ẍ e P + C n b And g n P? Under the assumption that L α = L β : ( Ω b ebω b eb + Ω ) b eb L b β Cb n aβ b (5) gp n g α n + gβ n 2 ( ) a b gp n Ce n Ẍ P e Cb n α + aβ b 2 (6) (7) Clément ROUSSEL ISPRS / CIPA Workshop April 17, / 24

14 Performance assessment & Filtering Numerical simulations (Monte-Carlo) g n P C n e Ẍ e P C n b f : λ P, ϕ P, h P δ, χ, η a α, a β ( a b α + a b β f multivariate function mapping R 12 into R 3 2 ) (8) g P (9) Clément ROUSSEL ISPRS / CIPA Workshop April 17, / 24

15 Performance assessment & Filtering Numerical simulations (Monte-Carlo) λ P + ɛ λ, ϕ P + ɛ ϕ, h P + ɛ h f : δ + ɛ δ, χ + ɛ χ, η + ɛ η a α + ɛ α, a β + ɛ β g P + ɛ g (10) ɛ θ ɛ g N random draws additive noise term, θ = λ, ϕ, h, δ, χ, η error on gravity vector E[ɛ g ] = 1 N ɛ g,i (11) N i=1 σ[ɛ g ] = E [(ɛ g E[ɛ g ]) 2 ] (12) Clément ROUSSEL ISPRS / CIPA Workshop April 17, / 24

16 Performance assessment & Filtering Numerical simulations (Monte-Carlo) Reference gravity field derives from a geological model of oceanic crust bathymetric survey (2.70 g.cm 3 ) distribution of mineral blocks (3.85 g.cm 3 ) Reference trajectory of the AUV derives from a test mission carried out by the IFREMER deterministic models polynomial & periodic functions 12 profiles, each about 3, 600 m long, h P = 2200 m Clément ROUSSEL ISPRS / CIPA Workshop April 17, / 24

17 Performance assessment & Filtering Numerical simulations (Monte-Carlo) 3 Clément ROUSSEL ISPRS / CIPA Workshop April 17, / 24

18 Performance assessment & Filtering Numerical simulations (Monte-Carlo): effect of the AUV positionning uncertainty uncertainty = 0.1 % of the travelled distance (manufacturer s informations) ɛ λ & ɛ ϕ are modelled by a double integration of a Gaussian White Noise process ɛ h is modelled by a simple Gaussian White Noise process Results are consistent with the fact that the uncertainty affecting the coordinates h P is more likely to perturb the restitution of the vertical component g u Clément ROUSSEL ISPRS / CIPA Workshop April 17, / 24

19 Performance assessment & Filtering Numerical simulations (Monte-Carlo): effect of the AUV attitude uncertainty uncertainty = 0.02 deg (δ) & 0.01 deg (χ & η) (manufacturer s informations) ɛ δ, ɛ χ & ɛ η are modelled by a Gaussian White Noise process Components g e & g n are more affected by the attitude uncertainty than the vertical component g u Clément ROUSSEL ISPRS / CIPA Workshop April 17, / 24

20 Performance assessment & Filtering Numerical simulations (Monte-Carlo): effect of the accelerometer uncertainties Stochastic processes are identified thanks to Allan Variance ɛ α & ɛ β are modelled by a White Noise and a first order Random Walk processes The low pass filtering has no effect on the noise reducing because of the non-stationnary nature of the first order random walk process. Clément ROUSSEL ISPRS / CIPA Workshop April 17, / 24

21 Performance assessment & Filtering Filtering strategy Kalman Filter only for linear system Extented Kalman Filter needs Jacobian matrix to be estimated (implementation errors & time-consuming) may lead to an improper estimation of the covariance matrix if the linearity hypothesis is not respected (divergence of the filter) Unscented Kalman Filter does not require the calculation of Jacobian matrix relies on a deterministic sampling method Clément ROUSSEL ISPRS / CIPA Workshop April 17, / 24

22 Performance assessment & Filtering Filtering strategy Clément ROUSSEL ISPRS / CIPA Workshop April 17, / 24

23 Further Work Calibration scale factor & bias of each accelerometer transformation matrix C b s Improve Unscented Kalman Filtering complex noise models spatial variability of the gravity field Test mission scheduled on March 2016 off the Mediterrean coasts in the south of France Clément ROUSSEL ISPRS / CIPA Workshop April 17, / 24

24 Thank you for your attention We are indebted to the French Ministry of Defence and the Pays de la Loire Region for their support of this work. Jérôme Verdun - supervisor Marcia Maïa - co-supervisor José Cali Jean François d Eu clement.roussel@cnam.fr Clément ROUSSEL ISPRS / CIPA Workshop April 17, / 24