Modelling and Simulation of Marine Craft KINEMATICS
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1 Modellig ad Simulatio of Marie Craft KINEMATICS Edi Omerdic Seior Research Fellow
2 Outlie Kiematics Referece Frames ECEF Frame NED Frame BODY Frame UNITY Frame UTM Trasformatios BODY NED Euler Agles Quaterios Quaterios from Euler Agles Euler Agles from Quaterios Trasformatios ECEF NED Lat-Lo from ECEF Coordiates ECEF Coordiates from Lat-Lo
3 GNC Sigal Flow Waypoits Weather routig program Weather data Waves wid ad ocea currets Trajectory Geerator Motio Cotrol System Fusio Cotrol Marie Craft INS (GNSS + AHRS) Allocatio OA Sesors (Radar Cameras etc.) GNC Sigal Flow Ostacle Avoidace Module Oserver Guidace System Cotrol System Estimated positios ad velocities Navigatio System
4 Kiematics Referece Frames ECEF Frame Symol e = (x e y e z e ) Defiitio The Earth-cetered Earth-fixed frame which rotate aroud North-South axis with agular speed w e = rad/s Type Right-haded Axes x e (from origi through itersectio of Prime Meridia ad Equator) Origi Assumptio Applicatio y e z e (from origi through North pole) Locatio of ECEF origi o e is fixed to the Cetre of the Earth. For marie craft movig at relatively low speed the Earth rotatio ca e eglected ad e ca e cosidered to e iertial such that Newto's laws apply. Typically used for gloal guidace avigatio ad cotrol (for example to descrie motio ad locatio of ships i trasit etwee cotiets).
5 Kiematics Referece Frames NED Frame λ Logitude φ Latitude z e x y z x e λ φ Equator y e Symol = (x y z ) Defiitio Taget plae o the surface of Earth referece ellipsoid WGS Type Right-haded Axes x = North y = East z = Dowwards ormal to the surface. Origi Locatio of NED origi o relative to ECEF frame e = (x e y e z e ) is determied usig Lat ad Lo. Assumptio is iertial such that Newto's laws apply. Applicatio The positio ad orietatio of the craft are descried relative to.
6 Kiematics Referece Frames BODY Frame Symol = (x y z ) Defiitio Movig frame fixed to the craft. Type Right-haded Axes x = logitudial axis (directed toward frot); y = trasversal axis (directed toward staroard); z = ormal axis (directed toward ottom) Origi Locatio of BODY origi o is usually chose to coicide with a poit midships i the water lie for marie crafts. Applicatio The liear ad agular velocities of the vessel are descried relative to.
7 Kiematics Referece Frames UNITY Frame Symol u = (x u y u z u ) Defiitio Frame used for 3D visualisatio i Uity 3D. Type Left-haded Axes x u = East y u = Up z u = North. Origi Locatio of UNITY origi o u relative to ECEF frame e = (x e y e z e ) is determied usig Lat ad Lo. Applicatio The positio ad orietatio of all vessels are set to Uity 3D for visualisatio.
8 UTM Kiematics Referece Frames Symol UTM = (x UTM y UTM z UTM ) Defiitio Type Axes Origi The UTM system divides the Earth ito 60 zoes ad uses a secat trasverse Mercator projectio i each zoe (coformal projectio) to give locatio o the surface of the Earth. Each zoe is segmeted ito 20 latitude ads. Right-haded x UTM = Northig y UTM = Eastig z UTM = Dowwards ormal to the surface. Locatio of UTM origi o UTM is shifted such that Northig ad Eastig coordiates are always positive. For this purpose artificial offset False Eastig is added i Norther hemisphere while oth False Eastig ad False Northig are used i Souther hemisphere. I vertical plae the UTM origi is at sea level.
9 Kiematics Referece Frames UTM (cot.)
10 Kiematics Trasformatios etwee Frames Notatio p UTM /UTM = v / = θ = x UTM y UTM z UTM u v w R P Y Positio of vessel o i UTM Liear velocity of o with respect to {} expressed i {} Attitude (Euler agles) etwee {} ad {} UTM p /UTM Positio of NED origi o i UTM R θ Trasformatio matrix of liear velocity vector v / v / i {} from {} to p / w / = x y z Positio of vessel o i NED v / Liear velocity of o with respect to {} expressed i {} = ሶ θ = p q r Rሶ P ሶ Y ሶ Agular velocity of with respect to {} expressed i {} Euler rate vector T θ θ Trasformatio matrix of agular velocity vector w / from {} to Euler rate vector ሶ θ i {}
11 Kiematics Trasformatios etwee Frames Euler s Rotatio Theorem: Geometry perspective Ay displacemet of a rigid-ody i 3D such that a poit o the rigid ody remais fixed is equivalet to a sigle rotatio aout some axis that passes through the fixed poit. Liear Algera perspective Ay two Cartesia coordiate systems i 3D with a commo origi are related y a rotatio aout some fixed axis passig through the origi.
12 ሶ Kiematics Trasformatios BODY-NED Euler Agles Liear Velocity Trasformatio p Τ = v Τ = R θ v Τ = R zy R yp R xr v Τ R xr = cr sr 0 sr cr R yp = cp 0 sp sp 0 cp R zy = cy sy 0 sy cy R θ 1 = R θ T R θ = cycp sycr + cyspsr sysr + cycrsp sycp cycr + syspsr cysr + spsycr sp cpsr cpcr
13 ሶ Kiematics Trasformatios BODY-NED Euler Agles Agular Velocity Trasformatio θ = T θ θ w Τ 1 srtp crtp = 0 cr sr 0 s RΤc P c RΤc P w Τ Sigularity for P=±90
14 Kiematics Trasformatios BODY-NED Euler Agles 6 DOF Kiematic Equatios ሶ p Τ ሶ θ = R θ T θ θ v Τ w Τ ሶ η = J Θ η ν Simulatio Model Attitude represetatio: Euler Agles r q p w v u / / w v v θ p η / Y P R z y x θ p η / Gai Itegrator / θ p η KINEMATICS
15 Kiematics Trasformatios BODY-NED Quaterios H R 4 Basis: 1ijk Scalar part q = a1 + i + cj + dk Real quaterios: a1 + 0i + 0j + 0k Pure quaterios: 01 + i + cj + dk Vector part Additio: Scalar Multiplicatio: Quaterio Multiplicatio: a i + c 1 j + d 1 k + a i + c 2 j + d 2 k = a 1 + a i + c 1 + c 2 j + d 1 + d 2 k α a1 + i + cj + dk = αa1 + αi + αcj + αdk a i + c 1 j + d 1 k a i + c 2 j + d 2 k = a 1 a c 1 c 2 d 1 d a a 2 + c 1 d 2 d 1 c 2 i + a 1 c 2 1 d 2 + c 1 a 2 + d 1 2 j + a 1 d c 2 c d 1 a 2 k
16 Kiematics Trasformatios BODY-NED Quaterios Scalar part q = r Ԧv Vector part Additio: r 1 Ԧv 1 + r 2 Ԧv 2 = r 1 + r 2 Ԧv 1 + Ԧv 2 Scalar Multiplicatio: α r Ԧv = αr α Ԧv Quaterio Multiplicatio: r 1 Ԧv 1 r 2 Ԧv 2 = r 1 r 2 Ԧv 1 Ԧv 2 r 1 Ԧv 2 + r 2 Ԧv 1 + Ԧv 1 Ԧv 2
17 Kiematics Trasformatios BODY-NED Quaterios Scalar part q = r Ԧv = r1 + v x i + v y j + v z k Cojugate: Vector part q = r Ԧv = r1 v x i v y j v z k qq = r Ԧv r Ԧv = r 2 + Ԧv 2 0 = r 2 + v 2 x + v 2 y + v 2 z 1 + 0i + 0j + 0k q q = r Ԧv r Ԧv = r 2 + Ԧv 2 0 = qq q 1 q 2 = q 2 q 1 Norm: q = qq = r 2 + Ԧv 2 = r 2 + v x 2 + v y 2 + v z 2 pq 2 = pq pq = pqq p = p q 2 p = pp q 2 = p 2 q 2 Iverse: q 1 = q q 2 q 1 q = qq 1 = 1
18 Kiematics Trasformatios BODY-NED Quaterios Uit Quaterio: q = 1 r 2 + Ԧv 2 = 1 θ 0 π : cos θ = r si θ = Ԧv cos θ si θ q = 1 q = r Ԧv = r + Ԧv Ԧv Ԧv = cos θ + si θ Ԧ v = e 0θ v Polar Decompositio: q = q Uit Quaterio Uit Quaterio q q = q cos θ + si θ Ԧ v = q e Ԧ v Uit Quaterio 0θ v Power: q p = q p e 0 pθ v = q p cos pθ + si pθ Ԧ v Euler Formula (Geeral Case): e rv = e r0 + 0 v v = e r0 e 0θ v = e r e 0θ v = e r cos θ si θ Ԧ v
19 Kiematics Trasformatios BODY-NED Quaterios Euler Formula ( Pure Quaterio): q = 0 Ԧv q 2 = 0 Ԧv 0 Ԧv = Ԧv 2 0 q 3 = Ԧv Ԧv = 0 Ԧv 2 Ԧv q 4 = 0 Ԧv 2 Ԧv 0 Ԧv = Ԧv 4 0 q 5 = Ԧv Ԧv = 0 Ԧv 4 Ԧv q 6 = 0 Ԧv 4 Ԧv 0 Ԧv = Ԧv 6 0 q 7 = Ԧv Ԧv = 0 Ԧv 6 Ԧv e q = 1 + q + q2 2! + q3 3! + q4 4! + q5 5! + q6 6! + q7 7! e 0v = Ԧv + Ԧv 2 0 2! e 0v = 1 Ԧv 2 2! + Ԧv 4 4! + 0 Ԧv 2 Ԧv 3! 1 Ԧv 2 3! + Ԧv 4 5! + Ԧv 4 0 4! Ԧv Ԧv + 0 Ԧv 4 Ԧv 5! = cos Ԧv si Ԧv Ԧ v
20 Kiematics Trasformatios BODY-NED Quaterios Special case: Rotatio Aout Perpedicular Axis Ԧv = cos θ Ԧv + si θ Ԧv Ԧv = 1 0 Ԧv θ Ԧv Ԧv si θ Ԧv 0 θ cos θ Ԧv Ԧv Ԧv Solutio usig Quaterios: v = 0 Ԧv = 0 v = 0 0 Ԧv = 0 Ԧv 2 = 0 0 = 1 0 v = 0 Ԧv = 0 cos θ Ԧv + si θ Ԧv = cos θ si θ 0 Ԧv = e 0θ 0 Ԧv = e 0θ v v = e 0θ v
21 Kiematics Trasformatios BODY-NED Quaterios Rotatio Aout Geeral Axis Ԧv = Ԧv si θ cos θ Ԧv Ԧv = cos θ + si θ si θ 0 θ cos θ Ԧv 0 θ Ԧv = Ԧv + = Ԧv + cos θ + si θ
22 Kiematics Trasformatios BODY-NED Quaterios Vector Projectio oto Lie Ԧv Ԧv Ԧv = α Ԧv = Ԧv + = Ԧv Ԧv = Ԧv α = 0 Ԧv α = 0 Ԧv α = 0 α = Ԧv = T Ԧv T Ԧv = α = Ԧv = T Ԧv T = T Ԧv T = T T Ԧv Scalar Projectio Matrix
23 Kiematics Trasformatios BODY-NED Quaterios Special Case: = = 1 Ԧv Ԧv Ԧv = Ԧv = Ԧv 2 = Ԧv Scalar = Ԧv = T T Ԧv = T 2 Ԧv = T Ԧv Projectio Matrix
24 Kiematics Trasformatios BODY-NED Quaterios Special Case: = Ԧq = si θ 2 Ԧq = si θ 2 Ԧv Ԧv Ԧv = Ԧq Ԧv Ԧq Ԧq Ԧq Ԧv Ԧq Ԧv Ԧq = Ԧq 2 Ԧq = si 2 θ 2 Ԧq Scalar = Ԧq = si θ 2 Ԧv = Ԧq ԦqT Ԧq T Ԧq Ԧv = Ԧq ԦqT Ԧq 2 Ԧv = Ԧq ԦqT si 2 θ 2 Ԧv Projectio Matrix
25 Kiematics Trasformatios BODY-NED Quaterios Rotatio Aout Geeral Axis 1 cos θ Ԧv cos θ Ԧv si θ Ԧv = Ԧv 0 Ԧv θ Ԧv Ԧv = cos θ + si θ si θ 0 θ cos θ Ԧv Ԧv = Ԧv + = Ԧv + cos θ + si θ = Ԧv + cos θ Ԧv Ԧv + si θ Ԧv Ԧv 0 = 1 cos θ Ԧv + cos θ Ԧv + si θ Ԧv si θ Ԧv Ԧv = Ԧv Ԧv Rodrigues Rotatio Formula: Ԧv = cos θ Ԧv + 1 cos θ Ԧv + si θ Ԧv
26 Kiematics Trasformatios BODY-NED Quaterios Rotatio Aout Geeral Axis: Solutio usig Quaterios v = 0 Ԧv = 0 Lema 1: e 0θ v = v e v = 0 Ԧv v = 0 Lema 2: e 0θ v = v e v = 0 Ԧv 0θ 0 θ Ԧv = Ԧv Ԧv Ԧv Ԧv = Ԧv + = Ԧv + v = 0 Ԧv = 0 Ԧv + = 0 Ԧv + v = v + v = v + e 0θ v Ԧq = si θ 2 0 θ v = e 0θ 2 e 0 θ 2 v + e 0θ 2 e 0θ 2 v = e 0θ 2 v e 0 θ 2 + e 0θ 2 v e 0 θ 2 Lema 1 Lema 2 v = e 0θ 2 v + v e 0 θ 2 = e 0θ 2 ve 0 θ 2 Ԧv = Ԧq Ԧv Ԧq q Ԧq
27 Kiematics Trasformatios BODY-NED Quaterios Rodrigues Rotatio Formula: Ԧv = cos θ Ԧv + 1 cos θ Ԧv + si θ Ԧv Ԧv = 1 cos θ Ԧv Ԧv + Ԧv + cos θ Ԧv + si θ Ԧv = 1 cos θ Ԧv Ԧv + 1 cos θ + cos θ Ԧv + si θ Ԧv Ԧv = Ԧv + 1 cos θ Ԧv Ԧv + si θ Ԧv si θ Ԧv Ԧv 1 cos θ Ԧv Ԧv = Ԧv Ԧv Ԧv S v Ԧv = Ԧv Ԧv = S S v = S 2 v = T I 3 3 v = T v I 3 3 v = v v = v 0 θ v = I cos θ S 2 + si θs v R θ
28 Rotatio Formula usig Quaterios: Kiematics Trasformatios BODY-NED Quaterios q = e 0θ 2 = cos θ 2 si θ 2 = q 0 Ԧq q Ԧq 2 = 1 v = e 0θ 2 v + v e 0 θ 2 = e 0θ 2 ve 0 θ 2 = qvq v = 0 I cos θ S 2 + si θs Ԧv = = 0 I si 2 θ 2 S2 + 2 si θ 2 cos θ S Ԧv 2 v = 0 I S 2 si θ cos θ 2 S si θ Ԧv 2 2q 0 S Ԧq Ԧv Ԧv 2S 2 Ԧq Ԧv v = 0 I S 2 Ԧq + 2q 0 S Ԧq Ԧv R q = R q0 q Ԧv = Ԧv Ԧv S 2 Ԧq = Ԧq Ԧq T Ԧq 2 I 3 3 Ԧq = si θ 2 0 θ
29 Kiematics Trasformatios BODY-NED Quaterios Wrog Formula: v = e 0θ v + v = e 0θ v + e 0θ v Ԧv Ԧv Ԧv Ԧv Ԧv Correct Formula: v = e 0θ 2 v + v e 0 θ 2 = 0 Ԧv Ԧp θ Ԧv Ԧv = e 0θ 2 v e 0 θ 2 + e 0θ 2 v e 0 θ 2 Ԧp Ԧv = Ԧv Ԧp Ԧp Ԧv Ԧv Step 1: 0 Ԧp Step 2: 0 Ԧp
30 Rotatio Formula usig Quaterios: Kiematics Trasformatios BODY-NED Quaterios Quaterio Rotatio Operator: q = e 0θ 2 = cos θ 2 si θ 2 = q 0 Ԧq q Ԧq 2 = 1 v = e 0θ 2 v + v e 0 θ 2 = e 0θ 2 ve 0 θ 2 = qvq v = L q v qvq = q 0 Ԧq 0 Ԧv q 0 Ԧq = 0 q 0 2 Ԧq 2 Ԧv + 2 Ԧq Ԧv Ԧq + 2q 0 Ԧq Ԧv q 0 2 Ԧq 2 Ԧv = q Ԧq 2 2 Ԧq 2 Ԧv = 1 2si 2 θ 2 Ԧv = cos θ Ԧv 1 cos θ Ԧv = 2si 2 θ 2 = 2si 2 θ 2 Ԧq Ԧv si 2 θ 2 Ԧq Ԧv Ԧq q Ԧq = 2si2 θ 2 Ԧq = 2 Ԧq Ԧv Ԧq = 2 Ԧq Ԧq T Ԧv = 2 Ԧq Ԧq T Ԧv Scalar v = 0 q 0 2 Ԧq 2 I Ԧq Ԧq T + 2q 0 S Ԧq Ԧv Ԧq Ԧv 2q 0 S Ԧq Ԧv Ԧq 2 Ԧq = 2 Ԧq Ԧq T Ԧv = 2 Ԧq Ԧv Ԧq Projectio Matrix q 0 2 Ԧq 2 Ԧv = Ԧv Ԧv Ԧq = si θ 2 0 θ Ԧv Ԧv
31 ሶ Kiematics Trasformatios BODY-NED Quaterios Liear Velocity Trasformatio p Τ = v Τ = R q v Τ R q = R q0 q = I S 2 Ԧq + 2q 0 S Ԧq q = q 0 Ԧq = q q 1 i + q 2 j + q 3 k q 0 = cos θ 2 Ԧq = si θ 2 R q 1 = R q T R q = 1 2 q q 3 2 q 1 q 2 q 3 q 0 2 q 1 q 3 + q 2 q 0 2 q 1 q 2 + q 3 q q q 1 2 q 2 q 3 q 1 q 0 2 q 1 q 3 q 2 q 0 2 q 2 q 3 + q 1 q q q 2
32 ሶ ሶ Kiematics Trasformatios BODY-NED Quaterios Agular Velocity Trasformatio R q = R q S w Τ q = T q q w Τ q = q 0 Ԧq = q q 1 i + q 2 j + q 3 k q 0 = cos θ 2 T q T q T q q = 1 4 I 3 3 Ԧq = si θ 2 T q q = 1 2 Ԧq T q 0 I S Ԧq T q q = 1 2 q 1 q 2 q 3 q 0 q 3 q 3 q 0 q 2 q 1 q 2 q 1 q 0
33 Kiematics Trasformatios BODY-NED Quaterios versus Euler Agles Agular Velocity Trasformatio T q q = 1 2 Quaterios q 1 q 2 q 3 q 0 q 3 q 3 q 0 q 2 q 1 T θ θ = q 2 q 1 q 0 Euler Agles 1 srtp crtp 0 cr sr 0 s RΤc P c RΤc P Sigularity for P=±90
34 Kiematics Trasformatios BODY-NED Quaterios 6 DOF Kiematic Equatios ሶ p Τ qሶ = R q T q q v Τ w Τ ሶ η = J q η ν Simulatio Model Attitude represetatio: Quaterios r q p w v u / / w v v q p η / / q q q q z y x q p η Gai Itegrator / q p η KINEMATICS
35 Kiematics Trasformatios BODY-NED Quaterios from Euler Agles Euler Agles Quaterios q = q 0 q 1 q 2 q 3 = c R/2 c P/2 c Y/2 + s R/2 s P/2 s Y/2 s R/2 c P/2 c Y/2 c R/2 s P/2 s Y/2 c R/2 s P/2 c Y/2 + s R/2 c P/2 s Y/2 c R/2 c P/2 s Y/2 s R/2 s P/2 c Y/2
36 Kiematics Trasformatios BODY-NED Euler Agles from Quaterios Quaterios Euler Agles θ = R P Y = ata2 2 q 2 q 3 + q 1 q q q 2 2 asi 2 q 0 q 2 q 3 q 1 ata2 2 q 0 q 3 + q 1 q q q 3 2
37 ሶ ሶ e p Τe Kiematics Trasformatios ECEF-NED Lat-Lo Trasformatios NED ECEF = R e θ e R θ v Τ λ Logitude φ Latitude e z e = R zλ R y φ π 2 x R e θ e y R zy = p Τ R y φ π 2 cλ sλ 0 sλ cλ = c φ π 2 0 s φ π s φ π 2 0 c φ π 2 x e z φ λ Equator y e
38 ሶ ሶ ሶ ሶ ሶ ሶ ሶ ሶ ሶ ሶ Kiematics Trasformatios ECEF-NED Lat-Lo Trasformatios NED ECEF e p Τe = R e θ e ሶ p Τ ECEF NED ECEF NED p Τ = R e θ e ሶ e p Τe NED ECEF Speed & Headig = cλsφ sλ cλcφ sλsφ cλ sλcφ cφ 0 sφ = R e T θ e ሶ R e θ e e p Τe = x y z p Τ North Velocity East Velocity Dow Velocity cλsφ sλsφ cφ sλ cλ 0 cλcφ sλcφ sφ R e θ e x e y e z e e p Τe Speed = xሶ yሶ Headig = ta 1 y x
39 Kiematics Trasformatios ECEF-NED Geodetic Latitude vs Geocetric Latitude Geodetic Latitude vs Geocetric Latitude Raw GNSS (GPS GLONASS ad Gallileo) output e p Τe is give i Cartesia ECEF frame. It is preseted to users i terms of logitude λ latitude φ ad height h relative to WGS-84 ellipsoid. φ Geodetic Latitude ψ Geocetric Latitude ψ φ = ta 1 1 e 2 ta φ North Pole Normal h WGS-84 parameters: a = m semi major axis Taget = m semi mior axis Ellipsoid ψ φ N e = a2 2 a 2 eccetricity a e = a2 2 2 f = a /a "flatteig" parameter e 2 = 2f f 2
40 Kiematics Trasformatios ECEF-NED Height Datums 1. Mea Sea Level (MSL) Datum (height relative to geoid) 2. GPS height (height aove WGS-84 ellipsoid) h ellipsoidal height geodetic H orthometric height MSL M geoid separatio udulatio ε d deflectio of the vertical 0 M 100m h H + M h H ε d M Earth s surface Geoid WGS-84 Ellipsoid
41 Kiematics Trasformatios ECEF-NED ECEF Coordiates from Lat-Lo Lat-Lo ECEF Coordiates N = a 2 a 2 cos 2 φ + 2 si 2 φ Prime Vertical of Curvature (m) e p Τe = x e y e z e = N + h cos φ cos λ N + h cos φ si λ 2 a 2 N + h si λ N
42 Kiematics Trasformatios ECEF-NED Lat-Lo from ECEF Coordiates ECEF Coordiates Lat-Lo p = x e 2 + y e 2 θ = ta 1 z ea p λ = ta 1 y e x e φ = ta 1 z e + e 2 si 3 θ p e 2 acos 3 θ h = p cos φ N
43
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