Chapter 2 - Kinematics 2.1 Kinematic Preliminaries 2.2 Transformations between BODY and NED 2.3 Transformations between ECEF and NED 2.4 Transformations between ECEF and Flat-Earth Coordinates 2.5 Transformations between BODY and FLOW “The study of dynamics can be divided into two parts: kinematics, which treats only geometrical aspects of BODY motion, and kinetics, which is the analysis of the forces causing the motion” Overall Goal of Chapters 2 to 10 Represent the 6-DOF equations of motion in a compact matrix-vector form according to: 1 Lecture Notes TTK 4190 Guidance, Navigation and Control of Vehicles (T. I. Fossen) Chapter Goals • Understand the geographic reference frame NED, the Earth-centered reference frame ECEF and the body-fixed reference frame BODY • Understand what FLOW axes are and why we use these axes for marine craft and aircraft • Be able to write down the differential equations relating BODY velocities to NED positions (Euler angles and unit quaternions) • Be able to: • Transform ECEF (x, y, z) positions to (longitude, latitude, height) and vice versa • Transform (longitude, latitude, height) to flat-Earth positions (x, y, z) and vice versa • Be able to define and visualize: • Angle of attack • Sideslip angle • Crab angle • Heading angle • Course angle 2 Lecture Notes TTK 4190 Guidance, Navigation and Control of Vehicles (T. I. Fossen) 2.1 Vector Notation (Fossen 2021) Coordinate-free vector (arrow notation) n n n !u " u1!n1 # u2!n2 # u3!n3 !ni !i " 1,2,3" are the unit vectors that define #n$ Coordinate form of !u in ! n " (bold notation) n n n n ! u ! !u1,u2,u3" ‘‘ The coordinate-free vector is expressed in {n}’’ 3 Lecture Notes TTK 4190 Guidance, Navigation and Control of Vehicles (T. I. Fossen) 2.1 Six-DOFs Motions The notation is adopted from: SNAME (1950). Nomenclature for Treating the Motion of a Submerged Body Through a Fluid. The Society of Naval Architects and Marine Engineers, Technical and Research Bulletin No. 1-5, April 1950, pp. 1-15. 4 Lecture Notes TTK 4190 Guidance, Navigation and Control of Vehicles (T. I. Fossen) 2.1 Generalized Coordinates For a marine craft not subject to any motion constraints Number of independent (generalized) coordinates = DOFs The term generalized coordinates refers to the parameters that describe the configuration of the craft relative to some reference configuration. For marine craft, the generalized position and velocity vectors are These quantities are all formulated in NED. It is advantageous to express the velocities of the craft in the BODY frame. In other words The relationship between the velocity vector in NED and BODY will be derived on the subsequent pages. 5 Lecture Notes TTK 4190 Guidance, Navigation and Control of Vehicles (T. I. Fossen) 2.1 Summary: 6-DOFs Vectors 6-DOF generalized position, velocity and force vectors expressed in {b} 6 Lecture Notes TTK 4190 Guidance, Navigation and Control of Vehicles (T. I. Fossen) zi, ze 2.1 Reference Frames we Earth-Centered Reference Frames ECI {i}: The Earth-centered inertial (ECI) frame {i} = (xi, yi, zi) is an inertial frame for terrestrial navigation, that is a nonaccelerating reference frame in which New- ton’s laws of ECEF/ECI we t motion apply. ye yi ECEF {e}: The Earth-centered Earth-fixed (ECEF) reference frame {e} = xi (xe, ye, ze) has its origin oe fixed to the center of the Earth but the axes rotate relative to the inertial frame ECI, which is fixed in space. xe xe - axis in the equatorial plane pointing towards the zero/prime meridian; same longitude as the Greenwich observatory ye - axis in the equatorial plane completing the right-hand frame ze - axis pointing along the Earth’s rotational axis −5 The Earth rotation is ωie = 7.2921 × 10 rad/s and the Earth’s rotational vector expressed in {e} is 7 Lecture Notes TTK 4190 Guidance, Navigation and Control of Vehicles (T. I. Fossen) 2.1 Reference Frames (cont.) ze BODY NED Geographic Reference Frames (Tangent Planes) N E D Geographical reference frames are usually chosen as tangent planes on the surface of the Earth. µ l ye ECEF/ECI • Terrestrial navigation: The tangent plane on the surface of the Earth moves with the craft and its location is specified by time- varying longitude-latitude values (l, μ). The tangent frame is xe usually rotated such that its axes points in the NED directions. • Local navigation: The tangent plane is fixed at constant values NED {n}: North-East-Down frame; defined (l0 , μ0 ) and the position is computed with respect to a local relative to the Earth’s reference coordinate origin. The axes of the tangent plane are usually ellipsoid (WGS 84), usually as the chosen to coincide with the NED axes. tangent plane to the ellipsoid “Flat-Earth navigation”: position is accurate to a smaller geographical area (10 km × 10 km). xb - axis points towards true North yb - axis points towards East zb - axis points downwards normal to the Earth’s surface 8 Lecture Notes TTK 4190 Guidance, Navigation and Control of Vehicles (T. I. Fossen) 2.1 Reference Frames (cont.) ze Body-Fixed Reference Frames N NED E The body-fixed reference frame {b} = (xb, yb, zb) D BODY with origin ob is a moving coordinate frame that is fixed to the craft. µ l y BODY {b}: For marine craft, the body axes are ECEF/ECI e chosen as: xb- longitudinal axis (directed from aft to fore) xe yb- transversal axis (directed to starboard) zb-normal axis (directed from top to bottom) FLOW {f}: Flow axes are used to align the x-axis with the craft’s velocity vector such that lift is perpendicular to the relative flow and drag is parallel. The transformation from FLOW to BODY axes is defined by two principal rotations where the rotation angles are the angle of attack α and the sideslip angle β. The main purpose of the flow axes is to simply the computations of lift and drag forces. 9 Lecture Notes TTK 4190 Guidance, Navigation and Control of Vehicles (T. I. Fossen) 2.1 Body-Fixed Reference Points The most important reference point: The following time-varying points are expressed with respect to the CO The CF is located a distance LCF from the CO in the x-direction The center of flotation is the centroid of the water plane area Awp in calm water. The vessel will roll and pitch about this point. 10 Lecture Notes TTK 4190 Guidance, Navigation and Control of Vehicles (T. I. Fossen) 2.2 Transformations between BODY and NED Orthogonal matrices of order 3: O!3" ! #R|R ! !3!3, RR" ! R!R ! I$ Special orthogonal group of order 3: SO!3" ! #R|R ! !3!3, R is orthogonal and detR !1$ Rotation matrix: Example: RR! ! R!R ! I, det R ! 1 Since R is orthogonal, R!1 ! R! 11 Lecture Notes TTK 4190 Guidance, Navigation and Control of Vehicles (T. I. Fossen) 2.2 Transformations between BODY and NED (cont.) Cross-product operator as matrix-vector multiplication: ! ! a :! S!!"a 0 !!3 !2 !1 ! S!!" ! !S !!" ! !3 0 !!1 , ! " !2 !!2 !1 0 !3 where S ! ! S ! is a skew-symmetric matrix The inverse operator is denoted: 12 Lecture Notes TTK 4190 Guidance, Navigation and Control of Vehicles (T. I. Fossen) 2.1 Transformations between BODY and NED (cont.) Euler’s theorem on rotation: 2 R!,"! I3#3 ! sin" S!"" ! !1 ! cos"" S !"" n n b n vb/n ! Rbvb/n, Rb :! R!," # ! ! ! !!1,!2,!3" , |!| ! 1 where 2 R11 ! !1 ! cos!" "1 " cos ! 2 R22 ! !1 ! cos!" "2 " cos ! ! 2 R33 ! !1 ! cos!" "3 " cos ! R12 ! !1 ! cos!" "1"2 ! "3 sin! R21 ! !1 ! cos!" "2"1 " "3 sin! R23 ! !1 ! cos!" "2"3 ! "1 sin! R32 ! !1 ! cos!" "3"2 " "1 sin! R31 ! !1 ! cos!" "3"1 ! "2 sin! R13 ! !1 ! cos!" "1"3 " "2 sin! 13 Lecture Notes TTK 4190 Guidance, Navigation and Control of Vehicles (T. I. Fossen) 2.1 Euler Angle Transformation x 3 x2 y u2 u U Three principal rotations: 3 (1) Rotation over yaw angle y about z3. Note that w =w . ! ! !0, 0, 1"! ! ! " 32 ! ! !0, 1, 0"! ! ! " v ! 3 y ! ! !1, 0, 0" ! ! " y 3 v2 y2 x1 u1 c! !s! 0 u q x 2 Rz,! ! s! c! 0 (2) Rotation over pitch 2 angle q about y2. w 0 0 1 U 2 Note that v21 =v . w1 c! 0 s! Ry,! ! 0 1 0 z1 z2 !s! 0 c! v 1 y 1 0 0 f 1 y0b =y v=v2 Rx,! ! 0 c! !s! w=w (3) Rotation over roll 0 angle f about x . 0 s c 1 ! ! w Note that u =u f 1 12 U . 14 z0b =z z1 Lecture Notes TTK 4190 Guidance, Navigation and Control of Vehicles (T. I. Fossen) 2.2 Euler Angle Transformation (cont.) Linear velocity transformation (zyx convention): Euler angle rotation matrix Small-angle approximation: 15 Lecture Notes TTK 4190 Guidance, Navigation and Control of Vehicles (T. I. Fossen) 2.2 Euler Angle Transformation (cont.) NED positions (continuous time and discrete time) Euler’s method with sampling time h Component form 16 Lecture Notes TTK 4190 Guidance, Navigation and Control of Vehicles (T. I. Fossen) 2.2 Euler Angle Transformation (cont.) Angular velocity transformation (zyx convention): where Small angle approximation: Notice that: o 1 0 !" 1. Singular point at ! ! " 90 !1 ! T!!!!nb" ! 0 1 "!# T! !!nb" " T!!!nb" 0 !# 1 17 Lecture Notes TTK 4190 Guidance, Navigation and Control of Vehicles (T. I. Fossen) 2.2 Euler Angle Transformation (cont.) Euler angle attitude representations ODE for Euler angles ODE for rotation matrix Component form where Must be combined with an algorithm for computation of the Euler angles from the rotation matrix 18 Lecture Notes TTK 4190 Guidance, Navigation and Control of Vehicles (T.