Case 2:17-cv-00140-RWS-RSP Document 66-5 Filed 02/23/18 Page 1 of 45 PageID #: 1684
`Case 2:17-cv—00140-RWS—RSP Document 66-5 Filed 02/23/18 Page 1 of 45 PageID #: 1684
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`EXHIBIT E
`EXHIBIT E
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`

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`Case 2:17-cv-00140-RWS-RSP Document 66-5 Filed 02/23/18 Page 2 of 45 PageID #: 1685
`
`Chapter 2
`Basic Navigational Mathematics,
`Reference Frames and the Earth’s
`Geometry
`
`Navigation algorithms involve various coordinate frames and the transformation of
`coordinates between them. For example, inertial sensors measure motion with
`respect to an inertial frame which is resolved in the host platform’s body frame.
`This information is further transformed to a navigation frame. A GPS receiver
`initially estimates the position and velocity of the satellite in an inertial orbital
`frame. Since the user wants the navigational information with respect to the Earth,
`the satellite’s position and velocity are transformed to an appropriate Earth-fixed
`frame. Since measured quantities are required to be transformed between various
`reference frames during the solution of navigation equations, it is important to
`know about the reference frames and the transformation of coordinates between
`them. But first we will review some of the basic mathematical techniques.
`
`2.1 Basic Navigation Mathematical Techniques
`
`This section will review some of the basic mathematical techniques encountered in
`navigational computations and derivations. However, the reader is referred to
`(Chatfield 1997; Rogers 2007 and Farrell 2008) for advanced mathematics and
`derivations. This section will also introduce the various notations used later in the
`book.
`
`2.1.1 Vector Notation
`
`In this text, a vector is depicted in bold lowercase letters with a superscript that
`indicates the coordinate frame in which the components of the vector are given.
`The vector components do not appear in bold, but they retain the superscript. For
`example, the three-dimensional vector r for a point in an arbitrary frame k is
`depicted as
`
`A. Noureldin et al., Fundamentals of Inertial Navigation, Satellite-based
`Positioning and their Integration, DOI: 10.1007/978-3-642-30466-8_2,
`Ó Springer-Verlag Berlin Heidelberg 2013
`
`21
`
`

`

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`2 Basic Navigational Mathematics, Reference Frames
`
`35
`
`xk
`yk
`zk
`
`24
`
`rk ¼
`
`ð2:1Þ
`
`22
`
`In this notation, the superscript k represents the k-frame, and the elements
`ðxk; yk; zkÞ denote the coordinate components in the k-frame. For simplicity, the
`superscript is omitted from the elements of the vector where the frame is obvious
`from the context.
`
`2.1.2 Vector Coordinate Transformation
`
`Vector transformation from one reference frame to another is frequently needed in
`inertial navigation computations. This is achieved by a transformation matrix.
`A matrix is represented by a capital letter which is not written in bold. A vector of
`any coordinate frame can be represented into any other frame by making a suitable
`transformation. The transformation of a general k-frame vector rk into frame m is
`given as
`
`rm ¼ Rm
`k rk
`where Rm
`k represents the matrix that transforms vector r from the k-frame to the
`m-frame. For a valid transformation, the superscript of the vector that is to be
`transformed must match the subscript of the transformation matrix (in effect they
`cancel each other during the transformation).
`The inverse of a transformation matrix Rm
`k describes a transformation from the
`
`
`m-frame to the k-frame
`ð2:3Þ
`rm ¼ Rk
`rk ¼ Rm
`mrm
`If the two coordinate frames are mutually orthogonal, their transformation
`matrix will also be orthogonal and its inverse is equivalent to its transpose. As all
`the computational frames are orthogonal frames of references, the inverse and the
`transpose of their transformation matrices are equal. Hence for a transformation
`matrix Rm
`
`
`
`
`k we see that
`ð2:4Þ
`T¼ Rk
`k ¼ Rk
`Rm
`A square matrix (like any transformation matrix) is orthogonal if all of its
`vectors are mutually orthogonal. This means that if
`
`ð2:2Þ
`
`ð2:5Þ
`
`1
`
`k
`
`1
`
`m
`
`m
`
`35
`
`r11
`r21
`r31
`
`r12
`r22
`r32
`
`r13
`r23
`r33
`
`24
`
`R ¼
`
`

`

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`23
`
`ð2:6Þ
`
`ð2:7Þ
`
`2.1 Basic Navigation Mathematical Techniques
`
`35
`
`r13
`r23
`r33
`
`24
`
`35
`
`; r3 ¼
`
`r12
`r22
`r33
`
`24
`
`; r2 ¼
`
`35
`
`r11
`r21
`r31
`
`24
`
`r1 ¼
`
`where
`
`then for matrix R to be orthogonal the following should be true
`r1 r2 ¼ 0; r1 r3 ¼ 0; r2 r3 ¼ 0
`
`2.1.3 Angular Velocity Vectors
`
`The angular velocity of the rotation of one computational frame about another is
`represented by a three component vector x: The angular velocity of the k-frame
`relative to the m-frame, as resolved in the p-frame, is represented by xp
`mk as
`
`ð2:8Þ
`
`35
`
`xx
`xy
`xz
`
`24
`
`mk ¼
`xp
`
`where the subscripts of x denote the direction of rotation (the k-frame with respect
`
`
`to the m-frame) and the superscripts denote the coordinate frame in which the
`components of the angular velocities xx; xy; xz
`are given.
`The rotation between two coordinate frames can be performed in two steps and
`expressed as the sum of the rotations between two different coordinate frames, as
`shown in Eq. (2.9). The rotation of the k-frame with respect to the p-frame can be
`performed in two steps: firstly a rotation of the m-frame with respect to the
`p-frame and then a rotation of the k-frame with respect to the m-frame
`
`pk ¼ xkpm þ xk
`xk
`For the above summation to be valid, the inner indices must be the same (to
`cancel each other) and the vectors to be added or subtracted must be in the same
`reference frame (i.e. their superscripts must be the same).
`
`ð2:9Þ
`
`mk
`
`2.1.4 Skew-Symmetric Matrix
`
`The angular rotation between two reference frames can also be expressed by a
`skew-symmetric matrix instead of a vector. In fact this is sometimes desired in
`order to change the cross product of two vectors into the simpler case of matrix
`multiplication. A vector and the corresponding skew-symmetric matrix forms of an
`angular velocity vector xp
`mk are denoted as
`
`

`

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`2 Basic Navigational Mathematics, Reference Frames
`xz
`xy
`0
`xx
`) Xpmk ¼
`
`xz
`0
`|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
`xy xx
`0
`Skewsymmetric form of angular the velocity vector
`
`35
`
`24
`
`ð2:10Þ
`
`35
`
`24
`
`xx
`mk ¼
`xp
`xy
`|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}
`xz
`
`Angular velocity vector
`
`35
`
`24
`
`vx
`vp ¼
`vy
`|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}
`vz
`
`24
`
`as
`
`Similarly, a velocity vector vp can be represented in skew-symmetric form Vp
`
`ð2:11Þ
`
`35
`
`24
`
`vz
`0
`vy
`0 vx
`) Vp ¼
`vz
`|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
`vy
`0
`vx
`Skewsymmetric form of the velocity vector
`
`Velocity vector
`
`Note that the skew-symmetric matrix is denoted by a non-italicized capital
`letter of the corresponding vector.
`
`2.1.5 Basic Operations with Skew-Symmetric Matrices
`
`Since a vector can be expressed as a corresponding skew-symmetric matrix, the
`rules of matrix operations can be applied to most vector operations. If a, b and c are
`three-dimensional vectors with corresponding skew-symmetric matrices A, B and
`C, then following relationships hold
`a b ¼ aT b ¼ bT a
`ð2:12Þ
`ð2:13Þ
`a  b ¼ Ab ¼ BT a ¼ Ba
`ð2:14Þ
`
`
` Ab½ Š½ Š ¼ AB BA
`
`ð2:15Þ
`
`Þ c ¼ a b  cð
`Þ ¼ aTBc
`
`a  bð
`ð2:16Þ
`
`a  b  cð
`Þ ¼ ABc
`ð2:17Þ
`ð
`a  b
`Þ  c ¼ ABc BAc
`where Ab½ Š½ Š in Eq. (2.14) depicts the skew-symmetric matrix of vector Ab:
`
`
`
`
`
`
`
`2.1.6 Angular Velocity Coordinate Transformations
`
`Just like any other vector, the coordinates of an angular velocity vector can be
`transformed from one frame to another. Hence the transformation of an angular
`velocity vector xmk from the k-frame to the p-frame can be expressed as
`
`

`

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`2.1 Basic Navigation Mathematical Techniques
`
`25
`
`ð2:18Þ
`mk ¼ Rpkxk
`
`xp
`mk
`The equivalent transformation between two skew-symmetric matrices has the
`special form
`
`mk ¼ RpkXk
`
`Xp
`mkRk
`p
`
`ð2:19Þ
`
`2.1.7 Least Squares Method
`
`The method of least squares is used to solve a set of equations where there are
`more equations than the unknowns. The solution minimizes the sum of the squares
`of the residual vector. Suppose we want to estimate a vector x of n parameters
`ðx1; x2; . . .; xnÞ from vector z of m noisy measurements ðz1; z2; . . .; zmÞ such that
`m [ n: The measurement vector is linearly related to parameter x with additive
`error vector e such that
`
`ð2:20Þ
`z ¼ Hx þ e
`where H is a known matrix of dimension m  n; called the design matrix, and it is
`of rank n (linearly independent row or columns).
`In the method of least square, the sum of the squares of the components of the
`residual vector ðz HxÞ is minimized in estimating vector x, and is denoted by ^x:
`Hence
`
`minimize z H^x
`ð2:21Þ
`k2¼ ðz H^xÞT1mðz H^xÞm1
`
`k
`This minimization is achieved by differentiating the above equation with
`respect to ^x and setting it to zero. Expanding the above equation gives
`z H^x
`k2¼ zT z zT H^x ^xT HT z þ ^xT HT H^x
`k
`Using the following relationships
`oðxT aÞ
`ox
`
`ð2:22Þ
`
`ð2:23Þ
`
`¼
`
`oðaT xÞ
`ox
`
`¼ aT
`
`and
`
`oðxT AxÞ
`ox
`the derivative of the scalar quantity represented by Eq. (2.22) is obtained as
`follows
`
`¼ ðAxÞT þ xT A
`
`ð2:24Þ
`
`

`

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`26
`
`
`
`
`
`2 Basic Navigational Mathematics, Reference Frames
`
`k
`
`o
`
`k
`
`o
`
`
`o
`
`k
`
`z H^x
`o^x
`z H^x
`o^x
`z H^x
`o^x
`To obtain the maximum, the derivative is set to zero and solved for ^x
`2ðzT H ^xT HT HÞ ¼ 0
`^xT HT H ¼ zT H
`ð^xT HT HÞT ¼ ðzT HÞT
`HT H^x ¼ HT z
`
`k2
`
`
`
`¼ 0 zT H ðHT zÞT þ ðHT H^xÞT þ ^xT HT H
`
`
`
`k2
`
`k2
`
`¼ zT H zT H þ ^xT HT H þ ^xT HT H
`
`¼ 2ðzT H þ ^xT HT HÞ
`
`ð2:25Þ
`
`ð2:26Þ
`
`ð2:27Þ
`
`and finally
`
`^x ¼ ðHT HÞ1HT z
`ð2:28Þ
`We can confirm that the above value of ^x produces the minimum value of the
`cost function (2.22) by differentiating Eq. (2.25) once more that results in 2HT H
`which is positive definite.
`
`2.1.8 Linearization of Non-Linear Equations
`
`The non-linear differential equations of navigation must be linearized in order to
`be usable by linear estimation methods such as Kalman filtering. The non-linear
`system is transformed to a linear system whose states are the deviations from the
`nominal value of the non-linear system. This provides the estimates of the errors in
`the states which are added to the estimated state.
`Suppose we have a non-linear differential equation
`ð2:29Þ
`_x ¼ fðx; tÞ
`and that we know the nominal solution to this equation is ~x and we let dx be the
`error in the nominal solution, then the new estimated value can be written as
`x ¼ ~x þ dx
`ð2:30Þ
`The time derivative of the above equation provides
`_x ¼ _~x þ d _x
`Substituting the above value of _x in the original Eq. (2.29) gives
`
`ð2:31Þ
`
`

`

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`2.1 Basic Navigation Mathematical Techniques
`
`27
`
`ð2:32Þ
`_~x þ d _x ¼ fð~x þ dx; tÞ
`Applying the Taylor series expansion to the right-hand side about the nominal
`value ~x yields
`
`fð~x þ dx; tÞ ¼ fð~x; tÞ þ
`
`ofðx; tÞ
`ox
`
`

`
`x¼~x
`
`dx þ HOT
`
`ð2:33Þ
`
`where the HOT refers to the higher order terms that have not been considered.
`Substituting Eq. (2.32) for the left-hand side gives
`ofðx; tÞ
`ox
`
`_~x þ d _x  fð~x; tÞ þ
`
`

`
`dx
`
`x¼~x
`
`ð2:34Þ
`
`and since ~x also satisfies Eq. (2.29)
`_~x ¼ fð~x; tÞ
`
`substituting this in Eq. (2.34) gives
`
`_~x þ d _x  _~x þ
`
`

`
`ofðx; tÞ
`ox
`
`ð2:35Þ
`
`ð2:36Þ
`
`dx
`
`x¼~x
`The linear differential equations whose states are the errors in the original states
`is give as
`
`

`
`d _x 
`
`ofðx; tÞ
`ox
`
`x¼~x
`After solving this, we get the estimated errors that are added to the estimated
`state in order to get the new estimate of the state.
`
`dx
`
`ð2:37Þ
`
`2.2 Coordinate Frames
`
`Coordinate frames are used to express the position of a point in relation to some
`reference. Some useful coordinate frames relevant to navigation and their mutual
`transformations are discussed next.
`
`2.2.1 Earth-Centered Inertial Frame
`
`An inertial frame is defined to be either stationary in space or moving at constant
`velocity (i.e. no acceleration). All inertial sensors produce measurements relative
`to an inertial frame resolved along the instrument’s sensitive axis. Furthermore,
`
`

`

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`28
`
`2 Basic Navigational Mathematics, Reference Frames
`
`we require an inertial frame for the calculation of a satellite’s position and velocity
`in its orbit around the Earth. The frame of choice for near-Earth environments is
`the Earth-centered inertial (ECI) frame. This is shown in Fig. 2.1 and defined1 by
`(Grewal et al. 2007; Farrell 1998) as
`
`a. The origin is at the center of mass of the Earth.
`b. The z-axis is along axis of the Earth’s rotation through the conventional
`terrestrial pole (CTP).
`c. The x-axis is in the equatorial plane pointing towards the vernal equinox.2
`d. The y-axis completes a right-handed system.
`
`In Fig. 2.1, the axes of the ECI frame are depicted with superscript i as xi; yi; zi;
`and in this book the ECI frame will be referred to as the i-frame.
`
`2.2.2 Earth-Centered Earth-Fixed Frame
`
`This frame is similar to the i-frame because it shares the same origin and z-axis as
`the i-frame, but it rotates along with the Earth (hence the name Earth-fixed). It is
`depicted in Fig. 2.1 along with the i-frame and can be defined as
`
`a. The origin is at the center of mass of the Earth.
`b. The z-axis is through the CTP.
`c. The x-axis passes through the intersection of the equatorial plane and the
`reference meridian (i.e. the Greenwich meridian).
`d. The y-axis completes the right-hand coordinate system in the equatorial plane.
`
`In Fig. 2.1 the axes of the Earth-Centered Earth-Fixed Frame (ECEF) are shown
`as Xe; Y e; Ze and ðt t0Þ represents the elapsed time since reference epoch t0: The
`term xe
`ie represents the Earth’s rotation rate with respect to the inertial frame resolved
`in the ECEF frame. In this book the ECEF frame will be referred to as the e-frame.
`
`2.2.3 Local-Level Frame
`
`A local-level frame (LLF) serves to represent a vehicle’s attitude and velocity
`when on or near the surface of the Earth. This frame is also known as the local
`geodetic or navigation frame. A commonly used LLF is defined as follows
`
`1 Strictly speaking this definition does not satisfy the requirement defined earlier for an inertial
`frame because, in accordance with Kepler’s second law of planetary motion, the Earth does not
`orbit around the sun at a fixed speed; however, for short periods of time it is satisfactory.
`2 The vernal equinox is the direction of intersection of the equatorial plane of the Earth with the
`ecliptic (the plane of Earth’s orbit around the sun).
`
`

`

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`2.2 Coordinate Frames
`
`29
`
`Fig. 2.1 An illustration of the ECI and ECEF coordinate frames
`
`a. The origin coincides with the center of the sensor frame (origin of inertial
`sensor triad).
`b. The y-axis points to true north.
`c. The x-axis points to east.
`d. The z-axis completes the right-handed coordinate systems by pointing up,
`perpendicular to reference ellipsoid.
`
`This frame is referred to as ENU since its axes are aligned with the east, north and
`up directions. This frame is shown in Fig. 2.2. There is another commonly used LLF
`that differs from the ENU only in that the z axis completes a left-handed coordinate
`system and therefore points downwards, perpendicular to the reference ellipsoid.
`This is therefore known as the NED (north, east and down) frame. This book will use
`the ENU convention, and the LLF frame will be referred to as the l-frame.
`
`2.2.4 Wander Frame
`
`In the l-frame the y-axis always points towards true north, so higher rotation rates
`about the z-axis are required in order to maintain the orientation of the l-frame in
`the polar regions (higher latitudes) than near the equator (lower latitudes). As is
`apparent in Fig. 2.3b, the l-frame must rotate at higher rates to maintain its ori-
`entation when moving towards the pole, reaching its maximum when it crosses the
`north pole. This rate can even become infinite (a singularity condition) if the
`l-frame passes directly over the pole. The wander frame avoids higher rotation
`rates and singularity problems. Instead of always pointing northward, this rotates
`
`

`

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`2 Basic Navigational Mathematics, Reference Frames
`
`30
`
`Fig. 2.2 The local-level ENU reference frame in relation to the ECI and ECEF frames
`
`Fig. 2.3 a The wander frame shown with respect to the local-level frame. b Rotation of the
`y-axis of the local-level frame (shown with red/dark arrows) for a near polar crossing trajectory
`at various latitudes
`
`about the z-axis with respect to the l-frame. The angle between the y-axis of the
`wander frame and north is known as the wander angle a: The rotation rate of this
`angle is given as
`
`ð2:38Þ
`_a ¼ _k sin u
`The wander frame (in relation to the l-frame) is shown in Fig. 2.3a, and is
`defined as
`
`a. The origin coincides with the center of the sensor frame (origin of inertial
`sensor triad).
`
`

`

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`2.2 Coordinate Frames
`
`31
`
`Fig. 2.4 The body frame
`of a moving platform
`
`b. The z-axis is orthogonal to the reference ellipsoid pointing upward.
`c. The y-axis rotates by an angle a anticlockwise from north.
`d. The x-axis is orthogonal to the y and z axes and forms a right-handed coor-
`dinate frame.
`
`In this book the wander frame is referred to as the w-frame.
`
`2.2.5 Computational Frame
`
`For the ensuing discussion, the computational frame is defined to be any reference
`frame used in the implementation of the equations of motion. It can be any of the
`abovementioned coordinate frames, and is referred to as the k-frame.
`
`2.2.6 Body Frame
`
`In most applications, the sensitive axes of the accelerometer sensors are made to
`coincide with the axes of the moving platform in which the sensors are mounted.
`These axes are usually known as the body frame.
`The body frame used in this book is shown in Fig. 2.4, and is defined as
`
`a. The origin usually coincides with the center of gravity of the vehicle (this
`simplifies derivation of kinematic equations).
`b. The y-axis points towards the forward direction. It is also called the roll axis as
`the roll angle is defined around this axis using the right-hand rule.
`c. The x-axis points towards the transverse direction. It is also called the pitch
`axis, as the pitch angle corresponds to rotations around this axis using the right-
`hand rule.
`d. The z-axis points towards the vertical direction completing a right-handed
`coordinate system. It is also called the yaw axis as the yaw angle corresponds to
`rotations around this axis using the right-hand rule.
`
`In this book the body frame is referred to as b-frame.
`
`

`

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`
`32
`
`Fig. 2.5 A depiction of a vehicle’s azimuth, pitch and roll angles. The body axes are shown in
`red
`
`2.2.6.1 Vehicle Attitude Description
`
`Apart from a vehicle’s position, we are also interested in its orientation in order to
`describe its heading and tilt angles. This involves specifying its rotation about the
`vertical (z), transversal (x) and forward (y) axes of the b-frame with respect to
`the l-frame. In general, the rotation angles about the axes of the b-frame are called
`the Euler angles. For the purpose of this book, the following convention is applied
`to vehicle attitude angles (Fig. 2.5)
`
`a. Azimuth (or yaw): Azimuth is the deviation of the vehicle’s forward (y) axis
`from north, measured clockwise in the E-N plane. The yaw angle is similar, but
`is measured counter clockwise from north. In this book, the azimuth angle is
`denoted by ‘A’ and the yaw angle by ‘y’. Owing to this definition, the vertical
`axis of the b-frame is also known as the yaw axis (Fig. 2.4).
`b. Pitch: This is the angle that the forward (y) axis of the b-frame makes with the
`E-N plane (i.e. local horizontal) owing to a rotation around its transversal (x)
`axis. This axis is also called the pitch axis, the pitch angle is denoted by ‘p’ and
`follows the right-hand rule (Fig. 2.5).
`c. Roll: This is the rotation of the b-frame about its forward (y) axis, so the
`forward axis is also called the roll axis and the roll angle is denoted by ‘r’ and
`follows the right-hand rule.
`
`2.2.7 Orbital Coordinate System
`
`This is a system of coordinates with Keplerian elements to locate a satellite in
`inertial space. It is defined as follows
`
`a. The origin is located at the focus of an elliptical orbit that coincides with the
`center of the mass of the Earth.
`
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`
`2.2 Coordinate Frames
`
`33
`
`Fig. 2.6 The orbital coordinate system for a satellite
`
`b. The y-axis points towards the descending node, parallel to the minor axis of the
`orbital ellipse.
`c. The x-axis points to the perigee (the point in the orbit nearest the Earth’s center)
`and along the major axis of the elliptical orbit of the satellite.
`d. The z-axis is orthogonal to the orbital plane.
`
`The orbital coordinate system is illustrated in Fig. 2.6. It is mentioned here to
`complete the discussion of the frames used in navigation (it will be discussed in
`greater detail in Chap. 3).
`
`2.3 Coordinate Transformations
`
`The techniques for transforming a vector from one coordinate frame into another
`can use direction cosines, rotation (Euler) angles or quaternions. They all involve a
`rotation matrix which is called either the transformation matrix or the direction
`cosine matrices (DCM), and is represented as Rl
`k where the subscript represents the
`frame from which the vector originates and the superscript is the target frame. For
`example, a vector rk in a coordinate frame k can be represented by another vector
`rl in a coordinate frame l by applying a rotation matrix Rl
`k as follows
`
`

`

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`2 Basic Navigational Mathematics, Reference Frames
`
`34
`
`ð2:39Þ
`rl ¼ Rl
`krk
`If Euler angles are used, these readily yield the elementary matrices required to
`construct the DCM.
`
`2.3.1 Euler Angles and Elementary Rotational Matrices
`
`A transformation between two coordinate frames can be accomplished by carrying
`out a rotation about each of the three axes. For example, a transformation from the
`reference frame a to the new coordinate frame b involves first making a rotation of
`angle c about the z-axis, then a rotation of an angle b about the new x-axis, and
`finally a rotation of an angle a about the new y-axis. In these rotations, a; b and c
`are the Euler angles.
`
`To transform a vector ra ¼ xa; ya; za½
`Š from frame a to frame d where the two
`frames are orientated differently in space, we align frame a with frame d using the
`three rotations specified above, each applying a suitable direction cosine matrix.
`The individual matrices can be obtained by considering each rotation, one by one.
`First we consider the x-y plane of frame a in which the projection of vector r
`(represented by r1) makes an angle h1 with the x-axis. We therefore rotate frame a
`around its z-axis through an angle c to obtain the intermediate frame b; as illus-
`trated in Fig. 2.7.
`According to this figure, the new coordinates are represented by xb; yb; zb and
`can be expressed as
`
`ð2:40Þ
`Þ
`xb ¼ r1 cos h1 cð
`
`ð2:41Þ
`yb ¼ r1 sin h1 cð
`Þ
`
`Since the rotation was performed around the z-axis, this remains unchanged
`zb ¼ za
`ð2:42Þ
`Using the following trigonometric identities
`sinðA  BÞ ¼ sin A cos B  cos A sin B
`cosðA  BÞ ¼ cos A cos B  sin A sin B
`Equations (2.40) and (2.41) can be written as
`ð2:44Þ
`xb ¼ r1 cos h1 cos c þ r1 sin h1 sin c
`ð2:45Þ
`yb ¼ r1 sin h1 cos c r1 cos h1 sin c
`The original coordinates of vector r1 in the x-y plane can be expressed in terms
`of angle h1 as
`
`ð2:43Þ
`
`

`

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`
`2.3 Coordinate Transformations
`
`35
`
`Fig. 2.7 The first rotation of
`frame ‘a’ about its za-axis
`
`xa ¼ r1 cos h1
`ya ¼ r1 sin h1
`Substituting the above values in Eqs. (2.44) and (2.45) produces
`xb ¼ xa cos c þ ya sin c
`yb ¼ xa sin c þ ya cos c
`
`and we have shown that
`
`ð2:46Þ
`ð2:47Þ
`
`ð2:48Þ
`ð2:49Þ
`
`ð2:50Þ
`
`ð2:51Þ
`
`ð2:52Þ
`
`zb ¼ za
`In matrix form, the three equations above can be written as
`
`
`
`¼ cos c sin c
`0
`
`35
`
`xb
`yb
`zb
`
`¼ Rb
`
`a
`
`35
`
`xa
`ya
`za
`
`24
`35
`
`sin c
`0
`cos c 0
`0
`1
`
`35
`
`xa
`ya
`za
`
`24
`
`24
`
`24
`
`35
`
`xb
`yb
`zb
`
`24
`
`where Rba is the elementary DCM which transforms the coordinates xa; ya; za to
`
`xb; yb; zb in a frame rotated by an angle c around the z-axis of frame a:
`
`

`

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`
`36
`
`Fig. 2.8 The second rotation
`of rotated frame ‘b’ about
`xb-axis
`
`For the second rotation, we consider the y-z plane of the new coordinate frame b,
`and rotate it by an angle b around its x-axis to an intermediate frame c as shown in
`Fig. 2.8.
`In a similar fashion it can be shown that the new coordinates xc; yc; zc can be
`expressed in terms of xb; yb; zb as follows
`
`35
`
`xb
`yb
`zb
`
`24
`35
`
`24
`
`1
`0
`0
`cos b
`sin b
`0
`0 sin b cos b
`
`35
`
`xb
`yb
`zb
`
`24
`
`¼ Rc
`
`b
`
`35
`
`xc
`yc
`zc
`
`24
`
`¼
`
`35
`
`xc
`yc
`zc
`
`24
`
`ð2:53Þ
`
`ð2:54Þ
`
`ð2:55Þ
`
`where Rcb is the elementary DCM which transforms the coordinates xb; yb; zb to
`
`xc; yc; zc in a frame rotated by an angle b around the x-axis of frame b:
`For the third rotation, we consider the x-z plane of new coordinate frame c; and
`rotate it by an angle a about its y-axis to align it with coordinate frame d as shown
`in Fig. 2.9.
`The final coordinates xd; yd; zd can be expressed in terms of xc; yc; zc as follows
`0 sin a
`¼ cos a
`1
`0
`cos a
`0
`
`35
`
`xc
`yc
`zc
`
`24
`35
`
`0
`sin a
`
`24
`
`35
`
`xd
`yd
`zd
`
`24
`
`

`

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`
`2.3 Coordinate Transformations
`
`37
`
`Fig. 2.9 The third rotation of
`rotated frame ‘c’ about
`yc-axis
`
`ð2:56Þ
`
`35
`
`xc
`yc
`zc
`
`24
`
`¼ Rd
`
`c
`
`35
`
`xd
`yd
`zd
`
`24
`
`where Rdc is the elementary DCM which transforms the coordinates xc; yc; zc to
`
`xd; yd; zd in the final desired frame d rotated by an angle a around the y-axis of
`frame c:
`We can combine all three rotations by multiplying the cosine matrices into a
`single transformation matrix as
`
`ð2:57Þ
`
`ð2:58Þ
`
`35
`
`35
`
`24
`35
`
`24
`35
`
`cos c
`sin c
`a ¼ cos a
`0
`1
`0
`0
` sin c
`cos c
`cos b
`sin b
`0
`0
`0 sin b cos b
`0
`0
`1
`cos a sin c þ cos c sin b sin a cos b sin a
`a ¼ cos a cos c sin b sin a sin c
` cos b sin c
`cos b cos c
`sin b
`cos c sin a þ cos a sin b sin c
`sin a sin c cos a cos c sin b
`cos b cos a
`ð2:59Þ
`
`a ¼ RdcRcbRb
`
`
`Rd
`a
`The final DCM for these particular set of rotations can be given as
`0 sin a
`1
`0
`cos a
`0
`
`0
`sin a
`
`24
`
`24
`
`Rd
`
`Rd
`
`The inverse transformation from frame d to a is therefore
`
`

`

`38
`
`
`
`
`d ¼ Rd
`Ra
`
`a
`
`a
`
`a
`
`
`
`T
`
`
`T Rd
`c
`
`T
`
`ð2:60Þ
`
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`
`
`
`
`1¼ Rd
`T ¼ Rd
`
`
`
`
`
`cRcbRba
`
`¼ Rb
`T Rc
`b
`It should be noted that the final transformation matrix is contingent upon the
`
`
`order of the applied rotations, as is evident from the fact that RcbRba 6¼ RbaRcb: The
`
`
`order of rotations is dictated by the specific application. We will see in Sect. 2.3.6
`that a different order of rotation is required and the elementary matrices are
`multiplied in a different order to yield a different final transformation matrix.
`For small values of a; b and c we can use the following approximations
`cos h  1 ; sin h  h
`ð2:61Þ
`Using these approximations and ignoring the product of the small angles, we
`can reduce the DCM to
`
`375
`
`264
`
`a 
`Rd
`
`a ¼
`Rd
`
`c a
`1
`c
`b
`1
`a b
`1
`1
`0 0
`0
`1 0
`0
`0 1
`a ¼ I W
`Rd
`where W is the skew-symmetric matrix for the small Euler angles. For the small-
`angle approximation, the order of rotation is no longer important since in all cases
`the final result will always be the matrix of the Eq. (2.62). Similarly, it can be
`verified that
`
`ð2:62Þ
`
`375
`
`0 c
`a
`0 b
`c
`a
`b
`0
`
`264
`
`
`
`375
`
`264
`
`T
`
`ð2:63Þ
`
`375
`
`264
`
`d 
`Ra
`
`c a
`1
`c
`b
`1
`a b
`1
`d ¼ I WT
`Ra
`
`2.3.2 Transformation Between ECI and ECEF
`
`The angular velocity vector between the i-frame and the e-frame as a result of the
`rotation of the Earth is
`
`
`xe
`
`ie ¼ 0; 0; xeð
`
`ÞT
`
`ð2:64Þ
`
`

`

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`
`2.3 Coordinate Transformations
`
`39
`
`Fig. 2.10 Transformation
`between the e-frame and the
`i-frame
`
`where xe denotes the magnitude of the Earth’s rotation rate. The transformation
`from the i-frame to the e-frame is a simple rotation of the i-frame about the z-axis
`by an angle xet where t is the time since the reference epoch (Fig. 2.10). The
`
`rotation matrix corresponds to the elementary matrix Rba; and when denoted Rei
`which can be expressed as
`
`
`
`35
`
`24
`
`Re
`
`ð2:65Þ
`
`ð2:66Þ
`
`sin xet
`i ¼ cos xet
`0
` sin xet
`cos xet
`0
`1
`0
`0
`Transformation from the e-frame to the i-frame can be achieved through Ri
`e; the
`inverse of Re
`
`
`
`
`i : Since rotation matrices are orthogonal
`1¼ Re
`e ¼ Re
`Ri
`
`i
`
`T
`
`i
`
`2.3.3 Transformation Between LLF and ECEF
`
`From Fig. 2.11 it can be observed that to align the l-frame with the e-frame, the
`l-frame must be rotated by u 90 degrees around its x-axis (east direction) and
`then by 90 k degrees about its z-axis (up direction).
`For the definition of elementary direction cosine matrices, the transformation
`from the l-frame to the e-frame is
`
`
`ðl ¼ Rba k 90
`Re
`
`
`
`Þ
`
`ð2:67Þ
`
`
`
`ðÞRcb u 90
`
`

`

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`
`40
`
`Fig. 2.11 The LLF in
`relation to the ECEF frame
`
`1
`0
`Þ
`cos u 90ð
`
`0
`
`0 sin u 90ð
`
`Þ
`
`24
`35
`
`Þ
`
`Þ
`sin k 90ð
`
`
`cos k 90ð
`Þ
`0
`
`0
`0
`1
`
`24
`
`Þ
`
`l ¼ cos k 90ð
`
` sin k 90ð
`0
`
`Re
`
`24
`35
`
`24
`
`24
`
`l ¼
`Re
`
`l ¼
`Re
`
` sin k cos k 0
`1
`cos k sin k
`0
`0
`0
`0
`1
`0
` sin k sin u cos k
`cos k sin u sin k
`cos u
`0
`
`
`35
`
`0
`Þ
`sin u 90ð
`
`
`cos u 90ð
`Þ
`ð2:68Þ
`
`ð2:69Þ
`
`ð2:70Þ
`
`ð2:71Þ
`
`35
`
`0
`0
`sin u cos u
`cos u
`sin u
`
`35
`
`cos u cos k
`cos u sin k
`sin u
`
`
`
`T
`
`
`The reverse transformation is
`e ¼ Re
`Rl
`
`l
`
`
`1¼ Re
`
`l
`
`2.3.4 Transformation Between LLF and Wander Frame
`
`The wander frame has a rotation about the z-axis of the l-frame by a wander angle
`a; as depicted in Fig. 2.12. Thus the transformation matrix from the w-frame frame
`a with an angle a; and is
`to the l-frame corresponds to the elementary matrix Rb
`expressed as
`
`

`

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`
`2.3 Coordinate Transformations
`
`41
`
`Fig. 2.12 The relationship
`between the l-frame and the
`w-frame (the third axes of
`these the frames are not
`shown because they coincide
`and point out of the page
`towards the reader)
`
`ð2:72Þ
`
`ð2:73Þ
`
`ð2:74Þ
`
`24
`
`24
`
`Rl
`
`Rl
`
`Þ
`w ¼ cos að
` sin að
`Þ
`0
`w ¼ cos a sin a
`
`sin a
`0
`
`cos a
`0
`
`Þ
`sin að
`cos að
`Þ
`0
`
`35
`
`0
`0
`1
`
`35
`
`0
`0
`1
`
`and
`
`
`
`
`l ¼ Rl
`Rw
`
`w
`
`
`1¼ Rl
`
`w
`
`
`
`T
`
`2.3.5 Transformation Between ECEF and Wander Frame
`
`This transformation is obtained by first going from the w-frame to the l-frame and
`then from the l-frame to the e-frame
`
`Re
`
`ð2:75Þ
`
`ð2:76Þ
`
`35
`
`24
`35
`
`w ¼ Rel Rl
`w
`
` sin k sin u cos k cos u cos k
`cos k sin u sin k
`cos u sin k
`cos u
`sin u
`0
`
`cos a sin a
`sin a
`cos a
`0
`0
`
`0
`0
`1
`
`24
`
`w ¼
`Re
`
`

`

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`
`35
`
`sin k sin a cos k sin u cos a
` sin k cos a cos k sin u sin a
`cos k cos u
`cos k cos a sin k sin u sin a cos k sin a sin k sin u cos a sin k cos u
`cos u sin a
`cos u cos a
`sin u
`ð2:77Þ
`
`24
`
`42
`
`w ¼
`Re
`
`The inverse is
`
`
`e ¼ Re
`Rw
`
`w
`
`
`
`
`1¼ Re
`
`w
`
`
`
`T
`
`ð2:78Þ
`
`2.3.6 Transformation Between Body Frame and LLF
`
`One of the important direction cosine matrices is Rl
`b; which transforms a vector
`from the b-frame to the l-frame, a requirement during the mechanization process.
`This is expressed in terms of yaw, pitch and roll Eul