Fundamentals of Navigation Systems - PowerPoint Presentation

Slide1

Fundamentals of Navigation Systems

Uğur Doğan GÜLSlide2

Outline

Coordinate

Frames

Earth-Centered Inertial Frame

Earth-Centered Earth-Fixed Frame

Local Navigation Frame

Body

-Fixed

Frame

Kinematics

Attitude

Angular Rate

Cartesian

Position

Velocity

Acceleration

Earth Surface and Gravity

Models

The Ellipsoid Model of the Earth’s Surface

Curvilinear Position

Earth

Rotation

Specific Force, Gravitation, and

Gravity

Frame Transformations

Inertial and Earth Frames

Earth and Local Navigation Frames

Inertial and Local Navigation Frames

Transposition of Navigation SolutionsSlide3

Coordinate Frames

Earth-Centered

Inertial Frame

Denoted

by the symbol

i, Centered at the Earth’s center of mass,Oriented with respect to the Earth’s spin axis and the stars,Slide4

Coordinate Frames

Earth-Centered

Inertial Frame

The

z-axis always points along the Earth’s axis of rotation from the center to the North Pole (true, not magnetic

),The x-and y-axes lie within the equatorial plane,They do not rotate with the Earth, but the y-axis always lies 90 degrees ahead of the x-axis in the direction of rotation. Slide5

Coordinate Frames

Earth-Centered

Earth-Fixed

Frame

D

enoted by the symbol e.Similar to the ECI frame, except that all axes remain fixed with respect to the Earth. The z-axis always points along the Earth’s axis of rotation from the center to the North Pole(true, not magnetic). Slide6

Coordinate Frames

Earth-Centered

Earth-Fixed

Frame

The x-axis points from the center to the intersection of the equator with the IERS reference meridian (IRM), or conventional zero meridian (CZM), which defines 0 degree longitude.

The y-axis completes the right-handed orthogonal set, pointing from the center to the intersection of the equator with the 90deg east meridian. The Earth frame is important in navigation because it is wanted to know the position relative to the Earth, so it is commonly used as both a reference frame and a resolving frame. Slide7

Coordinate Frames

Local Navigation Frame

Denoted by the symbol

n

,

It’s origin is the point a navigation solution is sought for. The z-axis, also known as the down (D) axis, is defined as the normal to the surface of the reference ellipsoid, pointing roughly toward the center of the Earth. The x-axis, or north (N) axis, is the projection in the plane orthogonal to the z-axis of the line from the user to the North Pole.

By

completing the orthogonal set, the y-axis always points east and is hence known as the east (E) axis. Slide8

Coordinate Frames

Local Navigation Frame

The

local navigation frame is important in navigation because it is wanted to know the attitude relative to the north, east, and down directions. For position and velocity, it provides a convenient set of resolving axes, but is not used as a reference frame.Slide9

Coordinate Frames

Body-Fixed Frame

Denoted by the symbol

b

,

Comprises the origin and orientation of the object for which a navigation solution is sought. The origin is coincident with that of the local navigation frame, but the axes remain fixed with respect to the body and are generally defined as x=forward, z=down, y=right, completing the orthogonal set. Slide10

Coordinate Frames

Body-Fixed Frame

For angular motion, the x-axis is the roll axis, the y-axis is the pitch axis, and the z-axis is the yaw axis. Hence, the axes of the body frame are sometimes known as roll, pitch, and yaw.

The body frame is essential in navigation because it describes the object that is navigating. All strap down inertial sensors measure the motion of the body frame (with respect to a generic inertial frame).Slide11

Kinematics

In navigation, the linear and angular motion of one coordinate frame must be described with respect to another. Most kinematic quantities, such as position, velocity, acceleration, and angular rate, involve three coordinate frames:

The frame whose motion is described, known as the object frame, α;

The frame with which that motion is respect to, known as the reference frame, β;

The set of axes in which that motion is represented, known as the resolving frame, γ.

To describe these quantities the following notation is used for Cartesian position, velocity, acceleration, and angular rate:

Where the vector, x, describes a kinematic property of frame

α

with respect to frame β, expressed in the frame γ axes. For attitude, only the object frame, α, and reference frame, β, are involved; there is no resolving frame.

 Slide12

Kinematics

Euler Attitude

Representation

In Euler attitude representation, the attitude is broken down into three successive rotations, namely yaw (

), pitch (

), and roll (

) rotation.

The

Euler rotation from frame β to frame α may be denoted by the

vector

Where the Euler angles are listed in the reverse order to that in which they are applied

.

 Slide13

Kinematics

Coordinate Transformation

Matrix

Representation

The coordinate transformation matrix is a 3x3 matrix, denoted by

. Coordinate transformation matrix can be used:

To transform a vector from one set of resolving axes to another (the lower index represents the “from” coordinate frame,α, and the upper index represents the “to” frame,β).

To represent the attitude (the lower index represents the object frame,α, and the upper index represents the reference frame,β).

A set of Euler angles is converted to a coordinate transformation matrix by first representing the roll, pitch and yaw rotations as a matrix and then multiplying, noting that with matrices the order of the operation is important and, the first operation is placed on the right. Here ZYX order is

used

.

 Slide14

Kinematics

Coordinate Transformation Matrix

 Slide15

Kinematics

Why to use coordinate transformation matrix representation instead

of

Euler attitude representation

?

Rotation cannot be reversed simply by reversing the sign of the Euler angles:

However

transpose of the coordinate transformation matrix is used to reverse the rotation:

Successive rotations cannot be expressed simply by adding the Euler angles:

However to perform successive rotations coordinate transformation matrices are simply multiplied:

 Slide16

Kinematics

Angular Rate

The angular rate vector,

, is the rate of rotation of the α-frame axes with respect to the β-frame axes, resolved about the γ-frame axes. The angular rate tensor is the skew-symmetric matrix of the angular rate vector:

It can be shown, using the small angle approximation applied in the limit δt→0, that the time derivative of the coordinate transformation matrix is:

 Slide17

Kinematics

Cartesian Position

The Cartesian position of the origin of frame α with respect to the origin of frame β, resolved about the axes of frame γ,

is

:

where

x, y, and z are the components of position in the x, y, and z axes of the γ frame.

Position may be resolved in a different frame by applying a coordinate transformation matrix:

 Slide18

Kinematics

Velocity

Velocity is defined as the rate of change of the position of the origin of an object frame with respect to the origin and axes of a reference frame. This may, in turn, be resolved about the axes of a third frame. Thus, the velocity of frame α with respect to frame β, resolved about the axes of frame γ, is

Velocity may be transformed from one resolving frame to another using the appropriate coordinate transformation matrix:

is not equal to the time derivative of

unless there is no angular motion of the resolving frame, γ, with respect to the reference frame, β.

 Slide19

Kinematics

Acceleration

Acceleration is defined as the second time derivative of the position of the origin of one frame with respect to the origin and axes of another frame. Thus, the acceleration of frame α, with respect to frame β, resolved about the axes of frame γ, is:

Acceleration may be resolved about a different set of axes by applying the appropriate coordinate transformation matrix:

 Slide20

Kinematics

Acceleration

Acceleration is not the same as the time derivative of

or the second time derivative of

. These depend on the rotation of the resoling frame, γ, with respect to the reference frame, β:

The first term on the right-hand side is the centrifugal acceleration given by,

And the second term on the right-hand side is the Coriolis acceleration, given by

Therefore the second time derivative of

is

 Slide21

Earth Surface and Gravity Models

The Ellipsoid Model of the Earth’s Surface

The surface of the Earth can be approximated as an ellipsoid fitted in the main sea level. The ellipsoid is commonly defined in terms of the equatorial radius

an

d

the eccentricity of the ellipsoid, e. The eccentricity is defined by

Where

is

the length of semi-major axis

and

is the

length of semi-minor axis

.

According to World Geodetic System 1984 (WGS84):

= 6,378,137.0 m,

= 6,356,752.3141 m,

= 0.0818191908425

 Slide22

Earth Surface and Gravity Models

Curvilinear position

Position with respect to the Earth’s surface is described using three mutually orthogonal coordinates, aligned with the axes of the local navigation

frame

:

The distance from the body described to the surface alone the normal to that surface is the height or altitude (h),The

north-south axis coordinate of the point on the surface where that normal intersects is the

latitude

(L),

T

he

coordinate of that point in the east-west axis is the

longitude

(λ).Slide23

Earth Surface and Gravity Models

Curvilinear position

The

curvilinear position is obtained from the Cartesian ECEF position by following equations:

where R

E

is the radius of curvature for east-west motion known as the transverse radius of curvature, and given by

where

e

is the eccentricity of the ellipsoid model of earth.

 Slide24

Earth Surface and Gravity Models

Earth Rotation

The Earth rotates, with respect to space, clockwise about the common z-axis of the ECI and ECEF frames. The Earth-rotation vector resolved in these axes is given by

According to WGS 84 the value of the Earth’s angular rate is

 Slide25

Earth Surface and Gravity Models

Specific Force, Gravitation, and

Gravity

Specific force

is the non-gravitational force per unit mass on a body, sensed with respect to an inertial frame.

Gravitation is the fundamental mass attraction force and it does not incorporate any centripetal components.Specific force, f, varies with acceleration, a, and the acceleration due to the gravitation force, γ, as

Specific force is the quantity measured by accelerometers. The measurements are made in the body frame of the accelerometer triad; thus the sensed specific force is

.

The specific force sensed when stationary with respect to the Earth frame is the reaction to what is known as the acceleration due to

gravity

, which is thus defined by

Therefore the acceleration due to gravity is

 Slide26

Frame Transformations

Cartesian position, velocity, acceleration, and angular rate referenced to the same frame transform between resolving axes simply by applying:

 Slide27

Frame Transformations

Inertial and Earth Frames

The center and the z-axes of the ECI and ECEF frames are coincident, x and y-axes are coincident at time

, and the frames rotate about the z-axes at

. Thus,

,

Position transformation:

Velocity transformation:

Acceleration transformation:

Angular rate transformation:

,

 Slide28

Frame Transformations

Earth and Local Navigation Frames

The relative orientation of the earth and local navigation frames is determined by the geodetic latitude,

, and longitude,

, of the body frame whose center coincides with that of the local navigation frame.

,

Position

, velocity and acceleration referenced to the local navigation frame are meaningless as the body frame center coincides with the navigation frame center.

Angular

rate transformation:

 Slide29

Frame Transformations

Inertial and Local Navigation Frames

The inertial-local navigation frame coordinate transform is obtained by multiplying inertial-Earth and Earth-local frame coordinate transformation matrices:

,

Velocity

transformation:

Acceleration transformation:

Angular rate transformation:

 Slide30

Frame Transformations

Transposition of Navigation Solutions

Sometimes, there is a requirement to transpose a navigation solution from one position to another. To transpose a navigation solution from frame a to frame b, the position of frame a with respect to frame b,

, which is known as the lever arm or moment arm is required. The transformations are as follows:

Attitude transposition:

Cartesian position transposition:

Curvilinear position transposition:

Velocity transposition:

Assuming

is constant,

 Slide31

References

PAUL D. GROVES

PRINCIPLES OF GNSS, INERTIAL, AND MULTISENSOR INTEGRATED NAVIGATION SYSTEMS

TL798.N3 G76 2008