```Fundamentals of
Uğur Doğan GÜL
Outline
• Coordinate Frames
•
•
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•
Earth-Centered Inertial Frame
Earth-Centered Earth-Fixed 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
•
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•
Inertial and Earth Frames
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,
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.
Coordinate Frames
Earth-Centered Earth-Fixed Frame
•
Denoted 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).
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 righthanded orthogonal set, pointing
from the center to the
intersection of the equator with
the 90deg east meridian.
•
The Earth frame is important in
to know the position relative to
the Earth, so it is commonly used
as both a reference frame and a
resolving frame.
Coordinate Frames
• 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.
Coordinate Frames
• The local navigation frame is
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.
Coordinate Frames
Body-Fixed Frame
• Denoted by the symbol b,
• Comprises the origin and
orientation of the object for
is sought.
• The origin is coincident with
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.
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
the object that is navigating. All
strap down inertial sensors
measure the motion of the
body frame (with respect to a
generic inertial frame).
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.
Kinematics
Euler Attitude Representation
• In Euler attitude
representation, the attitude is
broken down into three
successive rotations, namely
yaw ( ), pitch ( ), and
roll (ɸ ) 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.

ɸ
=

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.
Kinematics
Coordinate Transformation Matrix
1
= 0
0
0
ɸ
−ɸ
0
ɸ
ɸ

=

−

0

0
1
0
−
0

−
0

0
−ɸ
+ɸ
ɸ
+ɸ
ɸ
+ɸ
ɸ
−ɸ
+ɸ
ɸ
0
0
1
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:
ɸ
−ɸ
≠ −

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

= ( )
• Successive rotations cannot be expressed simply by adding the Euler angles:
ɸ
ɸ + ɸ
≠  +

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

=
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 γframe axes. The angular rate tensor is the skew-symmetric matrix
of the angular rate vector:

0
−3 2

=

∧ =

3

−2
0

1

−1
0
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:

=

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:

=
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

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

ν
=  ν

ν 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.

=   +   =   + ν
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:

=
Kinematics
Acceleration

Acceleration is not the same as the time derivative of ν or the second time derivative of  .
These depend on the rotation of the resoling frame, γ, with respect to the reference frame, β:

ν =   +   =   +

=   +   + ν

=   + 2  + a
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

= −   +   − 2  +
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 and the eccentricity of
the ellipsoid, e. The eccentricity is
defined by
=
2
1− 2
0
Where 0 is the length of semi-major
axis and  is the length of semi-minor
axis. According to World Geodetic
System 1984 (WGS84):
• 0 = 6,378,137.0 m,
•  = 6,356,752.3141 m,
•  = 0.0818191908425
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
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),
• The coordinate of that point in
the east-west axis is the
longitude (λ).
Earth Surface and Gravity Models
Curvilinear position
The curvilinear position is obtained
from the Cartesian ECEF position by
following equations:
ℎ =

sin  = 1− 2   +ℎ

tan  =

2
2

+
−  ( )
cos
where RE is the radius of curvature for
east-west motion known as the
given by
0
=
1 −  2 2
where e is the eccentricity of the
ellipsoid model of earth.
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 Earthrotation vector resolved in these axes is given by
0

=
= 0

According to WGS 84 the value of the Earth’s angular rate
is  = 7.29211510−5   −1
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

= −

=0, =0
Therefore the acceleration due to gravity is

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

=
,
∈ , , ,
,  ∈ , , ,
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 0 ,
and the frames rotate about the z-axes at  . Thus,
cos  ( − 0 ) sin  ( − 0 ) 0

= −sin  ( − 0 ) cos  ( − 0 ) 0 ,  = ( )
0
0
1
Position transformation:

=

=
Velocity transformation:

ν =  (ν −
)

ν =  (ν +   )
Acceleration transformation:

a =  (a − 2
ν −

)

a =  (a + 2 ν +    )
Angular rate transformation:

=

0
− 0

,

=

0
+ 0

Frame Transformations
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.
−
−
=
−
−

−

0
,  = ( )
−
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:

=
−

)

=  (
+
Frame Transformations
The inertial-local navigation frame coordinate transform is obtained by multiplying
inertial-Earth and Earth-local frame coordinate transformation matrices:
=
−   ( + ( − 0 )) −   ( +   − 0 )
− ( +  ( − 0 ))
( +  ( − 0 ))
0
,
−   ( + ( − 0 )) −   ( +   − 0 ) −
= ( )
Velocity transformation:

ν =  (ν −
)

ν =  ν +

Acceleration transformation:

a =  (a − 2
ν −

)

a =  (a + 2
ν ) +

Angular rate transformation:

=
−

=  (
+
)
Frame Transformations
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:

1 + (  + ℎ )
0
0

≈  +
0
1/   + ℎ
0
ℎ
ℎ
0
0
−1
Velocity transposition:

ν = ν +

Assuming
is constant,

ν = ν +  (
)
References
• PAUL D. GROVES
• PRINCIPLES OF GNSS, INERTIAL, AND
MULTISENSOR INTEGRATED