Navigation Equations in the Earth Centered Earth Fixed Frame
Arland B. Thompson
Chief Scientist Advanced Technology Associates
(www.atacolorado.com)
I) Introduction
Practical (in terms of computer memory and computational speed
requirements), highly accurate navigation schemes are of great
importance in various military and commercial applications. Choice of
coordinates systems used to integrate the equations of motion, to a large
extent, drives the implementation of the navigation scheme. Many
applications use an Earth Centered Inertial (ECI) frame of reference
such as EME J2000 to integrate the equations of motion. This paper will
derive the equations of motion used for navigation in an Earth Centered
Earth Fixed (ECEF) reference frame, and demonstrate many advantages
for using such a reference frame for navigation.
II) Reference Frame Definitions
Two reference frames will be discussed in this paper; EME J2000
(Earth Centered Inertial) and ECEF (International Terrestrial Reference
System). The ITRF has the center as the center of mass of the earth
(including oceans and atmospheres). The x axis is the intersection of the
mean equator, and the mean prime meridian. The z axis is the mean spin
axis of the earth between 1900 until 1905 (Conventional International
Origin - CIO). EME J2000 is the Earth Mean Equator and Equinox of
epoch J2000. The transformation between ECEF and EMEJ2000 is
complex, and a full discussion is not intended here. The transformation is
composed of 4 parts: precession, nutation, Earth rotation, and polar
motion.
U t   NP Eq. 1
Where:
P – Precession
N – Nutation
 - Earth rotation
 - Polar motion
III) Derivation of ECEF Navigation Equations
The following notation will be used throughout the derivation:
C ab - Direction Cosine matrix (DCM) transformation from
some general reference frame a to some general reference
frame b.
C ab - First time derivative of Direction Cosine matrix (DCM)
transformation from some general reference frame a to
some general reference frame b.
b
Ca - Second time derivative of Direction Cosine matrix (DCM)
transformation from some general reference frame a to
some general reference frame b.
xa
x a
xa
- Three dimensional vector expressed (parameterized) in
some general reference frame a.
- First time derivative of three dimensional vector
expressed (parameterized) in some general reference
frame a.
- Second time derivative of three dimensional vector
expressed (parameterized) in some general reference
frame a.
baa
- Three dimensional angular rate vector of some general
reference frame a relative to some general reference
frame b expressed (parameterized) in some general
reference frame a.
a
 ba
- 3X3 skew symmetric matrix composed of three
dimensional angular rate vector of some general
reference frame a relative to some general reference
frame b expressed (parameterized) in some general
reference frame a.
 a - First Time derivative of 3X3 skew symmetric matrix

ba
composed of three dimensional angular rate vector of
some general reference frame a relative to some general
reference frame b expressed (parameterized) in some
general reference frame a.
Throughout the derivation, a subscript or superscript e will
designate the ECEF reference frame. A subscript or superscript i will
designate the ECI reference frame.
The total accelerations acting on the vehicle in the inertial frame is
expressed as follows:
ri  g i  a i
(Eq. 2)
Where:
a i - Acceleration due to thrust and aerodynamics in the inertial
frame of reference.
g i - Acceleration due to gravity in the inertial frame of reference.
Note that the first time derivative of the transformation DCM from
i
ECEF to inertial ( C e ) is as follows:
C ei  Cei iee (Eq. 3)
The angular rate vector of the ECEF frame relative to the ECI
e
frame used to populate the elements of  ie in Eq. 3 is:
0
iee   0 
(Eqs. 4)
 
Where:
 - Earth rotation rate (approximately 7.292e-5 radians/second)
Using the multiplication rule, the second time derivative of the
transformation DCM from ECEF to inertial is as follows:
i  C i 
 e i e
C
e
e ie  Ce ie
i  C i 
 e  C i e e
C
e
e ie
e ie ie
i  C i 
 e  e e
C
e
e

ie
ie
ie

(Eqs. 5)
It will be assumed that Earth rotation rate is constant. Therefore:
i  C i  e  e (Eq. 6)
C
e
e ie ie
The transformation of a position vector from ECEF to ECI is
expressed as follows:
r i  Cei r e
(Eq. 7)
Taking the derivative with respect to time to get the velocity:
ri  Cei r e  C ei r e
(Eq. 8)
Taking the second derivative with respect to time to get
acceleration:
i r e
ri  Cei re  C ei r e  C ei r e  C
e
i r e
ri  Cei re  2C ei r e  C
e
r  C r  2C  r  C   r
i
i
e
e
i
e
e
ie
e
i
e
e
ie
e
ie
e
(Eqs. 9)
e
Using Eq. 1 and solving for r :
   
 
r  C  g  a  2C  C  r  C  C   r
r  C  g  C  a  2C  C  r  C  C  
T
T
T
re  Cei ri  2 Cei Cei  iee r e  Cei Cei  iee  iee r e
e
i T
e
e
i T
e
i
i
i T
e
i
i T
e
i
i
e
e e
ie
i T
e
i
e
i T
e
e e
ie
i
e
i T
e
e
ie
e e
ie
i
e
e
ie
e e
ie
r (Eqs. 10)
re  g e  a e  2 iee r e   iee  iee r e
Note that the last equation in Eqs. 10 is an ordinary second order
differential which can be integrated once to get velocity in ECEF, and
then once more to get position in ECEF. The last equation in Eqs. 10 can
be transformed into a system of two first order ordinary differential
equations by introducing velocity as follows:
d e
r  r e
dt
d e
r  g e  a e  2 iee r e   iee  iee r e
dt
(Eqs. 11)
IV) Quaternions
Quaternions will be used to maintain the direction cosine matrix
(attitude) of the vehicle body frame relative to the Earth fixed frame
discussed in the next section. Therefore, it is briefly introduced here.
There exists a single judiciously chosen axis about which one frame of
reference (call it the “I” frame) can be rotated by some angle to bring it
into alignment with another frame of reference (call it the “F” frame).
This is illustrated by Figure 1.
F
I
I
e

Figure 1
One way of viewing the quaternion (there are others) is as a
coordinate frame transformation (In this case, from the “I” frame to the
“F” frame), and is equivalent to a direction cosine matrix. The form of
which can be taken directly from the axis of rotation and the angle of
rotation about that axis in figure 3. The quaternion is 4 dimensional and
is said to have a “vector portion” which is 3 dimensional, and a “scalar”
portion. For most engineering applications, the scalar portion seems to be
listed last and is the case here. For most academic applications, the scalar
seems to be listed first.

  
e[0] * sin  2  
 

 e[1] * sin    

 2 
q
 
e[ 2] * sin  

 2 


  
 cos  
2 

(Eq. 12)
Note the form. The vector portion (first three components) is the
coordinates of the axis of rotation in the “I” frame (and “F” frame for
that matter) multiplied by the sine of half the rotation angle. The reason
that it has half the rotation angle is not presented here. The scalar portion
is the cosine of half the rotation angle. Also note that a quaternion must
be of unit magnitude to be of use in coordinate frame transformation. As
a practical example, take the coordinate frame transformation for a
rotation angle of /4. The axis of rotation is the z axis.


4
e  0 0 1

  
0 * sin  8 
 

0 * sin   

 8 
q
 
1 * sin   

 8 

  
 cos  
8 

(Eq. 13)
(Eq. 14)
0




0


q
0.382683432


0.923879533
(Eq. 15)
The above example is generalized in the theory developed by the
great 18th century mathematician Leonard Euler that bears his name.

 
e[0] sin  2



 e[1] sin 


 2
q  

e[ 2] sin 


 2

  

 cos
 2 

Euler’s theorem and quaternions:


e
F













I Rotated
from t to t+t
I Frame at t
Euler’s theorem states that a rigid body can be brought from any arbit rary
initial orientation to an arbitrary final orientation by a single rotation about a
judiciously chosen axis fixed in both the initial and final frames by some angle.
Figure 2
When transforming a vector from one coordinate frame
parameterization to another, the quaternion can be used directly. This
method is not presented here. As previously stated the quaternion is
equivalent to a direction cosine matrix and can be converted to it using
the appropriate algorithm.
q00 = qUnit[0]*qUnit[0]
q11 = qUnit[1]*qUnit[1]
q22 = qUnit[2]*qUnit[2]
q33 = qUnit[3]*qUnit[3]
q01 = qUnit[0]*qUnit[1]
q02 = qUnit[0]*qUnit[2]
q03 = qUnit[0]*qUnit[3]
q12 = qUnit[1]*qUnit[2]
q13 = qUnit[1]*qUnit[3]
q23 = qUnit[2]*qUnit[3]
dcm[0][0] = q00 - q11 - q22 + q33
dcm[0][1] = 2.0*(q01 + q23)
(Eqs. 16)
dcm[0][2] = 2.0*(q02 - q13)
dcm[1][0] = 2.0*(q01 - q23)
dcm[1][1] = -q00 + q11 - q22 + q33
dcm[1][2] = 2.0*(q12 + q03)
dcm[2][0] = 2.0*(q02 + q13)
dcm[2][1] = 2.0*(q12 - q03)
dcm[2][2] = -q00 - q11 + q22 + q33
Then, the direction cosine matrix is used to perform the vector
transformation.
V) Mechanization of Navigation Equations in the Earth Fixed Frame
Airborne accelerometers measure the difference between inertial
kinetic acceleration a , and gravitational acceleration g . The difference
is called specific force and is essentially the sum of all contact forces
divided by the mass, or
f ag
(Eq. 17)
For an accelerometer in free fall in a vacuum,
a
= g , therefore
f 0.
For the accelerometer at rest in a gravitational field, a  0 therefore,
f   g . Therefore, acceleration due to gravity (mass attraction) must
be calculated during flight to navigate properly. Acceleration due to
gravity will be addressed in a subsequent section. Specific force is
essentially the on-board measurement of the accelerations due to thrust
and aerodynamics by three orthogonal (ideally) IMU (Inertial
Measurement Unit) axes. This is typically output from the IMU in the
form of delta V increments over an IMU measurement cycle ( t ), which
can be on the order of milliseconds.
V t  t   
b
t  t
t
a b (t )dt
(Eq. 18)
The superscript b in Eq. 16 implies that the integration is carried out in
the vehicle body frame. The notation is slightly misused in this instance.
The integration is actually carried out in the IMU frame, which can be
identical to the body frame. If they are not, they merely differ by an
orthogonal transformation for the purposes of transforming the V into
vehicle body coordinates.
In order to maintain the velocity (and position) of the vehicle
relative to an Earth-fixed frame, the V must be transformed from the
vehicle body frame (IMU frame) into the Earth-fixed frame. In order to
accomplish this, an accurate estimate if the orthogonal transformation
from the body frame to the Earth-fixed frame must be maintained by the
navigation system. The IMU contains three gyros mounted on three
orthogonal (ideally) axes which measure angular rates of the body frame
relative to the inertial frame. These are typically output from the IMU in
the form of three delta angles about the IMU axes over the IMU
measurement cycle.
 (t  t )  
b
ib
t  t
t
wibb (t )dt
(Eq. 19)
The angular rate of the vehicle body frame relative to the Earth fixed
frame can be computed using the following expression:
ebe  ibb  Ceb iee (Eq. 20)
Clearly Eq. 20 involves the orthogonal transformation that is to be
e
maintained. This difficulty can be overcome by realizing that ie is small
compared to ib . The average angular rate across the IMU measurement
interval can be treated as constant angular rate across the IMU
measurement interval. Also, the angular rate of the Earth fixed frame
relative to the inertial frame can be regarded as constant over the IMU
measurement interval. Using the orthogonal transformation and angular
rate of the Earth fixed frame relative to the inertial frame from the
previous IMU measurement interval, the delta angles of the body relative
to the Earth fixed frame can be computed as follows:
b
 ebb (t )   ebb (t )  Ceb (t  t )iee (t  t )t (Eq. 21)
A second order algorithm to maintain the quaternion that represents the
orthogonal transformation between the vehicle body frame and the Earth
fixed reference frame is as follows:


1

1
 
 1
qeb (t  t )  cos  ebb (t )  I 
sin   ebb (t )   qeb t 
(Eq. 22)
b
2
2





(
t
)
eb


Where:
 (t ) 
b
eb
 (t) 
b
eb
T
b
eb
(t )
(Eq. 23)
I – 4X4 identity matrix
 - 4X4 skew symmetric matrix composed of the elements of  ebb (t )
An alternate method for maintaining the orthogonal transformation
from the Earth-fixed frame to the vehicle body frame can be used for
vehicles with relatively short and known mission durations. Expendable
launch vehicles would fall into this category. The inertial to body
transformation can be maintained throughout flight using equations 22
and 23, except that the delta angles (  ib (t ) ), which are the output of the
IMU, would be used directly in Equations 22 and 23. Equation 21 is no
longer necessary. The transformation from the Earth-fixed frame to the
vehicle body frame would be computed as follows:
b


C t  C t   Ceb t  (Eq. 24)
b
i
e
i
T
This scheme would require that the initial transformation from
EMEJ2000 to ECEF at some epoch can be loaded into the navigation and
can be maintained in flight once again with equations 22 and 23,
e
C
Understanding that i t  must be converted to a quaternion. The delta
angles needed, can be computed as follows using Eq. 4 from above:
iee (t )  iee * t
(Eq. 25)
Error in the transformation from ECI to ECEF would build up with
time with this scheme because of Precession, Nutation, and polar motion
(movement of the ECEF rotation axis with respect to ECI). However a
more accurate estimate of  ie (t ) can be found by extracting the delta
quaternion from two quaternions which represent the exact
transformations from ECI to ECEF at time t and some time t  t later.
The delta quaternion is a quaternion that is multiplied by one quaternion
(ECI to ECEF at time t) to yield the second quaternion (ECI to ECEF at
e
time t  t ). On other words the delta quaternion represents the
transformation between ECEF at time t and ECEF and time t  t . Once
the delta quaternion is computed, the axis of rotation and angle about
e


that axis from the delta quaternion can be used to compute
ie (t ) .
Cie t   qie t 
Cie t  t   qei t  t 
qt   qie t  qei t  t 
*


  2.0 * a tan 2 1  q[3] * q[3], q[3]
q[0]
e[0] 
1  q[3] * q[3]
q[1]
e[1] 
(Eqs. 26)
1  q[3] * q[3]
q[2]
e[2] 
1  q[3] * q[3]
 iee (t )   * e
e
Where qi t  is the conjugate of q i t  found by negating the vector
*
portion of
e
q ie t  .
The above scheme can be used to achieve arbitrary accuracy by including
the required number of exact ECI to ECEF transformations spanning the
expected mission time with the required
transformations.
t
between successive exact
Realizing that 2ie r  ieie r is small and that acceleration due to
mass attraction is nearly constant over the IMU measurement cycle leads
to the following position and velocity update formulas:
e
e
e
e
e


 
t
1
1
r e t  t   r e t   re t t  g e  2iee re  iee iee r e tt  Ceb t  V b t t
2
2


 
r t  t   r t   g  2 r    r t  C t  V t 
e
e
e
e e
ie
e
ie
e e
ie
b
e
t
b
Eqs. 27
VI) Acceleration due to Gravity
One of the advantages of navigating in the Earth-fixed frame is that high
fidelity gravity models are readily expressed in this frame. A brief explanation of
one such model is presented here.
Gravitational potential in the Earth-fixed frame can be expressed as follows:
n

   n R
V r ,  ,    1     Pnm sin  Cnm cosm   S nm sin m 
r  n2 m0  r 

Eq. 28
Where:
r - Distance from geo center to object
 - Geocentric latitude of object
 - Longitude of object
R - Equatorial radius magnitude of Earth
“n” is referred to as the degree of the model, and “m” is referred to as the
order. Pnm are the associated Legendre functions with argument sin   , and will be
discussed later. Cnm and Snm are the spherical harmonic coefficients. In the above
series expansion, when the order equals 0 (m = 0), the coefficients are referred to as
zonal harmonics. These describe the variation in the Earth’s gravitational field due
to latitude. The largest variation is due to oblateness (J2).
It is orders of magnitude larger than the other coefficients (C20 = -1.0826266836e003; C21 = -2.414e-010), and for many applications, it is sufficient to only model J2.
When n = m, the coefficients are referred to as sectorial. Sectorial harmonics
account for the gravitational field variation in longitude. The coefficients are
referred to as tesseral harmonics when m  n  0. They divide the Earth sphere into a
checkerboard array or tiles (Hence the name, tesseral). Degree and order of the
models used in practice depend upon the application. The shuttle uses degree and
order 8 for on-board navigation. SEASAT uses degree and order 36 for precise orbit
determination.
V  r  V    V   
ag         
r   r     r     r 
T
T
T
n

V
  n
R
  2 1   n  1  Pnm sin  Cnm cosm   S nm sin m  Eqs. 29
r
r  n 2 m0
r

V  n  R  Pnm sin  
Cnm cosm   S nm sin m 
   
 n2 m0  r 

n
V   n  R 
    Pnm sin  mS nm cosm   Cnm sin m 
 r n  2 m 0  r 
n
r  x y z 
 , , 
r  r r r 
  xz  yz  z 2 
 2 , 2 , 1  2 
2
2
r  r 
x y  r


r

r
1

1
 y, x,0
2
2
x y
As stated earlier, Pnm are the associated Legendre functions. The associated
Legendre functions are obtained from the Rodriguez formula.


n
1 dn 2
Pn u   2
u

1
2 n! du n
m
Eqs. 30
2 m/2 d
Pnm u   1  u
Pn u 
du m


Expressions from the first formula in Eqs. 30 are referred to as Legendre
polynomials. Expressions obtained from the second formula are referred to as the
associated Legendre functions. The associated Legendre functions should be
computed recursively, in order to be efficient. One such method for doing so is as
follows (Reference: Satellite Orbits; Montenbruck and Gill):
P00  1
Pmm  2m  1 cos( ) Pm 1,m 1
Pm 1,m  2m  1sin(  ) Pmm
Pnm 
Eqs. 31
1
2n  1sin(  ) Pn1,m  n  m  1Pn2,m 
nm
For the second equation in Eqs. 31 the index ranges from m = 1 to m = desired
degree. For the third equation in Eqs. 31 the index ranges from m = 0 to m = desired
degree - 1. For the fourth equation in Eqs. 31 the index ranges from n = 2 to desired
degree and m = 0 to n – 2 for each n.
The partial derivatives with respect to  of the associated Legendre functions
must also be computed to support Eqs. 29. They can be computed recursively as
follows:

P00  0


Pmm  2m  1cos( ) Pm1,m1  (2m  1) sin(  ) Pm1,m1


Pm1,m  2m  1sin(  ) Pmm  2m  1cos( ) Pmm
1 

 2n  1sin(  ) Pn1,m  2n  1cos( ) Pn1,m  n  m  1Pn2,m 
Pnm 

nm
Eqs. 32
The index ranges are the same as above.
For a model of small degree and order (say 2X2), the gravitational equations
can be written explicitly for computational speed. This may be needed in a real time
system.
VII) Computation of Heading and Geodetic Latitude, Longitude and Altitude
Local North-East-Down (NED) frame of reference is defined as a frame of
reference directly below the vehicle at the surface of the Earth reference ellipse. The
+X axis (N) points due north (true north), the +Y axis (E) points due east, and the
+Z axis (D) points down along the local vertical (line perpendicular to the tangent to
the reference ellipse directly below the vehicle). Vehicle heading is most readily
computed in this frame of reference. The conversion from ECEF to NED is very
straight forward, and can be accomplished using the following equations:
 cos 
Z     sin  

0
sin  
cos 
0
0
0
1
  
 

 
cos       0  sin       
2

 
 2
 
  
Y         
0
1
0
 Eqs. 33
2



   
 

 
 sin    2     0 cos   2     

  
 
  
 

NED
C ECEF
 Y        * Z  

 2
Where:
 - Longitude of the vehicle (radians)
 - Geodetic latitude of the vehicle (radians)
NED
C ECEF
- Transformation direction cosine matrix from ECEF to NED
To compute vehicle heading from navigational data:
NED
vNED  CECEF
* r e
  a tan 2vNED [1], vNED [0]
Eqs. 34
Where:
 - Vehicle heading (angle of velocity vector from true north)
It should be noted that vNED has little meaning other than the parameterization
of the vehicle velocity in the NED frame.
Latitude, longitude, and altitude can be computed from navigational data to
arbitrary accuracy with the following iterative procedure:
Initial computations
  a tan 2r e [1], r e [0]
p
r e [0] * r e [0]  r e [1] * r e [1]
  r [ 2]   1.0  
 * 
 
p
1

e
*
e

 

  tan 1  
Eqs. 35
e
Loop through the following until arbitrary small change in latitude
N
h
RE 
2
RE 2 * cos 2    RP 2 * sin 2  
p
N
cos 
Eqs. 36



 e




r
[
2
]
1
1 

*
  tan 
N 
  p  
1 e * e *


N  h 


Where:
 - Longitude of the vehicle (radians)
 - Geodetic latitude of the vehicle (radians)
h – Altitude of the vehicle above the earth reference ellipse
RE – Earth equatorial radius
RP – Earth polar radius
e - Eccentricity of earth ellipse
RE 2  RP 2
RE
Note that the above converges in a few iterations, and should not present any
computational problems for a modern real-time system.
Also note that GPS raw data is supplied in the ECEF frame, which makes
implementing a blended INS/GPS navigation solution more straight forward in the
ECEF frame.
VIII) References
1) Introduction to Modern Navigation Systems; Esmat Bekir
2) Inertial Navigation Systems with Geodetic Applications; Christopher Jekeli
3) Satellite Orbits; Oliver Montenbruck and Eberhard Gill
4) Spacecraft Attitude Determination and Control; James R. Wertz
5) Fundamentals of Astrodynamics and Applications; David A. Vallado
6) Strapdown Inertial Navigation Technology; D. H. Titterton and J. L.
Weston