Global Positioning System Reference
In-Depth Information
tively. Following [25], the Euler angles may be found by first determining a
least-squares estimate of
T
:
(
)
−
1
T
T
TRR RR
=
(8.47)
ned
body
body
ned
followed by a solution of (8.46):
T
T
(
)
−
1
−
1
32
−
1
21
θ
=
sin
−
T
;
φ
=
sin
;
ψ
=
sin
cos
(8.48)
31
cos
θ
θ
where
T
ij
refers to the (
i
,
j
)th element of the matrix
T
.
From inspection of (8.48), it is apparent that this approach is not stable for pitch
angles approaching
90º. Nonetheless, the method outlined here is practical for a
number of applications where such an attitude is not encountered. The preferred
mechanization for platforms that may experience any attitude is the use of
quater-
nions
. A quaternion is a mathematical construct that essentially extends the notion
of a complex number to four dimensions. Whereas a complex number can be viewed
as a mapping from a 2-vector (
a
,
b
) to a complex number
a
±
+
i
b
, a four-vector
(
a
,
b
,
c
,
d
) can be mapped into a quaternion
a
+
i
b
+
j
c
+
k
d
(referred to as a
pure
quaternion
if
a
0), which has its own associated set of mathematical rules. An
excellent introduction to quaternions is provided in [37].
GPS attitude determination systems are often implemented with four antennas
or more, even though all three Euler angles can be determined with only three. Addi-
tional antennas provide redundancy. They are especially important for all-attitude
platforms that can rotate such that they block visibility of one or more of the anten-
nas to the visible GPS satellites. Whereas four satellites are normally required for
carrier-phase positioning, only two satellites are needed for attitude determination,
provided that a common receiver is utilized for phase measurements for each
antenna and that the baseline lengths between the antennas are precisely known
[38]. The common receiver results in cancellation of the receiver clock bias when
single differences of carrier-phase measurements between antennas are formed.
Although each antenna has a different analog path to the receiver, which results
from differing electrical path lengths, these
line biases
can be mostly removed
through calibration procedures.
Multipath is the error source that limits performance for most GPS attitude
determination systems. Typical 1-sigma accuracy for each Euler angle in radians is
the 1-sigma single-difference carrier phase multipath error divided by the antenna
baseline length (with both the 1-sigma multipath error and the baseline length in the
same units of length) [38]. Additional error sources that may be significant for GPS
attitude determination applications that have not been previously discussed include
structural flexing and tropospheric refraction. Structural flexing is the bending of
the platform on which the multiple GPS antennas are installed due to applied forces
or temperature changes. If flexing is nonnegligible, its effects can be mitigated
through estimation or modeling. Tropospheric refraction is the bending of GPS sig-
nals as they pass through the troposphere. The very slight bending of each GPS sig-
nal path does not significantly alter pseudorange and carrier-phase measurements
but may introduce unacceptable Euler angle errors for some applications. Tropo-
=
Search WWH ::
Custom Search