惯性导航的基本误差方程
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在惯性导航领域，误差方程是理解和分析系统性能的关键。"Error equations of inertial navigation"这篇论文由George Arshal撰写，发表于1987年7-8月的VOL.10, NO.4期。文章的核心在于，它推导出适用于所有正确构建的惯性导航系统的通用误差方程，这些方程不仅限于特定参考框架。
惯性导航基于积分方程，通过矢量分析方法进行推导。这一分析的一个主要成果是一组基础的误差传播方程，这些方程对于绝对导航误差的处理至关重要，而传统的速度和位置误差被认为是传递错误。这揭示了惯性导航系统中误差传播的新视角。
惯性导航系统的基本构造包括一个能够测量加速度和地球旋转速率的装置，这些数据经过连续积分计算，得出物体的位置、速度和姿态信息。然而，由于传感器的内在误差、漂移以及积分过程中的累积效应，会导致导航结果出现偏差。
在传统的误差分析中，通常会假设一个局部水平参考框架来简化问题。然而，Arshal的文章突破了这个限制，使用经典矢量方法直接处理矢量误差，从而得到适用于所有惯性导航系统的通用误差方程。这意味着无论导航系统如何设计或在何种环境中运行，这些误差方程都能提供精确的误差模型。
新提出的绝对误差方程关注的是导航结果的绝对精度，而不仅仅是相对于某一特定参考框架的误差。这对于评估整个导航任务的准确性和长期稳定性具有重要意义，尤其是在长时间飞行、深海探索或太空导航等对精度要求极高的应用中。
此外，论文还可能探讨了如何将这些误差方程应用于误差校正策略，以提高惯性导航系统的整体性能。例如，通过集成全球定位系统（GPS）或其他外部参照信号，可以减少惯性导航的累积误差，并增强其导航解的可靠性。
"Error equations of inertial navigation"这篇文章对于深入理解惯性导航系统的误差特性，以及开发更有效的误差补偿算法具有重要的理论价值和实践意义。通过这些新的误差方程，工程师和研究人员能够更好地预测和控制导航系统的误差，从而提升系统的整体性能。
VOL.
10, NO. 4,
JULY-AUGUST 1987
J.
GUIDANCE
351
Error
Equations
of
Inertial
Navigation
George Arshal
Federal
Way,
Washington
This
paper derives basic
error
equations
of
inertial
navigation
which apply
to any
properly constructed
inertial
navigator.
The
equations
are
deduced
from
the
integral equations
of
inertial navigation
by a
vectorial analysis.
A
major
result
of
this analysis
is a
set
of
fundamental
error
propagation equations
that
has
apparently been
missed. These equations
regard
the
absolute navigational errors.
The
conventional velocity
and
position
errors
are
shown
to be
transfer errors.
Introduction
T
HE
error propagation equations
of
inertial navigation
are
typically derived
in
relation
to a
specific frame
of
reference,
most often
a
locally level
frame.
1
'
2
They
are de-
rived
here
without
such confinement. Classical vectorial
methods
3
are
used
to
treat
the
vector errors directly. General
error
equations
applicable
to all
inertial navigators
are ob-
tained, including
a new set of
absolute error equations.
The
discussion begins
by
reviewing
the
constructions
of in-
ertial navigation.
The
absolute
errors
carried
by
these con-
structions
are
identified
and are
traced
through
the
naviga-
tional processes
to
obtain their equations
of
propagation.
The
manner
in
which
an
Earth-bound measurement
of
posi-
tion
and its
derivatives
can
relate
to
these errors
is
then
ex-
amined. This exercise yields
a
second
set of
error quantities,
termed
"transfer
errors,"
which
are
equally
as
descriptive
of
the
navigational error state
as the
absolute errors.
The
equa-
tions propagating
the
transfer
errors
are
also derived. These
equations generalize prior art. Both sets
of
error equations
are
subsequently reduced
to the
"Schuler-tuned"
case
in
which
the
vertical channel
of the
navigator
is
independently
constrained. This
is
followed
by a
note
on the
practice
of
navigational updates
and by
remarks comparing
the
different
error equations, including those
of the
prior art.
Navigational
Constructions
The
basic equation
of
inertial navigation
is
/?=
f
f
(,4-G)df
2
or
Js Js
dt
2
(1)
A
represents
the
sensed acceleration
at a
point defined
by a
cluster
of
accelerometers;
R
represents
the
position
of
this
point
from
Earth
center;
G
represents
the
specific force
of
reaction
to
gravitation
at
this point;
and S
represents
a
space-
fixed
reference
frame.*
The
subscript
on the
integral
sign
identifies
the
reference frame
in
which
the
integration pro-
ceeds,
i.e.,
the
reference frame
to
which
the
components
of the
integrand
are
produced
for
integration, either functionally
or
actually. Similarly,
the
subscript
on the
differential operator
d
identifies
the
reference frame
in
which
a
vector
is
noticed
for
its
change
of
components. These notations
are
generally
ap-
plied, i.e.,
the
operations
of
rate
of
change
and
integration
over
time relative
to a
reference frame
Q are
noted
as
d
Q
/dt
and
\
Q
dt.
Received
Sept.
9,
1985; revision received April
27,
1986. Copyright
©
1987
by G.
Arshal.
Published
by the
American Institute
of
Aeronautics
and
Astronautics,
Inc.,
with permission.
The
space-fixed
reference
frame
or
simply
space
is
represented
in a
rigid body whose angular velocity
is
measured
by
gyroscopes
as
zero.
It is
frequent practice
to
structure
the
navigational com-
putation
so
that
it
gives
its
output velocity
as
velocity relative
to the
Earth.
In
this scheme
del?
d
F
R
dt
dt
(2)
where
E
represents
an
Earth-fixed
reference
frame,
<*
E
represents
the
angular velocity
of the
Earth,
and
V
E
=
d
E
R/dt
represents
the
velocity relative
to the
Earth.
d
s
2
R/dt
2
is ex-
pressed
from
Eq. (2) by
1)
Differentiating
V
E
relative
to
space
in a
reference
frame
F, as
(3)
dt
dt
where
o>
F
represents
the
angular velocity
of F in
space.
2)
Differentiating
<*>
E
xR
as
d
s
~~—
((»)
E
X
jK)
—
(ji
E
X
V
E
~}-
WE-
X
(co
E
XR)
dt
The sum of
these results equals
A
—
G, so
that
dt
Also
d
s
R
d
F
R
dt
dt
whereby
d
F
R
dt
(4)
(5)
(6)
(7)
V
E
and R are
then produced
by
integrations reciprocal
to
their
expressed
rates
of
change
as
=\-
J
=
\
J F
[Y
E
-(<»
F
-u
E
)xR]dt
(8)
(9)
These computations
are
complemented
by an E
frame
(or S
frame)
that maintains alignment
with
the
Earth.
The E
frame
sustains
the
vector
&
E
and
forms
the
means
of
expressing
R in
Earth coordinates.
E
also enters into
the
statement
of G. The
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