SAVAGE 239
executed by the basic updating function with the high-speed opera-
tion containing comparatively few calculations per high-speed cy-
cle. The net result is that computer through put is conserved because
only the high-speed computations must be operated at the rate re-
quired to process high-frequency angular dynamics accurately; the
basic navigation parameter update frequency can be set based on
other less demanding requirements (e.g., to ensure that the maxi-
mum attitude change per update cycle is small, thereby protecting
small angle approximations in the processing algorithms). When
the basic algorithms (the dominant navigation solution) are formu-
lated from closed-form analytical expressions that are exact under
particular input conditions (e.g., constant strapdown angular rate
and specific force acceleration) documentation is straightforward,
validation is precise (by exact comparison with comparable exact
solution truth models), and one set of algorithms can be used gener-
ically for all applications.
The two-speed structure for attitude updating was originated by
the author in 1966 using a second-order basic updating algorithm
with the high-speed digital integration function derived from a first-
order differential equation.
1
In 1969, Jordan proposed a two-speed
attitude computation structure in which the form of the basic up-
dating operation was based on the correct attitude solution under
rotation about a fixed (nonrotating) axis.
2
For this condition, the
algorithm input is the direct integral of angular rate vector com-
ponents provided by angular rate sensors. For general angular mo-
tion (in which the axis of rotation is changing direction), Jordan
proposed that the analytical form of the basic algorithm should re-
main the same, but that the Euler rotation vector be used for input
rather than integrated angular rate. The high-speed operation then
became an integration of rotation vector rate-of-change over the at-
titude update cycle. The Jordan high-speed integration algorithm for
the rotation vector was based on an approximate first order applica-
tion of the Goodman/Robinson theorem.
3
In 1971, Bortz
4
proposed that the high speed integration operation
in the two-speed attitude updating structure be based on the exact
rotation vector rate equation derived in 1949 by Laning.
5
Having the
exact rotation vector rate equation as a base provided a framework
for simplified algorithm development and accuracy evaluation by
comparison with the exact form. The exact Jordan structure coupled
with the integrated Bortz/Laning exact rotation vector rate equation
has been the basis for continuing modern day strapdown attitude
algorithm development.
This paper presents a new concept for strapdown velocity and
position updating using a Jordan two-speed attitude computation
structure. In direct analogy to Jordan attitude updating, the analytical
form of thevelocity/positionalgorithms is based on solutionsthat are
exact under particular input conditions, in this case constant angular
rate and specific force acceleration (Ref. 6 and Ref. 7, Secs. 7.2.2.2.1
and 7.3.3.1). For constant angular-rate/specific-force, the algorithm
input is the direct integral of angular-rate/specific-force vector com-
ponents provided by angular-rate sensors and accelerometers. The
exact constant-input solutions were originally derived as an expan-
sion of rotation compensation terms present in first-order two-speed
algorithms designed for general (nonconstant) inputs (Refs. 6, 8,
and Ref. 7, Secs. 7.2.2.2 and 7.3.3). For the new concept, ex-
act general-input velocity/position algorithms are synthesized us-
ing the Jordan approach of having the same analytical form under
general input as the exact constant-input solutions, but with veloc-
ity/position translation vectors (analogous to the rotation vector)
replacing the integrated angular-rate/specific-force used under
constant input. The translation vectors are calculated by inte-
grating translation vector rate equations over a velocity/position
update cycle. Differential equations are derived in the paper for
the velocity/position translation vectors that are exact under gen-
eral motion (analogous to the Laning rotation vector rate equa-
tion). The new velocity/position updating concept coupled with
the Jordan/Bortz/Laning attitude updating approach provides a uni-
fied mathematical framework for strapdown integration algorithm
design.
Strapdown computation algorithms can be designed within the
unified framework using the exact closed-form equations directly
(without approximation) for attitude/velocity/position updating. In-
puts to the updating algorithms would be high-speed numerical in-
tegration routines based on simplified integral versions of the exact
rotation/translation vector rate equations. The paper provides ex-
amples of simplified rotation/translation vector rate equations and
associated performance characteristics under generalized angular-
rate/specific-force maneuver and vibration profiles. The perfor-
mance investigations and some of the high speed routines are based
on a Picard expansion solution
9
to the exact rotation/translation
vector rate equations, derived here in powers of integrated angular-
rate/specific-force.
A general discussion is included in the paper on potential design
approaches for digital algorithms used for high-speed integration
of the simplified rotation/translation vector rate equations over the
attitude/velocity/position update cycle. The Appendix provides a
description of digital simulation studies conducted to numerically
verify the accuracy of the principal analytical results.
Unified Mathematical Framework Based on Constant
Angular Rate/Specific Force
The following generalized differential equations describe time
rates of change of attitude and specific-force acceleration induced
velocity/position in a coordinate frame B
m −1
representing the ori-
entation of a generalized rotating coordinate frame B at time t
m −1
(Ref. 7, Secs. 4.1, 4.3, and 4.4.1.2):
˙
C
B
m −1
B(t)
= C
B
m −1
B(t)
(ω×)
˙
v
B
m −1
SF
(t) = C
B
m −1
B(t)
a
SF
˙
R
B
m −1
SF
(t) = v
B
(m −1)
SF
(t) (1)
A generalized structure for updating attitude/velocity/position in a
strapdown inertial navigation system is obtained from the cumula-
tive integral of Eqs. (1) over strapdown computer update cycles m,
referenced to a nonrotating coordinate frame N and including the
effect of gravity,
C
B
m −1
B(t)
= I +
t
t
m −1
˙
C
B
m −1
B(t)
dtC
N
B
m
= C
N
B
m −1
C
B
m −1
B(t
m
)
v
N
m
= v
N
m −1
+ v
N
g
(t
m
) + C
N
B
m −1
v
B
m −1
SF
(t
m
)
v
N
g
(t) =
t
t
m −1
g
N
dt v
B
m −1
SF
(t) =
t
t
m −1
˙
v
B
m −1
SF
dt
R
N
m
= R
N
m −1
+ v
N
m −1
T
m
+ R
N
g
(t
m
) + C
N
B
m −1
R
B
m −1
SF
(t
m
)
R
N
g
(t) =
t
t
m −1
v
N
g
(t) dt R
B
m −1
SF
(t) =
t
t
m −1
˙
R
B
m −1
SF
(t) dt
(2)
In a strapdown inertial navigation system, the angular rate vector
ω is measured by strapdown angular rate sensors, the specific force
acceleration vector a
SF
is measured by strapdown accelerometers,
the B frame represents a coordinate frame that maintains align-
ment with the rotating strapdown sensors (the body frame), and the
N frame represents navigation coordinates for output reporting. In
many systems, the N frame is slowly rotated (e.g., to maintain one
axis vertical in the presence of vehicle motion and Earth’s rotation
rate) in which case Eqs. (2) would have additional terms (Refs. 6, 8,
10, and Ref. 7, Secs. 4.1, 4.3, and 4.4.1.2). Note that in some systems
the C
N
B
direction cosine matrix is replaced by an equivalent attitude
quaternion (using the quaternion equivalent to C
B
m −1
B(t
m
)
in Eqs. (2) for
input), and position location R is represented by altitude and angu-
lar location over the Earth’s surface [using the C
N
B
m −1
R
B
m −1
SF
(t
m
)
term in Eqs. (2) for input (Refs. 6, 8, 10, and Ref. 7, Secs. 4.1, 4.4,
7.1.2, and 7.3)]. Equations (2) and subsequent results in this paper
can be easily modified to incorporate these alternative navigation
parameter representations.
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