autonomous monitoring system to meet nonprecision
landing approach requirements.
ACKNOWLEDGMENTS
[13]
[14]
The authors would like to thank Prof. I. Y.
Bar-Itzhack of the Technion for his comments and
suggestions.
REN DA
CHING-FANG LIN
American GNC Corp.
P.O. Box 10987
Canoga Park, CA 91304
[15]
[16]
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[17]
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[4]
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[10]
[11]
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methods–A survey.
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IEEE Transactions on Aerospace and Electronic Systems,
23, 1 (1987), 83—119.
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Survey of model based failure detection and isolation in
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IEEE Control Systems Magazine, 8, 6 (1988), 3—11.
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Englewood Cliffs, NJ: Prentice-Hall, 1991.
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Advanced Control System Design.
Englewood Cliffs, NJ: Prentice-Hall, 1994.
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Stochastic Models, Estimation, and Control, Vol. 1.
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Real-time failure detection: A static nonlinear
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(1977), 509—536.
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Fault-tolerant multisensor navigation system design.
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GPS integrity channel.
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Improved Command to Line-of-Sight for Homing
Guidance
The command to line-of-sight (CLOS) guidance is known
to require very high acceleration commands as the distance
between a missile and its target becomes closer. Therefore, the
CLOS is generally used for midcourse guidance. In order to use
the CLOS for homing, an additional feedforward acceleration
command (FFC) is necessary. In this correspondence an improved
CLOS (ICLOS) technique which includes FFC is proposed.
Its performance improvement is demonstrated by applying the
technique to a surface-to-air missile (SAM).
Manuscript received July 14, 1994; revised September 20, 1994.
IEEE Log No. T-AES/31/1/08037.
This work was partially supported by the Agency for Defense
Development, Korea.
c 1995 IEEE
0018-9251/95/$10.00 °
IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 31, NO. 1
JANUARY 1995
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Fig. 1. Guidance of missile to target using CLOS.
Fig. 2. CLOS guidance command displacement error.
I. INTRODUCTION
The principle of the command to line-of-sight
(CLOS) is to force a missile to fly along an
instantaneous line-of-sight (LOS) between a launcher
and a target. If the missile can continuously stay
on the LOS, it will eventually hit the target [1—3].
However, Fig. 1 shows that a missile requires large
lateral guidance commands as it approaches its target.
Therefore, it is difficult to employ the CLOS for
homing guidance.
This correspondence presents an improved
command to line-of-sight (ICLOS) technique that
is suitable for homing guidance. To use a CLOS
for homing guidance, an additional feedforward
acceleration command (FFC) is necessary to
compensate the high-acceleration requirements at
the final stage. In previous attempts, the FFC was
computed under the assumption that the accelerations
of a target and a rotating LOS have the same direction
[3, 4]. The main idea of our proposed method is that
the motion of a target is not circular but straight
in the duration of computing a guidance command.
The method to reduce errors generated from this
assumption is to compensate errors of the FFC with
a new weight function, resulting in decreasing miss
distance. An improvement in the ICLOS is confirmed
via a simulation of a surface-to-air missile (SAM).
And it should be noted that the other system errors
influencing the performance of CLOS like radar,
autopilot, and actuator errors are not considered in
the simulation. The main purpose of this work is to
demonstrate an improvement in calculating a CLOS
guidance command.
II. IMPROVED CLOS
The FFCs are the components perpendicular to
the LOS between the launcher and the missile under
the assumption that a missile stays on the LOS [3, 4].
Because the x-axis acceleration in the missile body
frame is not controllable, the FFCs deal with two
accelerations perpendicular to the x-axis. A general
FFC Af is readily obtained from (1) whose derivation
Fig. 3. Incoming target (Case I).
is in the Appendix
μ
¶ μ
_ Ã_ cosμ ¶
afy
R Ã̈ cosμ ¡ 2R Ã_ μ_ sinμ + 2R
=
Af =
_ μ_ ¡ R μ̈ ¡ R Ã_ 2 cosμ sinμ
¡2 R
afz
(1)
where afy and afz are the y- and z-axis commands,
respectively. R is the range from the launcher to the
missile on the LOS, and μ and à are the elevation and
the azimuth angle of the LOS.
The FFCs are computed from the rotating LOS
without considering the actual motion of a target.
Consequently, when the target motion is not circular,
there can be a displacement error. For instance, if a
target moves along a straight line as in Fig. 2, a target
velocity vector V is divided into Vc and Vo . In the
CLOS, the missile acceleration command is computed
using only the target acceleration component
along Vc . Since the acceleration component along
Vo is neglected, the predicted LOS has an angle
displacement error ". In midcourse guidance, the effect
of " is negligible. However, in homing guidance, " can
be a critical error resulting in a large miss distance in
the final stage.
This " can be removed by adding a weight function
K to afy and afz . The process of obtaining the weight
function K falls into one of the following three cases.
Case I: incoming target (refer to Fig. 3), Case II:
outgoing target (refer to Fig. 4), and Case III: crossing
target (refer to Fig. 5).
It is assumed that the target moves along a straight
line and maintains a constant velocity in the duration
of computing a guidance command. Also it is assumed
that the missile is on the LOS between the launcher
and the target.
CORRESPONDENCE
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507
Fig. 6. Systemic flow of missile.
given by
K = LN=LM
Fig. 4. Outgoing target (Case II).
= sin(® + Á+ )=(sin(® + Á) cos(Á+ ¡ Á)):
(6)
For the case of the outgoing target, LN is less than
LM as in Fig. 4, and K is less than 1. Equations (7)
and (8) show LM and LN, respectively. The equations
corresponding to LM and LN are the same as the
previous case (Case I). Therefore, K is exactly the
same as before
LM = V ¢t cos¯
= V ¢t sin(® + Á)
Fig. 5. Crossing target (Case III).
LN = V ¢t sin°= sin(¼ ¡ ¯ ¡ °)
For the incoming target as in Fig. 3, if Á¡ , Á, R ¡ ,
and R are known, the direction angle of a target ®, at
time t, is expressed by
μ
¶
R(t) sinÁ(t) ¡ R(t ¡ ¢t) sinÁ(t ¡ ¢t)
® = tan ¡1
:
R(t ¡ ¢t) cosÁ(t ¡ ¢t) ¡ R(t) cosÁ(t)
(2)
The predicted angle of the LOS, Á+ , at time t + ¢t,
is readily obtained by
μ
¶
2R(t) sinÁ(t) ¡ R(t ¡ ¢t) sinÁ(t ¡ ¢t)
+
¡1
Á = tan
:
2R(t) cosÁ(t) ¡ R(t ¡ ¢t) cosÁ(t ¡ ¢t)
(3)
Employing the guidance command Af computed
from (1), the missile will be located on the LOS
at position M after time ¢t. If the target velocity
vector is divided into two components as in Fig. 3,
the length of LM is a target acceleration component
perpendicular to the LOS. However, since the real
target is located at position N, the discrepancy has
to be compensated. The lengths of LM and LN are
calculated by
LM = V ¢t cos¯
= V ¢t sin(® + Á)
(4)
LN = V ¢t sin(¼ ¡ °)= sin(³)
= V ¢t sin(® + Á+ )= cos(Á+ ¡ Á):
(5)
The weight function K should be multiplied to Af
so that the missile moves to the position N and this
is given by LN=LM. Hence, using (4) and (5), K is
508
(7)
= V ¢t sin(® + Á+ )= cos(Á+ ¡ Á):
(8)
If we compute K from Fig. 5, the following results
can be obtained. It should be noted that K equals 1.
This coincides with the previous case, i.e., (6)
® + Á = ¼=2
(9)
® + Á+ = (® + Á) + (Á+ ¡ Á)
= ¼=2 + (Á+ ¡ Á)
(10)
K = sin(® + Á+ )=(sin(® + Á) cos(Á+ ¡ Á))
= sin(¼=2 + (Á+ ¡ Á))=(sin(¼=2) cos(Á+ ¡ Á))
= cos(Á+ ¡ Á)= cos(Á+ ¡ Á)
= 1:
(11)
As mentioned above, the new guidance command
An is expressed by
μ ¶ μ
¶
any
Ky afy
=
(12)
An =
a nz
Kz afz
where Ky and Kz are weight functions computed from
(6).
III. SIMULATION RESULTS
To compare the ICLOS guidance with a
conventional CLOS, a simulation is executed. A SAM
system in Fig. 6 is selected for simulation.
Fig. 7 shows the initial position and velocity of a
target for the simulation, where the initial height is
0[m], initial range of x-axis is 3,500[m], and initial
velocity is Mach 1. Three different cases of 30± , 45± ,
and 60± target directions to the y-axis of the inertial
frame were carried out. Miss distance is selected as
IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 31, NO. 1
JANUARY 1995
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Fig. 7. Initial position and velocity of target for simulation.
TABLE I
Comparison of Conventional CLOS and Improved CLOS
Fig. 8. Calculation of feedforward acceleration.
The x-component is due to the change in length of R.
The y-component comes from the change in direction
of the X ¡ Y component of R due to the swinging of R
with Ã. The z-component is due to the swinging of R
with μ.
The acceleration of the particle is determined from
the vector changes in the three velocity components
in Fig. 8 as these components change in magnitude
the indicator of the performance of homing guidance.
and direction. During time dt, the component ºR
The results of these cases are shown in Table I. It
undergoes changes dºR , ºR cosμ dÃ, and ºR dμ in the
can be observed from the table that the ICLOS is
x-, y-, and z-axes, respectively. The y-component plays
consistantly superior to that of the conventional CLOS
a role only in the X ¡ Y projection of ºR . During the
with FFC. The table also shows that the case of 45±
same interval, ºÃ undergoes changes dºÃ in the y-axis
provides higher improvement than the other two cases,
and ºÃ dà parallel to the X ¡ Y plane. The change
where the intercepting angle is almost perpendicular.
of ºÃ consists of two components, ºÃ dà cosμ in the
negative x-axis and ºÃ dà sinμ in the positive z-axis.
IV. CONCLUSIONS
Finally, ºμ has the vector changes dºμ , ºμ sinμ dÃ, and
ºμ dμ in the z-, minus y-, minus x-axes, respectively.
For homing guidance, a technique to improve the
The y-component is due only to the swinging of the
CLOS guidance is proposed in this paper. In order to
X ¡ Y projection of ºμ . By collecting these terms, the
reduce the displacement error induced in calculating
three components of the vector change in velocity are
the FFC, a weight function K is employed. The
found to be
weight function K is geometrically derived under the
jdºjx = dºR ¡ ºÃ cosμ dà ¡ ºμ dμ
assumption that a target moves along a straight line
and maintains a constant velocity during a computation
jdºjy = ºR cosμ dà + dºÃ ¡ ºμ sinμ dÃ
(14)
step of the guidance command. To compare the ICLOS
with a conventional CLOS, a SAM model is used
jdºjz = ºR dμ + ºÃ sinμ dà + dºμ :
for simulation. The simulation results show that the
guidance performance of the ICLOS is superior to that Upon substitution of the expressions for ºR , ºÃ , and
ºμ , dividing by dt, and rearranging the terms, the
of a conventional CLOS. Therefore, the ICLOS will
enhance the missile performance in the case of homing acceleration and its components become
guidance.
(15)
a = ax + ay + az
where
jajx = R̈ ¡ R μ_ 2 ¡ R Ã_ 2 cos2 μ
APPENDIX
The location of a particle at A may be described by
spherical coordinates as shown in Fig. 8. The velocity
of the particle may also be expressed in terms of its
three components,
º = ºx + ºy + ºz
where
_
jºjx = R
jºjy = R Ã_ cosμ
_
jºjz = R μ:
(13)
cosμ d 2 _
(R Ã) ¡ 2R Ã_ μ_ sinμ
R dt
1 d 2_
(R μ) + R Ã_ 2 sinμ cosμ:
jajz =
R dt
jajy =
Out of the three components, jajx is of no interest
in the CLOS guidance. The last two components are
only utilized, and they are given in (16)
μ
¶ μ
_ Ã_ cosμ ¶
afy
R Ã̈ cosμ ¡ 2R Ã_ μ_ sinμ + 2R
Af =
=
:
_ μ_ ¡ R μ̈ ¡ R Ã_ 2 cosμ sinμ
¡2R
afz
(16)
CORRESPONDENCE
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509
GYU TAEK LEE
JANG GYU LEE
Automatic Control Research Center
Dept. of Control and Instrumentation Engineering
College of Engineering
Seoul National University
San 56-1
Shinrim-Dong, Kwanak-Ku
Seoul 151-742
Korea
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
Siegal, J., and Lee, J. G. (1978)
Evaluation of command to line-of-sight guidance for
medium range missiles.
Report TR-1053-2, The Analytic Sciences Corp., June
1978.
Ha, I-J., and Chong, S. (1992)
Design of a CLOS guidance law via feedback linearization.
IEEE Transactions on Aerospace and Electronic Systems,
28, 1 (1992), 61—63.
Garnell, P., and East, D. J. (1977)
Guided Weapon Control Systems.
Elmsford, NY: Pergamon, 1977, 140—152.
Lin, C.-F. (1991)
Modern Navigation, Guidance and Control Processing.
Englewood Cliffs, NJ: Prentice-Hall, 1991, 318—319.
Blakelock, J. H. (1991)
Automatic Control of Aircraft and Missiles (2nd ed.).
New York: Wiley, 1991, 273—279.
Kain, J. E., and Yost, D. J. (1977)
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Comments on “Analysis of Geolocation by TDOA”
Reference [1] gives an alternative solution
algorithm for the time difference of arrival (TDOA)
problem and makes a number of unjustified claims.
The paper shows the following.
1) With an emitter on a spherical Earth, the
problem of three-receiver geolocation is reduced to
the solution of a quartic equation.
2) Four-receiver geolocation can be obtained by
solving a quadratic equation.
Both of these results were already obtained in [2],
which shows that even more general cases can be
reduced to the quartic solution. The algorithm in [1]
is in Earth-fixed coordinates, which may have some
minor computational advantage, but certainly does not
convey as much physical insight as the results of [2].
These are relatively minor matters. More objectionable
are the following claims made in [1].
1) The solution method can be easily extended to
the extra-receiver geolocation problem.
2) Previous error analysis formulations such as that
given in [3] are invalid for the correlated TDOA data.
Also, the position variance from [3] is for iterative
solution only, not for exact solution.
Claim 1 follows from the proposal in [1] to use
pseudoinversion to solve a set of “over-determined
equivalent observation equations” similar to (7)
and then substitute the result into one of (3). This
procedure only yields a solution of questionable
accuracy for the following reasons:
1) The procedure favors the particular (3) that is
used, as this equation is satified exactly, while (7) is
satisfied approximately. An injudicious choice of the
particular (3) may result in a poor solution.
2) Equation (7) is highly correlated and not of
uniform accuracy, a straightforward pseudoinversion
does not produce an “optimum” solution.
Claim 2 is erroneous as even a cursory reading of
[3] will reveal that the formulation does not exclude
correlated data, and can be used to compute the
“minimum variance” (Cramèr-Rao bound). The
statement that the position variance from [3] is for
iterative solution only, not for exact solution makes no
sense. If the iterative solution converges, it should not
be different from the so-called exact solution, which is
also obtained numerically.
The commentator would also like to take this
opportunity to make one observation and to correct
a minor error concerning the result in [2]. The
observation is that the Global Positioning System
(GPS) navigation is really equivalent to the TDOA
geolocation problem, as the difference of two
pseudoranges is a TDOA. Therefore results in TDOA
geolocation analysis are applicable to GPS navigation
and vice versa. The correction is that the surface of
the ocean is approximately an oblate spheroid rather
than an ellipsoid of revolution. The corresponding
geolocation is still given by the solution of a quartic,
but not that given by [2, eq. (21)].
BERTRAND T. FANG
TASC
12100 Sunset Hills Road
Reston, VA 22090
REFERENCES
[1]
[2]
Manuscript received March 24, 1994.
IEEE Log No. T-AES/31/1/08033.
c 1995 IEEE
0018-9251/95/$10.00 °
510
[3]
Ho, K. C., and Chan, Y. T. (1993)
Analysis of geolocation by TDOA.
IEEE Transactions on Aerospace and Electronic Systems,
29, 4 (Oct. 1993), 1311—1322.
Fang, B. T. (1990)
Simple solutions for hyperbolic and related position fixes.
IEEE Transactions on Aerospace and Electronic Systems,
26, 5 (Sept. 1990), 748—753.
Torrieri, D. J. (1984)
Statistical theory of passive location systems.
IEEE Transactions on Aerospace and Electronic Systems,
AES-20, 2 (Mar. 1984), 183—198.
IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 31, NO. 1
JANUARY 1995
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