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An analysis of the lead-angles necessary for tracking and pointing at distant targets
An Analysis Tracking
of the Lead-angles
and Pointing
Necessav
at Distant
for
Targets7
by ANTHONYN.PAYNE Electronics Engineering University of California,
Department, Lawrence Livermore Livermore, CA 94.550, U.S.A.
National
Laboratorv,
ABSTRACT : A rigorous analysis is made of the lead-aheadproblem that arises when a moving observer must track and point at a moving target at long range. The efSect of errors in the
predict-ahead time and in the range upon lead-angles is studied. For general target and observer motions in three dimensions, s@cient conditions are exhibited under which the relative error in the lead-angles is, to good approximation, directly proportional to the relative error in range. It is jtirther shown how lead-angle requirements impose a constraint upon the maximum allowable uncertainty in range. Finally, the analysis is applied to a numerical example.
I. Zntvoduction Consider the problem in which a moving observer must track a moving target at long range (e.g. a ballistic missile) and deposit a beam of energy having narrow divergence (e.g. a high power laser) on the target. In order for the beam to hit the target, the observer must “lead” the target. That is, it must predict where the target will be when the beam arrives and direct its line-of-fire accordingly. Thus a leadaheadproblem arises in which the lead-angles defining the line-of-sight between the observer and the target at some time in the future must be predicted. The predict-ahead time consists of two components. First, it must include the time-delays incurred by the operations of the observer’s tracking and pointing system. These time-delays originate from such sources as sensor read-out, signal and image processing of sensor data and pointing servo lags. Second, for large ranges between the target and the observer, the light-propagation-delays contribute significantly to the predict-ahead time. The transit times of energy from the target sensed by the observer and of the observer’s beam directed to the target must both be included in the predict-ahead time. The predict-ahead time is thus a function of the target-observer range. Since the lead-angles depend upon predict-ahead time and thus upon range, an important practical issue concerns the effect that errors in range estimates have upon lead-angles. Such a consideration is important for the following reason. The t This work was performed Lawrence Livermore National
under the auspices of the U.S. Department of Energy Laboratory under contract number W-7405-ENG-48.
The FranklinInstitute 001&0032~89$3.00+0.00
by
435
A. N. Payne
1
beam divergence imposes a direct constraint upon the allowable error in the leadangles, which in turn restricts the maximum range error that is permissible. We need to know, therefore, how sensitive the lead angles are to uncertainties in range. In this paper, we make a simple, yet rigorous, analysis of the lead-ahead problem for general target and observer motions in three dimensions. We develop expressions for the relative error in lead-angles caused by uncertainty in predictahead time and by uncertainty in range. As a main result, we show that, under suitable conditions, the relative error in lead-angle A$/$ is well-approximated in terms of the relative error in range by
where T,~is the component of the predict-ahead time contributed by the system delay and zP corresponds to the light-propagation-delay component. Moreover, we demonstrate that if A$ is the maximum magnitude of error allowable in the lead-angle, then independent of zs, the range error must satisfy the constraint
WI<* r
(1.2)
’ I*“1 ’
where i+Gp is the portion of lead-angle arising from light-propagation-delay alone. We organize the remainder of the paper as follows. In Section II, we develop expressions for the lead-angle and its relative error in terms of the predict-ahead time and the uncertainty or error in predict-ahead time. In the process, we establish the fact that the relative error in lead-angle is approximately equal to the relative uncertainty in the predict-ahead time under certain conditions, In Section III, we express these sufficient conditions in terms of bounds which are functions of the range and the relative velocity and acceleration of the target and observer. Then, in Section IV, we characterize the predict-ahead time and its relative error in terms of range and the uncertainty in range. Combining these results with the findings of Sections II and III, we obtain (1.1). Finally, we develop (1.2) and demonstrate the use of our analysis with a numerical example in Section V.
II. A Characterization
of Lead-angle and its Relative Error
Let /I(.) denote an angle associated with the line-of-sight between the observer and the target and measured with respect to a non-rotating reference frame attached to the observer. If the target and the observer motions are coplanar, then p may correspond simply to the bearing angle. For general motions in three dimensions, /I represents one of the two independent angles necessary to specify uniquely the line-of-sight. For example, /? may be the target azimuth or elevation relative to the observer, or it may be a direction-angle of the line-of-sight. Suppose that z is the predict-ahead time. Then, at time t, the lead-angle associated with /I is Journal
436
of the Franklin Pergamon
Institute Press plc
Lead-angles for Tracking and Pointing at Distant $s(t,z)
4 B(t+4
Targets
-P(t).
(2.1)
Now let AZ be the error or uncertainty in the predict-ahead (2. l), the corresponding error in the lead-angle is
time z. Then,
from
A$b(t, r, Ar) g $s(t, r + Ar) - $,r(t, r)
(2.2)
= p(t+z+AT>-B(t+z). We have the following characterization of rja and its relative error A$a/$O. Proposition 2.1. Suppose that T > 0 and that p(s) is twice-continuously entiable on the interval [t, t+ T], with /3(t) # 0. Further, define sg(t, T) 2 T,$;;,
Ik+s)/2k)I.
differ-
(2.3)
Then, for t, t+ Ar E [0, T], there exist ~,(t, z) and EZ(t, z, AZ) such that, at time t, the lead-angle and the relative error in the lead-angle arising from an error AZ in the predict-ahead time z satisfy $s(t, r) = P(t)r(l
W,dt,
7,
+&I(& r)),
(2.4)
AT> AT 1 + E2(t, z, AZ)
$0 (4 r)
-
z (
If&,(&T)
(2.5)
-
and
IE,(t,~)/ d +Xt, T),
(2.6)
Mt, z, A4 < 3+(t, 0. Proof. Consider (2.1) and (2.2). By Taylor’s Theorem of the remainder, there exist I,, I, E [0, l] such that $fl(t,z)
(2.7) with the Lagrange
= &t)z+:B(t+A,+2
form
(2.8)
and At,b,&,qAx) Again,
by Taylor’s
= #(t+z)Ar+tB’(t+~+&At)(Az)*.
Theorem,
there exists a i3 E [0, l] such that
&t+r) Substituting
(2.9)
= /Q)+B’(t+&+.
(2.10)
(2.10) into (2.9), we obtain A$@(t,qAr)
= lj(t)Az+b(t+&z)zAz+:B’(t+z+&Ar)(A$2.
(2.11)
Defining E,(&Z) 4 i&+&+/B(t), Ez(t,qAT) Eq. (2.8) becomes Vol. 326, No 3, pp. 435448, Printed in Great Britain
A [B(t+~3t)Z+tB(t+Z+;IZAZ)AZI/P(t),
(2.12) (2.13)
(2.4) and (2.11) becomes 1989
437
A. N. Payne
A$,j(t, z, AZ) = b(t)Ar(l +cz(t, z, AT)).
(2.14)
Then (2.4) and (2.14) imply (2.5). It remains to show that (2.6) and (2.7) hold. By assumption z, z+Az E [0, T], so that 0 d r < Tand 0 < IArl < T. Now, using (2.12), we obtain Is,(t,r)l Similarly,
d rIJi(t+&r)/28(t)I
d +(t, T).
(2.15)
from (2.13), we have
d %(t, T),
(2.16)
and the proposition is proven. 0 Note that this result implies that in the limiting case when +(t, T) = 0, the relative error in the lead-angle is equivalent to the relative error in the predict-ahead we time. Moreover, if 0 < +(t, T) but +(t, T) << 1, then as a good approximation may still say that tia = P(t)?
(2.17) (2.18)
It is easy to show from (2.3) that a necessary condition for +(t, T) to be small is that the maximum relative change in p(e) on the interval [t, t + T] be small. While our present approach is more general, we can also arrive at (2.17) and (2.18) if we pursue a somewhat more intuitive, approximate approach, which proceeds as follows. Consider coplanar target and observer motions and let r be the range from the observer to the target and vg be the transversal component of the reiative velocity. If r and up are approximately constant over the predict-ahead interval, then for small lead-angles (2.19) and
The division of (2.20) by (2.19) leads immediately to (2.18). Bounds on the relative error made in assuming that (2.17)-(2.18) hold are given as follows. Corollary 2.1. Under the same assumptions as Proposition 2.1, if e&t, T) < 1, then (2.21) and
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Journal of the Franklin Institute Pergamon Press plc
Lead-angles for Tracking and Pointing at Distant
IAt,ba(t,z,Az),h,bg(t,z) -Add
6
Proof. The bound From (2.9,
‘%(t, T) 1 -c+(t, T) ’
IWl
given by (2.21) is a direct consequence
wa
AZ
__-~ *a
Ar
s2-.s1
(2.22) of (2.4) and (2.6).
(2.23)
1 +s, >
r -( r
Targets
and thus (2.24) where the last inequality follows from (2.6)-(2.7) and from the fact that I&,(
we have
II+&21
1+
l&21
ll+s,j’
I-_I.s,J’
1 + 3Ep(t, T) l--Eg(t,T)
’
(2.26)
This result and (2.5) imply (2.25). q In the next section, we develop an upper bound on cg(t, T) in terms of the parameters which define the dynamics of the relative motion between the target and the observer in three-space. This bound may then be used to establish sufficient conditions on target-observer motion which guarantee that (2.17) and (2.18) are good approximations.
ZZZ.Bounds for General Motion in Three Dimensions Let r be the position vector of the target relative to the observer, as depicted in Fig. 1. The state equations governing the relative motion between the observer and the target are
where a(.) is the relative “0,. 326, NO. 3, pp. 435.448, 19x9 Prmted
in Great
Britain
i(t) = v(t),
(3.1)
v(t) = a(t),
(3.2)
target-observer
acceleration
arising
from forces on the 439
A. N. Payne
FIG. 1. Observer-target
geometry. XYZ is a non-rotating observer.
reference frame attached to the
target and the observer exerted by gravity, thrust, etc. Now we need two independent angles to specify uniquely the line-of-sight between the observer and the target. For this purpose, we employ azimuth $ and elevation 8 (see Fig. 1). We let Ydenote the range; that is, r = Irj, where for vector arguments 1.1denotes the usual Euclidean norm. Representing (3.1) and (3.2) in the range-azimuthelevation coordinate system, it can be shown (1) that f= i=v
rd2+rfj2cos29+a,,
(3.3)
i-9
(3.4) (3.5)
(I=;&, 8’= - ys-&2sinBcos8+
(3.6) T,
(3.7)
where (zI,, v~, us) and (a,, a$, a*) are representations of v and a in the r-q&0 coordinate system. We can now bound +(f, T) and .s+(t, T) in terms of the range r(t), the magnitude of velocity Iv(t) [,4(t) and d(t) and an upper bound on the magnitude of acceleration la(l+s)l for SE [0, T]. Proposition 3.1. Let r(a), 4(m) and Q(s) satisfy Eqs (3.3)-(3.8) and assume that T > 0. Suppose that 440
Journal
of the Franklin Pergamon
Institute Press plc
Lead-angles for
le(t+S)I
Tracking and Pointing at Distant
< 8<;
forallsE[O,
and that d(t) # 0 and d(t) # 0. Further, such that
(3.9)
that there exists an cc(t, T) > 0
suppose
< a(t, T)
la(t+s)l
T],
Targets
for all SE [0, T]
(3.10)
and that 0 < p(t, T) k r(t)-lv(t)IT-&(t,
T)T*.
(3.11)
Then, under these conditions,
E&t>T) B
Tv(t, T,
e>
(3.12)
2l&(t)I case’
Tv(t, T, 8) ~(t, T) d 21d(t)l ,
(3.13)
where
A& O2
q(t,T,@
a(r, T>
A (l+tan8)---,+--At, T)
(3.14)
At> T)
with p(t, T) 4 Iv(t)1 +a(t, T)T. Proof. Let SE [0, T] and consider
(3.15)
(3.1) and (3.2). By Taylor’s
Theorem,
I r(t+s)
= r(t)+v(t)s+s*
(1 -A)a(t+As)
dA
(3.16)
s0 and I v(t+s)
= v(t) fs
a(t + As) dA.
(3.17)
s0 Thus, utilizing
(3.10), we obtain u(t+s)
> r(t)-
Iv(t)js-&(t,
> r(t) - (v(t)\ T-$x(t,
T)s* T)T*,
(3.18)
and
Iv(t+.9l 6 Iv(t)1 +cr(t, T)s < Iv(t)1 +or(t, T)T.
(3.19)
Hence, we have shown that p(t, T) < r(t+s),
for all SE [0, T]
(3.20)
and Vol. 326. No. 3, pp. 435-448, Printed in Great Britam
1989
441
A. N. Payne Iv(t+s)l Now from (3.4)-(3.8), IJ(t+s)I
G p(t, T)
forall
sE[O, r].
(3.21)
we obtain
2lv,(t+s)ug(t+s)I d r(t+s)2~COSe(t+s)~
+ 2Iz+(t+s)~~(t+s)I
ltand(t+s)I
r(t+s)2~COSQ(t+-
lq(t+4l
(3.22)
+ r(tfs)lcosB(t+s)/ and
le(t+s)l d
~I~,(~+-JM~+~)I+ ~~(~+~)l*Itan~(~+~)I+ ladt+s>l r(t+s)2
r(t+s)2
r(t+s)
We will now obtain bounds uniform in s for the terms making sides of (3.22) and (3.23). Note that (3.9) implies that 0 < cosB<
(3.23)
.
up the right-hand
lcosCqt+s)l
(3.24)
and ltanB(t+s))
(3.25)
d tang.
Note also that 0 ,<
(l~rl-l~~l>2= I~r12-21wgl+l~~/2,
(3.26)
/V,U~l d t(lv,l’+l&$l’) < tlvl’.
(3.27)
so that
Applying
(3.21) we then have d &(t, q2.
(3.28)
Iv,(t+s)vs(t+s)l
=G$(t,
T)2
(3.29)
Iv0(t+s)zQ(t+s)(
d :#u
(3.30)
Iv,(t+s)v&+s)/ It may be shown in a similar fashion
that
and
We also have Iv,(t+s)12
< Jv(t+s)12
(3.31)
< p(t, q2.
And finally, using (3.10), we obtain lap(t+s>l
d la(t+s)l
Now, applying (3.24))(3.32), (3.22) and (3.23) to obtain
B = 4,0.
(3.32)
from above the right-hand
sides of
d a(& T)
we can bound
for
Journal
442
of the Franklin Pergamon
Institute Press plc
Lead-angles for Tracking and Pointing at Distant a(t, T) p(t, T) cos e
Targets
(3.33)
and l&t+,)1
< (1 +tan8)
At, T>’ 4t, T>
Y p(t, T)
+ ~ p(t> T) ’
(3.34)
From these two inequalities and (2.3), the upper bounds on E4(t, T) and co(t, T) defined by (3.12)-(3.13) follow. 0 We now make several observations concerning this result. First, the constraint (3.9) is imposed to avoid infinitely large angular velocity and acceleration in azimuth. Such a restriction would be needed in practical pointing and tracking systems. Moreover, it poses no limitation on the present result as we may generally orient the coordinate frame so that (3.9) is satisfied in the time-interval of interest. Second, observe that the constant p(t, T) defined by (3.11) is merely the minimum range possible in the time interval [t, t + T], and the constant p(t, T) given by (3.15) is the maximum relative speed possible in the same time interval. The constants y(t, g and .. T, e> and q(t, T, @/cos 0 are upper bounds on the angular accelerations 4, respectively, in the interval [t, t + T]. Third, we note that Propositions 3.1 and 2.1 imply that the conditions Q(t,
T, 0) << 1
(3.35) ’
21C#$t)lcosB
Tv(t, T>01<< 1
(3.36)
2@(t)l -are sufficient
for us to say that, as a good approximation, (3.37)
As a final remark, we mention that a more detailed analysis of (3.4)-(3.8) reveals that if 10~1and 1~~1are not much greater than [asIT and lagI T, respectively, then it may not be possible to insure that Ed << 1 and E*<< 1. In this case, it may no longer be safe to assume that (3.37) holds. Bounds derived from Corollaries 2.1 and 2.2 may then be useful. It appears, however, that conditions (3.35)-(3.36) are usually satisfied for large relative ranges and for the relative velocities and accelerations typical of ballistic or thrusting targets and observers. This is illustrated by an example in Section V.
IV. The Effect of Uncertainty
in Range
It remains to establish the relationship of the relative error in lead-angle to range and the error in range. We shall do this by first characterizing the effect of range uncertainty upon predict-ahead time. voi 326, NO. 3, pp. 43s448, Printed in Great Britam
19x9
443
A. N. Payne
Consider then the sequence of events which define the predict-ahead time z. Energy is reflected or radiated from the target at time t and received by the observer’s sensor at time t,. The transit time is thus t, - t, and the distance traversed by the energy wavefront is the relative range at time t,. Thus, r(&f) = c(ta - 0,
(4.1)
where c is the speed of light. Let the system processing delay be z,?, so that the observer fires its beam at time t,+z,. The beam arrives on target at time t,,. The distance traveled by the beam is then r(tJ Therefore,
the total predict-ahead
= c(t* - t, -7,).
(4.2)
time is r = tb-t.
(4.3)
Note that (4.1)-(4.3) define r implicitly as a function of t, z,~and y(m). It is not clear at first glance, however, that a z(t, t,, Y(S)) satisfying (4. I)-(4.3) exists or is unique for a general range function r(a). Nevertheless, for real targets and observers, it is true that Ji(t)/cJ < 1, and this turns out to be a sufficient condition to guarantee existence and uniqueness. We state and prove this fact formally as follows. Proposition 4.1. If r(a) is differentiable and ji(t)l/c < 6 < 1 for all t, then there exists a unique z(t, z,, r(m)) satisfying (4.1)-(4.3). Proof Observe that (4.1) and (4.2) can be written as fixed-point equations :
Consider then the function f(s) = r(-)/c+s. Note that f(q) is a mapping of the nonnegative reals into the nonnegative reals. Moreover, by the Mean-value Theorem, there exists a u such that
If(ul)--f(~2)I = Ir(ul>/c-r(uJ/cl =
I~I -UZI 6 Jlu, -u2I.
(4.6)
Thus, since 6 < 1, f(*) is a contraction mapping. This is sufficient to ensure that there exist unique fixed points t, and tb satisfying (4.4) and (4.9, respectively (see, for example, the fixed-point theorem for contractions in (2)). Therefore, it follows that a z(t, TV,r(e)) satisfying (4.1)-(4.3) exists and is unique. 0 Having established that z is well-defined for all r(s) of practical interest, we now proceed to show the influence that uncertainty in range has upon the predict-ahead time. Let Ar(*) be the error or uncertainty in the range function r(e). Then, the predict-ahead time required for the range function r(s) +Ar(*) is z(t, z,, r(B) + Ar(-)) and the error in the predict-ahead time is Az(t, z,, r(m),Ar(.>> h z(t, z,, r(e) + Art)) - z(t, z,, r(a)). We then have the following 444
representation
(4.7)
of z and the relative error AZ/Z. Journal
of the Franklin Pergamon
Institute Press plc
Lead-angles for Trucking
and Pointing at Distant
Proposition 4.2. Let 0 d 6 <: 1 and let R(6) denote functions r(e) such that
Targets
the class of differentiable
(4.8) Then for r(a), r(m)+ Ar(*) E R(6), there exist 6 ,(t, rS, v(*>) and s,(t, zS, r(e), AY(*)) such that, at time t, the predict-ahead time and the relative error in the predict-ahead time arising from an error Ar(.) in range satisfy r(l, r,, r(s)) = (rP +r,)(l
(4.9)
+a,)
where 2r(t) TP
=
(4.11)
.-__ c
and (4.12) Proof. Employing
(4.1))(4.3),
we obtain r(t+z) _c
7=zs+ By the Mean-value
Theorem,
I r(tJ c
(4.13)
(4.13) becomes (4.14)
for some &, E [t, tbJ and t, E [t, toI. Thus, z satisfies (4.9) with al(G 7S,r(.)) = (zs+2,)-‘(i(fb)~/c+i(5,)(t,
- W).
(4.15)
Note that since t,- t -c z and r(‘) E R(6), we find, using (4.15) and (4.9), that
IS~(t,~,,r(-))l d (~,+~,)Y’~~ G 60 +I~I(t,~.s,r(*))l).
(4.16)
Hence
l~~(t,~,,r(~))lG
&
for all r(v) ER(~).
(4.17)
[1+6,(t,~,,r(*)+Ar(*))].
(4.18)
Likewise, z+Az
=
z,+z,+
2&@ >
Vol. 326, No. 3, pp. 435448, Pnnted in Great Britam
1989
445
A. N. Payne Setting (4.19)
&(t, L r(e), Ar(.)) 2 6, (t, r,, r(*) +ArC)) and subtracting
(4.9) from (4.1 S), gives us Az
=
2Ar(t)
----(1+~*>+(~,-~,)(~,+~,). C
(4.20)
Hence Ar - = 2(2,+2,)z which is equivalent
62-6,
__
+ 1+6,
(4.21)
’
to (4.10). The inequality
(4.12) follows from (4.17) and (4.19). 0 Now for targets and observers of practical concern, a 6 << 1 can always be chosen for (4.8). Hence, from (4.9) and (4.10), we obtain as very good approximations z =
z,+z,
(4.22) (4.23)
Furthermore,
under conditions
(3.35)-(3.36), (4.24)
is a good approximation of the relationship of relative error in lead-angles error. Note that, in the limiting case when 2, = 0, (4.24) becomes
to range
(4.25) Equation (4.24) establishes the sensitivity of the lead-angles to errors in the range. It is a “large-change” sensitivity result in the sense that it is valid without the requirement that Ar(t) be infinitesimal.
V. Application A practical use of the preceding analysis is the determination of the permissible range error given a constraint on the maximum allowable lead-angle error. A constraint of this form is necessary, for example, to ensure that the observer’s beam hits the target. Suppose that we can allow the magnitude of lead-angle error in both azimuth and elevation to be at most A$. That is, (5.1)
446
Journal of the Franklin lnst~tute Pergamon Press plc
r
.
Lead-angles for Tracking and Pointing at Distant
6 Via (4.24), this constraint range :
imposes the following
constraint
Targets
on the relative error in
(5.2) where Gs satisfies (2.17). Of course, for this Ed<< 1 and+<< 1. It may not be apparent, but the right-hand of the system delay 7,. To see this, note that arises solely from the light propagation delay $i; = li(t)r,
result
to be valid,
we require
that
side of (5.2) is actually independent the component of lead-angle which z,, is
B = #,(I.
(5.3)
But (2.17) and (4.22) imply that $B = *jr/r,. Substituting
(5.4)
this result into (5.2), we obtain (5.5)
which is clearly independent of z,~. To illustrate the application of our results, we provide the following example. Consider target and observer motions for which, at time t, r(t) = 1000 km, Iv(t)/ = 8 km/s, Jo,++(t)1 = 4 km/s, Ivs(t)l = 5 km/s and e(t) = 20”. Suppose that the relative acceleration between the target and the observer never exceeds 0.1 km/s* in magnitude. We are interested in (i) the lead-angles required for system delays t, of 0 s and of 0.02 s, and (ii) the allowable range uncertainty given that the maximum permissible error in the lead-angles is A$ = I prad. We first establish that em and s0 are very small. For this purpose, we choose lo?= 30” and T = 0.2 s. By assumption, we may choose a(t, r) = 0.1 km/s*. Then by (3.11), (3.14) and (3.15), we find p(t, T) = 998.4 km, p(t, T) = 8.02 km/s and q(t, T, 0) = 2.02 x lop4 rad/s*. Using (3.6) and (3.8), we compute I$(t)i = 4.26x lop3 rad/s and Id(t)1 = 5x lop3 rad/s. Applying Proposition 3.1, we find ~~(t, T) < 5.45 x 10e3 and +,(t, T) d 4.02 x lo-‘. Since Ed and Ed are both more than 100 times smaller than unity, we may confidently apply (2.17) and (5.5). For the case when r,, = 0 s, (4.11) and (4.22) give us r = 6.67 x lop3 s (with c = 3 x lo5 km/s). The lead-angles given by (2.17) are then I,!I$= 28.4 prad and $I1 = 33.3 prad. For the case when z,, = 0.02 s, we have r = 2.667 x lo- 3 s, and tis = 113.6 prad and tiO = 133.33 prad. In either case, (5.3) gives us I& = 28.4 prad and $# = 33.3 ,urad. Thus, independent of system delay, (5.5) implies that IAr(t)l/r(t) d 0.03. In other words, the range at time t must be known to within 30 km (3%) to ensure lead-angle errors no greater than 1 ,urad. VI. Conclusion We have made a somewhat rigorous analysis of the lead-ahead problem arising when a moving observer must track and point at a distant moving target. In Vol. 3x. No. 3. pp 435448, Prmted m Great Britain
19x9
447
A. N. Payne
r
.
.
particular, we have established the effect that errors in the estimate of relative range have upon the lead-angles. We developed simple formulae which approximate the relative error in the lead-angles as a function of the relative error in the range. Moreover, we established the conditions which ensure that these approximations are valid. We have expressed these conditions in terms of parameters defining the relative target and observer motion in three dimensions. We further demonstrated how lead-angle error constraints, which typically arise from precision tracking and pointing requirements, restrict the admissible error in the range. Finally, the application of our theoretical results is illustrated with an example.
References (1) A. N. Payne, “Equations of Relative Motion for Moving Target-Moving Observer Systems”, Lawrence Livermore National Laboratory, Tech. Rep. UCRL, 1989. (2) R. G. Bartle, “The Elements of Real Analysis”, Wiley, New York, 1964.
448
Journal of the Franklin Inmtute Pergamon Press plc
of the Lead-angles
and Pointing
Necessav
at Distant
for
Targets7
by ANTHONYN.PAYNE Electronics Engineering University of California,
Department, Lawrence Livermore Livermore, CA 94.550, U.S.A.
National
Laboratorv,
ABSTRACT : A rigorous analysis is made of the lead-aheadproblem that arises when a moving observer must track and point at a moving target at long range. The efSect of errors in the
predict-ahead time and in the range upon lead-angles is studied. For general target and observer motions in three dimensions, s@cient conditions are exhibited under which the relative error in the lead-angles is, to good approximation, directly proportional to the relative error in range. It is jtirther shown how lead-angle requirements impose a constraint upon the maximum allowable uncertainty in range. Finally, the analysis is applied to a numerical example.
I. Zntvoduction Consider the problem in which a moving observer must track a moving target at long range (e.g. a ballistic missile) and deposit a beam of energy having narrow divergence (e.g. a high power laser) on the target. In order for the beam to hit the target, the observer must “lead” the target. That is, it must predict where the target will be when the beam arrives and direct its line-of-fire accordingly. Thus a leadaheadproblem arises in which the lead-angles defining the line-of-sight between the observer and the target at some time in the future must be predicted. The predict-ahead time consists of two components. First, it must include the time-delays incurred by the operations of the observer’s tracking and pointing system. These time-delays originate from such sources as sensor read-out, signal and image processing of sensor data and pointing servo lags. Second, for large ranges between the target and the observer, the light-propagation-delays contribute significantly to the predict-ahead time. The transit times of energy from the target sensed by the observer and of the observer’s beam directed to the target must both be included in the predict-ahead time. The predict-ahead time is thus a function of the target-observer range. Since the lead-angles depend upon predict-ahead time and thus upon range, an important practical issue concerns the effect that errors in range estimates have upon lead-angles. Such a consideration is important for the following reason. The t This work was performed Lawrence Livermore National
under the auspices of the U.S. Department of Energy Laboratory under contract number W-7405-ENG-48.
The FranklinInstitute 001&0032~89$3.00+0.00
by
435
A. N. Payne
1
beam divergence imposes a direct constraint upon the allowable error in the leadangles, which in turn restricts the maximum range error that is permissible. We need to know, therefore, how sensitive the lead angles are to uncertainties in range. In this paper, we make a simple, yet rigorous, analysis of the lead-ahead problem for general target and observer motions in three dimensions. We develop expressions for the relative error in lead-angles caused by uncertainty in predictahead time and by uncertainty in range. As a main result, we show that, under suitable conditions, the relative error in lead-angle A$/$ is well-approximated in terms of the relative error in range by
where T,~is the component of the predict-ahead time contributed by the system delay and zP corresponds to the light-propagation-delay component. Moreover, we demonstrate that if A$ is the maximum magnitude of error allowable in the lead-angle, then independent of zs, the range error must satisfy the constraint
WI<* r
(1.2)
’ I*“1 ’
where i+Gp is the portion of lead-angle arising from light-propagation-delay alone. We organize the remainder of the paper as follows. In Section II, we develop expressions for the lead-angle and its relative error in terms of the predict-ahead time and the uncertainty or error in predict-ahead time. In the process, we establish the fact that the relative error in lead-angle is approximately equal to the relative uncertainty in the predict-ahead time under certain conditions, In Section III, we express these sufficient conditions in terms of bounds which are functions of the range and the relative velocity and acceleration of the target and observer. Then, in Section IV, we characterize the predict-ahead time and its relative error in terms of range and the uncertainty in range. Combining these results with the findings of Sections II and III, we obtain (1.1). Finally, we develop (1.2) and demonstrate the use of our analysis with a numerical example in Section V.
II. A Characterization
of Lead-angle and its Relative Error
Let /I(.) denote an angle associated with the line-of-sight between the observer and the target and measured with respect to a non-rotating reference frame attached to the observer. If the target and the observer motions are coplanar, then p may correspond simply to the bearing angle. For general motions in three dimensions, /I represents one of the two independent angles necessary to specify uniquely the line-of-sight. For example, /? may be the target azimuth or elevation relative to the observer, or it may be a direction-angle of the line-of-sight. Suppose that z is the predict-ahead time. Then, at time t, the lead-angle associated with /I is Journal
436
of the Franklin Pergamon
Institute Press plc
Lead-angles for Tracking and Pointing at Distant $s(t,z)
4 B(t+4
Targets
-P(t).
(2.1)
Now let AZ be the error or uncertainty in the predict-ahead (2. l), the corresponding error in the lead-angle is
time z. Then,
from
A$b(t, r, Ar) g $s(t, r + Ar) - $,r(t, r)
(2.2)
= p(t+z+AT>-B(t+z). We have the following characterization of rja and its relative error A$a/$O. Proposition 2.1. Suppose that T > 0 and that p(s) is twice-continuously entiable on the interval [t, t+ T], with /3(t) # 0. Further, define sg(t, T) 2 T,$;;,
Ik+s)/2k)I.
differ-
(2.3)
Then, for t, t+ Ar E [0, T], there exist ~,(t, z) and EZ(t, z, AZ) such that, at time t, the lead-angle and the relative error in the lead-angle arising from an error AZ in the predict-ahead time z satisfy $s(t, r) = P(t)r(l
W,dt,
7,
+&I(& r)),
(2.4)
AT> AT 1 + E2(t, z, AZ)
$0 (4 r)
-
z (
If&,(&T)
(2.5)
-
and
IE,(t,~)/ d +Xt, T),
(2.6)
Mt, z, A4 < 3+(t, 0. Proof. Consider (2.1) and (2.2). By Taylor’s Theorem of the remainder, there exist I,, I, E [0, l] such that $fl(t,z)
(2.7) with the Lagrange
= &t)z+:B(t+A,+2
form
(2.8)
and At,b,&,qAx) Again,
by Taylor’s
= #(t+z)Ar+tB’(t+~+&At)(Az)*.
Theorem,
there exists a i3 E [0, l] such that
&t+r) Substituting
(2.9)
= /Q)+B’(t+&+.
(2.10)
(2.10) into (2.9), we obtain A$@(t,qAr)
= lj(t)Az+b(t+&z)zAz+:B’(t+z+&Ar)(A$2.
(2.11)
Defining E,(&Z) 4 i&+&+/B(t), Ez(t,qAT) Eq. (2.8) becomes Vol. 326, No 3, pp. 435448, Printed in Great Britain
A [B(t+~3t)Z+tB(t+Z+;IZAZ)AZI/P(t),
(2.12) (2.13)
(2.4) and (2.11) becomes 1989
437
A. N. Payne
A$,j(t, z, AZ) = b(t)Ar(l +cz(t, z, AT)).
(2.14)
Then (2.4) and (2.14) imply (2.5). It remains to show that (2.6) and (2.7) hold. By assumption z, z+Az E [0, T], so that 0 d r < Tand 0 < IArl < T. Now, using (2.12), we obtain Is,(t,r)l Similarly,
d rIJi(t+&r)/28(t)I
d +(t, T).
(2.15)
from (2.13), we have
d %(t, T),
(2.16)
and the proposition is proven. 0 Note that this result implies that in the limiting case when +(t, T) = 0, the relative error in the lead-angle is equivalent to the relative error in the predict-ahead we time. Moreover, if 0 < +(t, T) but +(t, T) << 1, then as a good approximation may still say that tia = P(t)?
(2.17) (2.18)
It is easy to show from (2.3) that a necessary condition for +(t, T) to be small is that the maximum relative change in p(e) on the interval [t, t + T] be small. While our present approach is more general, we can also arrive at (2.17) and (2.18) if we pursue a somewhat more intuitive, approximate approach, which proceeds as follows. Consider coplanar target and observer motions and let r be the range from the observer to the target and vg be the transversal component of the reiative velocity. If r and up are approximately constant over the predict-ahead interval, then for small lead-angles (2.19) and
The division of (2.20) by (2.19) leads immediately to (2.18). Bounds on the relative error made in assuming that (2.17)-(2.18) hold are given as follows. Corollary 2.1. Under the same assumptions as Proposition 2.1, if e&t, T) < 1, then (2.21) and
438
Journal of the Franklin Institute Pergamon Press plc
Lead-angles for Tracking and Pointing at Distant
IAt,ba(t,z,Az),h,bg(t,z) -Add
6
Proof. The bound From (2.9,
‘%(t, T) 1 -c+(t, T) ’
IWl
given by (2.21) is a direct consequence
wa
AZ
__-~ *a
Ar
s2-.s1
(2.22) of (2.4) and (2.6).
(2.23)
1 +s, >
r -( r
Targets
and thus (2.24) where the last inequality follows from (2.6)-(2.7) and from the fact that I&,(
we have
II+&21
1+
l&21
ll+s,j’
I-_I.s,J’
1 + 3Ep(t, T) l--Eg(t,T)
’
(2.26)
This result and (2.5) imply (2.25). q In the next section, we develop an upper bound on cg(t, T) in terms of the parameters which define the dynamics of the relative motion between the target and the observer in three-space. This bound may then be used to establish sufficient conditions on target-observer motion which guarantee that (2.17) and (2.18) are good approximations.
ZZZ.Bounds for General Motion in Three Dimensions Let r be the position vector of the target relative to the observer, as depicted in Fig. 1. The state equations governing the relative motion between the observer and the target are
where a(.) is the relative “0,. 326, NO. 3, pp. 435.448, 19x9 Prmted
in Great
Britain
i(t) = v(t),
(3.1)
v(t) = a(t),
(3.2)
target-observer
acceleration
arising
from forces on the 439
A. N. Payne
FIG. 1. Observer-target
geometry. XYZ is a non-rotating observer.
reference frame attached to the
target and the observer exerted by gravity, thrust, etc. Now we need two independent angles to specify uniquely the line-of-sight between the observer and the target. For this purpose, we employ azimuth $ and elevation 8 (see Fig. 1). We let Ydenote the range; that is, r = Irj, where for vector arguments 1.1denotes the usual Euclidean norm. Representing (3.1) and (3.2) in the range-azimuthelevation coordinate system, it can be shown (1) that f= i=v
rd2+rfj2cos29+a,,
(3.3)
i-9
(3.4) (3.5)
(I=;&, 8’= - ys-&2sinBcos8+
(3.6) T,
(3.7)
where (zI,, v~, us) and (a,, a$, a*) are representations of v and a in the r-q&0 coordinate system. We can now bound +(f, T) and .s+(t, T) in terms of the range r(t), the magnitude of velocity Iv(t) [,4(t) and d(t) and an upper bound on the magnitude of acceleration la(l+s)l for SE [0, T]. Proposition 3.1. Let r(a), 4(m) and Q(s) satisfy Eqs (3.3)-(3.8) and assume that T > 0. Suppose that 440
Journal
of the Franklin Pergamon
Institute Press plc
Lead-angles for
le(t+S)I
Tracking and Pointing at Distant
< 8<;
forallsE[O,
and that d(t) # 0 and d(t) # 0. Further, such that
(3.9)
that there exists an cc(t, T) > 0
suppose
< a(t, T)
la(t+s)l
T],
Targets
for all SE [0, T]
(3.10)
and that 0 < p(t, T) k r(t)-lv(t)IT-&(t,
T)T*.
(3.11)
Then, under these conditions,
E&t>T) B
Tv(t, T,
e>
(3.12)
2l&(t)I case’
Tv(t, T, 8) ~(t, T) d 21d(t)l ,
(3.13)
where
A& O2
q(t,T,@
a(r, T>
A (l+tan8)---,+--At, T)
(3.14)
At> T)
with p(t, T) 4 Iv(t)1 +a(t, T)T. Proof. Let SE [0, T] and consider
(3.15)
(3.1) and (3.2). By Taylor’s
Theorem,
I r(t+s)
= r(t)+v(t)s+s*
(1 -A)a(t+As)
dA
(3.16)
s0 and I v(t+s)
= v(t) fs
a(t + As) dA.
(3.17)
s0 Thus, utilizing
(3.10), we obtain u(t+s)
> r(t)-
Iv(t)js-&(t,
> r(t) - (v(t)\ T-$x(t,
T)s* T)T*,
(3.18)
and
Iv(t+.9l 6 Iv(t)1 +cr(t, T)s < Iv(t)1 +or(t, T)T.
(3.19)
Hence, we have shown that p(t, T) < r(t+s),
for all SE [0, T]
(3.20)
and Vol. 326. No. 3, pp. 435-448, Printed in Great Britam
1989
441
A. N. Payne Iv(t+s)l Now from (3.4)-(3.8), IJ(t+s)I
G p(t, T)
forall
sE[O, r].
(3.21)
we obtain
2lv,(t+s)ug(t+s)I d r(t+s)2~COSe(t+s)~
+ 2Iz+(t+s)~~(t+s)I
ltand(t+s)I
r(t+s)2~COSQ(t+-
lq(t+4l
(3.22)
+ r(tfs)lcosB(t+s)/ and
le(t+s)l d
~I~,(~+-JM~+~)I+ ~~(~+~)l*Itan~(~+~)I+ ladt+s>l r(t+s)2
r(t+s)2
r(t+s)
We will now obtain bounds uniform in s for the terms making sides of (3.22) and (3.23). Note that (3.9) implies that 0 < cosB<
(3.23)
.
up the right-hand
lcosCqt+s)l
(3.24)
and ltanB(t+s))
(3.25)
d tang.
Note also that 0 ,<
(l~rl-l~~l>2= I~r12-21wgl+l~~/2,
(3.26)
/V,U~l d t(lv,l’+l&$l’) < tlvl’.
(3.27)
so that
Applying
(3.21) we then have d &(t, q2.
(3.28)
Iv,(t+s)vs(t+s)l
=G$(t,
T)2
(3.29)
Iv0(t+s)zQ(t+s)(
d :#u
(3.30)
Iv,(t+s)v&+s)/ It may be shown in a similar fashion
that
and
We also have Iv,(t+s)12
< Jv(t+s)12
(3.31)
< p(t, q2.
And finally, using (3.10), we obtain lap(t+s>l
d la(t+s)l
Now, applying (3.24))(3.32), (3.22) and (3.23) to obtain
B = 4,0.
(3.32)
from above the right-hand
sides of
d a(& T)
we can bound
for
Journal
442
of the Franklin Pergamon
Institute Press plc
Lead-angles for Tracking and Pointing at Distant a(t, T) p(t, T) cos e
Targets
(3.33)
and l&t+,)1
< (1 +tan8)
At, T>’ 4t, T>
Y p(t, T)
+ ~ p(t> T) ’
(3.34)
From these two inequalities and (2.3), the upper bounds on E4(t, T) and co(t, T) defined by (3.12)-(3.13) follow. 0 We now make several observations concerning this result. First, the constraint (3.9) is imposed to avoid infinitely large angular velocity and acceleration in azimuth. Such a restriction would be needed in practical pointing and tracking systems. Moreover, it poses no limitation on the present result as we may generally orient the coordinate frame so that (3.9) is satisfied in the time-interval of interest. Second, observe that the constant p(t, T) defined by (3.11) is merely the minimum range possible in the time interval [t, t + T], and the constant p(t, T) given by (3.15) is the maximum relative speed possible in the same time interval. The constants y(t, g and .. T, e> and q(t, T, @/cos 0 are upper bounds on the angular accelerations 4, respectively, in the interval [t, t + T]. Third, we note that Propositions 3.1 and 2.1 imply that the conditions Q(t,
T, 0) << 1
(3.35) ’
21C#$t)lcosB
Tv(t, T>01<< 1
(3.36)
2@(t)l -are sufficient
for us to say that, as a good approximation, (3.37)
As a final remark, we mention that a more detailed analysis of (3.4)-(3.8) reveals that if 10~1and 1~~1are not much greater than [asIT and lagI T, respectively, then it may not be possible to insure that Ed << 1 and E*<< 1. In this case, it may no longer be safe to assume that (3.37) holds. Bounds derived from Corollaries 2.1 and 2.2 may then be useful. It appears, however, that conditions (3.35)-(3.36) are usually satisfied for large relative ranges and for the relative velocities and accelerations typical of ballistic or thrusting targets and observers. This is illustrated by an example in Section V.
IV. The Effect of Uncertainty
in Range
It remains to establish the relationship of the relative error in lead-angle to range and the error in range. We shall do this by first characterizing the effect of range uncertainty upon predict-ahead time. voi 326, NO. 3, pp. 43s448, Printed in Great Britam
19x9
443
A. N. Payne
Consider then the sequence of events which define the predict-ahead time z. Energy is reflected or radiated from the target at time t and received by the observer’s sensor at time t,. The transit time is thus t, - t, and the distance traversed by the energy wavefront is the relative range at time t,. Thus, r(&f) = c(ta - 0,
(4.1)
where c is the speed of light. Let the system processing delay be z,?, so that the observer fires its beam at time t,+z,. The beam arrives on target at time t,,. The distance traveled by the beam is then r(tJ Therefore,
the total predict-ahead
= c(t* - t, -7,).
(4.2)
time is r = tb-t.
(4.3)
Note that (4.1)-(4.3) define r implicitly as a function of t, z,~and y(m). It is not clear at first glance, however, that a z(t, t,, Y(S)) satisfying (4. I)-(4.3) exists or is unique for a general range function r(a). Nevertheless, for real targets and observers, it is true that Ji(t)/cJ < 1, and this turns out to be a sufficient condition to guarantee existence and uniqueness. We state and prove this fact formally as follows. Proposition 4.1. If r(a) is differentiable and ji(t)l/c < 6 < 1 for all t, then there exists a unique z(t, z,, r(m)) satisfying (4.1)-(4.3). Proof Observe that (4.1) and (4.2) can be written as fixed-point equations :
Consider then the function f(s) = r(-)/c+s. Note that f(q) is a mapping of the nonnegative reals into the nonnegative reals. Moreover, by the Mean-value Theorem, there exists a u such that
If(ul)--f(~2)I = Ir(ul>/c-r(uJ/cl =
I~I -UZI 6 Jlu, -u2I.
(4.6)
Thus, since 6 < 1, f(*) is a contraction mapping. This is sufficient to ensure that there exist unique fixed points t, and tb satisfying (4.4) and (4.9, respectively (see, for example, the fixed-point theorem for contractions in (2)). Therefore, it follows that a z(t, TV,r(e)) satisfying (4.1)-(4.3) exists and is unique. 0 Having established that z is well-defined for all r(s) of practical interest, we now proceed to show the influence that uncertainty in range has upon the predict-ahead time. Let Ar(*) be the error or uncertainty in the range function r(e). Then, the predict-ahead time required for the range function r(s) +Ar(*) is z(t, z,, r(B) + Ar(-)) and the error in the predict-ahead time is Az(t, z,, r(m),Ar(.>> h z(t, z,, r(e) + Art)) - z(t, z,, r(a)). We then have the following 444
representation
(4.7)
of z and the relative error AZ/Z. Journal
of the Franklin Pergamon
Institute Press plc
Lead-angles for Trucking
and Pointing at Distant
Proposition 4.2. Let 0 d 6 <: 1 and let R(6) denote functions r(e) such that
Targets
the class of differentiable
(4.8) Then for r(a), r(m)+ Ar(*) E R(6), there exist 6 ,(t, rS, v(*>) and s,(t, zS, r(e), AY(*)) such that, at time t, the predict-ahead time and the relative error in the predict-ahead time arising from an error Ar(.) in range satisfy r(l, r,, r(s)) = (rP +r,)(l
(4.9)
+a,)
where 2r(t) TP
=
(4.11)
.-__ c
and (4.12) Proof. Employing
(4.1))(4.3),
we obtain r(t+z) _c
7=zs+ By the Mean-value
Theorem,
I r(tJ c
(4.13)
(4.13) becomes (4.14)
for some &, E [t, tbJ and t, E [t, toI. Thus, z satisfies (4.9) with al(G 7S,r(.)) = (zs+2,)-‘(i(fb)~/c+i(5,)(t,
- W).
(4.15)
Note that since t,- t -c z and r(‘) E R(6), we find, using (4.15) and (4.9), that
IS~(t,~,,r(-))l d (~,+~,)Y’~~ G 60 +I~I(t,~.s,r(*))l).
(4.16)
Hence
l~~(t,~,,r(~))lG
&
for all r(v) ER(~).
(4.17)
[1+6,(t,~,,r(*)+Ar(*))].
(4.18)
Likewise, z+Az
=
z,+z,+
2&@ >
Vol. 326, No. 3, pp. 435448, Pnnted in Great Britam
1989
445
A. N. Payne Setting (4.19)
&(t, L r(e), Ar(.)) 2 6, (t, r,, r(*) +ArC)) and subtracting
(4.9) from (4.1 S), gives us Az
=
2Ar(t)
----(1+~*>+(~,-~,)(~,+~,). C
(4.20)
Hence Ar - = 2(2,+2,)z which is equivalent
62-6,
__
+ 1+6,
(4.21)
’
to (4.10). The inequality
(4.12) follows from (4.17) and (4.19). 0 Now for targets and observers of practical concern, a 6 << 1 can always be chosen for (4.8). Hence, from (4.9) and (4.10), we obtain as very good approximations z =
z,+z,
(4.22) (4.23)
Furthermore,
under conditions
(3.35)-(3.36), (4.24)
is a good approximation of the relationship of relative error in lead-angles error. Note that, in the limiting case when 2, = 0, (4.24) becomes
to range
(4.25) Equation (4.24) establishes the sensitivity of the lead-angles to errors in the range. It is a “large-change” sensitivity result in the sense that it is valid without the requirement that Ar(t) be infinitesimal.
V. Application A practical use of the preceding analysis is the determination of the permissible range error given a constraint on the maximum allowable lead-angle error. A constraint of this form is necessary, for example, to ensure that the observer’s beam hits the target. Suppose that we can allow the magnitude of lead-angle error in both azimuth and elevation to be at most A$. That is, (5.1)
446
Journal of the Franklin lnst~tute Pergamon Press plc
r
.
Lead-angles for Tracking and Pointing at Distant
6 Via (4.24), this constraint range :
imposes the following
constraint
Targets
on the relative error in
(5.2) where Gs satisfies (2.17). Of course, for this Ed<< 1 and+<< 1. It may not be apparent, but the right-hand of the system delay 7,. To see this, note that arises solely from the light propagation delay $i; = li(t)r,
result
to be valid,
we require
that
side of (5.2) is actually independent the component of lead-angle which z,, is
B = #,(I.
(5.3)
But (2.17) and (4.22) imply that $B = *jr/r,. Substituting
(5.4)
this result into (5.2), we obtain (5.5)
which is clearly independent of z,~. To illustrate the application of our results, we provide the following example. Consider target and observer motions for which, at time t, r(t) = 1000 km, Iv(t)/ = 8 km/s, Jo,++(t)1 = 4 km/s, Ivs(t)l = 5 km/s and e(t) = 20”. Suppose that the relative acceleration between the target and the observer never exceeds 0.1 km/s* in magnitude. We are interested in (i) the lead-angles required for system delays t, of 0 s and of 0.02 s, and (ii) the allowable range uncertainty given that the maximum permissible error in the lead-angles is A$ = I prad. We first establish that em and s0 are very small. For this purpose, we choose lo?= 30” and T = 0.2 s. By assumption, we may choose a(t, r) = 0.1 km/s*. Then by (3.11), (3.14) and (3.15), we find p(t, T) = 998.4 km, p(t, T) = 8.02 km/s and q(t, T, 0) = 2.02 x lop4 rad/s*. Using (3.6) and (3.8), we compute I$(t)i = 4.26x lop3 rad/s and Id(t)1 = 5x lop3 rad/s. Applying Proposition 3.1, we find ~~(t, T) < 5.45 x 10e3 and +,(t, T) d 4.02 x lo-‘. Since Ed and Ed are both more than 100 times smaller than unity, we may confidently apply (2.17) and (5.5). For the case when r,, = 0 s, (4.11) and (4.22) give us r = 6.67 x lop3 s (with c = 3 x lo5 km/s). The lead-angles given by (2.17) are then I,!I$= 28.4 prad and $I1 = 33.3 prad. For the case when z,, = 0.02 s, we have r = 2.667 x lo- 3 s, and tis = 113.6 prad and tiO = 133.33 prad. In either case, (5.3) gives us I& = 28.4 prad and $# = 33.3 ,urad. Thus, independent of system delay, (5.5) implies that IAr(t)l/r(t) d 0.03. In other words, the range at time t must be known to within 30 km (3%) to ensure lead-angle errors no greater than 1 ,urad. VI. Conclusion We have made a somewhat rigorous analysis of the lead-ahead problem arising when a moving observer must track and point at a distant moving target. In Vol. 3x. No. 3. pp 435448, Prmted m Great Britain
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particular, we have established the effect that errors in the estimate of relative range have upon the lead-angles. We developed simple formulae which approximate the relative error in the lead-angles as a function of the relative error in the range. Moreover, we established the conditions which ensure that these approximations are valid. We have expressed these conditions in terms of parameters defining the relative target and observer motion in three dimensions. We further demonstrated how lead-angle error constraints, which typically arise from precision tracking and pointing requirements, restrict the admissible error in the range. Finally, the application of our theoretical results is illustrated with an example.
References (1) A. N. Payne, “Equations of Relative Motion for Moving Target-Moving Observer Systems”, Lawrence Livermore National Laboratory, Tech. Rep. UCRL, 1989. (2) R. G. Bartle, “The Elements of Real Analysis”, Wiley, New York, 1964.
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Journal of the Franklin Inmtute Pergamon Press plc