Nd Paper Yunes Incas 01 01 2018 [623867]

1 Guidance Optimization for Tactical Homing Missiles and
Air Defense Systems
Yunes, sh., Alqudsi*1, and Gamal, M., El -Bayoumi2
*Corresponding author
Cairo University, Giza, 12613, Egypt

Abstract: The aim of this paper is to develop a functional approach to optimize the engagement
effectiveness of the tactical homing missiles and air defense systems by utilizing the differential
geometric concepts. In this paper the engagement geometry of the interceptor and the target i s
developed and expressed in differential geometric terms in order to demonstrate the possibilities of the
impact triangles and specify the earliest interception based on the direct intercept geometry. Optimizing
the missile heading angle and suitable miss ile velocity against the target velocity is then examined to
achieve minimum missile latax, minimum time -to-go (time -to-hit) and minimum appropriate missile
velocity that is guaranteed a quick and precise interception for the given target. The study termin ates
with different scenarios of engagement optimization with two -dimensional simulation to demonstrate
the applicability of the DG approach and to show its properties.
Keywords : Homing Guidance; Guidance Optimization; Differential Geometry; Proportional
Navigation (PN); Intercept; Engagement; Latax; Air Defense Systems (ADS); Line -Of-Sight(LOS).

Nomenclature
Basic Latin Letters
am,at = Missile and Target lateral accelerations (m/sec2).
I = Impact point .
rs = Sight line range (m).
rt = Target range (m).
rm = Missile range (m).
sm,st = Length of missile and target trajectories .
t,n,b = Tangent, normal and bi -normal unit vectors .
ts,ns = Tangent and normal unit vectors of the LOS .
tm,nm = Tangent and normal unit vectors of the missile trajectory .
tt,nt = Tangent and normal unit vectors of the target trajectory .
vm ,vt = Missile and target velocities (m/sec) .

Greek Letters
κm ,κt = Missile and target curvatures (m-1).
θm = Missile heading angle .
θs = Sight line angle .

1 Teaching Assistant, Nile University, Egypt.
2 Professor of Flight Mechanics & Control – Aerospace Engineering Dept.

2 θmo ,θto = Initial heading angle of the missile and the target .
η = Missile to target velocity’s ratio .

List of Abbreviations
ADS = Air Defense Systems .
DG. = Differential Geometry.
LOS. = Line of sight.
PN. = Proportional Navigation.
SLR. = Sight line rate of change.
t2go. = Missile time -to-go.

1. INTRODUCTION
Nowadays, seeking for an effective air defense system (ADS) starting from the launching
phase until the intercept ion precisely occurs, became an essential concern for the developers
of the guidance algorithms so that generate an accurate and carefully chosen trajectories for
the interceptor homing missile against highly maneuverable targets [1-3]. For the short and
medium ranges, h oming guidance is considered as one of the effective approaches for tactical
missiles to intercept smart and stealthy targets .
Homing guidance systems or two -points guidance systems “missile and target” as the two
reference points, is a com mon expression referred to missile that steers and directs its motion
according to the commands of the missile’s onboard seeker which generates its commands
based on the reflected or emanating signals from the targets [4-9]. Since the concepts of
different ial geometric control theory provide useful tools for modelling, analysis and design
for nonlinear guidance and control systems [10-16], we will use the differential geometry
approach to develop an optimization algorithm for optimally steering the homing m issile to
the collision point and achieving certain and/or given requirements such as produce the fast
collision course guarantee the minimum required time -to-go. From another aspect, maybe for
design reasons, the maximum allowable lateral acceleration for the interceptor missile is
limited within a given range, in such case the guidance algorithm should take into
consideration the missile latax limits [17,18 ].
The objective is to develop optimized and better guidance algorithm to ensure intercepting
smaller and highly maneuverable targets with greater flexibility in controlling and choosing
the engagement trajectories in contrast with the Proportional Navigation (PN) guidance law
[19-21]. Furthermore, considering the challenges such as, the complexity and non linearity of
the missile system , consider the relative velocity between the missile and target , the imposed
restrictions on the missile normal acceleration , and the maximum flying time for the missile
based on the burned fuel of the missile .
2. ENGAGE MENT KINEMATIC EQUATIONS
In homing guidance, the sight line or line-of-sight (LOS) is an essential measure of the
target -missile relative geometry as depicted in figure ( 1), where t and 𝑛 represent the tangent
and normal unit vectors respectively . According to the postulate states that, the smallest
distance between two points in the space is the dir ect path i.e. the straight line, we will examine
the kinematics of the direct intercept ion for a target flying at a constant velocity .

3 The relative position and motion of the target -to-missile in addition to estimate the target
LaTax are determined using accurate sensor s, which mostly located in the nose of the homing
missile. For now , to derive the engagement kinematic equations , let’s consider the LOS
connecting the target c.g. with the missile c.g . and from figure (3.1) we find that:
𝑟𝑠=𝑟𝑡−𝑟𝑚; 𝑟𝑠=𝑟𝑠𝑡𝑠;
𝑟.
𝑠=𝑟.
𝑠𝑡𝑠+𝑟𝑠𝑡.
𝑠; 𝑡.
𝑠=𝜃.
𝑠𝑛𝑠; 𝑛.
𝑠=−𝜃.
𝑠𝑡𝑠;
𝑟.
𝑠𝑡𝑠+𝑟𝑠𝜃.
𝑠𝑛𝑠=𝑣𝑡𝑡𝑡−𝑣𝑚𝑡𝑚; (1)
Equation (1) illustrates the relative velocities of the missile as well as the target. By
projection onto the basis, 𝑡𝑠 ,𝑛𝑠 individually we then get the components of the relative
velocities along and normal to the LOS.
Along the LOS:
𝑟.
𝑠=𝑣𝑡𝑡𝑠.𝑡𝑡−𝑣𝑚𝑡𝑠.𝑡𝑚;
Or 𝑟.
𝑠=𝑣𝑡cos(𝜃ts)−𝑣𝑚cos(𝜃ms); (2)
Normal to the LOS:
𝑟𝑠𝜃.
𝑠=𝑣𝑡𝑛𝑠.𝑡𝑡−𝑣𝑚𝑛𝑠.𝑡𝑚; or 𝑟𝑠 𝜃.
𝑠=𝑣𝑡 sin(𝜃ts)−𝑣𝑚 sin(𝜃ms); (3)
Where 𝜃ts=𝜃𝑡−𝜃𝑠 and 𝜃ms=𝜃𝑚−𝜃𝑠
By differentiating (1) and substituting, 𝑡.
𝑠 ,𝑛.
𝑠 we get:
(𝑟..
𝑠−𝑟𝑠𝜃.
𝑠2)𝑡𝑠+(𝑟𝑠𝜃..
𝑠+2𝑟.
𝑠𝜃.
𝑠)𝑛𝑠= 𝑣𝑡2𝜅𝑡𝑛𝑡−𝑣𝑚2𝜅𝑚𝑛𝑚 (4)
Noticing that: From Frenet -Serret formula [11,19 ],
𝑡.
=𝜅 𝑣 𝑛=𝜃 .
𝑛; and 𝑛.=−𝜅 𝑣 𝑡=−𝜃 .
𝑡;
Figure 1: Homing guidance and kinematic geometry

4 Equation (4) shows the relative acceleration between the missile and the target. The missile
and target lat eral accelerations are
𝑎𝑚=𝑣𝑚2𝜅𝑚=𝑣𝑚𝜃.
𝑚;
𝑎𝑡=𝑣𝑡2𝜅𝑡=𝑣𝑡𝜃.
𝑡;
The components of the target -missile relative accelerations, along and normal to the sightline
are:
Along the LOS:
𝑟..
𝑠−𝑟𝑠𝜃.
𝑠2=𝑎𝑡𝑡𝑠.𝑛𝑡−𝑎𝑚𝑡𝑚.𝑛𝑚;
or 𝑟..
𝑠−𝑟𝑠𝜃.
𝑠2=−𝑎𝑡sin(𝜃ts)+ 𝑎𝑚sin(𝜃ms) (5)
Norma to the LOS:
𝑟𝑠𝜃..
𝑠+2𝑟.
𝑠𝜃.
𝑠=𝑎𝑡𝑛𝑠.𝑛𝑡−𝑎𝑚𝑛𝑚.𝑛𝑚;
or 𝑟𝑠𝜃..
𝑠+2𝑟.
𝑠𝜃.
𝑠=𝑎𝑡cos(𝜃ts)+𝑎𝑚cos(𝜃ms); (6)
Easily, we can refer equations (2 -6) to the inertial coordinate system x and y.
3. THE ENGAGEMENT GEOMETRY OF HOMING GUIDANCE
As the fact states that, the smallest distance between two points in the space is the direct path
i.e. the straight line thus, we will consider the direct intercept geometry for both the missil e
and target where they are flying at a constant velocity. Let's consider the sightline and the two
courses connecting the missile and the target with the anticipated impact point 𝐼 respectively,
these three sides establish what so called the impact tria ngle (MIT) as illustrated in figure (2).

𝛉𝐦𝐬
𝛉𝐭𝐬
Figure 2: Homing Guidance Configuration

5 So that determine the intercepting condition, consider the time 𝑻, whereas both the target
and missile travels form their initial positions to the impact point during this time. Such that
the target path length 𝒔𝒕 is equal to
𝒔𝒕=𝒗𝒕𝑻 (7)
Which mean that to guarantee occurring the impact, the missile should travel a distance 𝒔𝒎 at
the identical time period 𝑻 where
𝒔𝒎=𝒗𝒎𝑻 (8)
In other words, the missile should maneuver until the missile -to-target traj ectories -length -ratio
is equal to the missile -to-target velocities ratio as follows:
𝒔𝒎
𝒔𝒕=𝒗𝒎
𝒗𝒕=𝜼 (9)
Equation (9) shows the essential impact condition from ge ometric point of view.

By considering the intercept geometry showing in figure (3), to estimate the expected
positions of the impact point, let us consider the two triangles M𝑥I and TEI and utilizing
Pythagoras theorem then we can find that
𝑠𝑡2=(𝑥−𝑟𝑠cos(𝜃𝑠))2+(𝑦−𝑟𝑠sin(𝜃𝑠))2 (10)
𝑠𝑚2=𝑥2+𝑦2 (11)
𝑠𝑚
𝑠𝑡=𝜂 =cos(𝜃ts)
cos(𝜃ms) (12)
Then : 𝜂2=𝑥2+𝑦2
(𝑥−𝑟𝑠cos(𝜃𝑠))2+(𝑦−𝑟𝑠sin(𝜃𝑠))2
Figure 3: Intercept Geometry

6 After rearrangement and simplifications of (10), (11), and (12) we get
𝑥2+𝑦2−2𝐶cos(𝜃𝑠) 𝑥−2𝐶sin(𝜃𝑠) 𝑦+𝑟𝑠𝐶=0
or ( 𝑥−𝐶cos(𝜃𝑠) )2+( 𝑦−𝐶sin(𝜃𝑠) )2=𝐶2(1−𝑟𝑠
𝐶) (13)
Where: 𝐶=𝜂2 𝑟𝑠
1−𝜂2 ;
Equation (13) Shows that the anticipation positions of the impact points can be represented by
a circle equation with radius 𝑟 , where 𝑟= 𝐶 √1−𝑟𝑠
𝐶 and the circle is centered at
(𝐶 𝑐𝑜𝑠𝜃𝑠 ,𝐶 𝑠𝑖𝑛𝜃𝑠). In case of 𝜂=1 the equation yield s to
𝑥 𝑐𝑜𝑠(𝜃𝑠)+𝑦 𝑠𝑖𝑛(𝜃𝑠)=𝑟𝑠
2 (14)
As it shown in the figure (4), where the circles representing the loci of the anticipated impact
points such that, if 𝜂>1 , the circle enclose the target, which indicates that, the missile will
intercept the target whatsoever the target’s approach direction. In contrast with the case of 𝜂<
1, the interception will o ccur only if the target velocity vector intersects with the anticipated
impact circle, in such case the locus of the anticipated impact points encloses the missile.

Figure (4) also illustrates the earliest interception that can be achieved by the missile;
known its heading angle and pursuit’s velocity according to our treatment, which is implicitly
based on the shortest engagement trajectory utilizing the mentioned fact. Moreover, th e
intercepting geometry is useful also for the targets in order to attempt avoiding the anticipated
area of interception.
Now, let us demonstrate the possibilities of the impact triangles for certain engagement
scenarios; if we have a target flying in a st raight line and a constant velocity from the initial Figure 4: Positions of the expected impact points for different 𝜂

7 position at (0, 0) and constant heading angle 𝜃𝑡𝑜=0o, to the impact point 𝐼. Further, to illustrate
the possibilities for different missile -to-target ranges, the impact points will be located at
different ranges from the target, starting form 2 km until 12 km with step=2 km. thus the we
will figure out the third point which represents the missile’s positions for each range to
complete the impact triangle. Then figure (5) shows the missile position’s p ossibilities or the
possibilities of the impact triangle for each range and for different missile -target velocity
ratio 𝜂={0.5 ,1 ,1.5 ,2}.
Again, figure (5) where the circles represent the missile’s position possibilities also shows
that if the veloci ties ratio is greater than one then the interception will occur whatsoever the
missile position is since the circle enclosing the target. In contrast to the case if the missile –
target velocities ratio is less than one where the circle does not enclose the target which
indicated that the missile position’s possibilities are restricted, which mean the missile should
be in front of the target.

4. OPTIMIZATION OF THE MISSILE’S HEADING ANGLE AND THE
VELOCITY RATIO
In this section, we will use the differential geometric principles so as to optimizing the
engagement configuration seeking for the best trajectory, which guarantee the missile -target
intercept within a given requirement such as:
(a) The min imum missile latax. (b) The minimum time to go.
(c) The minimum missile velocity.
Now, let us consider real situations of missile -target intercept scenarios, whereas the
inputs to the section of the optimization in our guidance progr am including:
Figure 5: Impact Triangles possibilities for different 𝜂

8 (a) Target initial position. (b) Target flying velocity. (c) Target heading angle.
As we need to determine the missile parameters to acquire the best interception, we will
discuss and examine four different scenarios sep arately and illustrate their specific properties.
In these scenarios we will take into account the effect of changing the missile velocity, target
attacking angle, and the target -missile range on the missile -target intercept geometry as well
as the approxi mate time -to-go and the missile’s heading angle or LaTax .
A. Constant Velocity Ratio with Variable Target’s Angle
The target initial position is (12 km, 12 km) and the missile initial position is at the origin
point, where the target velocity is set to 310 m/sec, i.e. ( 𝜂=1.5). This scenario observing the
effect of the target’s angle of attack variation on the engagement configuration. Figure (6)
shows different cases of the intercept s cenarios for different target’s heading angles, whereas
the circles represent the expected impact points for the given range and velocity ratio. The
optimization algorithm chooses the best missile’s heading angle to intercept the given target
within a mini mum approximated time -to-hit.
Figure 6: Missile and target trajectories and the impact points

9 B. Constant Target ’s Angle with Variable Velocity Ratio ( 𝜼)
Let the target’s heading angle equal to 30o and its initial position is (0, 12 km), the missile is
launched from the origin point, and the target velocity is set to 310 m/sec. we will demonstrate
the engagement configuration where the ratios of the missile -to-target velocities are equal to
{2, 1.5, 1, 0.5}. This scenario examines the effect of the velocity ratio 𝜂 on the engagement
possibility. As the two cases of 𝜂={1,0.5} are critical, since they introduce a question states
that, is it possible for a missile to intercept a target even if its v elocity equal or less than the
target velocity? To figure out that, let us change the target’s heading angle to 290o with the
velocities ratios 𝜂={1,0.5} and see the properties of the engagement configuration.

Figure (7) displays the engagement configuration of different missile velocity ratios where
the simulation results determine the best missile’s heading angle for each missile velocity.
When 𝜂={1,0.5} the results show that no interception will occur for a target’s heading angle
of 30o. However, if the target’s heading angle was as shown in figure (8) then the interception
possibly occurs. Figure 7: Engagement configuration of different missile velocity ratios at 𝜃𝑡0=30o

10
C. Variable Range with a Constant Target ’s Heading Angle
In In this scenario we will determine the engagement parameters for different missile’s
velocities i.e. for 𝜂={2,1.5,1,0.5 } separately and remain the target’s angle and velocity
constant, where 𝑣𝑡 =310 m/sec. The range between the missile and the targe t will changed
from 12 km to 2 km with step=2 km, to examine the effect on the position of the impact points
due to changing the missile velocity and the range between the missile and the target.
Figure 9: Intercepting possibilities for different missile -target ranges and 𝜂 Figure 8: Engagement configuration of 𝜂={1,0.5} at 𝜃𝑡𝑜=290o

11 Figure (9) Obviously states that the smaller the range, th e lower the area of collision points
and the figure also illustrates that, when the missile velocity is greater than the target velocity;
the circle that represents the expected positions of the impact points enclose the target initial
position which mean the missile will hit the target what so ever the target’s angle. On the other
hand, when the missile velocity equal to or less than the target’s velocity then the interception
will occur if and only if the target direction intersects the circle of the impa ct points. Which
mean, if we require hitting a target with velocity greater or equal to the missile’s velocity then
the target’s heading angle should be considered precisely .
D. The Optimum Missile ’s Heading and Suitable Velocity Ratio
Now we will consider different scenarios with a variable target’s heading angle and a
constant missile -to-target range. Let the target initial position at (12, 12) km from the origin
point and the target velocity equal to 310 m/sec. The guidance optimization algorithm
“program” will determine the best missile heading angle that guarantee the minimum missile
lateral acceleration required to intercept the target as well as choosing the reasonable missile’s
velocity accord ing to design parameters or restrictions on either the time -to-go or the height
of the impact point, and so on .
Figure 10: Optimum missile’s velocity and heading angle.

12 The results of these scenarios illustrated in figure (10) and their summary are shown in
table (1).
Table 1: Target and missile angles & time -to-go for different 𝛈

5. SIMULATION EXAMPLE OF OPTIMIZED ENGAGEMENT SCENARIO
In this section, a missile -target intercepting scenario will be demonstrated to show the
optimization properties. In this scenario a target flying in an initial angle equal to 200o with a
velocity =310 m/sec and maneuvering by 1g.
A. Before the Optimization Algorithm:
In order to intercept the target within 22 sec., the missile requires to produce normal
acceleration starting from 24g and decreasing gradually as shown in figure ( 11). The missile
initial heading angle is equal to 30o and its velocity equal one and half of the target’s velocity.
The simulation program displays the convergence of Lyapunov function as well as the rate of
change of the LOS, and the missile curvature . 𝜽𝒕 30o 150o 180o 225o 330o
𝜽𝒎 34.35o 98.6o 124.14o 45o 5o
𝜼 1.4 1.2 0.72 0.5 1.5
t2g 133 sec. 65 sec. 65 sec. 36.5 sec. 61.55 sec
Figure 11: Non -Optimized missile velocity and heading angle {Trajectory,
Latax, Lyapunov Fn. , r/ 𝑠𝑡, SLR, missile curvature }

13 B. After the Optimization Algorithm:
The optimization algorithm offers availability to choose the minimum “suitable” missile
velocity, the best initial heading angle to guarantee minimum required latax. As shown in
figure (12), the required missile latax has been declined to begin at -2.2g with an initial heading
angle “based on the given in itial missile -target positions” equal to 171.55o. What is worth
mentioning here is that, the missile can intercept the target even if its velocity is less than the
target’s velocity (0.9 𝑣𝑡) according to the direct interception.
6. CONCLUSION
The study presented in this paper used the differential geometric concepts to develop an
optimization guidance algorithm for homing missiles and air defense systems. The
engagement geometry and available positions of the missile against the target have bee n
determined and illustrated based on the direct interception.
Distinct scenarios of the missile -target interception were presented and optimized from
different aspects, such as the best missile heading angle to minimize the required missile
normal acceler ation. Furthermore, the missile velocity compared to the target was also
optimized so as to guarantee intercepting the target by using missiles, which their maximum
velocity is as small as possible according to the prior design requirements, which mean it’ s not
always required missiles with high velocity to ensure intercept certain targets. Figure 12: Optimized missile velocity and heading angle {Trajectory,
Latax, Lyapunov Fn. , r/ 𝑠𝑡, SLR, missile curvature }

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