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Rot orcraft Pursuit-Evasion in Nap-of4he-Earth Flight
Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | | DOI: 10.2514/6.1990-3455
NASA Ames Research Center
Moffett Field, CA 94035
Two approaches for studying the pursuitevasion problem between rotorcraft executing napof-the-earth flight are presented. The first of these
employs a constant speed kinematic helicopter
model, while the second approach uses a three degree of freedom point-mass model. The candidate
solutions to the first differential game are generated by integrating the state-costate equations
backward in time. The second problem employs
feedback linearization to obtain guidance laws in
nonlinear feedback form. Both approaches explicitly use the terrain profile data. Sample extremals
are presented.
Recently, there has been significant research
activity in developing approaches for automating
the nap-of-the-earth flight regime for high performance helicopters1". This research is driven by a
need to decrease the pilot work load to acceptable
levels in such an operational mode. Technology requirements for achieving this objective include not
only the synthesis of guidance and control laws,
but also the development of various sensors for
gathering information about the terrain and obstacles. Several of these issues have been addressed
in Reference .I. Specifically, the development of
image-based ranging systems for obstacle and terrain detection have been discussed in References 2
*Member AIAA, School of Aerospace Engineering, Georgia Institute of technology, Atlanta. Mailing Address : FSN
Branch, MS 210-9
'Graduate Student, School of Aerospace Engineering,
Georgia institute of Technology, Atlanta
$Research Scientist, FSN Branch, MS 210-9
Copyright @ 1990 American Institute of Aeronautics and
Astronautics, Inc. All rights reserved.
and 3. A heuristic guidance law for trajectory selection in the presence of obstacles was outlined in
Reference 4, while the ~ r o b l e mof optimal trajectory planning using macroscopic terrain features
was examined in detail in References 5 and 6. The
research in References 5 and 6 have focussed on
planning methods useful for generating trajectories that optimize a linear combination of flight
time and terrain masking connecting initial conditions with specified terminal conditions. Present
research is an outgrowth of the work reported in
Reference 5.
This paper examines the trajectory planning
process wherein two vehicles with conflicting objectives are involved. The first participant in this
process, designated as the evader, is a rotorcraft
attempting to reach a specified terminal condition
while masking its trajectory using the terrain. The
second participant, designated as the pursuer, has
the objective of intercepting the evader before it
completes its mission, while masking its trajectory using the available terrain. The motivation
for maximizing terrain masking is to escape detection from each other and also from other ground
based or airborne detection devices. It is assumed
here that the evading rotorcraft has no offensive
capabilities. Both vehicles may attempt to accomplish their objectives in a time optimal fashion.
This problem is suitable for analysis via the
theory of differential games7's. Two differential
games incorporating models with increasing complexity are considered in this paper. The first of
these uses a kinematic model with local heading
angle as the control variable while the second employs a point-mass model with the rotor thrust,
pitch and yaw attitudes of vehicle as the three control variables. The performance index for the f i s t
Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | | DOI: 10.2514/6.1990-3455
game is a Linear combination of flight time and
terrain masking, while the second pursuit-evasion
game considers a quadratic form in the terminal
miss distance together with the square of the rotorcraft acceleration magnitude as the performance
The candidate trajectories for the differential game formulation using the kinematic vehicle model are obtained by backward integration
of the state-costate system. This approach was
motivated by the research discussed in Reference
9. The second differential game formulation is
patterned after the procedure discussed in Reference 10. In this problem, the solution is obtained
by first transforming the nonlinear vehicle models
into a linear, time invariant form using the theory
of feedback linearizati~n'~*".This process is followed by the solution of the differential game in
transformed coordinates. TEe resulting guidance
laws are transformed back to the original coordinates to obtain nonlinear feedback laws. Each of
these differential game solutions will be discussed
in the ensuing.
Rotorcraft Pursuit-Evasion with
a Kinematic Model
In Reference 5, a trajectory planning scheme
useful for nap-of-the-earth flight of a rotorcraft using optimal control theory12was discussed. In that
work, the terrain profile was used to perform coordinate transformations to reduce the dimension of
the problem from three to two dimensions. Secondly, by invoking a constant of motion in the
problem, the task of finding the optimal trajectory was reduced to that of determining the initial
value of the control variable. The computational
effort involved in this trajectory planning process
was also estimated. In this section, the same frame
work will be used to formulate the two sided trajectory planning problem.
As in Reference 5, let the composite profile
F(z,y) be given by
with h, being the desired clearance above the terrain profile, f (2, y) being the actual terrain profile
and P(x, y) the specified threat overlay. The kinematic models for the pursuer and evader moving
over this composite profile are given by5
Ve C O S X ~
VeFx, F,, sin X,
+ J(1+ F.,Z)(~ + Fxe2
+ Ffi2)
here, the symbol V denotes rotorcraft airspeeds
and x denotes the heading angle defined on the
local tangent plane. The subscript p and e denote
the pursuer and the evader respectively. The vehicle altitude rates being given by the equations
he = V'siny,, hp = vPsinyp
The pursuer-evader flight path angles yp, 7, are
given by the equations
It is assumed that the rotorcraft airspeeds
Vp, V, either remain constant throughout the game
duration or is specified at each terrain location.
Note that as in Reference 5, it is also possible to
consider a case in which the rotorcraft airspeeds
are considered to be control variables.
The variables F,, ,F,, ,FaP,
F,, are the composite terrain profile gradients at the instantaneous evader-pursuer locations. The heading angles xp and xe are the control variables in this
Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | | DOI: 10.2514/6.1990-3455
problem. These equations are obtained by resolving the rotorcraft velocity vector in the local
tangent plane using the terrain gradients to define a transformation between the local tangent
plane coordinate system with an inertial frame.
These coordinate systems are illustrated in Figure
1. For further details on this coordinate system,
the reader is directed to References 5 and 6.
The rotorcraft pursuit-evasion begins at a certain set of initial conditions when the participants
first become aware of one another. One of the participants in the game will be identified as the pursuer, while the other participant is assumed to be
the evader. Further, it is assumed here that this
designation continues unchanged throughout the
duration of the encounter. In the present game,
the objective of the evader is to reach a specified
set of final conditions without being captured by
the pursuer. The pursuer on the other hand, is
attempting to capture the evader before it gets to
the desired final conditions. Further, the participants in the pursuit-evasion encounter attempt to
accomplish their objectives in a time-optimal fashion while providing for terrain masking.
As in reference 5 and 6, it is assumed here
that the terrain masking will be accomplished if
the participants minimize their altitude above the
specified datum. Thus, the performance index employed here is
Here, Fp and Fe are the altitudes at the current pursuer-evader locations defined by the composite profile given in equation (1). Wp, We are
weighting factors for the pursuer and evader altitudes. These reflect the participant's concern
about degree of desired terrain masking. The negative sign in the second term explicitly recognizes
the fact that the evader is attempting to maximize
the performance index. At this stage, the definition of this differential game is incomplete because
no criterion has been laid down for the termination of the game. In this paper, the pwsuit-evasion
game is assumed to terminate the first instant the
pursuer succeeds in approaching the evader within
the firing range of its weapon system. The pursuer's weapon envelope is assumed to be spherical.
Note that alternate weapon envelope geometries
can also be employed in the present formulation.
It needs to be emphasized that the present
formulation does not allow for any offensive capabilities for the evader. Additionally, the game is
declared a draw if the pursuer-evader trajectories
leave a predefined region in the terrain at any inst ant during the encounter
For the case of a circular weapon envelope
centered at the pursuer, the capture condition can
be expressed by requiring that
The quantity d is a specified constant. The
condition (11) ensures that sufficient opportunity
exists for weapon usage. The time for capture is
then determined as the first time instant when the
inequality (10) is met as an equality.
The game termination constraint may be satisfied by augmenting the performance index with
equation (10) as an equality, i.e.,
The quantity v is an undetermined multiplier.
Given the performance index (12) together with
the differential constraints (2) - (5), one may derive necessary conditions that must be satisfied by
the optimal trajectories. To this end, define variational Hamiltonian12 as
+ A42ip
The Euler-Lagrange equations for the evader
can then be obtained as
- B1 sin ~e
+ (B2
sin xe + B3 cos Xe)X1 K
Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | | DOI: 10.2514/6.1990-3455
- B4 sin xe
-k (B5 sin xe 4- B6 cos xe)X1
These equations were derived using the symbol manipulation program MACSYMA13. The
Euler-Lagrange equations and the optimality condition for the pursuer has exactly the same form as
expressions (15) - (23), and can be written down
by inspection. In the interests of conserving space,
these will not be given here.
It may be verified that the form of the necessary conditions remain the same regardless of
the order of the minimization and maximization.
Next, the terminal conditions on the costates can
be obtained as :
Since the final time is open and the variational
Hamiltonian does not explicitly depend on time,
this problem has a constant of motion, Viz.,
H(t) = 0, 0 5 t 5 tf
Using this fact , together with the terminal
conditions on the various costates, the undetermined multiplier may be computed as:
B6 = -A23~xeFxeye
and the optimality condition is:
The denominator of this equation is simply
the negative of the product of range and range
rate at the final time, i. e., the closing rate. It
Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | | DOI: 10.2514/6.1990-3455
is possible to verify that as in the planar pursuitevasion game, the value of this parameter will turn
out to be positive if the terrain masking weights
were zero.
Extremals for the pursuit-evasion problem
can be constructed if the initial conditions on
the costates or the terminal conditions on the
states were known. Since the final condition to
be reached by the evader is normally known, in
this research, the extremals are obtained by integrating the state-cost ate system backward in time
until the trajectories reach the boundary of the admissible region. Since the last opportunity for the
pursuer to catch the evader is at its final position,
this represents the upper bound on the duration of
the game. The detailed procedure for constructing
the trajectories is as follows.
The pursuer's capture set is first superimposed at the evader's final position. The terminal condition of the pursuer is selected from this
set and the equations are integrated backwards till
the trajectories reach the boundary of the admissible region. The relative distance between the
two vehicles are continuously monitored along the
trajectories. Trajectories are constructed for several conditions in the capture set. From the final
time, if the relative distance between the vehicles
increase as the trajectories evolve in retrogressive
time, it indicates that the pursuer can catch the
evader only at the final time. On the other hand,
if the relative distance decreases or stays constant
while the trajectories are evolving in retro-time, it
indicates that capture is possible anywhere along
this segment of the trajectories. The results of
this exercise can be used to determine the set of
initial conditions for the pursuer and the evader
that will not result in capture at the specified final conditions. These computations are also useful
to delineate the set of initial conditions from which
capture is assured. Construction of these sets will
require extensive numerical computations and will
be a future research item.
Some sample trajectories will be illustrated
in the ensuing. Figure 2 shows the pursuer and
evader trajectories in the down-range cross-range
plane. The dotted lines and the lines with numerals attached are the constant altitude contours on
the terrain. A fixed terrain clearance of 50 feet
together with unity terrain masking weight were
used in this computation. The value of the parameter v corresponding to this set of final conditions
turned out to be 5,793-03.The trajectory duration was approximately 95 seconds. The intricate
behavior of the trajectories are apparent from this
figure. The corresponding altitude evolution is illustrated in Figure 3. Note that this trajectory is
given in retrogressive time and the tgo = 0 corresponds to the terminal instant. During the course
of the present investigation, several such trajectories were generated.
It may be verified that if the terrain gradients
were zero, the resulting pursuit-evasion game produces what is called a simple motion7>*. In this
case, the trajectories will turn out to be straight
lines joining the initial conditions and the intercept point.
Rotorcraft Pursuit-Evasion with
a Point-mass Model
From the previous discussions, it is clear that
the differential game solution is complex even in
the case of simple vehicle models. The usefulness of the foregoing analysis is limited to addressing the question of capturability since generating
large number of trajectories in real-time is not a
viable proposition. In practical situations, it is desirable to obtain feedback solutions. It has been
shown in Reference 10 that the theory of feedback
linearizationll can be used to solve a class of differential games with high order nonlinear dynamics and a quadratic payoff. The approach here is
to transform a nonlinear vehicle model into a linear, time invariant form using feedback. This approach has been subsequently extended to study
spacecraft pursuit-evasion problems14. In this section, this approach will be illustrated for rotorcraft
pursuit-evasion in nap-of-the-earth flight. More
recently, the analysis has been extended to include
a realistic weapon envelope and the time-of-flight
in the performance index15.
The point-mass model for a rotorcraft is given
by the following six, fist-order nonlinear differential equations :
. T sin6
V = --- - g sin 7
2 = T cos 0sinq5
mV cos 7
g T cos6 cos q5
- cos 7)
Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | | DOI: 10.2514/6.1990-3455
Here, z is the down range, y the cross range, h
altitude, V airspeed, y the flight path angle, x the
heading angle, T the main rotor thrust, m the vehicle mass and g is the acceleration due to gravity.
The coordinate system for this model is illustrated
in Figure 4. The control variables in this model are
the pitch attitude of the helicopter 8, the roll attitude q5 and the main rotor thrust T. Note that the
rotor-fuselage drag is not included in the foregoing set of equations. It is sometimes convenient to
decompose the rotor thrust into two componentsthe fist one being fixed, while the second one is
allowed to vary.
To ensure that the rotorcraft executes a terrain following flight, it is necessary that the vehicle
altitude follow the terrain profile. This requirement may be enforced in the present problem by
by differentiating the terrain profile equation (1)
twice with respect to time and substituting for
various quantities from the vehicle model. This
results in a state-control constraint of the form :
T cos 8 cos 4 = mV2 cos8 7 (FEEcos x
+Fwsin X)cos X+ (F,~cos X+ Fuusin x) sin X]
+(F, cos X- F, sinx)T COST cos 0 sin 4+mg cos 7
This constraint specifies the relationship between various state and control variables while the
vehicle is in terrain following flight. The equations
of motion and the terrain constraint for both vehicles have the same form. However, the model
parameters may be different.
If one attempts to formulate a pursuit-evasion
problem using the nonlinear model described in
the foregoing, the solution can only be obtained using numerical methods. The objective here, however, is to obtain a feedback law. To meet this objective, the nonlinear equations will be f i s t transformed to a linear time invariant form via a coordinate transformation.
Differentiating the down-range cross-range
equations once with respect to time, it is possible
to write the pursuer-evader equations of motion in
the following form.
= ael, ye = ae2
For example, the right hand sides of these
equations are given by
-jpVp sin 7, cos xp
sin ye sin xe
The variables V ,j,i for both the pursuer and
the evader may be eliminated from equations (37),
(38) using expressions (30) - (32).
If the variables apl, ap2, ael, ae2 are known,
the actual control variables can be computed as
4, = tan"
ap2 cos xp - apl sin xp
0, = tan-' [{sin q5,(aPl cos X,
+ ap2 sin ;yp
apl cos X, - apl sin xP
cos 0, sin 4,
Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | | DOI: 10.2514/6.1990-3455
model and the quadratic performance index, this
problem has a feedback solution. The control laws
emerging from such a differential game formulation
can be shown1' to be of the form
With the feedback gains Kl, K2,K3 being
computed using the formulae
Similar inverse transformations can be written for
the evading rotorcraft also.
Once the vehicle models are transformed into
linear time invariant form, the differential game
between the two rotorcraft may be formulated. If
the performance index is defined in terms of transformed quantities, a solution may be obtained in
a straightforward manner. If mini-maximization
of the relative distance at the final time is taken
to be the performance index, then the differential game may be solved as in Reference 10. In
order to ensure that the emerging strategies are
implementable, an acceleration constraint will be
imposed on the problem. Thus the objective of the
pursuit-evasion game may be defhed as :
where :
The variables a, a,P are three scalar weighting factors between relative distance at the h a 1 time,
and square of the pursuer-evader acceleration magnitudes. The superscript T denotes the transpose
operation. The final time t j is assumed to be
The necessary conditions for optimizing the
performance index (45) subject to the differential constraints (37) and (38) can be obtained
by proceeding formally by defining the variational
Hamiltonian12. Due to the linear nature of the
Note that the time-to-go should be available
in order to complete the solution. Although it
was assumed to be specified, due to inaccuracies
in modeling, this parameter has to be estimated
while implementing the pursuit-evasion guidance
law. It is possible to obtain one such estimate by
invoking a constant of motion H(t) = constant in
the problem as in Reference 14.
Once the control variables apl, ap2, ael, ac2 in
the transformed problem are computed, the control variable transformations (41)-(43) may be
used to obtain nonlinear pursuit-evasion guidance
A typical pursuit-evasion scenario using this
pursuit-evasion strategy is illustrated in Figure 5.
The vehicle trajectories are denoted with arrows
in this figure. The lines with numerals appended
are the terrain altitude contours. The pursuer is
initially behind the evader with its velocity vector in a direction perpendicular to that of the
evader. The weighting factors used in this study
were a = 1.0, a = 0.0007, P = 0.00001. The
engagement time corresponding to this initial condition was 100 seconds. The vehicle altitude histories corresponding to this encounter are given in
Figure 6. At the final time, the pursuer altitude
was same as that of the evader, although the flight
path angle and the heading angles were different.
The maximum load factor corresponding to this
maneuver was about 2 and the maximum'vehicle
roll-pitch attitudes were about 30 degrees.
This solution is currently being studied in further detail with a view t o testing on a manned
simulation of rotorcraft.
Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | | DOI: 10.2514/6.1990-3455
This paper presented a preliminary study of
the rotorcraft pursuit-evasion problem in nap-ofthe-earth flight. Two games of increasing complexity were discussed. The first of these uses
a kinematic vehicle model, while the second approach employs a full point-mass model. In the
first case, the solutions were obtained by integrating the state- costate system backwards in time
from the h a l position of the evader. The solution for the second differential game was obtained
via feedback linearization. in this case, the vehicle point-mass models were f i s t transformed into
a linear, time-invariant form and the differential
game was solved in transformed coordinates. The
solution was transformed back t o the original space
to yield nonlinear pursuit-evasion guidance laws.
Sample extremals for both these cases were presented.
The solution t o the first pursuit-evasion problem is useful for developing pilot decision aids for
rotorcraft pursuit-evasion, while the solution to
the second differential game is useful for onboard
implement a t ion.
Partial research support for the first two authors under NASA Grant NAG 2 - 463 from Ames
Research Center is gratefully acknowledged.
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Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | | DOI: 10.2514/6.1990-3455
[13] Bogen, R., et al., MACSYMA Reference Manual, The Mathlab Group Laboratory for Computer Science, M.I.T., Cambridge, MA, Version 10, 1983.
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Fig.1. The Coordinate System for
Kinematic Vehicle Models
Fig. 2. Pursuer-Evader Trajectories
....-..-.. Evdcr
Fig.3. Altitude Hbtorier for the Pursuer
and the Evader
Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | | DOI: 10.2514/6.1990-3455
Fig.4. The Coordinate System for
Point-Mass Vehicle Models
Fig.0. Altitude Histories for the Pursuer
and the Evader
Fig.6. Pursuer-Evader Trajectories
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