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Tentacle-Based Guidance for Entry Flight with
No-Fly Zone Constraint
Zixuan Liang1 and Zhang Ren
2
Beihang University, Beijing, 100191, China
I. Introduction
URING the past decades, the guidance problem for hypersonic entry vehicles has attracted great attention. In
order to fly vehicles safely from an initial condition to a desired terminal condition, entry guidance methods
have been deeply studied [1-6]. To date, the essential requirement of the entry flight has almost been fulfilled.
Nevertheless, the flight needs to be autonomous and intelligent in future missions. For instance, entry vehicles
should have the threat/collision avoidance capability in complex environments.
Conventional entry guidance methods focus on the path and terminal constraints that mainly relate to the
longitudinal trajectory. The lateral trajectory with weaker constraints is usually controlled by bank reversals. The
corridor-based bank reversal logic is capable to fly the vehicle to the terminal target [1, 7]. However, this logic is not
applicable to the threat avoidance mission where no-fly zones bring an additional constraint to the lateral trajectory
[8]. The no-fly zone is an area that the vehicle is not allowed to fly into [9]. To complete an autonomous entry flight
with the no-fly zone constraint, an advanced guidance algorithm is required.
There are two guidance approaches for the no-fly zone constrained entry flight. In one approach, an entry
trajectory is pre-designed under multiple constraints and tracked during the actual flight. Trajectory
optimization/planning algorithms are investigated in [10-13] to satisfy the no-fly zone constraint. The trajectory
design process, if performed in advance, relies on the prior knowledge of all the no-fly zones. Moreover, to ensure
that the pre-designed trajectory can be followed by the entry vehicle, a robust three-dimensional tracking law is
necessary. The other approach considers the no-fly zone in the lateral guidance logic. In order to avoid circular
no-fly zones, the vehicle’s velocity heading angle is controlled by a dynamic corridor in [8, 14-16]. The corridor
boundary is calculated according to the geographic relation of the vehicle, the no-fly zone, and the target. The
geographic relation usually relies on the tangency of the no-fly zone, which means that the dynamic corridor may
1 Postdoctoral Fellow, School of Astronautics, aliang@buaa.edu.cn.
2 Professor, School of Automation Science and Electrical Engineering, renzhang@buaa.edu.cn.
D
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not be applicable to no-fly zones in other shapes. In [8], a waypoint vector (containing both location and direction) is
configured between a pair of circular no-fly zones. Then the zone avoidance problem can be solved by the waypoint
constrained guidance method in [17]. However, for no-fly zones in complex shapes, the configuration of the
waypoint vector would be difficult.
In this study, the entry guidance problem for the no-fly zone avoidance is investigated through a tentacle-based
approach. In nature, some insects use tentacles for sense. Inspired by this idea, tentacles (virtual tentacles, to be
precise) are employed by mobile robots for obstacle avoidance [18-20]. Here, a tentacle represents one of a robot’s
possible paths starting from its current position. Given the robot’s velocity and acceleration, the tentacle can be
generated in an analytical or numerical approach. The tentacle is able to detect an obstacle in advance and suggest a
feasible path in which the obstacle can be avoided. To find the feasible path, a set of tentacles are generated, and the
appropriate one is followed. The tentacle-based strategy addresses the threat avoidance guidance of entry vehicles.
In contrast to mobile robots, entry vehicles have complex flight dynamics and multiple constraints. Thus, the
tentacle of an entry vehicle is hard to be generated in an analytical approach. Instead, it can be given by a
numerically propagated trajectory. However, numerically generating a set of tentacles in each guidance cycle may
beyond the capability of an onboard computer. To reduce the computation cost, only two tentacles are generated
each time. To be specific, a left and a right tentacle are propagated by assuming a negative and a positive bank angle
in dynamic equations, respectively. The two tentacles are utilized for sense of threats, i.e., no-fly zones. The
feedback of a tentacle is given by two variables: (1) the time to touch a no-fly zone, and (2) the remaining time to
avoid a no-fly zone. These variables are combined with the vehicle’s heading error (the error between the heading
angle and the line-of-sight angle towards the target) to develop the bank reversal logic that can fly the vehicle to the
target without entering any of the no-fly zones. Cooperated with the bank reversal logic, a conventional trajectory
tracking law is employed to follow the longitudinal profile.
The tentacle-based guidance method is employed to provide a feasible entry path under the no-fly zone
constraint. The guidance method propagates tentacles using the dynamic model of an entry vehicle, and calculates
guidance commands according to the feedback of tentacles, which is a model predictive control (MPC) approach [21,
22]. The MPC has been utilized to solve the trajectory tracking problem of entry vehicles. The work in [23] and [24]
calculates the magnitude of the bank angle according to the error between the predicted and reference trajectories. In
order to solve the obstacle avoidance problem, we apply the feedback of tentacles to determine the sign of the bank
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angle. The length of the tentacle (i.e., the control horizon interval) is not constant but determined by the flight state
and the no-fly zones. The tentacle-based guidance method has two advantages. Firstly, the method is applicable to
not only the circular no-fly zone but also zones in other shapes (including some non-convex shapes). Secondly, the
method does not rely on the prior knowledge of no-fly zones. The effectiveness of the tentacle-based guidance
method is verified on the high-performance common aero vehicle (CAV-H) in entry missions with various no-fly
zones.
II. Entry Guidance Problem
A. Dynamics
The three-dimensional point-mass dynamics of an entry vehicle over a rotating spherical Earth are given by [25]
sinrv
γ
=
(1)
2
sin cos (sin cos cos sin cos )
D
vg r
m
γ
ω
φγφ γφψ
=− − + −
(2)
2
2
1cos cos
[ cos 2 cos sin cos (cos cos sin sin cos )]
Lv
gvr
vm r
σγ
γγ
ω
φψ
ω
φγφ γφψ
=−++ + +
(3)
cos sin
cos
v
r
γψ
θφ
=
(4)
cos cosv
r
γψ
φ
=
(5)
2 2
1 sin cos sin tan
[ + 2 (tan cos cos sin ) sin cos sin ]
cos cos
Lv r
v
vm r
σγψφ ω
ψ
ω
γφψ φ φφψ
γγ
=−−+
(6)
where r is the radial distance from the Earth center to the vehicle, v is the velocity magnitude, γ is the flight-path
angle, θ is the longitude, ϕ is the latitude, ψ is the velocity heading angle, σ is the bank angle, ω is the Earth
self-rotation rate, m is the vehicle mass, and g is the gravitational acceleration. The lift force L and the drag force D
are affected by the angle of attack α through the lift and drag coefficients, respectively.
B. Constraints
Given an initial condition, the entry vehicle is expected to reach a desired terminal condition. With the
range-to-go as the independent variable, the terminal constraints are given by
()
f
f
hs h= (7)
()
f
f
vs v= (8)
4
()
f
f
s
ψψ
Δ≤Δ
(9)
where hf, vf, and sf are the desired terminal altitude, velocity, and range-to-go, respectively. The heading error ∆ψ at
the entry terminal should be limited to the interval [-∆ψf, ∆ψf]. The range-to-go from the vehicle to the target is
computed using the great-circle distance, which is given by
1
Ecos [cos cos cos( ) sin sin ]
f
ff
sR
φφ
θθ
φφ
−
=−+
(10)
where θf and ϕf are the longitude and latitude of the target, respectively. The radius of the Earth RE is a constant.
During the entry flight, inequality path constraints need to be satisfied. Typical path constraints are given by [26]
0.5 3.15
max
22
max
2
max
0.5
Q
QK v Q
LD
nn
m
qvq
ρ
ρ
=≤
+
=≤
=≤
(11)
where Qmax, nmax, and max
q are maximum limits for the heating rate, the aerodynamic load, and the dynamic
pressure, respectively.
Apart from the path and terminal constraints, an additional constraint needs to be considered in missions
containing no-fly zones. No-fly zones are areas where the vehicle must avoid flying into. Particularly, the constraint
of a circular no-fly zone requires the distance from the vehicle to the zone center to be no less than the zone radius.
For a general case, the constraint is expressed as
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NN N
min{ , , , } 0
n
SS S≥ (12)
where n is the number of no-fly zones. The distance N(1 )
i
Sin≤≤ is measured from the vehicle to the nearest
boundary of the i th zone. The positive and negative distances represent that the vehicle is outside and inside the
zone, respectively.
III. Tentacle-Based Guidance Algorithm
A. Guidance Strategy
Like most of the previous work, the entry guidance method employs a tracking law and a bank reversal logic for
the longitudinal profile and the lateral trajectory, respectively. The tracking law is designed based on the linear
quadratic regulator (LQR), which is given by [3, 17, 27]
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()
r
s
v
σ
α
γ
Δ
Δ
=Δ
Δ
Δ
K (13)
where Δ σ and Δ α are adjustments for the bank angle and the angle of attack with respect to the reference profiles,
respectively. The reference angle of attack profile αref (v) is designed as a piecewise linear function. The reference
bank profile σref (v) is optimized to satisfy the terminal and path constraints. The optimization problem is solved
offline by the sequential quadratic programming (SQP) method. The inputs Δ r, Δ v, and Δ γ are tracking errors for
the altitude, the velocity, and the flight-path angle, respectively. The feedback matrix K (s) is solved offline from the
Algebraic Riccati equation, and scheduled onboard with respect to the range-to-go [27].
The bank reversal logic provides the sign of the bank angle. The conventional logic only considers the heading
error or the crossrange [1, 7], which cannot satisfy the no-fly zone constraint. To complete a threat avoidance
mission against no-fly zones, a tentacle-based algorithm is employed. Figure 1 illustrates a vehicle with a pair of
tentacles. The tentacles are applied to the early warning of no-fly zones, and thereafter actions can be taken for the
threat avoidance. The generation and application of the tentacles are given in the subsequent subsections.
Fig. 1 Vehicle with left and right tentacles
The longitudinal profile tracking law tries to achieve a zero tracking error for each concerned state. Because the
law works for an equality constraint, it should be continuously performed in each guidance cycle. However, the
lateral guidance logic is employed to satisfy an inequality constraint. Thus, the real-time requirement of the lateral
guidance logic is relatively low. Let ∆T be the update time for the longitudinal guidance cycle. Then, the update time
for the lateral guidance cycle is set as
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1
Lat
1
, if
, if
kT s s
TTss
Δ>
Δ=
Δ≤
(14)
where k is an integer generally greater than 1. The parameter s1 is slightly larger than sf, and represents a range-to-go
near the terminal zone. The lateral guidance logic works more frequently in the final phase to meet the heading error
constraint in Eq. (9). Note that k = 1 is allowed as well. In that case, the lateral guidance logic would work in each
guidance cycle together with the longitudinal tracking law.
B. Generation of Tentacle
A tentacle is given by a numerically propagated trajectory. In contrast to the numerical predictor-corrector
algorithm [6, 28], the tentacle only concerns the lateral trajectory which is constrained by no-fly zones. Thus, to
improve the propagation speed of the tentacle, integrations are conducted for the lateral dynamics instead of the
three-dimensional dynamics given by Eqs. (1-6). Assuming r = RE, cos γ = 1, and ω = 0 in Eqs. (4-6), the simplified
lateral dynamics are given by
E
sin
cos
v
R
ψ
θ
φ
=
(15)
E
cosv
R
ψ
φ
=
(16)
sin sin tan
E
Lv
mv R
σ
ψφ
ψ
=+
(17)
In Eqs. (15-17), variables related to the longitudinal dynamics are the velocity v and the lift force L. The
longitudinal trajectory tracking law can ensure that both v and L follow their reference profiles. Therefore, during
the integration process, v and L are scheduled from the reference profiles according to the range-to-go. Similarly, in
Eq. (17), the magnitude of the bank angle is scheduled from the reference bank profile σref (v) according to the
scheduled velocity. The sign of the bank angle is determined by the tentacle. To be specific, the negative and
positive bank angles are utilized for the left and right tentacles, respectively.
With the vehicle’s current state as the initial condition, the tentacle is generated by integrations. The break
condition of integrations affects the tentacle length which is of great importance to the guidance performance. On
one hand, generating a significantly long tentacle is time-consuming and would reduce the real-time performance.
On the other hand, a short tentacle cannot touch and consider a far no-fly zone, which may result in a wrong flight
path. With these considerations, the break condition is designed as follows.
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Condition 1: t ≥ tmax, which means that the flight time t exceeds its maximum limit.
Condition 2:N0 (1 )
i
Sin≤≤≤, which means that a no-fly zone is touched or entered.
Condition 3: |∆ψ| ≥ 90 deg, which means that the target would be missed.
Condition 1 is a common break condition for trajectory integrations. In Condition 2, the trajectory integration
stops when a no-fly zone is touched by the tentacle. Even though there are other zones, the tentacle no longer needs
to be propagated since the nearest threat has been found. In Condition 3, when the heading error is greater than 90
deg, the tentacle no longer extends towards the target. The further propagated tentacle cannot lead the vehicle to the
target, and thus would be useless. Once one of the three conditions is met, the integration stops. Conditions 1-3
produce a useful tentacle with a relatively long length, which helps to avoid a wrong path given by a local solution.
Note that in each cycle for the lateral guidance logic, the integration process needs to be conducted twice to obtain
the left and right tentacles.
C. Threat Evaluation
The vehicle’s two tentacles are generated for the threat evaluation of no-fly zones. The feedback of a tentacle is
given by two variables. The first variable is the time-to-touch which is defined by
LL
touch touch start
RR
touch touch start
Ttt
Ttt
=−
=−
(18)
where tstart is the start time of the tentacle generation. The terms L
touch
t and R
touch
t are the time when the left and
right tentacles touch a no-fly zone, respectively. In general, the touch time corresponds to the break time in
Condition 2 for the tentacle generation. For cases that no zones are touched, the integration would break under
Conditions 1 or 3, and then tmax is used as the time-to-touch.
The other variable is the time-for-avoid Tavoid. It is the remaining time for one tentacle to avoid the no-fly zone
touched by the other tentacle. Assuming the vehicle flies with a positive bank angle, the tentacles at various
positions are shown in Fig. 2. In this figure, tentacles are just plotted for analysis and provide no feedback. At Points
A, B, and C, three pairs of tentacles are generated. The three left tentacles are each generated under the assumption
of a negative bank profile. The right tentacle, which corresponds to a positive bank profile, is unchanged and
followed by the vehicle. In practice, the right tentacle may slightly change under follow errors. The minimum
distance between the left tentacle and the no-fly zone decreases during the flight from Point A to Point C. The left
tentacle does not touch the no-fly zone until Point C is reached. At this point, no time remains for the vehicle to
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avoid the zone along the left tentacle, and thus no feasible path exists. Therefore, to complete the avoidance mission,
the time-for-avoid needs to be considered as a feedback. Let L
N
S be the minimum distance between the left tentacle
and the no-fly zone. The time-for-avoid is calculated by
LL L
LNN N
avoid L
N
/ , if 0
, if 0
SS S
TS
−<
=+∞ ≥
(19)
where the derivative of L
N
Scan be calculated through a difference method, which is given by
LL
LNN Lat
N
Lat
() ( )St St T
ST
−−Δ
=Δ
(20)
Fig. 2 Tentacles generated at various positions
D. Bank Reversal Logic
The bank reversal logic needs to provide an appropriate sign to the vehicle’s bank angle so that all the no-fly
zones are avoided and the target is finally reached. Thus, the logic is developed by combining the following three
factors:
(1) The vehicle’s heading error ∆ψ which determines whether the target can be reached or not.
(2) The time-to- touch Ttouch which describes the threat level of no-fly zones with respect to a tentacle.
(3) The time-for-avoid Tavoid which describes the avoidance chance of a tentacle with respect to no-fly zones.
The vehicle’s initial heading error is zero or relatively small because the initial velocity generally points towards
the target. Therefore, in the first guidance cycle, the sign of the bank angle command is determined by Ttouch and
Tavoid. In order to fly the vehicle in a path with a weaker threat, the bank angle sign is given by
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RL
touch touch
RL RL
touch touch N N
1, if
sgn( ) 1, if and
1, otherwise
TT
TT SS
σ
>
==≥
−
(21)
During the entry flight, all of the three factors mentioned before need to be considered to determine a bank
reversal. In addition, unnecessary bank reversals are not expected since frequent bank reversals have negative effects
on the attitude control system. Assuming a positive bank angle command is utilized in the previous guidance cycle,
two criterions are established for performing a bank reversal in the current guidance cycle. The two criterions are
expressed as
LR
touch touch
L
avoid
T
TT
T
>
≤Δ
(22)
LR
touch touch
L
N
()
S
TT
S
v
ψ
ψ
≥
≥Δ
Δ>Δ
(23)
where ∆T > 0 is a threshold of the time-for-avoid, ∆S > 0 is a threshold of the distance to a no-fly zone, and ∆ψ (v) is
the threshold of the heading error. With the velocity as the independent variable, ∆ψ (v) is designed by [1]:
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221
323 2
2
3
, if
, if
() ( ), if
, if
f
f
f
f
cvv
cvvv
vv
vcccvvv
vv
cvv
ψ
≥
≤<
−
Δ=
+− <<
−
≤
(24)
where v1 and v2 are velocities at the two piecewise points. Parameters c1, c2, and c3 are positive constants. To satisfy
the terminal constraint in Eq. (9), c3 = 0.8∆ψf is employed. Higher values should be set for c1 and c2 to avoid
frequently bank reversals.
Equation (22) describes the first criterion where a larger time-to-touch can be achieved after the reversal, and the
remaining time for avoidance actions is going to be insufficient. Under these conditions, a bank reversal must be
performed immediately. For instance, consider the case in Fig. 2. A bank reversal at Point A can increase the touch
time but is unnecessary because sufficient time remains for the avoidance actions. The reversal should be delayed
until the time-for-avoid enters its threshold. In the second criterion given by Eq. (23), the heading error exceeds its
threshold, which means that a bank reversal requirement arises. Meanwhile, if the bank angle is reversed to negative,
the time-to-touch would not be reduced and a safe distance would exist. Under these conditions, performing a bank
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reversal is necessary and also allowed.
Similarly, when a negative bank angle command is utilized in the previous guidance cycle, the criterions in Eqs.
(22) and (23) are replaced by
RL
touch touch
R
avoid
T
TT
T
>
≤Δ
(25)
RL
touch touch
R
N
()
S
TT
S
v
ψ
ψ
≥
≥Δ
Δ<−Δ
(26)
The bank reversal logic is activated in first guidance cycle together with the longitudinal tracking law. If a no-fly
zone is touched by both of the two tentacles (generated in the first guidance cycle), the current bank reversal logic
would be hard to complete the threat avoidance mission, and a trajectory re-planning approach would be required to
update the reference profiles. Otherwise, a feasible path exists at the beginning of the flight. Then, during the flight,
the bank reversal logic would try to avoid the case that a no-fly zone is touched by the two tentacles, i.e., the case
that no feasible path exists. On the other hand, if a no-fly zone is not touched by any tentacle during the flight, it is
actually not a threat to the vehicle. Note that the proposed bank reversal logic is also applicable to entry missions
without no-fly zones. In those missions, the threshold of the heading error in Eqs. (23) and (26) would keep working
to fly the vehicle to the target.
Because the guidance logic does not rely on the prior knowledge of no-fly zones, it can also be used for an entry
mission where no-fly zones are detected during the flight instead of at the beginning. In that mission, the initial sign
of the bank angle is set to be positive, and the vehicle is flown without considering no-fly zones. Once some zones
are detected, Eq. (21) is utilized to give a reasonable bank angle sign. Thereafter, bank reversals are performed under
the criterions established before. Note that the detection time cannot be so late that no avoidance path exists.
IV. Numerical Results
The CAV-H model is utilized to verify the tentacle-based guidance method. The vehicle’s mass and reference area
are 907.2 kg and 0.484 m2, respectively. The database of aerodynamic coefficients (the lift coefficient, the drag
coefficient, and the lift-to-drag ratio) can be found in Ref. [29]. The vehicle achieves its maximum lift-to-drag ratio
of 3.5 at Mach 10 with the angle of attack being 10 deg. Four avoidance missions in two categories are considered.
Missions 1 and 2 have a circular and rectangle no-fly zone, respectively. Each of these two missions considers only
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one no-fly zone. To complete the avoidance mission, the vehicle must fly in a path on the left or right side of the
zone. Missions 3 and 4 consider a straight and curvy passageway formed by no-fly zones, respectively. The vehicle
must fly in a path inside each passageway. In the four missions, locations of the no-fly zones are assumed to be
known to the guidance system at the beginning of the flight. Main parameters utilized in the testing are configured
as follows.
1) Nominal states and dispersions for the initial condition are given by Table 1, where 3-sigma represents three
times the standard deviation of dispersions.
2) The desired terminal condition is hf = 30 km, vf = 2000 m/s, sf = 100 km, and ∆ψf = 5 deg. The target’s
location is θf = 96 deg and ϕf = 0 deg.
3) The path constraints are Qmax = 4 MW/m2, nmax = 3 g, and max
q= 60 kPa.
4) The maximum limit of the flight time is tmax = 3000 s. The guidance cycle update time is ∆T = 1 s.
5) In Eq. (14), k = 3 and s1 = sf + 20 km are set for the cycle update time of the lateral guidance logic. Thus, ∆TLat
= 3 s is utilized during most of the flight, and ∆TLat = 1 s is utilized only when the range-to-go is less than 120
km.
6) The tentacle is generated by the Adams integration method with a fixed step size of 10 s. Compared with the
Runge-Kutta integration, the Adams integration has higher computational efficiency to achieve the same
accuracy.
7) The threshold of the time-for-avoid is ∆T = 50 s, and the threshold of the distance to a no-fly zone is ∆S = 50
km. Parameters used in the heading error threshold are c1 = 13 deg, c2 = 20 deg, c3 = 4 deg, v1 = 6000 m/s,
and v2 = 4000 m/s.
Table 1 Initial condition and dispersions
Parameters Nominal 3-sigma
Altitude, km 70 1
Velocity, m/s 6800 50
Flight-path angle, deg 0 0.3
Longitude, deg 2 0.3
Latitude, deg 0 0.3
Heading angle, deg 90 0.3
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A. Nominal Case
The guidance method is tested in Missions 1-4 in the nominal case. Figure 3 shows the vehicle’s ground tracks
together with the no-fly zone constraints. In Missions 1 and 2, the vehicle is flown in the path on the right and left
sides of the zone, respectively. The flight path in each mission is selected according to the vehicle’s tentacles. Take
Mission 1 for instance. Early tentacles demonstrate that both paths are feasible. However, the path on the right side
is selected because the right tentacle can achieve a larger distance to the no-fly zone. The larger distance can benefit
the avoidance performance under dispersions. In Missions 3 and 4, the vehicle is successfully flown to the target
through the passageways. With the aid of tentacles, the vehicle does not touch the passageway boundary, which
means that the no-fly zone constraint is satisfied. Figure 4 illustrates the guidance commands utilized in Missions
1-4. In each mission, the angle of attack profile is almost unchanged as no dispersion is considered. The initial sign
of the bank angle is determined by Eq. (21) to select a safer path. Five bank reversals are performed in Missions 1
and 2, while six bank reversals are performed in Missions 3 and 4 which are more challenging. In fact, there are
an infinite number of feasible paths for each mission, and the guidance method just provides an appropriate one. For
instance, Fig. 3 shows that straighter paths exist in Mission 3. However, a straighter path would require a higher
frequency of bank reversals during the flight, which is generally not expected.
Note that the guidance method does not require the shape of a no-fly zone to be convex, which is also indicated
by the result for Mission 4. Nevertheless, the guidance method is not applicable to all kinds of no-fly zones. For
instance, in Mission 4, both the curvature and the width of the passageway (i.e., the area between the left and right
no-fly zones) have influences on the avoidance performance. When the curvature is significantly high or the
passageway is significantly narrow, there would be no feasible path.
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Fig. 3 Ground tracks for Missions 1-4 in nominal case
14
Fig. 4 Guidance commands for Missions 1-4 in nominal case
B. Dispersed Cases
Dispersions of the initial condition and aerodynamic parameters are considered to verify the robustness of the
guidance method. Three times the standard deviation for dispersions of the lift coefficient, the drag coefficient, the
aerodynamic density, and the vehicle mass are 10%, 10%, 15%, and 3% with respect to their reference values,
respectively. Under these dispersions, Monte Carlo simulations are conducted for Missions 1-4. In each mission,
1000 trajectories are simulated.
In dispersed cases, ground tracks of the entry trajectories are shown in Fig. 5. The vehicle’s ground tracks are
highly dispersed during the flight. Nevertheless, no-fly zones in all of the four missions are successfully avoided.
Figure 6 illustrates the ground tracks in the final phase. All the ground tracks reach the terminal zone whose radius is
sf = 100 km. In addition, results show that the terminal heading errors are no greater than 5 deg. Therefore, the
lateral guidance logic is demonstrated to be effective in dispersed cases. The longitudinal profiles in the four
missions are shown in Fig. 7. The reference altitude-velocity profile is closely followed by the actual trajectories.
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The statistics show that the mean terminal altitude is 30.01 km with the standard deviation being 0.13 km. The mean
terminal velocity is 2999.70 m/s with the standard deviation being 6.86 m/s. Results indicate the good performance
of the tracking law. Besides, the heating rate, the dynamic pressure, and the aerodynamic load are found all within
their limits. In summary, the guidance performance on the no-fly zone avoidance, the terminal precision, and the
path constraints demonstrates the robustness of the guidance method. To improve the robustness, advanced control
laws, such as the active-disturbance-rejection control, can be employed for the longitudinal profile tracking.
Moreover, to reduce the sensitivity to uncertainties, the desensitized optimal control (DOC) can be utilized for the
reference profile optimization [30, 31].
Fig. 5 Ground tracks for Missions 1-4 in dispersed cases
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Fig. 6 Ground tracks for Missions 1-4 in dispersed cases (final phase)
Fig. 7 Altitude-velocity profiles for Missions 1-4 in dispersed cases
C. Computational Performance
The tentacle-based guidance method would not be time-consuming in theory, because simplified lateral dynamics
are used in the tentacle generation, and only two tentacles are generated in each lateral guidance cycle. Numerically,
the computational performance of the guidance method can be evaluated through two indexes: the maximum and
mean time used to calculate the guidance commands in a cycle. Simulations in this study are conducted on a PC
(CPU: 3.4 GHz) using the software MATLAB. In the Monte Carlo simulations, the CPU computation time
(measured by the “tic-toc” function of MATLAB) is recorded in each guidance cycle. The statistics of the real-time
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computations for guidance cycle commands are given in Table 2. The maximum time spent in Missions 1-4 are
0.0311 s, 0.0324 s, 0.0178 s, and 0.0187 s, respectively. The mean time spent in Missions 1-4 are 0.0045 s, 0.0047 s,
0.0033 s, and 0.0036 s, respectively. For the same mission, the 3-sigma values show that both the maximum time
and the mean time do not change much in different cases. Less time is used in Missions 3 and 4 than Missions 1 and
2 because shorter tentacles are generated under the passageway constraint. In summary, the computational
performance indicates that the guidance method is rapid and possible to be employed onboard.
In fact, the computation time for guidance commands relates to two parameters used by the guidance method: (1)
the update time for the lateral guidance cycle, and (2) the integration step size for the tentacle propagation. The
guidance method is further tested by setting various values for these two parameters. Results show that the guidance
cycle update time has little impact on the maximum computation time. In contrast, the impact on the mean
computation time is obvious. The mean computation time used for guidance commands in the nominal case is
illustrated in Fig. 8. For each mission, the mean time decreases with the guidance cycle update time because longer
update time represents a lower frequency of the tentacle generation. Unlike the cycle update time, the step size of the
tentacle affects both the maximum and mean computation time. The maximum time under various step sizes is
illustrated in Fig. 9. The maximum time decreases with the step size because a larger step size generally results in
fewer steps to generate a tentacle.
Figures 8 and 9 indicate that the computational performance can be improved by increasing either the guidance
cycle time or the integration step size. However, when significantly long cycle time is used for the lateral guidance,
avoidance actions are likely to be delayed. When a significantly large step size is used, the tentacle would not be
accurate enough to evaluate the threat of no-fly zones. To discover the impacts, the guidance method using various
cycle time and step sizes is tested in Mission 4 which is relatively challenging. A flight is supposed to fail if the
vehicle breaks through the passageway boundary, i.e., the no-fly zone constraint is violated. The number of failures
among 1000 trajectories is given in Table 3. Cases 1-4 consider larger cycle time, and Cases 5-8 consider larger step
sizes. The number of failures reaches 5 in Case 4 where the cycle time is 15 s, and reaches 7 in Case 8 where the
step size is 30 s. The avoidance performance is affected when a significantly large value is configured for the cycle
time or the step size. Therefore, apart from the computational performance, the avoidance performance should also
be considered during the configuration of the two parameters.
18
Table 2 Statistics of real-time computations for guidance cycle commands
Mission
Maximum time (s) Mean time(s)
Mean 3-sigma Mean 3-sigma
1 0.0311 0.0011 0.0045 0.0001
2 0.0324 0.0008 0.0047 0.0001
3 0.0178 0.0010 0.0033 0.0001
4 0.0187 0.0008 0.0036 0.0002
Fig. 8 Mean computation time under various lateral guidance cycle update time
Fig. 9 Maximum computation time under various integration step sizes for tentacle propagation
19
Table 3 Statistics of failure cases (Mission 4)
Case Guidance cycle time (s) Integration step size (s) Number of failures
1 6 10 0
2 9 10 1
3 12 10 2
4 15 10 5
5 3 15 0
6 3 20 0
7 3 25 1
8 3 30 7
D. Testing for Threat Detection Time
In previous testing, locations of the no-fly zones are known to the vehicle at the beginning of each flight. Actually,
the tentacle-based guidance method is still applicable when no-fly zones are detected not at the beginning but
somewhere during the flight. Let Tdetection be the detection time of no-fly zones. Figure 10 shows the ground tracks in
Mission 2 under various Tdetection. The corresponding bank angle histories are illustrated in Fig. 11. In the case of
Tdetection = 100 s, the flight path on the left side of the no-fly zone is found to be safer, and thus a bank reversal is
performed once the no-fly zone is detected. However, when the detection time is 210 s, the advantage of the left path
is lost. In this case, the sign of the bank angle remains unchanged, which keeps the vehicle on the right side of the
no-fly zone. When the detection time delays to 400 s, the tentacles show that only the path on the right side is
feasible. In the case of Tdetection = 1200 s, the avoidance flight fails because the no-fly zone is entered before being
detected. Therefore, results show that the detection time of the no-fly zone can be delayed, but a feasible flight path
must exist when the zone is detected.
20
Fig. 10 Ground tracks under various detection time of no-fly zone (Mission 2)
21
Fig. 11 Bank angle histories under various detection time of no-fly zone (Mission 2)
V. Conclusions
A tentacle-based guidance method has been proposed for entry flights with the no-fly zone constraint. Compared
with previous methods designed for circular no-fly zones, the proposed method is applicable to zones in general
shapes. Simulations have been conducted for four entry missions with various no-fly zones. Results show that the
tentacle is capable of suggesting a feasible path for the vehicle. The lateral guidance logic, which combines the
tentacle feedback and the heading error, is able to fly the vehicle to the target without entering the no-fly zones. The
longitudinal tracking law performs well in following the reference trajectory. Therefore, the proposed guidance
method can satisfy both the conventional constraints and the additional no-fly zone constraint. Moreover, the
guidance method is demonstrated to be effective when the detection of the threat is delayed, which means that it
does not rely on the prior knowledge of no-fly zones. To guarantee the computational performance for guidance
commands, the guidance method generates only two tentacles in each lateral guidance cycle. The method can benefit
from the parallel implementation which allows to calculate more tentacles. More tentacles would improve the
22
possibility of finding a feasible path under challenging conditions. The logic of how to generate and use more
tentacles is a meaningful topic in further research.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (Grant No. 61333011) and the
China Postdoctoral Science Foundation (Grant No. 2017M610034). The authors would like to thank Prof. Craig A.
Kluever and the reviewers for their valuable comments and suggestions.
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