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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]

5

()

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

6

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]:

11

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.

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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

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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.

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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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