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Tentacle-Based Guidance for Entry Flight with No-Fly Zone Constraint

Tentacle-Based Guidance for Entry Flight with
No-Fly Zone Constraint
Zixuan Liang1 and Zhang Ren
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,
2 Professor, School of Automation Science and Electrical Engineering,
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
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
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]
sin cos (sin cos cos sin cos )
vg r
φγφ γφψ
=− − +
1cos cos
[ cos 2 cos sin cos (cos cos sin sin cos )]
vm r
φγφ γφψ
=−++ + +
cos sin
cos cosv
2 2
1 sin cos sin tan
[ + 2 (tan cos cos sin ) sin cos sin ]
cos cos
Lv r
vm r
σγψφ ω
γφψ φ φφψ
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
hs h= (7)
vs v= (8)
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
Ecos [cos cos cos( ) sin sin ]
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
QK v Q
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
min{ , , , } 0
SS S (12)
where n is the number of no-fly zones. The distance N(1 )
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]
 
 
 
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
, if
, if
kT s s
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
sin sin tan
mv R
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.
Condition 1: ttmax, which means that the flight time t exceeds its maximum limit.
Condition 2:N0 (1 )
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
touch touch start
touch touch start
where tstart is the start time of the tentacle generation. The terms L
t and R
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
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
S be the minimum distance between the left tentacle
and the no-fly zone. The time-for-avoid is calculated by
avoid L
/ , if 0
, if 0
=+∞ ≥
where the derivative of L
Scan be calculated through a difference method, which is given by
() ( )St St T
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
(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
touch touch
touch touch N N
1, if
sgn( ) 1, if and
1, otherwise
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
touch touch
touch touch
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]:
323 2
, if
, if
() ( ), if
, if
+− <<
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 = 0.8ψf 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
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
touch touch
touch touch
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
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
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
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
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.
Fig. 3 Ground tracks for Missions 1-4 in nominal case
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.
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
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
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.
Table 2 Statistics of real-time computations for guidance cycle commands
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
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.
Fig. 10 Ground tracks under various detection time of no-fly zone (Mission 2)
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
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.
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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... To date, the essential requirements for traditional landing guidance are almost fulfilled, which is also indicated by the successful flight of some entry vehicles [7]. In such case, more and more attentions are paid to the penetration capability of target reconnaissance [8,9], threat avoidance [10,11], and cooperative attack [12,13]. ...
... Let sf be the radius of the terminal zone, i.e., the desired range-to-go. The terminal constraints are [10] () f ...
... The entry vehicle needs to be delivered to the terminal zone with the constraints in Eqs. (8)(9)(10)(11) and (13) satisfied. The path constraint in Eq. (8) can be considered in the longitudinal profile planning before the entry starts. ...
Full-text available
A simultaneous attack mission requires the entry vehicle to be capable of controlling the terminal time. To satisfy the terminal time constraint and meanwhile to cooperate with the previous longitudinal guidance algorithms, a lateral entry guidance method is developed using a closed-loop trajectory predictor for the terminal time, a bank reversal logic based on a double-corridor strategy, and a terminal time determination approach. The double-corridor contains a time error corridor used to activate the modification of the terminal time, and an adjustable heading corridor, when activated, used to reduce the terminal time error. The desired terminal time is calculated according to a feasibility index which describes the tolerance of aerodynamic dispersions. The simulations of a time constrained entry flight show that the lateral guidance method can achieve a high accuracy of the terminal time control while satisfying the conventional entry constraints. The advantage of the desired terminal time given by the feasibility index is also demonstrated.
... Based on the prerequisite penetration condition, a penetration trajectory optimization algorithm for an air-breathing hypersonic vehicle was designed in [2], which took the control costs as the objective function to minimize the fuel consumption and maneuver range. The penetration trajectory optimization of hypersonic vehicles considering the waypoint and no-fly zone constraints was discussed in [12][13][14]. The penetration trajectory optimization of hypersonic vehicles striking a stationary target by a spiral-diving maneuver was studied in [15]. ...
... According to Assumption 2, θ ∈ (−π/2, π/2) and ψ ∈ (π/2, 3π/2) are ensured. Thus, the following inequality can be established according to (12). ...
Full-text available
The penetration trajectory optimization problem for the hypersonic gliding vehicle (HGV) encountering two interceptors is investigated. The HGV penetration trajectory optimization problem considering the terminal target area is formulated as a nonconvex optimal control problem. The nonconvex optimal control problem is transformed into a second-order cone programming (SOCP) problem, which can be solved by state-of-the-art interior-point methods. In addition, a penetration strategy that only requires the initial line-of-sight angle information of the interceptors is proposed. The convergent trajectory obtained by the proposed method allows the HGV to evade two interceptors and reach the target area successfully. Furthermore, a successive SOCP method with a variable trust region is presented, which is critical to balancing the trade-off between time consumption and optimality. Finally, the effectiveness and performance of the proposed method are verified by numerical simulations.
... Based on the prerequisite penetration condition, a penetration trajectory optimization algorithm for an air-breathing hypersonic vehicle was designed in [2], which took the control costs as the objective function to minimize the fuel consumption and maneuver range. The penetration trajectory optimization of hypersonic vehicles considering the waypoint and no-fly zone constraints was discussed in [12][13][14]. The penetration trajectory optimization of hypersonic vehicles striking a stationary target by a spiral-diving maneuver was studied in [15]. ...
... According to Assumption 2, θ ∈ (−π/2, π/2) and ψ ∈ (π/2, 3π/2) are ensured. Thus, the following inequality can be established according to (12). ...
Full-text available
The penetration trajectory optimization problem for the hypersonic gliding vehicle (HGV) encountering two interceptors is investigated. The HGV penetration trajectory optimization problem considering the terminal target area is formulated as a nonconvex optimal control problem. The nonconvex optimal control problem is transformed into a second-order cone programming (SOCP) problem, which can be solved by state-of-the-art interior-point methods. In addition, a penetration strategy that only requires the initial position information of the interceptors is proposed. The convergent trajectory obtained by the proposed method allows the HGV to evade two interceptors and reach the target area successfully. Furthermore, a successive SOCP method with a variable trust region is presented, which is critical to balance the trade-off between time consumption and optimality. Finally, the effectiveness and performance of the proposed method are verified by numerical simulations.
The conventional re-entry guidance method considerably limits the maneuverability of hypersonic glide vehicles under pre-given angle-of-attack profiles. To address this issue, based on a previous three-dimensional (3D) profile guidance method, we propose a lateral-profile-first 3D profile optimization design strategy to solve the trajectory generation problem under complex conditions involving multiple constraints. Unlike the traditional method, which prioritizes the design of the longitudinal drag acceleration profiles, our method first converted the re-entry flight constraints into a lateral flight corridor, generated the initial lateral profile via parameterization, and solved the 3D profile using the coupling relationship between the longitudinal and lateral motions. The design of the 3D profile requires simultaneous consideration of waypoints, no-fly zones, and altitude profiles; hence, a new 3D profile optimization design model was constructed, and a customized sequential quadratic programming algorithm was employed to solve it. The advantages of the proposed method were demonstrated based on landing footprint results obtained using the lateral profile-first strategy. Finally, simulation results based on the common aero vehicle-high-performance model verified the feasibility and effectiveness of the strategy.
Aimed at complex distributed no-fly zones avoidance problems, a novel adaptive lateral reentry guidance algorithm is proposed. Firstly, by introducing the improved attractive and repulsive potential fields, an improved artificial potential field method is developed. Combined with the proposed judgment criterion for whether a no-fly zone has been avoided, the proposed improved artificial potential field method effectively solves the reference heading angle determination problem under the constraints of complex distributed no-fly zones. Then, based on the proposed no-fly zone’s threat quantitative evaluation method and the reference heading angle determined by the proposed improved artificial potential field method, the heading corridor is improved to increase its sensitivity to the threat changes of the no-fly zones. Finally, for satisfying the requirements of complex distributed no-fly zones avoidance, a novel guidance logic via improved heading corridor is proposed to update the reference heading corridor adaptively in real time according to the threat and constraint changes of the no-fly zones, and the bank reversal logic is employed to control the lateral motion. The simulation results for nominal and dispersed cases indicate that the proposed guidance algorithm has high robustness, stability, and applicability, and is feasible and effective to deal with the complex distributed no-fly zones avoidance problems.
This paper proposes a high-level path decision modeling approach of hypersonic vehicles for no-fly zones avoidance with complex distribution. The current algorithms of trajectory planning and guidance can achieve reliable convergence practically dependent on the initial trajectory. However, in the complex no-fly zones scenario under consideration, it is difficult to guarantee a priori a good initial guess, which requires the vehicle to have high-level decision-making ability to ensure trajectory's global performance. To deal with this, this paper establishes the path decision model and integrates physical level information. The path decision problem is formulated based on graph modeling by formulating new policies and rules. To the best of our knowledge, this paper is the first to study the path decision problem and model for hypersonic vehicles. The path decision approach is proposed by incorporating low-level waypoint-follow guidance simulation information in the path evaluation. This operation takes hypersonic dynamics into account so that it avoids the evaluation inconsistency between high-level decision and low-level guidance. Numerical simulation results show that this paper effectively realizes the path decision of the hypersonic vehicle for no-fly zones avoidance and improves the global performance of the trajectory.
Dynamic optimisation for hypersonic cruise vehicle (HCV) is a challenge for the complex nonlinear constraints with waypoint and no-fly zone during a global strike mission. Because the passing time of each waypoint is unknown, it is difficult to guarantee the precision of waypoint constraints on non-collocation points for traditional pseudospectral method. A novel optimal control approach is therefore proposed, where waypoints are firstly transformed into variable optimisation parameters of pseudospectral method, which can satisfy the waypoint constraints completely. Then, a novel non-uniform pseudospectral approach with mesh refinement is further proposed by introducing the adaptive gradient analysis to provide a time grid iteratively. With this refinement, the proposed approach needs fewer computational time to achieve better performance of accuracy compared with traditional uniform refinement pseudospectral method. The proposed approaches are finally applied to a classic HCV problem with waypoints and no-fly zone constraints and compared with other methods reported in literature in detail. The research results validate both the effectiveness and the time saving benefit of the proposed methods.
This article studies the drag‐tracking guidance design problem of entry vehicles with low lift‐to‐drag ratio. Taking issues of uncertainty and input saturation into account, we develop a reduced‐order observer‐based robust output feedback guidance law, making the drag‐tracking error converge near zero. Our study not only achieves robust drag‐tracking guidance against inherently existing uncertainty, but also removes the redundant drag estimation in the literature. The Monte Carlo simulation is done to illustrate the advantage of the developed method.
Achieving a desired terminal approach angle is required by an advanced entry vehicle in missions of landing or defense penetration. An entry guidance method is developed to satisfy the terminal approach angle constraint as well as the conventional constraints. The guidance method consists of a lateral guidance logic based on multiple aiming points and a two-phase trajectory tracking law. The aiming points are selected from a reference ground track and used to determine bank reversals for the lateral guidance logic. With the aid of aiming points, the vehicle can reach the entry terminal in a desired direction. The two-phase trajectory tracking law, corresponding to a two-dimensional and a three-dimensional trajectory, are designed to cooperate with the lateral guidance logic. Numerical results show that the guidance method is capable of flying the vehicle to the terminal zone with various desired approach angles, and meanwhile, robust to typical entry dispersions. The advantage of the guidance strategy is also verified by comparisons.
Full-text available
This study proposes a multiphase convex programming approach for rapid reentry trajectory generation that satisfies path, waypoint and no-fly zone (NFZ) constraints on Common Aerial Vehicles (CAVs). Because the time when the vehicle reaches the waypoint is unknown, the trajectory of the vehicle is divided into several phases according to the prescribed waypoints, rendering a multiphase optimization problem with free final time. Due to the requirement of rapidity, the minimum flight time of each phase index is preferred over other indices in this research. The sequential linearization is used to approximate the nonlinear dynamics of the vehicle as well as the nonlinear concave path constraints on the heat rate, dynamic pressure, and normal load; meanwhile, the convexification techniques are proposed to relax the concave constraints on control variables. Next, the original multiphase optimization problem is reformulated as a standard second-order convex programming problem. Theoretical analysis is conducted to show that the original problem and the converted problem have the same solution. Numerical results are presented to demonstrate that the proposed approach is efficient and effective.
Full-text available
In this paper, an adaptive, disturbance-based sliding-mode controller for hypersonic-entry vehicles is proposed. The scheme is based on high-order sliding-mode theory, and is coupled to an extended sliding-mode observer, able to reconstruct online the disturbances. The result is a numerically stable control scheme, able to adapt online to reduce the error in the presence of multiple uncertainties. The transformation of a high-order sliding-mode technique into an adaptive law by using the extended sliding-mode observer is, together with the multi-input/multi-output formulation for hypersonic-entry vehicles, the main contribution of this paper. The robustness is verified with respect to perturbations in terms of initial conditions, atmospheric density variations, as well as mass and aerodynamic uncertainties. Results show that the approach is valid, leading to an accurate disturbance reconstruction, to a better transient, and to good tracking performance, improved of about 50% in terms of altitude and range errors with respect to the corresponding standard sliding-mode-control approach.
Full-text available
This paper presents a novel three-dimensional autonomous entry guidance for relatively high lift-to-drag ratio vehicles satisfying geographic constraints and other path constraints. The guidance is composed of onboard trajectory planning and robust trajectory tracking. For trajectory planning, a longitudinal sub-planner is introduced to generate a feasible drag-versus-energy profile by using the interpolation between upper boundary and lower boundary of entry corridor to get the desired trajectory length. The associated magnitude of the bank angle can be specified by drag profile, while the sign of bank angle is determined by lateral sub-planner. Two-reverse mode is utilized to satisfy waypoint constraints and dynamic heading error corridor is utilized to satisfy no-fly zone constraints. The longitudinal and lateral sub-planners are iteratively employed until all of the path constraints are satisfied. For trajectory tracking, a novel tracking law based on the active disturbance rejection control is introduced. Finally, adaptability tests and Monte Carlo simulations of the entry guidance approach are performed. Results show that the proposed entry guidance approach can adapt to different entry missions and is able to make the vehicle reach the prescribed target point precisely in spite of geographic constraints.
In this paper, a model predictive path integral control algorithm based on a generalized importance sampling scheme is developed and parallel optimization via sampling is performed using a graphics processing unit. The proposed generalized importance sampling scheme allows for changes in the drift and diffusion terms of stochastic diffusion processes and plays a significant role in the performance of the model predictive control algorithm. The proposed algorithm is compared in simulation with a model predictive control version of differential dynamic programming on nonlinear systems. Finally, the proposed algorithm is applied on multiple vehicles for the task of navigating through a cluttered environment. The current simulations illustrate the efficiency and robustness of the proposed approach and demonstrate the advantages of computational frameworks that incorporate concepts from statistical physics, control theory, and parallelization against more traditional approaches of optimal control theory.
While early applications of model predictive control (MPC) were restricted only to industrial processes, where the system dynamics were so slow that long computation times could be tolerated, recent progress in microprocessor technology, as well as advances in the available solution tools have motivated applications of MPC for fast dynamic systems, such as aerospace platforms, including aircrafts, unmanned aerial vehicles and spacecrafts. Motivated by recent aerospace applications, this paper presents an overview of MPC applications for aerospace systems, and analyses trends of current and future perspectives. The basic formulation of MPC for linear and nonlinear systems is followed by a survey of solution methods and tools which are most commonly used in aerospace applications. Furthermore, this paper gives a summary of available mathematical models of aerospace vehicles; discusses the reliability and verifiability aspects of MPC; and presents the challenges and possibilities in real-time implementation. In addition, several problems related to robustness, fault tolerance and systems with time delays are covered. Potential future directions of MPC are also discussed and some possible contributions to aerospace systems are also proposed.
No-fly zones are geographic constraints for the atmospheric entry flight. Two lateral guidance methods are presented for the entry flight constrained by multiple no-fly zones. The first method employs a dynamic heading corridor to control the vehicle's velocity heading angle. The corridor is generated based on a chain mode strategy that takes heading limits for all the no-fly zones and the target into account. The dynamic corridor is adaptively updated during the flight according to the vehicle's state and the actual constraints. The second method is designed based on a waypoint vector which is located between each pair of no-fly zones. This waypoint vector contains both the location and the direction constraints for the flight trajectory. The vehicle is capable of flying the avoidance trajectory by passing through one or more expected waypoints in specified directions. The two guidance methods are finally tested in four entry missions constrained by various no-fly zones. Results indicate that these methods are effective for no-fly zone avoidance missions in both the nominal and dispersed cases.
The process, methodology, and results of a two-year effort are presented in this paper on verification of an advanced entry guidance algorithm, called Fully Numerical Predictor-corrector Entry Guidance (FNPEG). FNPEG is a model-based numerical guidance algorithm capable of performing both direct (orbital or sub-orbital) entry and skip entry missions. Few vehicle-dependent adjustments are necessary, and no reference trajectory or mission-dependent planning is required. The algorithm is applicable to a wide range of vehicles with different lift-to-drag ratios, and includes state-of-the-art capability to effectively control g-load and damp out phugoid oscillations, without adversely affecting the guidance precision. FNPEG has undergone extensive testing and evaluation in the high-fidelity simulation environment for the Orion spacecraft at NASA Johnson Space Center. In this paper the verification methodology and process are described. The metrics for verification are defined. Extensive testing and simulation results on FNPEG and the comparison with the primary entry guidance algorithm for Orion, PredGuid, are provided. The outcome of this effort has clearly demonstrated the capability, strong robustness, and excellent performance of FNPEG, even in the presence of dispersions and uncertainties significantly higher than the design level.
A guidance algorithm for the waypoint constrained atmospheric entry is presented. To guarantee that the vehicle is able to reach all the waypoints and the final target accurately, the flyby direction constraint for each waypoint is investigated. The controllable and reachable sets for the vehicle's velocity heading angle are defined and calculated. The expected heading angle is obtained from these sets and used as a direction constraint for the corresponding waypoint. Under the location and direction constraints, a bank reversal strategy based on the trajectory prediction is developed. With this strategy, a lateral trajectory that satisfies the waypoint constraint is generated online. Tracking laws for the longitudinal and lateral trajectories are designed. Finally, the guidance algorithm is tested on the Common Aero Vehicle model in highly constrained flights. Results show that the conventional path constraint, the terminal constraint and the additional waypoint constraint are all well satisfied, which indicates the effectiveness of the proposed guidance algorithm.
This study proposes an on-line predictor-corrector reentry guidance algorithm that satisfies path and no-fly zone constraints for hypersonic vehicles with a high lift-to-drag ratio. The proposed guidance algorithm can generate a feasible trajectory at each guidance cycle during the entry flight. In the longitudinal profile, numerical predictor-corrector approaches are used to predict the flight capability from current flight states to expected terminal states and to generate an on-line reference drag acceleration profile. The path constraints on heat rate, aerodynamic load, and dynamic pressure are implemented as a part of the predictor-corrector algorithm. A tracking control law is then designed to track the reference drag acceleration profile. In the lateral profile, a novel guidance algorithm is presented. The velocity azimuth angle error threshold and artificial potential field method are used to reduce heading error and to avoid the no-fly zone. Simulated results for nominal and dispersed cases show that the proposed guidance algorithm not only can avoid the no-fly zone but can also steer a typical entry vehicle along a feasible 3D trajectory that satisfies both terminal and path constraints.
This paper describes the design of the entry guidance for the Space Shuttle Orbiter. This guidance provides the steering commands for trajectory control from initial penetration of the Earth's atmosphere until the terminal area guidance is activated at an Earth-relative speed of 2500 fps. At this point, the Orbiter is at a distance of about 50 nmi form the runway threshold, and at an altitude of about 80,000 ft. The entry guidance design is based on an analytic solution of the equations of motion defining the drag acceleration profile that meets the terminal criteria of the entry flight while maintaining the flight within systems and operational constraints. Guidance commands, which are based on a control law that ensures damping of oscillatory type trajectory motion, are computed to steer the Orbiter to this drag acceleration profile.