aerospace

Article

Stealth-Maneuver Generation for Non-Stealth Aircraft:

A Control Barrier Function Approach
Mustafa Demir 1,2, * , Ege C. Altunkaya 1 , Akın Çatak 1 , Fatih Erol 1 , Emre Koyuncu 1 , İbrahim Özkol 1
and Uğur Zengin 2

1 Aerospace Research Center, Istanbul Technical University, Istanbul 34469, Türkiye;

altunkaya16@itu.edu.tr (E.C.A.); catak15@itu.edu.tr (A.Ç.); erolfa@itu.edu.tr (F.E.);
emre.koyuncu@itu.edu.tr (E.K.); ozkol@itu.edu.tr (İ.Ö.)
2 Turkish Aerospace, Ankara 06980, Türkiye; ugur.zengin@tai.com.tr
* Correspondence: demirmust@itu.edu.tr

Abstract: Aiming to address the vulnerability of non-stealth aircraft to radar detection

due to inherent design limitations, this paper proposes a method to generate maneuvers
that reduce an aircraft’s radar cross-section (RCS) value below a specified threshold. The
proposed method employs control barrier functions and leverages the relationship between
control inputs and the RCS. Due to confidentiality concerns, the required RCS database for
the F-16 aircraft was generated through analyses performed using the created geometry.
The results are compared with a virtual path that excludes RCS constraints and does not
alter the aircraft’s attitude. Simulations reveal that 89.6% of the cases using the proposed
method achieve a mean RCS value below the threshold, compared to only 1.26% for the
virtual path. Moreover, the ratio of the time during which the RCS constraint is successfully
met to the total simulation time averages over 78% across all simulations, demonstrating
the method’s effectiveness in reducing the RCS value below the specified threshold.

Keywords: low-observability; radar cross-section; survivability enhancement; stealth

motion planning

Academic Editor: Konstantinos

Kontis 1. Introduction
Received: 17 March 2025 In modern aerial warfare, stealth technology has become essential for maintaining a
Revised: 25 April 2025
tactical edge by allowing aircraft to evade sophisticated radar systems and operate effec-
Accepted: 13 May 2025
Published: 28 May 2025
tively in heavily defended airspace [1]. At the core of stealth capabilities lies the reduction
in radar cross-section (RCS), a critical factor determining radar detectability. By minimizing
Citation: Demir, M.; Altunkaya, E.C.;
Çatak, A.; Erol, F.; Koyuncu, E.; Özkol,
RCS, stealth technology addresses the growing challenge of radar detection, as evidenced
İ.; Zengin, U. Stealth-Maneuver by platforms like the F-117 and B-2. These aircraft show the substantial advantages of
Generation for Non-Stealth Aircraft: inherent stealth features, particularly in reducing the effectiveness of integrated air defense
A Control Barrier Function Approach. systems (IADSs) [1,2]. However, achieving stealth is not solely reliant on structural or mate-
Aerospace 2025, 12, 478. https:// rial technologies. Dynamic trajectory planning that minimizes RCS in real time also offers
doi.org/10.3390/aerospace12060478
a viable approach to achieving stealth capabilities, especially for non-stealth platforms
Copyright: © 2025 by the authors. operating in contested environments.
Licensee MDPI, Basel, Switzerland.
This article is an open access article 1.1. Related Works
distributed under the terms and
Historically, stealth capabilities have been achieved through aero-structural design,
conditions of the Creative Commons
Attribution (CC BY) license
material technologies, and signature management tactics [3]. For example, radar absorbent
(https://creativecommons.org/ materials (RAMs) and shaping techniques reduce the reflected radar energy [1]. While
licenses/by/4.0/). highly effective for purpose-built stealth platforms, these methods are not applicable to

Aerospace 2025, 12, 478 https://doi.org/10.3390/aerospace12060478

Aerospace 2025, 12, 478 2 of 32

legacy or non-stealth aircraft. Additionally, such structural approaches cannot adapt to

changing operational scenarios or threat dynamics. Another approach involves low-altitude
flight to exploit terrain masking, effectively leveraging the Earth’s curvature and natural
obstructions to avoid radar detection [2,4]. However, low-altitude flight carries its own
risks, including increased vulnerability to ground-based threats and terrain collision haz-
ards. Moreover, this tactic is not always feasible in highly urbanized or geographically
complex theaters of operation. In addition, many aircraft in active service, including fighter
jets and transport aircraft, lack these inherent capabilities. For these platforms, operational
survivability hinges on the development of innovative techniques to dynamically adapt
their trajectories and reduce their radar visibility. Furthermore, the growing sophistication
of radar systems has further intensified the need for advanced countermeasures. Modern
radar systems employ multi-band detection, adaptive tracking algorithms, and data fu-
sion across sensor networks to overcome traditional stealth methods [2,5]. This evolution
presents a formidable challenge: how can non-stealth aircraft effectively mitigate their
radar signature while maintaining mission effectiveness and flight safety? These limitations
highlight the need for dynamic and adaptive stealth methodologies that can be deployed
on existing non-stealth platforms. The emergence of computationally driven solutions, par-
ticularly in trajectory optimization and control systems, has paved the way for innovative
stealth strategies.
In the existing literature, dynamic trajectory optimization has emerged as a powerful
tool for minimizing radar exposure by exploiting the variability in radar detection parame-
ters. This strategy combines intelligent control with optimized route planning, enabling
aircraft to adapt in real time to radar threats [2]. Practical motion planning methods for
stealth aircraft extend these capabilities by focusing on high-risk maneuvers, such as precise
turning and altitude adjustments, to reduce radar visibility [6]. Techniques like modified
A-Star and sparse A-Star algorithms minimize exposure by employing bidirectional sector
expansions and variable step sizes, respectively. These methods allow unmanned aerial
vehicles (UAVs) and stealth aircraft to navigate complex radar fields while maintaining
stability and survivability by dynamically adjusting orientation to exploit low-detectability
angles and optimize distance from threat radars [7,8]. Multi-phase control models are
central to trajectory optimization in stealth aircraft, providing a structured framework to
address multi-objective challenges such as minimizing radar detection, fuel consumption,
and flight time.
To further enhance low observability (LO), advanced models such as multi-phase
control schemes balance competing objectives—minimizing radar detection, fuel consump-
tion, and flight time. Hybrid heuristic and adaptive pseudo-spectral methods have been
developed to jointly model radar signal characteristics and flight dynamics, optimizing
the trajectory in discrete phases. These techniques are particularly valuable in high-threat
environments where brief radar exposures can critically affect mission success. Machine
learning models also contribute to this objective by optimizing evasive maneuvers, enhanc-
ing agility while maintaining stealth [9]. An integrated approach to radar cross-section
(RCS) reduction merges static LO technologies with adaptive motion planning, enabling
dynamic responses to evolving radar threats in real time [1,10]. By combining LO features
with advanced motion planning, aircraft can maintain minimal detectability while navi-
gating complex and evolving threat landscapes [2,11]. Algorithms derived from robotic
motion planning, such as potential field methods, further optimize stealth by coordinating
vehicle orientation and trajectory to avoid detection [11–13]. These approaches integrate
guidance and control systems, allowing the aircraft to adaptively reduce RCS while navigat-
ing through contested and radar-monitored airspace, thereby supporting mission success
under dynamic threat conditions.
Aerospace 2025, 12, 478 3 of 32

By integrating RCS dynamics with trajectory planning, multi-phase control provides

an effective way to evade radar detection without compromising on other mission-critical
requirements [14,15]. Radar detection depends not only on distance but also on the relative
orientation, elevation, and RCS profile of the target [5,14]. For instance, a target’s aspect
angle can significantly influence its detectability, with certain orientations producing much
lower RCS values than others [11,16]. By dynamically adjusting an aircraft’s trajectory to
maintain low-RCS orientations relative to radar threats, it is possible to reduce detectability
without the need for structural modifications. Recent advancements in radar modeling
have further enhanced the potential of trajectory optimization. High-fidelity radar models
now account for complex factors such as terrain masking, radar multipath effects, and
environmental attenuation [4,9]. These models enable the development of more accurate
and effective algorithms for stealth maneuver generation. While trajectory optimization
provides a theoretical foundation, practical implementation requires real-time adaptability.
In contested environments, radar threats are dynamic, with detection systems frequently
shifting positions, scanning patterns, and operational modes [6,15].
Furthermore, dynamic radar environments demand adaptive path planning for sus-
tained stealth. Adaptive algorithms incorporate factors such as radar position, angle, and
power into UAV path planning to enhance survivability [15]. Additionally, machine learn-
ing methods like reinforcement learning further optimize real-time trajectory adjustments,
allowing UAVs to adapt flight paths effectively in response to radar threats [6]. These
algorithms provide a significant advantage in hostile environments where radar detection
variables constantly shift. Accurate RCS modeling is fundamental for stealth planning,
allowing for more precise avoidance of radar detection. Machine learning techniques offer
robust predictions by factoring in radar wave properties and incident angles, reducing
uncertainties in RCS values and improving flight path reliability [16]. These predictive
models enable aircraft to navigate radar fields with greater confidence, optimizing path
planning and evasion strategies.
The reviewed literature underscores the complexity and necessity of integrating multi-
ple technologies and methodologies to achieve effective RCS reduction for stealth aircraft.
While LO technologies such as RAM and aircraft shaping play foundational roles in reduc-
ing radar signatures, modern stealth requirements demand adaptive control and real-time
path adjustments. By combining machine learning, multi-phase trajectory planning, and
intelligent control algorithms, current research provides a comprehensive framework for
minimizing radar observability. This multi-dimensional approach enables stealth aircraft
to achieve higher survivability in hostile environments, adapting in real time to radar
detection threats and optimizing flight paths for minimal RCS.
The existing literature on stealth trajectory or maneuver optimization often employs
simplified or reduced-order models that fail to capture the intricate dynamics necessary for
precise trajectory or maneuver generation. Many studies rely on basic kinematic models
that do not account for detailed aerodynamic effects, thus limiting their applicability
to real-world scenarios [4,7,8,11,15]. By focusing solely on trajectory geometry, these
approaches fall short of addressing the critical need for maneuver adaptability in highly
dynamic and operationally complex airspace [5,12]. A significant limitation of many
studies is their reliance on 2D or pseudo-3D trajectory models, which fail to exploit the
full spatial dimensions required for effective stealth maneuvers [1,9,14]. Moreover, the
fidelity of RCS database in the studies often falls short of the requirements for accurate
stealth maneuver generation. Many works employ static or simplified RCS models, relying
on approximations that fail to capture the real-time variability of RCS with respect to
aircraft orientation and radar angle [7,8,15]. Although some studies incorporate a relatively
high-fidelity RCS database, they often do so without validating these models against
Aerospace 2025, 12, 478 4 of 32

experimental or high-fidelity databases [4,16] and they often limit the planning domain
to fixed altitudes or simplified geometries, thereby reducing their relevance in complex
operational scenarios [6]. Consequently, these studies do not provide sufficient fidelity for
generating realistic penetration trajectories, particularly for non-stealth platforms operating
in variable threat landscapes [2,13]. The absence of a comprehensive integration of RCS
dynamics with motion planning further weakens the practicality of these approaches, as
they cannot adapt to radar threats that depend on rapid changes in aircraft orientation [9,12].
Another point of concern, the control surface deflections, a critical factor in achieving precise
stealth maneuvers, is neglected in most studies [1,6,13,14]. Without explicit consideration
of control surface deflections, these methods cannot generate stealthy trajectories that are
both realistic and executable under operational constraints. Finally, previous studies have
predominantly focused on precomputed path planning rather than real-time maneuver
generation, which is a critical limitation in dynamic radar environments. Algorithms like
A-Star and sparse A-Star offer optimal solutions for static or semi-dynamic scenarios but
lack the adaptability needed for continuously changing radar threats [7,8,15]. By focusing
primarily on static optimization, these methods fail to address the need for continuous
trajectory adjustment, a cornerstone of effective stealth operations in modern contested
airspace [9,12].

1.2. Contributions and Organization

This study aims to develop a comprehensive framework that overcomes the limitations
of existing stealth maneuver generation methods by employing Control Barrier Functions
(CBFs) to dynamically enforce radar observability constraints. This framework bridges the
gap between theoretical advancements and practical implementation. Ultimately, this work
sets a foundation for adaptive stealth strategies that enhance the operational effectiveness of
non-stealth aircraft in contested environments. The contributions of this study are itemized;
• This study introduces a CBF-pilot design to generate stealthy maneuvers based on a
high-fidelity flight dynamics model that captures the complex behavior of non-stealth
platforms, contrary to most of the existing studies using simplified kinematic flight
dynamics model. The utilization of a high-fidelity flight dynamics model provides
an accurate representation of flight dynamics, allowing for better assessment of radar
observability under various and realistic operational conditions.
• By incorporating the effects of control surface deflections on RCS, the study ensures
that these factors are properly accounted for in stealth motion planning. This integra-
tion enhances the realism of the model and improves the ability to generate effective
stealthy maneuvers.
• The framework adapts in real time, dynamically adjusting flight maneuvers to main-
tain stealth characteristics. This real-time adaptability ensures that non-stealth plat-
forms can continuously optimize their flight paths to minimize radar detectability
while meeting operational constraints.
The remainder of the paper is organized as follows. First, Section 2 explains the
problem, then providing the necessary background for the study including modeling of
nonlinear flight dynamics, design of flight control laws. In Section 3, the radar cross-section
analysis methodology is presented. Section 4 presents the design of the stealth maneuver
generator using control barrier functions. In Section 5, the proposed strategy is evaluated
through various scenarios and Monte Carlo simulations. Finally, the results and potential
future research directions are discussed in Section 6.
Aerospace 2025, 12, 478 5 of 32

2. Problem Description and Preliminaries

Radar-penetration maneuvers are operationally critical but inherently risky for non-
stealth platforms due to their elevated radar visibility. This limitation poses a direct threat to
survivability in contested airspace. Addressing this challenge requires deep understanding
of how radar cross-section characteristics interact with aircraft maneuvering capabilities—a
central focus of this study. The key research question is: how can non-stealth aircraft execute
tactical maneuvers that minimize their detectability by radar systems while simultaneously
enhancing their survivability? Answering this involves exploring ways to temporarily
transform such platforms into low-observable assets during mission-critical phases. Thus,
the illustration in Figure 1 principally depicts the radar-penetration maneuver scenario as
the foundation of the proposed method.

Figure 1. The illustration of a radar-penetration maneuver scenario, emphasizing the virtual (or ideal)
path and alternative RCS-constrained maneuvers.

The scenario illustration demonstrates that the aircraft must pass through the radar
coverage zone, assuming that the radar location is pre-known, which is a reasonable
assumption consistent with the existing literature [4,6–9]. In the absence of radar, the
aircraft would have followed a straight flight path to pass over the zone. Consequently,
this straight flight trajectory is considered the virtual (or ideal) path, meaning that the
angular rates remain zero throughout the path, i.e., ωref = 03 . However, the presence of
radar necessitates reshaping the virtual path to account for the aircraft’s radar cross-section.
The problem can thus be formulated as an optimization problem for motion planning,
aiming to adhere to radar cross-section constraints while staying as close as possible to the
virtual path.
The methodology underlying this approach is grounded in the use of an RCS database,
belonging to a non-stealth aircraft. Building upon this foundation, the study leverages
control barrier functions to enforce constraints on RCS values during maneuver execution
by commanding angular rates, ωcmd .

2.1. Notations
Throughout this study, the time derivative of a continuously differentiable function
f : Rn → R is represented as f˙. Vectors are indicated using bold notation, i.e., v, and the
cross product of two vectors x and y is denoted by x × y. The notations s(∗), c(∗), and t(∗)
correspond to the sine, cosine, and tangent functions of (∗), respectively. A control affine
system is described as

ẋ = f ( x) + g ( x)u (1)

where x ∈ Rn is the state vector and u ∈ Rm is the control input vector. Nonlinear mappings
of f : Rn −
→ Rn and g : Rn − → Rn×m are locally Lipschitz continuous functions. Finally, for
∞
a C function Q : R −n → R and g : Rn× p −→ Rn , the Lie derivative is denoted by
Aerospace 2025, 12, 478 6 of 32

∂Q
L g Q( x) = ( x) g( x) (2)
∂x

2.2. Control Barrier Functions

The RCS-constrained motion planning algorithm is formulated by leveraging control
barrier functions. Primarily, using the safe set definition from [17], the zero-superlevel set
C is composed of

C ={ x ∈ D ⊂ Rn : h( x) ≥ 0}
∂C ={ x ∈ D ⊂ Rn : h( x) = 0} (3)
n
Int(C) ={ x ∈ D ⊂ R : h( x) > 0}

where C is safe set, ∂C is the boundary, and Int(C) is the interior of the safe set. In addition,
h( x) ≥ 0 defines the safe region while h( x) < 0 defines the unsafe region.

Definition 1 ([17,18]). Function h : D ⊂ Rn −

→ R is the control barrier function (CBF) if the
following conditions hold:
• A zero-superlevel set C exists for the function h( x).
• h( x) satisfies the inequality

sup{L f h( x) + L g h( x)u + α(h( x)) ≥ 0} (4)

u ∈U

where class κ∞ function α(h( x)) for a dynamic system described in Equation (1). If such safe
set C exists, then the control set ensuring the safety for ∀ x ∈ D can be given as
n o
Kcb f := u ∈ U : L f h( x) + L g h( x)u + α(h( x)) ≥ 0 (5)

2.3. Flight Dynamics Model

The baseline aircraft is an F-16, whose nonlinear flight dynamics model is adopted
from [19]. The major components of the flight dynamics model are detailed in the subse-
quent sections.

2.3.1. Equations of Motion

The axes frame of the baseline aircraft is depicted in Figure 2.

Figure 2. An illustration of the baseline aircraft with its body axis and wind axis frames.
Aerospace 2025, 12, 478 7 of 32

The nonlinear flight dynamics equations, including translational and rotational dy-
namics, and translational and rotational kinematics are, respectively,

u̇ = ∑ FX /m + rv − qw


v̇ = ∑ FY /m + pw − ru

ẇ = ∑ FZ /m + qu − pv



 ṗ = qr ( Iyy − Izz )/Ixx + (ṙ + pq) Ixz /Ixx + ∑ L/Ixx


q̇ = pr ( Izz − Ixx )/Iyy + (r2 − p2 ) Ixz /Iyy + ∑ M/Iyy

ṙ = pq( Ixx − Iyy )/Izz + ( ṗ − qr ) Ixz /Izz + ∑ N/Izz


 (6)
 ẋE = ucθcψ + v(sϕsθcψ − cϕsψ) + w(cϕsθcψ + sϕsψ)


ẏE = ucθcψ + v(sϕsθcψ + cϕcψ) + w(cϕsθsψ − sϕsψ)


żE = −usθ + vsϕcθ + wcϕcθ


ϕ̇ = p + tθ (qsϕ + rcϕ)


θ̇ = qcϕ − rsϕ


ψ̇ = (qsϕ + rcϕ)/cθ


where u, v, and w denote the components of the body velocity, while p, q, and r represent
the angular rate components in the body frame. The navigational position is specified
by xE , yE , and zE , and the orientation is described by the Euler angles ϕ, θ, and ψ. The
force components acting on the body frame are given as FX , FY , and FZ , with L, M, and N
representing the roll, pitch, and yaw moments, respectively. Additionally, m signifies the
mass of the aircraft, while Ixx , Iyy , Izz , and Ixz define the aircraft’s moments of inertia.

2.3.2. Aerodynamics and Actuators

The aerodynamic database and the corresponding formulations are directly adopted
from [19]. There are three body force coefficients, i.e., CX , CY , and CZ , and three moment
coefficients, i.e., Cl , Cm , and Cn . The aerodynamic coefficients depend on relevant flight
states, such as the angle of attack (α), sideslip angle (β), and control surface deflections,
including horizontal tail deflection (δHT ), aileron deflection (δA ), and rudder deflection
(δR ). The non-dimensional aerodynamic coefficients are converted to dimensional force
and moment expressions using the dynamic pressure and relevant geometric properties,
as follows: FX = q̄∞ SCX , FY = q̄∞ SCY , FZ = q̄∞ SCZ , L = q̄∞ SbCl , M = q̄∞ Sc̄Cm , and
N = q̄∞ SbCn .
Additionally, the actuator dynamics are modeled as a first-order system that accounts
for time constants, rate limitations, and position saturation constraints, as outlined in [19].
Specifically, the time constant for each control surface is 0.0495 s. The rate saturation limits
are 60◦ /s for the horizontal tails, 80◦ /s for the ailerons, and 120◦ /s for the rudder. Likewise,
the position saturation limits are ±25◦ for the horizontal tails, ±21.5◦ for the ailerons, and
±30◦ for the rudder [19].

2.4. Flight Control Law Design

The control augmentation system employs a single-loop angular rate control law based
on incremental nonlinear dynamic inversion (INDI). The derivation of this control law is
simplified by the control-affine structure of Euler’s equations of motion, as expressed in a
decomposed form
Aerospace 2025, 12, 478 8 of 32

 
b
ω̇ = − J −1 (ω × Jω) + J −1 q̄∞ S c̄ Φ |{z} (7)
 
δ
b u
| {z }
g ( x)

where q̄∞ , S, b, and c̄ represent the dynamic pressure, wing area, wing span, and mean
aerodynamic chord, respectively. Additionally, Φ ∈ R3×n denotes the control effectivity
matrix, which contains the moment coefficient derivatives with respect to the control
surface deflections at the current instant, with n indicating the number of control surfaces.
The INDI control law for regulating the angular rates is derived as

u = g ( x0 )−1 [ω̇c − ω̇0 ] + u0 (8)

where the subscript ‘0’ denotes the current state and ω̇c ∈ R3 represents the virtual input
to be designed. The final form of the control law is provided as
 
( b ) −1
−1
δ = J q̄∞ S c̄ Φ [ω̇c − ω̇0 ] + δ0 (9)
 
b

where δ ∈ R3 denotes the control surface deflections, corresponding to the horizontal

tail, aileron, and rudder, respectively. Additionally, since there are three control surfaces
(n = 3), the control effectivity matrix is a square matrix, and it remains invertible unless it
is rank-deficient. Furthermore, the virtual input ω̇c is given by
  
Kp pcmd − p
ω̇c =  Kq  qcmd − q  (10)
  
Kr rcmd − r

where K p , Kq , and Kr are the gains for the roll, pitch, and yaw channels, respectively. The
provided expressions outline the necessary generation of control surface deflections in
response to the pilot commands pcmd , qcmd , and rcmd .

3. Radar Cross-Section Quantification

The radar cross-section (RCS) measures a target’s ability to reflect radar signals back
toward the radar receiver. RCS compares the strength of the signal reflected by a target to
that of a perfectly smooth sphere with a cross-sectional area of 1 m2 . With RCS denoted as
σ, the value of a target is defined as

Ss
σ = lim 4πr2 (11)
r →∞ Si

where Si is the incident power density measured at the target, Ss is the scattered power
density seen at a distance r away from the target, r is the distance from target. Additionally,
RCS is often expressed on a logarithmic scale for clarity and practicality, defined as

RCS (dBsm) = 10 log10 (σ ) (12)

where σ represents the RCS measured in square meters (m2 ), while dBsm provides a
logarithmic representation of the RCS in decibels.

3.1. Methodology
The RCS database for a specific aircraft may not be available due to confidentiality
issues. Therefore, developing an approach regarding the observability during a radar
Aerospace 2025, 12, 478 9 of 32

penetration maneuver primarily requires generating a sufficiently accurate representation

of the RCS characteristics through detailed analyzes. The reliability of these analyzes
depends on several factors, such as the accuracy of the modeled aircraft geometry, the
quality of the mesh, and the solver type. It is important to note that even if the geometry,
including mesh options, and analysis parameters are chosen appropriately, the generated
RCS characteristics may still differ from the exact profile. This discrepancy can be attributed
to factors such as incomplete knowledge of the materials used in the aircraft, surface
treatments, and the effects of coatings [1,3]. Yet, in this study, the primary goal of generating
the RCS database is to achieve a close approximation of the real RCS profile, enabling the
demonstration of the proposed maneuver generation approach’s capabilities. For this
purpose, primarily the digital F-16 geometry with movable control surfaces was modeled
in Blender® v4.1, a computer-aided design software, and then it was meshed using a total
of 20,000 elements. The digital geometry and the generated mesh are demonstrated in
Figure 3.

Figure 3. Digital F-16 geometry and mesh.

The RCS analyzes were performed in ANSYS® v2021 R2 using the Shooting and
Bouncing Rays (SBR) method at 3 GHz (S-band radar) [20] with the generated mesh. The
S-band radar (3 GHz) represents a balance: high enough to capture detailed scattering be-
havior but still within reach of affordable GPU-accelerated platforms. It enables validation
of stealth characteristics against typical radar threats without requiring supercomputing
resources. RCS analyzes are characterized by the azimuth and elevation angles. These
angles represent the relative orientation of the aircraft with respect to the radar position, as
depicted in Figure 4.

Figure 4. The azimuth (Φ) and elevation (Θ) representation: the relative orientation of the aircraft
with respect to the radar position.

By definition, the range of the Φ angle is [−180◦ , 180◦ ], whereas the range of the Θ
angle is [0◦ , 180◦ ]. Therefore, a comprehensive RCS analysis should cover all combinations
within these ranges. Furthermore, however, analyzing only the possible combinations of Θ
and Φ angles is insufficient to construct a high-fidelity RCS database, as the control surfaces
are movable. Thereby, the deflection in the control surfaces changes the spatial geometry
and so do the RCS characteristics. To address the concern regarding the fidelity of the
constructed database, the geometry with control surface deflections must also be modeled,
Aerospace 2025, 12, 478 10 of 32

and the incremental addition of the control surface deflections should be included in the
RCS database. Consequently, the RCS analysis points should be defined as follows: Φ
within the range [−180◦ , 180◦ ], Θ within the range [0◦ , 180◦ ], and the aileron (δA ), horizontal
tail (δHT ), and rudder deflections (δR ) within the range [−30◦ , 30◦ ] The elevation and
azimuth angles are discretized in 15◦ increments, whereas the control surface deflections
are discretized into maximum, minimum, and neutral positions.

3.2. F-16 Radar Cross-Section Characteristics

Building on the previously discussed analysis methodology, the variation in the RCS
characteristics is illustrated in Figure 5 by keeping either the elevation or azimuth constant
at values of 90◦ and 0◦ , respectively.

(a) (b)
Figure 5. F-16 RCS characteristics with the elevation, Θ, and azimuth, Φ: the RCS profiles indicate
that the peak values are observed in the azimuth angles of −90◦ and 90◦ , while the elevation is
constant at 90◦ . Additionally, the peaks are trackable at the elevation angles of 0◦ and 180◦ , while
the azimuth is constant at 0◦ . The RCS characteristics and values at different orientations of F-16
clearly exhibit the toughness of generating stealthy maneuvers. The computational cost of using a
high-resolution angular increment (i.e., 0.1◦ ) increases dramatically; therefore, a relatively coarse
resolution (i.e., 15◦ ), which still captures the RCS characteristics appropriately, is utilized throughout
the study. (a) RCS variation while the elevation, Θ, remains constant at 90◦ ; (b) RCS variation while
the azimuth, Φ, remains constant at 0◦ .

The polar representation of the corresponding RCS variation is depicted in Figure 6

for the sake of creating visual intuition.

(a) (b)
Figure 6. F-16 RCS characteristics in polar map. (a) RCS variation while the elevation, Θ, remains
constant at 90◦ ; (b) RCS variation while the azimuth, Φ, remains constant at 0◦ .

The effects of the horizontal tail deflection on the RCS characteristics are depicted in
Figure 7 comparatively.
Aerospace 2025, 12, 478 11 of 32

(a) (b)
Figure 7. F-16 RCS variation with the horizontal tail deflection. (a) Positive horizontal tail deflection;
(b) Negative horizontal tail deflection.

The results indicate that while there is minimal variation in RCS values when the
aircraft is tracked from the frontal aspect of the radar, a noticeable increase occurs when
the radar rays encounter the horizontal tail at a perpendicular angle. Specifically, for a 25◦
horizontal tail angle, the RCS value increases by approximately 10 dBsm at an elevation
angle of 155◦ . Similarly, for a −25◦ horizontal tail angle, the RCS value rises by about
15 dBsm at an elevation angle of 205◦ . As an adjunct instance, the rudder deflection effects
on the RCS characteristics are illustrated in Figure 8.

Figure 8. F-16 RCS variation with the rudder deflection.

Subsequently, the generated RCS database is embedded in a 5D look-up table, where

each dimension corresponds to Θ, Φ, δA , δHT , and δR , with linear interpolation.
As the final step, the conversion between the aircraft orientation (ϕ, θ, ψ) and the
relative orientation (Φ, Θ) should be introduced since the RCS database is a function of the
relative orientation. Assuming the radar is always facing the aircraft, a conversion method
can be developed by using the positions of the F-16 and the radar, as well as the attitude
angles of the F-16. First, the distances between the radar and the F-16 are calculated and
converted to the body frame of the aircraft, defined as

Pb = Rib (Pa − Pr ) (13)

where Pa is the position of the aircraft, Pr is the position of the radar, and Rib represents
the direction cosine matrix from the body frame of the F-16 to the inertial frame with ZYX
order rotation and is formulated as
 
cθcψ cθsψ −sθ
Rib = sϕsθcψ − cϕsψ sϕsθsψ + cϕcψ sϕcθ  (14)
 
cϕsθcψ + sϕsψ cϕsθsψ − sϕcψ cϕcθ
Aerospace 2025, 12, 478 12 of 32

where bank angle ϕ, pitch angle θ, and yaw angle ψ represent the attitude angles. Then, the
Θ and Φ angles are determined by converting from the cartesian to the spherical coordinate
system using Equation (15).

Py
Θ = arctan
P
qx (15)
Px2 + Py2
Φ = arctan
Pz

where Px Py and Pz , describe the components of Pb in Equation (13).

At the end of the section, the methodology for generating the RCS database and its
direct correspondence to the aircraft’s motion are elaborated, providing a foundation for
the subsequent section, which introduces the stealth-maneuver generator design.

4. Stealth-Maneuver Generator
The primary principle of stealth-maneuver generator design is to maintain the aircraft’s
RCS below a predetermined maximum allowable threshold during radar penetration. In
this regard, the stealth-maneuver generator is required to generate pcmd , qcmd , and rcmd ;
therefore, the RCS should be formulated and decomposed in a manner that ensures that
angular rates are observable. Fortunately, the RCS is a function of aircraft attitude angles
(ϕ, θ, and ψ), implying that the angular rates can be revealed provided that an appropriate
barrier function is established. A barrier function is then designed as

h(σ ) = σmax − σ (ϕ, θ, ψ) (16)

where σmax ∈ R is the predetermined maximum allowable RCS value. It is obvious that
h(σ ) > 0, ∀σ ∈ R<σmax , and h(σ ) = 0 ↔ σ = σmax . Thus, the time derivative of the barrier
function is ḣ(σ) = −σ̇. At this point, an expansion of the time derivative of the barrier
function should be given as

∂σ (ϕ, θ, ψ) ∂σ (ϕ, θ, ψ) ∂σ (ϕ, θ, ψ)

σ̇ = ϕ̇ + θ̇ + ψ̇ (17)
∂ϕ ∂θ
| {z } ∂ψ
| {z } | {z }
σϕ ∈R σθ ∈R σψ ∈R

where σϕ , σθ , and σψ represent the RCS derivatives with respect to the bank angle, pitch
angle, and yaw angle, respectively. Provided that the RCS database is available, these
partial derivatives can be calculated using either the central difference method—preferred
in this study [21]—or formulating the RCS as a polynomial function.
Subsequently, to derive the angular rates from the RCS dynamics, the bank, pitch,
and yaw dynamics should be expressed as ϕ̇ = f (ϕ) + g(ϕ)ω, θ̇ = f (θ ) + g(θ )ω, and
ψ̇ = f (ψ) + g(ψ)ω, respectively. It is quite straightforward since the attitude dynamics
are represented through a transformation matrix and angular rates, i.e., Ω̇ = Rϕθψ ω. By
recalling the rotational kinematics given in Equation (6), the required decomposition for
the bank, pitch, and yaw dynamics can be performed as
Aerospace 2025, 12, 478 13 of 32

 
h i p
ϕ̇ = |{z}0 + 1 tan θ sin ϕ tan θ cos ϕ  q 
 
f (ϕ) | {z } r
g(ϕ) |{z}
ω
 
h i p
0 + 0 cos ϕ − sin ϕ  q 
θ̇ = |{z}
 
} r (18)
f (θ ) | {z
g(θ ) |{z}
ω
 
  p
sin ϕ cos ϕ  
ψ̇ = |{z}0 + 0 q
cos θ cos θ
f (ψ) | {z } r
g(ψ) |{z}
ω

Attentive eyes will notice that the components of g(ϕ), g(θ ), and g(ψ) are row elements
of the transformation matrix Rϕθψ . Subsequently, the RCS dynamics can be represented in
the form of σ̇ = f (σ ) + g(σ )ω, defined as
 
h i p
σ̇ = |{z}
0 + σϕ g(ϕ) σθ g(θ ) σψ g(ψ)  q  (19)
 
f (σ) | {z } r
g(σ) |{z}
ω

Consequently, the CBF constraint can be described as

− f ( σ ) − g ( σ ) ω + γσ h ( σ ) ≥ 0 (20)
| {z }
L f h(σ)+L g h(σ)ωcmd

where γσ ∈ R>0 is the design parameter to be chosen properly. As a consequence, the

final form of the stealth-maneuver generator formulation for commanding angular rates is
presented as

u⋆ = argmin ||ωref − ωcmd ||22

ωcmd ∈R3
(21)
s.t. − f (σ) − g(σ)ωcmd + γσ h(σ) ≥ 0
ωmin ≤ ωcmd ≤ ωmax

The constructed formulation enables the stealth-maneuver generator to command

angular rates (pcmd , qcmd , and rcmd ) that closely follow the reference angular rates (pref ,
qref , and rref ) corresponding to the virtual path while ensuring compliance with the radar
cross-section constraint. Furthermore, the generated angular rate commands must remain
within the interval [ωmin , ωmax ], considering the admissible and allowable angular rate
limits specific to the aircraft. The overall proposed architecture is depicted in Figure 9.
Aerospace 2025, 12, 478 14 of 32

Figure 9. General framework of the proposed method: (1) Stealth-maneuver generator through
CBF-pilot, (2) CAS including INDI, and (3) A/C dynamics. The reference commands pref , qref , and
rref from virtual path are subjected to RCS constraint. Each reference angular rate signals are adjusted
by CBF constraint, and pcmd , qcmd , and rcmd are generated, if necessary. Otherwise, the reference
angular rate commands are passed through. Note that the demonstrated framework is activated in
autopilot mode only when the radar penetration maneuver is intended to be initiated; otherwise,
the pilot’s commands are directly transferred to the control augmentation system as pcmd , qcmd , and
rcmd . However, since the scope of the study is limited to the generation of stealth radar penetration
maneuver, the autopilot mode takes the control over rather than pilot commands.

5. Simulations and Results

The simulation scenarios and methodologies employed to evaluate the proposed CBF-
based framework for stealth-maneuver generation for non-stealth aircraft are presented.
The simulations are conducted on a standard personal computer equipped with a processor
running at 3.3 GHz. Additionally, the high-fidelity nonlinear flight dynamics model, con-
structed using the MATLAB® v2023b Simulink environment, is utilized for the simulations
running at 100 Hz. Finally, Sequential Quadratic Programming (SQP) is employed to solve
the optimization problems. For the purpose of assessment, two primary scenarios are
designed: a radar-penetration maneuver and a radar-evasive maneuver, each developed
to simulate operationally realistic and dynamically challenging conditions. Additionally,
Monte Carlo simulations are conducted to assess the robustness and consistency of the
proposed approach across a range of initial conditions. This structured evaluation provides
a comprehensive analysis of the proposed methodology.

5.1. Scenario-1: Radar-Penetration Maneuver

This scenario simulates a case where radar penetration is unavoidable and ends when
the aircraft reaches the same position with the radar on the North axis. The objective of
this scenario could be to gather information around the radar base for a reconnaissance
mission or to neutralize targets near the radar position. The cruising aircraft’s initial speed
is 0.8 Mach, with a 0◦ path angle and a 30◦ heading, at an altitude of 2000 m. The radar
position is set at [5000 m, 0 m, 0 m] with a range of 5000 m. The maximum allowable RCS
value is set at 0 dBsm, and the design parameter γσ is set to 1 × 103 . The resulting RCS
history is presented in Figure 10.
The proposed methodology is compared with the virtual path, which has no RCS
constraints and does not change its attitude, as shown in Figure 10. For both results, the
initial RCS values exceed the threshold for a brief period; however, the proposed method
successfully manages to satisfy the constraint through the generated maneuvers. Over
10 s, the F-16 is able to maintain its RCS value below 0 dBsm, with only minor violation
observed afterward. In contrast, the virtual path consistently produces higher RCS values.
For the virtual path, the F-16’s RCS value reaches nearly 25 dBsm, while for the proposed
RCS-constrained maneuver generation approach, it remains close to 0 dBsm. Additionally,
the Φ and Θ angles resulting from this scenario are illustrated. The blue regions represent
Aerospace 2025, 12, 478 15 of 32

the combinations of Φ and Θ where the RCS values are below 0 dBsm, while the red regions
indicate RCS values exceeding 0 dBsm. The black circles denote the angle combinations at
each time step of the simulation. It can be observed that the constraints are quickly satisfied,
and the RCS value remains below 0 dBsm until the end of the scenario, with the angles
staying within the blue region. The fluctuations observed toward the end of the scenario
can be visually explained through Figure 10, as the portrait approaches the border of the
blue region. The rate commands pcmd , qcmd , and rcmd , along with the aircraft’s states, are
shown in Figure 11.

Figure 10. Radar cross-section history for the radar-penetration maneuver: (1) RCS trajectories for
both the virtual path and RCS-constrained maneuver on the left, (2) RCS map with azimuth and
elevation angles, along with the aircraft’s orientation trajectory on the right.

Figure 11. CBF-pilot commands for the radar-penetration maneuver.

Angular rate commands are issued when the RCS value exceeds the threshold. The ratio-
nale behind this behavior is to satisfy the constraint while minimizing the objective function
of the optimization. Since the optimization minimizes the difference between ωre f and ωcmd ,
the result of the optimization yields zero angular rate commands when the constraints are
already satisfied. The resulting attitude angles for this case are shown in Figure 12.
Aerospace 2025, 12, 478 16 of 32

Figure 12. Attitude trajectories of the aircraft during the radar-penetration maneuver.

The aircraft successfully reduces its RCS value by employing a negative roll rate,
effectively orienting its canopy toward the radar. For approximately 13 s, the attitude
remains almost unchanged as the rate commands are zero. In the final phase of this scenario,
the aircraft continuously adjusts its orientation to satisfy the RCS constraint. When the
orientations result in an RCS value smaller than the threshold, the rate commands are zero,
and the attitudes remain constant until the satisfaction of the constraint requires further
angular rate commands. The resulting trajectory of this case is depicted with three planar
views and one isometric view in Figure 13, providing a clearer visual representation of the
generated maneuvers.

Figure 13. Three-dimensional visualization of the radar-penetration maneuver scenario.

Aerospace 2025, 12, 478 17 of 32

The changes in the initial orientation and the constant attitude over approximately
10 s are visualized in Figure 13. The adjustments made to the attitude at the end of the
engagement to satisfy the RCS threshold can also be observed in the resulting trajectory.
Throughout the simulation, the aircraft attempts to orient its canopy toward the radar, with
the dive at the end aimed at maintaining the same Φ angle.

5.2. Scenario-2: Radar-Evasive Maneuver

This scenario simulates a case where a radar-evasive maneuver is essential. The
objective is to avoid radar detection as much as possible by remaining below the predefined
RCS threshold and subsequently exiting the radar coverage zone. The aircraft begins
cruising at a speed of 0.8 Mach, with a path angle and heading both set to 0◦ , at an altitude
of 2000 m. The radar is positioned at [4000 m, 1000 m, 0 m], with a detection range of
4000 m. The maximum allowable RCS value is set at 0 dBsm, and the design parameter γσ
is set to 1 × 103 . The resulting RCS history is presented in Figure 14.

Figure 14. Radar cross-section history for the radar-evasive maneuver: (1) RCS trajectories for both
the virtual path and RCS-constrained maneuver on the left, (2) RCS map with azimuth and elevation
angles, along with the aircraft’s orientation trajectory on the right.

The proposed methodology is compared with the virtual path, which has no RCS
constraints and maintains a constant attitude, as shown in Figure 14. The initial orientation
of the aircraft results in an RCS of approximately 2 dBsm. Subsequently, with agile interven-
tion, the CBF-pilot commands an orientation that reduces the resultant RCS to quite below
0 dBsm, reaching nearly −20 dBsm. For a prolonged period, the RCS remains below 0 dBsm
as the aircraft maneuvers to avoid detection. Upon approaching and passing over the radar,
the RCS exhibits a peak. This phenomenon is also visualized in the RCS map shown on the
right of Figure 14. Obviously, the ability to remain below the threshold is only achievable
through ceaseless contact of the blue regions, which represent the attitude combinations
yielding an RCS below 0 dBsm. Based on this observation, the baseline non-stealth aircraft
is inherently incapable of sustaining a stealth maneuver even though the CBF-pilot is in
charge of keeping stealth maneuver. Fortunately, the detectable period of the maneuver is
relatively short, after which the aircraft is reoriented to maintain stealthy flight. The rate
commands pcmd , qcmd , and rcmd , along with the aircraft’s states, are shown in Figure 15.
Aerospace 2025, 12, 478 18 of 32

Figure 15. CBF-pilot commands for the radar-evasive maneuver.

Again, the angular rate commands are only issued when the RCS value exceeds the
threshold. Thereby, the resulting attitude angles for this case are shown in Figure 16.

Figure 16. Attitude trajectories of the aircraft during the radar-evasive maneuver.

In this case as well, the aircraft successfully and rapidly reduces its RCS by employing
a negative roll rate. For approximately 12 s, the attitude remains almost unchanged as the
rate commands are zero. During the period of the subsequent peak in the RCS history,
the CBF-pilot applies impulsive adjustment commands. Depending on the aircraft’s RCS
characteristics, a significant portion of these adjustments during this period are effective,
while some fail to sustain the stealth maneuver, as its reason has been discussed previously.
Finally, the resulting trajectory for this case is illustrated in Figure 17 with three planar
views and one isometric view.
Aerospace 2025, 12, 478 19 of 32

Figure 17. Three-dimensional visualization of the radar-evasive maneuver scenario.

Seemingly, the aircraft is reoriented to exhibit low-observable performance at a certain

attitude. Subsequently, it maintains this attitude for approximately 12 s prior to encounter-
ing and passing over the radar. Finally, the CBF-pilot reorients the aircraft to keep the RCS
below the threshold, enabling it to exit the radar coverage zone successfully.

5.3. Sensitivity Analysis

In this section, three distinct sensitivity analyses are presented to reveal the maneuver-
generating characteristics under varying radar cross-section profiles and different values of
the control barrier function design parameter, γσ .

5.3.1. Dependency on Arbitrarily Increased Radar Cross-Section Characteristics

To evaluate the sensitivity of the proposed algorithm, the original F-16 radar cross-
section characteristics are arbitrarily deteriorated by increasing the RCS values for random
elevation–azimuth pairs within the range of [5 dBm, 30 dBm]. The simulation case is
identical to Scenario-1 presented in Section 5.1, i.e., the radar-penetration maneuver. The
RCS trajectory corresponding to this simulation is shown in Figure 18.
Aerospace 2025, 12, 478 20 of 32

Figure 18. Radar cross-section history for the radar-penetration maneuver: (1) RCS trajectories for
both the virtual path and RCS-constrained maneuver on the left, (2) RCS map with azimuth and
elevation angles, along with the aircraft’s orientation trajectory on the right.

For the sake of comparison, the RCS map in Figure 18 is significantly more challenging
than the one shown in Figure 10. Nevertheless, the proposed maneuver generation rationale
remains effective in producing stealth maneuvers compared to the virtual path, despite the
more demanding RCS characteristics. The commands of stealth maneuver generator are
shown in Figure 19.

Figure 19. CBF-pilot commands for the radar-penetration maneuver.

The yielding aircraft states are demonstrated in Figure 20.

Aerospace 2025, 12, 478 21 of 32

Figure 20. Attitude trajectories of the aircraft during the radar-penetration maneuver.

Finally, the resulting trajectory of this case is depicted with three planar views and one
isometric view in Figure 21, providing a clearer visual representation of the generated ma-
neuvers.

Figure 21. Three-dimensional visualization of the radar-penetration maneuver scenario.

The adjustments made to the attitude throughout the engagement to satisfy the RCS
threshold can also be observed in the resulting trajectory.
Aerospace 2025, 12, 478 22 of 32

An additional assessment involves duplicating Scenario-2 presented in Section 5.2,

i.e., the radar-evasive maneuver, using the same increased RCS characteristics. The RCS
trajectory corresponding to this simulation is shown in Figure 22.

Figure 22. Radar cross-section history for the radar-evasive maneuver: (1) RCS trajectories for both
the virtual path and RCS-constrained maneuver on the left, (2) RCS map with azimuth and elevation
angles, along with the aircraft’s orientation trajectory on the right.

Seemingly, the proposed maneuver generation rationale remains effective in produc-

ing stealth maneuvers compared to the virtual path, despite the more demanding RCS
characteristics. The commands of stealth maneuver generator are shown in Figure 23.

Figure 23. CBF-pilot commands for the radar-evasive maneuver.

The yielding aircraft states are demonstrated in Figure 24.

Aerospace 2025, 12, 478 23 of 32

Figure 24. Attitude trajectories of the aircraft during the radar-evasive maneuver.

Finally, the resulting trajectory of this case is depicted with three planar views and one
isometric view in Figure 25, providing a clearer visual representation of the generated ma-
neuvers.

Figure 25. Three-dimensional visualization of the radar-evasive maneuver scenario.

The adjustments made to the attitude throughout the engagement to satisfy the RCS
threshold can also be observed in the resulting trajectory.
Aerospace 2025, 12, 478 24 of 32

5.3.2. Dependency on Arbitrarily Reduced Radar Cross-Section Characteristics

To assess the algorithm’s sensitivity, the baseline F-16 radar cross-section profile is
intentionally degraded by systematically decreasing its RCS values by 15 dB within a
specific angular region. This reduction is applied for azimuth angles in the range of
[−130◦ , −45◦ ] and elevation angles in the range of [15◦ , 75◦ ]. The simulation case is
identical to Scenario-1 presented in Section 5.1, i.e., the radar-penetration maneuver. The
RCS trajectory corresponding to this simulation is shown in Figure 26.

Figure 26. Radar cross-section history for the radar-penetration maneuver: (1) RCS trajectories for
both the virtual path and RCS-constrained maneuver on the left, (2) RCS map with azimuth and
elevation angles, along with the aircraft’s orientation trajectory on the right.

The reduced RCS map in Figure 26 has more azimuth and elevation angle combinations
below the 0 dB threshold compared to the one shown in Figure 10. As expected, the
maneuver-generation method continues to produce effective stealth maneuvers compared
to the virtual path as well as the proposed maneuver generation results given in Figure 10.
However, the resulting RCS trajectory differs due to the altered RCS distribution. The
commands of the stealth maneuver generator are shown in Figure 27.

Figure 27. CBF-pilot commands for the radar-penetration maneuver.

Aerospace 2025, 12, 478 25 of 32

The yielding aircraft states for reduced RCS map are demonstrated in Figure 28.

Figure 28. Attitude trajectories of the aircraft during the radar-penetration maneuver.

The resulting trajectory, shown in three planar perspectives and one isometric view in
Figure 29, offers a clear visual representation of the generated maneuvers.

Figure 29. Three-dimensional visualization of the radar-penetration maneuver scenario.

An additional assessment using the same reduced RCS characteristics involves du-
plicating Scenario-2 presented in Section 5.2, i.e., the radar-evasive maneuver. The RCS
trajectory corresponding to this simulation is shown in Figure 30.
Aerospace 2025, 12, 478 26 of 32

Figure 30. Radar cross-section history for the radar-evasive maneuver: (1) RCS trajectories for both
the virtual path and RCS-constrained maneuver on the left, (2) RCS map with azimuth and elevation
angles, along with the aircraft’s orientation trajectory on the right.

Consistent with prior assessments, the proposed maneuver generation rationale re-
mains highly effective in producing stealth maneuvers compared to the virtual path, given
that a larger region falls below the RCS threshold. Also, the resulting commands success-
fully maintain the RCS value below the threshold continuously after the initial transition
maneuvers. The commands of stealth maneuver generator are shown in Figure 31.

Figure 31. CBF-pilot commands for the radar-evasive maneuver.

The yielding aircraft states are demonstrated in Figure 32.

Aerospace 2025, 12, 478 27 of 32

Figure 32. Attitude trajectories of the aircraft during the radar-evasive maneuver.

The resulting trajectory of this case is depicted with three planar views and one
isometric view in Figure 33, providing a clearer visual representation of the generated
maneuvers.

Figure 33. Three-dimensional visualization of the radar-evasive maneuver scenario.

The resulting trajectory clearly shows the attitude changes applied during the engage-
ment to stay within the RCS threshold.
Aerospace 2025, 12, 478 28 of 32

5.3.3. Dependency on the Control Barrier Function Design Parameter, γσ

To evaluate the sensitivity of the proposed method to the control barrier function
design parameter γσ , three different parameter settings (γσ = 1 × 101 , 1 × 102 , and 1 × 103 )
are assessed for a radar-evasive maneuver identical to Scenario-2 presented in Section 5.2.
The RCS trajectory corresponding to this assessment is shown in Figure 34.

Figure 34. Radar cross-section history for the radar-evasive maneuver: (1) RCS trajectories for both
the virtual path and RCS-constrained maneuver on the left, (2) RCS map with azimuth and elevation
angles, along with the aircraft’s orientation trajectory on the right.

It is evident that the setting of the design parameter has a significant impact on the
performance of the stealth maneuver generator. An increase in the parameter value extends
the duration during which the aircraft’s observability remains below the 0 dBsm threshold.
Additionally, an increase in the design parameter induces a more responsive behavior in
the RCS dynamics, as evidenced by the reduced settling time toward or below the threshold
RCS value. The setting of γσ = 1 × 103 enables a significant reduction in observability,
reaching as low as −20 dBsm. Furthermore, each setting of the design parameter results in a
distinct RCS profile, reflecting differences in the commands issued by the stealth maneuver
generator. Nevertheless, each design configuration ultimately aims to generate a maneuver
that keeps the radar cross-section below the threshold, albeit with varying characteristics.
Finally, the resulting trajectory, shown in three planar perspectives and one isometric view
in Figure 35, offers a clear visual representation of the generated maneuvers.
Consequently, the radar-evasive operation is successfully accomplished with varying
characteristics, despite differences in the design parameter settings.

5.4. Monte Carlo Simulations

Monte Carlo simulations were employed to evaluate the performance of the proposed
stealth-maneuver generation framework under varying initial conditions considering
radar-penetration maneuvers, thereby simulations were terminated at the time the aircraft
encountered the radar. Each simulation was conducted under two distinct conditions: one
where the RCS threshold of 0 dBsm was enforced, and another where no RCS threshold
was applied. For each condition, 6355 simulations were carried out, resulting in a total
of 12,710 simulations. This extensive set of simulations enabled the statistical evaluation
of the system’s performance. The aircraft’s altitude was randomized within the interval
[1000 m, 3000 m], while the radar’s position was varied with its north coordinate rang-
Aerospace 2025, 12, 478 29 of 32

ing within [2000 m, 5000 m] and its east coordinate ranging within [−2000 m, 2000 m].
The performance of the framework was assessed using three key metrics as depicted in
Figure 36.

Figure 35. Three-dimensional visualization of the radar-evasive maneuver scenario.

Figure 36. Monte Carlo simulation performance metrics: (1) The mean RCS value is −4.8739 dBsm
and 3.9508 dBsm for the CBF-pilot and the virtual path, respectively; (2) The percentage of the mean
RCS below the threshold is 89.6% for the CBF-pilot and 1.26% for the virtual path; (3) The RCS value
remains below the threshold for 78.28% and 20.52% of the simulation duration for the CBF-pilot and
the virtual path, respectively.

The first metric, σavg , represents the mean RCS value observed during each simulation,
providing an overall measure of radar visibility. The distribution of σavg values for both
the proposed framework and the virtual path is shown in the upper portion of Figure 36.
The concentration of σavg values is below the 0 dBsm threshold for the CBF-pilot, while it
remains above the threshold for the virtual path, as expected. Additionally, the maximum
Aerospace 2025, 12, 478 30 of 32

σavg values for the CBF-pilot are smaller than those for the virtual path, reflecting the efforts
to reduce visibility. The second metric focuses on the percentage of total simulations in
which the mean RCS value was below the 0 dB threshold, highlighting the framework’s
effectiveness in maintaining low observability across different scenarios. This metric
reveals a significant difference between the results of the CBF-pilot and the virtual path.
Specifically, the virtual path achieves only 1.26% of cases with σavg below the threshold,
whereas the proposed CBF-pilot method successfully keeps 89.6% of total simulations below
the threshold. The third metric evaluates the percentage of RCS values that remain below
0 dB within each simulation, offering insights into the consistency of stealth performance.
Relying solely on σavg for performance measurement can be misleading, as cases where the
RCS value exceeds the threshold for a certain duration can still result in an average below
the threshold if the RCS achieves very low dBsm values for a short time. Therefore, using
this metric provides deeper insights into true performance. The results indicate that, for
the majority of cases, the RCS value remains below the threshold for over 78% of the total
simulation time in the CBF-pilot results. Conversely, the virtual path lacks this characteristic
due to the absence of CBF constraints. These three metrics combined demonstrate the
effectiveness of the proposed maneuver generation approach under various conditions and
solidify its impact on reducing the aircraft’s visibility.

6. Conclusions
This study presents a novel framework for generating maneuvers using CBF to dy-
namically manage RCS constraints. The approach utilizes a high-fidelity flight dynamics
model of an F16 aircraft in contrast to existing studies based on simplified kinematic flight
dynamics models to assess radar observability. Additionally, the generation of the RCS
dataset is achieved by incorporating the control surface deflections in addition to the
aircraft’s orientations to realistically model the exact RCS profile. The approach allows
non-stealth aircraft to reduce their radar observability by the generated maneuvers in real
time, ensuring compliance with RCS thresholds.
The effectiveness of the proposed method is evaluated through comparisons between
cases with CBF-pilot and virtual path which excludes a CBF-based maneuver generator. In
scenarios where virtual path is applied, the aircraft exhibit consistently higher RCS values,
making them more susceptible to radar detection. By contrast, the CBF-pilot approach
maintains RCS values below predefined thresholds for the majority of the mission timeline.
In realistic operational scenarios, such as radar-penetration and radar-evasive maneuvers,
the framework demonstrates its ability to dynamically adjust the aircraft’s orientation by
controlling angular rates to minimize radar exposure. During Monte Carlo simulations,
over 89.6% of cases with stealth maneuver generator achieve sustained low radar observ-
ability compared to just 1.26% of cases with virtual path. These results underline the critical
impact of dynamic and adaptive motion planning in achieving low detectability under
radar threat conditions. This analysis illustrates the performance of the CBF-based method
in enabling aircraft to evade radar detection and maintain survivability.
In conclusion, this study provides a practical and effective solution for enabling non-
stealth aircraft to dynamically evade radar detection through generated maneuvers. By
comparing cases with and without stealth maneuver generator, the results emphasize
the importance of dynamic maneuver generation in reducing radar observability. The
proposed method demonstrates strong potential for real-time implementation due to its
simple linearly-constrained quadratic programming formulation, the strong convergence
characteristics of the sequential quadratic programming algorithm, and its ability to operate
at high frequencies in the simulation environment. Since the proposed strategy is an
optimization-based motion planning algorithm, it can generate stealth maneuvers even
Aerospace 2025, 12, 478 31 of 32

against mobile radar threats, provided that a feasible solution exists. Therefore, the study is
also promising for various contested and hostile environments. Consequently, the proposed
framework sets a new benchmark for enhancing survivability in contested environments
and lays the groundwork for future innovations in stealth strategy and motion planning.
Thus, potential future work may involve assessing the framework under more challenging
scenarios, such as dynamic radar threats posed by aerial radar platforms.

Author Contributions: Conceptualization, methodology, software, writing—original draft prepara-

tion, M.D., E.C.A., A.Ç. and F.E.; writing—supervising, review and editing, E.K., U.Z. and İ.Ö. All
authors have read and agreed to the published version of the manuscript.

Funding: This research received no external funding.

Data Availability Statement: The raw data supporting the conclusions of this article will be made
available by the authors on request.

Acknowledgments: During the preparation of this manuscript/study, the authors used ChatGPT 4.0
for the purposes of the grammar and spell check. The authors have reviewed and edited the output
and take full responsibility for the content of this publication.

Conflicts of Interest: Author Uğur Zengin was employed by the company Turkish Aerospace. The
remaining authors declare that the research was conducted in the absence of any commercial or
financial relationships that could be construed as a potential conflict of interest.

Abbreviations
The following abbreviations are used in this manuscript:

CAS Control augmentation system

CBF Control barrier functions
INDI Incremental nonlinear dynamic inversion
LO Low observability
RAM Radar-absorbing material
RCS Radar cross-section
UAV Unmanned aerial vehicle

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