Stealth-Maneuver Generation for Non-Stealth Aircraft: A Control Barrier Function Approach

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.

1. Introduction

In modern aerial warfare, stealth technology has become essential for maintaining a tactical edge by allowing aircraft to evade sophisticated radar systems and operate effectively 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 RCS, stealth technology addresses the growing challenge of radar detection, as evidenced by platforms like the F-117 and B-2. These aircraft show the substantial advantages of inherent stealth features, particularly in reducing the effectiveness of integrated air defense systems (IADSs) [1,2]. However, achieving stealth is not solely reliant on structural or material technologies. Dynamic trajectory planning that minimizes RCS in real time also offers a viable approach to achieving stealth capabilities, especially for non-stealth platforms operating in contested environments.

1.1. Related Works

Historically, stealth capabilities have been achieved through aero-structural design, material technologies, and signature management tactics [3]. For example, radar absorbent materials (RAMs) and shaping techniques reduce the reflected radar energy [1]. While highly effective for purpose-built stealth platforms, these methods are not applicable to 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 hazards. 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 fusion 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, particularly 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 parameters. 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 consumption, 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, enhancing 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 navigating 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 [11,12,13]. These approaches integrate guidance and control systems, allowing the aircraft to adaptively reduce RCS while navigating through contested and radar-monitored airspace, thereby supporting mission success under dynamic threat conditions.

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 sustained stealth. Adaptive algorithms incorporate factors such as radar position, angle, and power into UAV path planning to enhance survivability [15]. Additionally, machine learning 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 multiple technologies and methodologies to achieve effective RCS reduction for stealth aircraft. While LO technologies such as RAM and aircraft shaping play foundational roles in reducing 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 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 integration 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 maintain stealth characteristics. This real-time adaptability ensures that non-stealth platforms 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.

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.

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 [4,6,7,8,9]. 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., ${\mathit{\omega }}_{\mathrm{ref}}={\mathbf{0}}_3$. 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, ${\mathit{\omega }}_{\mathrm{cmd}}$.

2.1. Notations

Throughout this study, the time derivative of a continuously differentiable function $f:{\mathbb{R}}^n\to \mathbb{R}$ is represented as $\dot{f}$. Vectors are indicated using bold notation, i.e., $\mathit{v}$, and the cross product of two vectors $\mathit{x}$ and $\mathit{y}$ is denoted by $\mathit{x}\times \mathit{y}$. The notations $s(\ast )$, $c(\ast )$, and $t(\ast )$ correspond to the sine, cosine, and tangent functions of $(\ast )$, respectively. A control affine system is described as

$$\dot{\mathit{x}}=\mathit{f}\left(\mathit{x}\right)+\mathit{g}\left(\mathit{x}\right)\mathit{u}$$

(1)

where $\mathit{x}\in {\mathbb{R}}^n$ is the state vector and $\mathit{u}\in {\mathbb{R}}^m$ is the control input vector. Nonlinear mappings of $\mathit{f}:{\mathbb{R}}^n\to {\mathbb{R}}^n$ and $\mathit{g}:{\mathbb{R}}^n\to {\mathbb{R}}^{n\times m}$ are locally Lipschitz continuous functions. Finally, for a $C^{\infty }$ function $Q:{\mathbb{R}}^n\to \mathbb{R}$ and $g:{\mathbb{R}}^{n\times p}\to {\mathbb{R}}^n$, the Lie derivative is denoted by

$${\mathcal{L}}_{g}Q\left(\mathit{x}\right)={\displaystyle \frac{\partial Q}{\partial x}}\left(\mathit{x}\right)g\left(\mathit{x}\right)$$

(2)

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 $\mathcal{C}$ is composed of

$$\begin{array}{cc}\hfill \mathcal{C}=& \{\mathit{x}\in D\subset {\mathbb{R}}^n:h\left(\mathit{x}\right)\ge 0\}\hfill \\ \hfill \partial \mathcal{C}=& \{\mathit{x}\in D\subset {\mathbb{R}}^n:h\left(\mathit{x}\right)=0\}\hfill \\ \hfill Int\left(\mathcal{C}\right)=& \{\mathit{x}\in D\subset {\mathbb{R}}^n:h\left(\mathit{x}\right)>0\}\hfill \end{array}$$

(3)

where $\mathcal{C}$ is safe set, $\partial \mathcal{C}$ is the boundary, and $Int\left(\mathcal{C}\right)$ is the interior of the safe set. In addition, $h\left(\mathit{x}\right)\ge 0$ defines the safe region while $h\left(\mathit{x}\right)<0$ defines the unsafe region.

Definition 1

([17,18]). Function $h:D\subset {\mathbb{R}}^n\to \mathbb{R}$ is the control barrier function (CBF) if the following conditions hold:

• A zero-superlevel set $\mathcal{C}$ exists for the function $h\left(\mathit{x}\right)$.
• $h\left(\mathit{x}\right)$ satisfies the inequality

  $$\underset{\mathit{u}\in U}{sup}\{{\mathcal{L}}_{f}h\left(\mathit{x}\right)+{\mathcal{L}}_{g}h\left(\mathit{x}\right)\mathit{u}+\alpha \left(h\left(\mathit{x}\right)\right)\ge 0\}$$

  (4)

  where class ${\kappa }_{\infty }$ function $\alpha \left(h\right(\mathit{x}\left)\right)$ for a dynamic system described in Equation (1). If such safe set $\mathcal{C}$ exists, then the control set ensuring the safety for $\forall \mathit{x}\in D$ can be given as

$$K_{cbf}:=\left\{\mathit{u}\in U:{\mathcal{L}}_{f}h\left(\mathit{x}\right)+{\mathcal{L}}_{g}h\left(\mathit{x}\right)\mathit{u}+\alpha \left(h\left(\mathit{x}\right)\right)\ge 0\right\}$$

(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 subsequent sections.

2.3.1. Equations of Motion

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

The nonlinear flight dynamics equations, including translational and rotational dynamics, and translational and rotational kinematics are, respectively,

$$\begin{array}{cc}\hfill & \left\{\begin{array}{c}\dot{u}=\sum F_{\mathrm{X}}/m+rv-qw\hfill \\ \dot{v}=\sum F_{\mathrm{Y}}/m+pw-ru\hfill \\ \dot{w}=\sum F_{\mathrm{Z}}/m+qu-pv\hfill \end{array}\right.\hfill \\ \hfill & \left\{\begin{array}{c}\dot{p}=qr(I_{\mathrm{yy}}-I_{\mathrm{zz}})/I_{\mathrm{xx}}+(\dot{r}+pq)I_{\mathrm{xz}}/I_{\mathrm{xx}}+\sum L/I_{\mathrm{xx}}\hfill \\ \dot{q}=pr(I_{\mathrm{zz}}-I_{\mathrm{xx}})/I_{\mathrm{yy}}+(r^2-p^2)I_{\mathrm{xz}}/I_{\mathrm{yy}}+\sum M/I_{\mathrm{yy}}\hfill \\ \dot{r}=pq(I_{\mathrm{xx}}-I_{\mathrm{yy}})/I_{\mathrm{zz}}+(\dot{p}-qr)I_{\mathrm{xz}}/I_{\mathrm{zz}}+\sum N/I_{\mathrm{zz}}\hfill \end{array}\right.\hfill \\ \hfill & \left\{\begin{array}{c}{\dot{x}}_{\mathrm{E}}=uc\theta c\psi +v(s\varphi s\theta c\psi -c\varphi s\psi )+w(c\varphi s\theta c\psi +s\varphi s\psi )\hfill \\ {\dot{y}}_{\mathrm{E}}=uc\theta c\psi +v(s\varphi s\theta c\psi +c\varphi c\psi )+w(c\varphi s\theta s\psi -s\varphi s\psi )\hfill \\ {\dot{z}}_{\mathrm{E}}=-us\theta +vs\varphi c\theta +wc\varphi c\theta \hfill \end{array}\right.\hfill \\ \hfill & \left\{\begin{array}{c}\dot{\varphi }=p+t_{\theta }(qs\varphi +rc\varphi )\hfill \\ \dot{\theta }=qc\varphi -rs\varphi \hfill \\ \dot{\psi }=(qs\varphi +rc\varphi )/c\theta \hfill \end{array}\right.\hfill \end{array}$$

(6)

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 $x_{\mathrm{E}}$, $y_{\mathrm{E}}$, and $z_{\mathrm{E}}$, and the orientation is described by the Euler angles $\varphi$, $\theta$, and $\psi$. The force components acting on the body frame are given as $F_{\mathrm{X}}$, $F_{\mathrm{Y}}$, and $F_{\mathrm{Z}}$, with L, M, and N representing the roll, pitch, and yaw moments, respectively. Additionally, m signifies the mass of the aircraft, while $I_{\mathrm{xx}}$, $I_{\mathrm{yy}}$, $I_{\mathrm{zz}}$, and $I_{\mathrm{xz}}$ 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., $C_{\mathrm{X}}$, $C_{\mathrm{Y}}$, and $C_{\mathrm{Z}}$, and three moment coefficients, i.e., $C_{\mathrm{l}}$, $C_{\mathrm{m}}$, and $C_{\mathrm{n}}$. The aerodynamic coefficients depend on relevant flight states, such as the angle of attack ($\alpha$), sideslip angle ($\beta$), and control surface deflections, including horizontal tail deflection (${\delta }_{\mathrm{HT}}$), aileron deflection (${\delta }_{\mathrm{A}}$), and rudder deflection (${\delta }_{\mathrm{R}}$). The non-dimensional aerodynamic coefficients are converted to dimensional force and moment expressions using the dynamic pressure and relevant geometric properties, as follows: $F_{\mathrm{X}}={\overline{q}}_{\infty }S{C}_{\mathrm{X}}$, $F_{\mathrm{Y}}={\overline{q}}_{\infty }S{C}_{\mathrm{Y}}$, $F_{\mathrm{Z}}={\overline{q}}_{\infty }S{C}_{\mathrm{Z}}$, $L={\overline{q}}_{\infty }Sb{C}_{\mathrm{l}}$, $M={\overline{q}}_{\infty }S\overline{c}{C}_{\mathrm{m}}$, and $N={\overline{q}}_{\infty }Sb{C}_{\mathrm{n}}$.

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 $0.0495\mathrm{s}$. The rate saturation limits are ${60}^{\circ }/\mathrm{s}$ for the horizontal tails, ${80}^{\circ }/\mathrm{s}$ for the ailerons, and ${120}^{\circ }/\mathrm{s}$ for the rudder. Likewise, the position saturation limits are $\pm {25}^{\circ }$ for the horizontal tails, $\pm 21.5^{\circ }$ for the ailerons, and $\pm {30}^{\circ }$ 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

$$\dot{\mathit{\omega }}=-J^{-1}(\mathit{\omega }\times J\mathit{\omega })+\underset{\begin{array}{c}\mathit{g}\left(\mathit{x}\right)\end{array}}{\underbrace{J^{-1}{\overline{q}}_{\infty }S\left[\begin{array}{ccc}b& & \\ & \overline{c}& \\ & & b\end{array}\right]\mathsf{\Phi }}} \underset{\begin{array}{c}\mathit{u}\end{array}}{\underbrace{\mathit{\delta }}}$$

(7)

where ${\overline{q}}_{\infty }$, S, b, and $\overline{c}$ represent the dynamic pressure, wing area, wing span, and mean aerodynamic chord, respectively. Additionally, $\mathsf{\Phi }\in {\mathbb{R}}^{3\times n}$ 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

$$\mathit{u}=\mathit{g}{\left({\mathit{x}}_0\right)}^{-1}[{\dot{\mathit{\omega }}}_c-{\dot{\mathit{\omega }}}_0]+{\mathit{u}}_0$$

(8)

where the subscript ‘0’ denotes the current state and ${\dot{\mathit{\omega }}}_c\in {\mathbb{R}}^3$ represents the virtual input to be designed. The final form of the control law is provided as

$$\mathit{\delta }={\left\{J^{-1}{\overline{q}}_{\infty }S\left[\begin{array}{ccc}b& & \\ & \overline{c}& \\ & & b\end{array}\right]\mathsf{\Phi }\right\}}^{-1}[{\dot{\mathit{\omega }}}_c-{\dot{\mathit{\omega }}}_0]+{\mathit{\delta }}_0$$

(9)

where $\mathit{\delta }\in {\mathbb{R}}^3$ 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 ${\dot{\mathit{\omega }}}_c$ is given by

$${\dot{\mathit{\omega }}}_c=\left[\begin{array}{ccc}{K}_p& & \\ & K_q& \\ & & K_r\end{array}\right]\left[\begin{array}{c}{p}_{\mathrm{cmd}}-p\\ q_{\mathrm{cmd}}-q\\ r_{\mathrm{cmd}}-r\end{array}\right]$$

(10)

where $K_p$, $K_q$, and $K_r$ 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 $p_{\mathrm{cmd}}$, $q_{\mathrm{cmd}}$, and $r_{\mathrm{cmd}}$.

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 $1{\mathrm{m}}^2$. With RCS denoted as $\sigma$, the value of a target is defined as

$$\sigma =\underset{r\to \infty }{lim}4\pi r^2{\displaystyle \frac{S_s}{S_i}}$$

(11)

where $S_i$ is the incident power density measured at the target, $S_s$ 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

$$\mathrm{RCS}\left(\mathrm{dBsm}\right)=10{log}_{10}\left(\sigma \right)$$

(12)

where $\sigma$ represents the RCS measured in square meters (${\mathrm{m}}^2$), while $\mathrm{dBsm}$ 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 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.

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 behavior 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.

By definition, the range of the $\mathsf{\Phi }$ angle is $[-{180}^{\circ },{180}^{\circ }]$, whereas the range of the $\mathsf{\Theta }$ angle is $[0^{\circ },{180}^{\circ }]$. Therefore, a comprehensive RCS analysis should cover all combinations within these ranges. Furthermore, however, analyzing only the possible combinations of $\mathsf{\Theta }$ and $\mathsf{\Phi }$ 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, 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: $\mathsf{\Phi }$ within the range $[-{180}^{\circ },{180}^{\circ }]$, $\mathsf{\Theta }$ within the range $[0^{\circ },{180}^{\circ }]$, and the aileron (${\delta }_{\mathrm{A}}$), horizontal tail (${\delta }_{\mathrm{HT}}$), and rudder deflections (${\delta }_{\mathrm{R}}$) within the range $[-{30}^{\circ },{30}^{\circ }]$ The elevation and azimuth angles are discretized in ${15}^{\circ }$ 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 ${90}^{\circ }$ and $0^{\circ }$, respectively.

The polar representation of the corresponding RCS variation is depicted in Figure 6 for the sake of creating visual intuition.

The effects of the horizontal tail deflection on the RCS characteristics are depicted in Figure 7 comparatively.

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 ${25}^{\circ }$ horizontal tail angle, the RCS value increases by approximately $10\mathrm{dBsm}$ at an elevation angle of ${155}^{\circ }$. Similarly, for a $-{25}^{\circ }$ horizontal tail angle, the RCS value rises by about $15\mathrm{dBsm}$ at an elevation angle of ${205}^{\circ }$. As an adjunct instance, the rudder deflection effects on the RCS characteristics are illustrated in Figure 8.

Subsequently, the generated RCS database is embedded in a 5D look-up table, where each dimension corresponds to $\mathsf{\Theta }$, $\mathsf{\Phi }$, ${\delta }_{\mathrm{A}}$, ${\delta }_{\mathrm{HT}}$, and ${\delta }_{\mathrm{R}}$, with linear interpolation.

As the final step, the conversion between the aircraft orientation ($\varphi$, $\theta$, $\psi$) and the relative orientation ($\mathsf{\Phi }$, $\mathsf{\Theta }$) 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

$${\mathit{P}}_{\mathit{b}}=R_{ib}\left({\mathit{P}}_{\mathit{a}}-{\mathit{P}}_{\mathit{r}}\right)$$

(13)

where ${\mathit{P}}_{\mathit{a}}$ is the position of the aircraft, ${\mathit{P}}_{\mathit{r}}$ is the position of the radar, and $R_{ib}$ 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

$$R_{ib}=\left[\begin{array}{ccc}c\theta c\psi & c\theta s\psi & -s\theta \\ s\varphi s\theta c\psi -c\varphi s\psi & s\varphi s\theta s\psi +c\varphi c\psi & s\varphi c\theta \\ c\varphi s\theta c\psi +s\varphi s\psi & c\varphi s\theta s\psi -s\varphi c\psi & c\varphi c\theta \end{array}\right]$$

(14)

where bank angle $\varphi$, pitch angle $\theta$, and yaw angle $\psi$ represent the attitude angles. Then, the $\mathsf{\Theta }$ and $\mathsf{\Phi }$ angles are determined by converting from the cartesian to the spherical coordinate system using Equation (15).

$$\begin{array}{cc}\hfill & \mathsf{\Theta }=arctan{\displaystyle \frac{P_y}{P_x}}\hfill \\ \hfill & \mathsf{\Phi }=arctan{\displaystyle \frac{\sqrt{P_x^2+P_y^2}}{P_z}}\hfill \end{array}$$

(15)

where $P_x$$P_y$ and $P_z$, describe the components of ${\mathit{P}}_{\mathit{b}}$ 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 $p_{\mathrm{cmd}}$, $q_{\mathrm{cmd}}$, and $r_{\mathrm{cmd}}$; 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 ($\varphi$, $\theta$, and $\psi$), implying that the angular rates can be revealed provided that an appropriate barrier function is established. A barrier function is then designed as

$$h\left(\sigma \right)={\sigma }_{\max }-\sigma (\varphi ,\theta ,\psi )$$

(16)

where ${\sigma }_{\max }\in \mathbb{R}$ is the predetermined maximum allowable RCS value. It is obvious that $h\left(\sigma \right)>0,\forall \sigma \in {\mathbb{R}}_{<{\sigma }_{\max }}$, and $h\left(\sigma \right)=0\leftrightarrow \sigma ={\sigma }_{\max }$. Thus, the time derivative of the barrier function is $\dot{h}\left(\sigma \right)=-\dot{\sigma }$. At this point, an expansion of the time derivative of the barrier function should be given as

$$\dot{\sigma }=\underset{\begin{array}{c}{\sigma }_{\varphi }\in \mathbb{R}\end{array}}{\underbrace{{\displaystyle \frac{\partial \sigma (\varphi ,\theta ,\psi )}{\partial \varphi }}}}\dot{\varphi }+\underset{\begin{array}{c}{\sigma }_{\theta }\in \mathbb{R}\end{array}}{\underbrace{{\displaystyle \frac{\partial \sigma (\varphi ,\theta ,\psi )}{\partial \theta }}}}\dot{\theta }+\underset{\begin{array}{c}{\sigma }_{\psi }\in \mathbb{R}\end{array}}{\underbrace{{\displaystyle \frac{\partial \sigma (\varphi ,\theta ,\psi )}{\partial \psi }}}}\dot{\psi }$$

(17)

where ${\sigma }_{\varphi }$, ${\sigma }_{\theta }$, and ${\sigma }_{\psi }$ 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 $\dot{\varphi }=f\left(\varphi \right)+g\left(\varphi \right)\mathit{\omega }$, $\dot{\theta }=f\left(\theta \right)+g\left(\theta \right)\mathit{\omega }$, and $\dot{\psi }=f\left(\psi \right)+g\left(\psi \right)\mathit{\omega }$, respectively. It is quite straightforward since the attitude dynamics are represented through a transformation matrix and angular rates, i.e., $\dot{\mathsf{\Omega }}=R_{\varphi \theta \psi }\mathit{\omega }$. By recalling the rotational kinematics given in Equation (6), the required decomposition for the bank, pitch, and yaw dynamics can be performed as

$$\begin{array}{cc}\hfill \dot{\varphi }& =\underset{\begin{array}{c}f\left(\varphi \right)\end{array}}{\underbrace{0}}+\underset{\begin{array}{c}g\left(\varphi \right)\end{array}}{\underbrace{\left[\begin{array}{ccc}1& tan\theta sin\varphi & tan\theta cos\varphi \end{array}\right]}}\underset{\begin{array}{c}\mathit{\omega }\end{array}}{\underbrace{\left[\begin{array}{c}p\\ q\\ r\end{array}\right]}}\hfill \\ \hfill \dot{\theta }& =\underset{\begin{array}{c}f\left(\theta \right)\end{array}}{\underbrace{0}}+\underset{\begin{array}{c}g\left(\theta \right)\end{array}}{\underbrace{\left[\begin{array}{ccc}0& cos\varphi & -sin\varphi \end{array}\right]}}\underset{\begin{array}{c}\mathit{\omega }\end{array}}{\underbrace{\left[\begin{array}{c}p\\ q\\ r\end{array}\right]}}\hfill \\ \hfill \dot{\psi }& =\underset{\begin{array}{c}f\left(\psi \right)\end{array}}{\underbrace{0}}+\underset{\begin{array}{c}g\left(\psi \right)\end{array}}{\underbrace{\left[\begin{array}{ccc}0& {\displaystyle \frac{sin\varphi }{cos\theta }}& {\displaystyle \frac{cos\varphi }{cos\theta }}\end{array}\right]}}\underset{\begin{array}{c}\mathit{\omega }\end{array}}{\underbrace{\left[\begin{array}{c}p\\ q\\ r\end{array}\right]}}\hfill \end{array}$$

(18)

Attentive eyes will notice that the components of $g\left(\varphi \right)$, $g\left(\theta \right)$, and $g\left(\psi \right)$ are row elements of the transformation matrix $R_{\varphi \theta \psi }$. Subsequently, the RCS dynamics can be represented in the form of $\dot{\sigma }=f\left(\sigma \right)+g\left(\sigma \right)\mathit{\omega }$, defined as

$$\dot{\sigma }=\underset{\begin{array}{c}f\left(\sigma \right)\end{array}}{\underbrace{0}}+\underset{\begin{array}{c}g\left(\sigma \right)\end{array}}{\underbrace{\left[\begin{array}{ccc}{\sigma }_{\varphi }g\left(\varphi \right)& {\sigma }_{\theta }g\left(\theta \right)& {\sigma }_{\psi }g\left(\psi \right)\end{array}\right]}}\underset{\begin{array}{c}\mathit{\omega }\end{array}}{\underbrace{\left[\begin{array}{c}p\\ q\\ r\end{array}\right]}}$$

(19)

Consequently, the CBF constraint can be described as

$$\underset{\begin{array}{c}{\mathcal{L}}_{f}h\left(\sigma \right)+{\mathcal{L}}_{g}h\left(\sigma \right){\mathit{\omega }}_{\mathrm{cmd}}\end{array}}{\underbrace{-f\left(\sigma \right)-g\left(\sigma \right)\mathit{\omega }}}+{\gamma }_{\sigma }h\left(\sigma \right)\ge 0$$

(20)

where ${\gamma }_{\sigma }\in {\mathbb{R}}_{>0}$ 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

$$\begin{array}{cc}\hfill u^{★}=\underset{{\mathit{\omega }}_{\mathrm{cmd}}\in {\mathbb{R}}^3}{argmin}& \left|\right|{\mathit{\omega }}_{\mathrm{ref}}-{\mathit{\omega }}_{\mathrm{cmd}}{\left|\right|}_2^2\hfill \\ \hfill \mathrm{s}.\mathrm{t}.& -f\left(\sigma \right)-g\left(\sigma \right){\mathit{\omega }}_{\mathrm{cmd}}+{\gamma }_{\sigma }h\left(\sigma \right)\ge 0\hfill \\ \hfill & {\mathit{\omega }}_{\min }\le {\mathit{\omega }}_{\mathrm{cmd}}\le {\mathit{\omega }}_{\max }\hfill \end{array}$$

(21)

The constructed formulation enables the stealth-maneuver generator to command angular rates ($p_{\mathrm{cmd}}$, $q_{\mathrm{cmd}}$, and $r_{\mathrm{cmd}}$) that closely follow the reference angular rates ($p_{\mathrm{ref}}$, $q_{\mathrm{ref}}$, and $r_{\mathrm{ref}}$) 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 $[{\mathit{\omega }}_{\min },{\mathit{\omega }}_{\max }]$, considering the admissible and allowable angular rate limits specific to the aircraft. The overall proposed architecture is depicted in Figure 9.

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, constructed 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 $0^{\circ }$ path angle and a ${30}^{\circ }$ heading, at an altitude of 2000 m. The radar position is set at $[5000\mathrm{m},0\mathrm{m},0\mathrm{m}]$ with a range of 5000 m. The maximum allowable RCS value is set at $0\mathrm{dBsm}$, and the design parameter ${\gamma }_{\sigma }$ is set to $1\times {10}^3$. 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 $0\mathrm{dBsm}$, 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 $25\mathrm{dBsm}$, while for the proposed RCS-constrained maneuver generation approach, it remains close to $0\mathrm{dBsm}$. Additionally, the $\mathsf{\Phi }$ and $\mathsf{\Theta }$ angles resulting from this scenario are illustrated. The blue regions represent the combinations of $\mathsf{\Phi }$ and $\mathsf{\Theta }$ where the RCS values are below $0\mathrm{dBsm}$, while the red regions indicate RCS values exceeding $0\mathrm{dBsm}$. 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 $0\mathrm{dBsm}$ 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 $p_{\mathrm{cmd}}$, $q_{\mathrm{cmd}}$, and $r_{\mathrm{cmd}}$, along with the aircraft’s states, are shown in Figure 11.

Angular rate commands are issued when the RCS value exceeds the threshold. The rationale behind this behavior is to satisfy the constraint while minimizing the objective function of the optimization. Since the optimization minimizes the difference between ${\mathit{\omega }}_{ref}$ and ${\mathit{\omega }}_{\mathrm{cmd}}$, 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.

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.

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 $\mathsf{\Phi }$ 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 $0^{\circ }$, at an altitude of 2000 m. The radar is positioned at $[4000\mathrm{m},1000\mathrm{m},0\mathrm{m}]$, with a detection range of 4000 m. The maximum allowable RCS value is set at $0\mathrm{dBsm}$, and the design parameter ${\gamma }_{\sigma }$ is set to $1\times {10}^3$. The resulting RCS history is presented in Figure 14.

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 $2\mathrm{dBsm}$. Subsequently, with agile intervention, the CBF-pilot commands an orientation that reduces the resultant RCS to quite below $0\mathrm{dBsm}$, reaching nearly $-20\mathrm{dBsm}$. For a prolonged period, the RCS remains below $0\mathrm{dBsm}$ 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 $0\mathrm{dBsm}$. 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 $p_{\mathrm{cmd}}$, $q_{\mathrm{cmd}}$, and $r_{\mathrm{cmd}}$, along with the aircraft’s states, are shown in Figure 15.

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.

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.

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 encountering 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, ${\gamma }_{\sigma }$.

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 $[5\mathrm{dBm},30\mathrm{dBm}]$. 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.

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.

The yielding aircraft states are demonstrated in Figure 20.

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 maneuvers.

The adjustments made to the attitude throughout the engagement to satisfy the RCS threshold can also be observed in the resulting trajectory.

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.

Seemingly, 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 23.

The yielding aircraft states are demonstrated in Figure 24.

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 maneuvers.

The adjustments made to the attitude throughout the engagement to satisfy the RCS threshold can also be observed in the resulting trajectory.

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 $[-{130}^{\circ },-{45}^{\circ }]$ and elevation angles in the range of $[{15}^{\circ },{75}^{\circ }]$. 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.

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.

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

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

An additional assessment using the same reduced RCS characteristics involves duplicating 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.

Consistent with prior assessments, the proposed maneuver generation rationale remains 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 successfully maintain the RCS value below the threshold continuously after the initial transition maneuvers. The commands of stealth maneuver generator are shown in Figure 31.

The yielding aircraft states are demonstrated in Figure 32.

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.

The resulting trajectory clearly shows the attitude changes applied during the engagement to stay within the RCS threshold.

5.3.3. Dependency on the Control Barrier Function Design Parameter, ${\gamma }_{\sigma }$

To evaluate the sensitivity of the proposed method to the control barrier function design parameter ${\gamma }_{\sigma }$, three different parameter settings (${\gamma }_{\sigma }$ = $1\times {10}^1$, $1\times {10}^2$, and $1\times {10}^3$) 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.

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 $0\mathrm{dBsm}$ 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 ${\gamma }_{\sigma }$ = $1\times {10}^3$ enables a significant reduction in observability, reaching as low as $-20\mathrm{dBsm}$. 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 $0\mathrm{dBsm}$ 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 $[1000\mathrm{m},3000\mathrm{m}]$, while the radar’s position was varied with its north coordinate ranging within $[2000\mathrm{m},5000\mathrm{m}]$ and its east coordinate ranging within $[-2000\mathrm{m},2000\mathrm{m}]$. The performance of the framework was assessed using three key metrics as depicted in Figure 36.

The first metric, ${\sigma }_{\mathrm{avg}}$, represents the mean RCS value observed during each simulation, providing an overall measure of radar visibility. The distribution of ${\sigma }_{\mathrm{avg}}$ values for both the proposed framework and the virtual path is shown in the upper portion of Figure 36. The concentration of ${\sigma }_{\mathrm{avg}}$ values is below the $0\mathrm{dBsm}$ threshold for the CBF-pilot, while it remains above the threshold for the virtual path, as expected. Additionally, the maximum ${\sigma }_{\mathrm{avg}}$ 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 ${\sigma }_{\mathrm{avg}}$ 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 ${\sigma }_{\mathrm{avg}}$ 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 $\mathrm{dBsm}$ 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 dynamically 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 observability 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 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.