Safe Autonomous UAV Target-Tracking Under External Disturbance, Through Learned Control Barrier Functions

Abstract

Ensuring the safe operation of Unmanned Aerial Vehicles (UAVs) is crucial for both mission-critical and safety-critical tasks. In scenarios where UAVs must track airborne targets, they need to follow the target’s path while maintaining a safe distance, even in the presence of unmodeled dynamics and environmental disturbances. This paper presents a novel collision avoidance strategy for dynamic quadrotor UAVs during target-tracking missions. We propose a safety controller that combines a learning-based Control Barrier Function (CBF) with standard sliding mode feedback. Our approach employs a neural network that learns the true CBF constraint, accounting for wind disturbances, while the sliding mode controller addresses unmodeled dynamics. This unified control law ensures safe leader-following behavior and precise trajectory tracking. By leveraging a learned CBF, the controller offers improved adaptability to complex and unpredictable environments, enhancing both the safety and robustness of the system. The effectiveness of our proposed method is demonstrated through the AirSim platform using the PX4 flight controller.

1. Introduction and Related Works

The majority of the systems that drive our daily lives whether it be robotic systems, transportation systems, or manufacturing systems are of the Cyber-Physical nature. These applications couple control, computation, and physical dynamics into one integrated system, and most of the time these systems need to be designed with safety requirements in mind, hence becoming safety-critical [1,2,3]. Unmanned Aerial Vehicles (UAVs) have emerged as versatile aerial robotic platforms in the realm of cyber-physical systems [4] with applications spanning various domains, including surveillance, warfare, search and exploration in cluttered environments [5], agriculture, environmental monitoring, and even package delivery [4]. One of the key challenges in deploying UAVs within these contexts is ensuring safe operation, particularly in scenarios involving tracking and following dynamic targets like other UAVs. The safe operation of UAVs is crucial to prevent collisions that can cause damage to the vehicles, property, and people, especially in complex or congested environments. Collision avoidance systems enable UAVs to detect obstacles and autonomously adjust their flight paths, ensuring mission success and protecting both assets and humans. Collision avoidance is a paramount concern, as it directly influences the safety and effectiveness of the UAVs deployed in such an environment. In cooperative and coordinated autonomous operations with multiple UAVs, collisions can be controlled and avoided with more certainty, as we have full access to the control parameters of all the UAVs and can perform deterministic operations, e.g., coordinated UAV swarm, platooning, etc. However, when uncertainty is involved in some UAVs, and the platooning or swarm operation is not cooperative, ensuring collision avoidance is non-trivial.

Modern quadrotor unmanned aerial vehicles (UAVs) are becoming more and more popular, which increases the possibility that they will be used for malevolent purposes. One such example is an unregistered UAV flying in restricted airspace, occasionally moving erratically. Dynamic UAVs can be used to monitor and follow harmful targets under tight safety limitations in order to avoid these situations. However because the target’s movement can be unpredictable, erratic, and random, the follower UAV must decide quickly and in real-time while still making sure it can follow the target safely [6].

Different collision avoidance techniques have previously been researched in the field of robotics. A classic technique is the dynamic window-based approach for obstacle avoidance [7], or the mixed-integer Linear Program-based controllers for safe online trajectory planning of unmanned vehicles [8]. A modern robotics approach to tracking and collision avoidance is by using state-of-the-art artificial intelligence techniques like Deep Neural Networks (DNN) and Reinforcement Learning (RL) [9,10,11]. A similar approach is also taken in [12] for implementing platooning control using the YOLO detector [13].

Classic collision avoidance methods like dynamic window approaches and mixed-integer linear programming can be computationally intensive, less scalable, and sensitive to model accuracy and sensor noise. AI-based techniques such as Deep Neural Networks and Reinforcement Learning offer adaptability but require large training data, high computation, and often lack formal safety guarantees. Geometric and potential field methods may suffer from local minima or overly conservative maneuvers, limiting efficiency in complex environments. A state-of-the-art nonlinear safety-critical control technique is Control Barrier Function (CBF) [1]. There has been numerous research conducted on collision avoidance in drones using CBF [4,6,14,15,16,17,18]. Drone platooning and swarm formation can also be achieved using CBFs as shown in [19,20,21,22]. CBFs are also useful in exploration in cluttered environments, as shown in [5,23]. Among other applications, CBFs have also been utilized for surveillance control for drone networks as shown in [24].

Various versions of CBFs like high order and adaptive CBFs have been shown to perform well in systems with high relative degree and uncertainty [17,25,26,27,28,29,30]. More recently modern data-driven and learning-based methods have been developed for finding CBFs with uncertainties [31,32,33], this allows the utilization of experimental data to find a CBF rather than manually formulating one which can be extremely difficult in practice. Estimators like neural networks and reinforcement learning have shown great success in estimating CBFs with uncertainties and input constraints [18,34,35,36].

Contribution. The main idea of our work is to design a robust safety controller for a quadrotor UAV that has to track and safely follow an unknown target UAV under various uncertainties in the form of unmodeled dynamics and external disturbances like wind. We summarize our contributions as follows:

• We introduce an offline, neural network-powered approach tailored for UAVs, to learn Control Barrier Function (CBF) constraints. This empowers the system to adeptly manage unmodeled dynamics and external disturbances prevalent in UAV flight.
• The proposed CBF is combined with a Sliding Mode Controller (SMC) to achieve robustness against model uncertainties. This ensures safe and reliable tracking even when the UAV’s behavior deviates from its nominal model.
• We thoroughly evaluate the effectiveness of our proposed technique through an AirSim platform enabled with the PX4 controller. The results demonstrate its ability to maintain safe target tracking despite external disturbances and modeling errors, which is suitable in real-time applications.

The paper is organized as follows. We introduce the mathematical model and formulate the problem in Section 2. Section 3 discusses the sliding mode control law and the proposed CBF method is presented in Section 4. The experimental results are presented in Section 5 followed by the conclusion.

2. System Dynamics and Problem Formulation

We consider the translational dynamics of a standard UAV for our work [37,38]. The position of the UAV evolves as:

$$\dot{p}=v$$

where $p={[x,y,z]}^T$ is the position vector in 3 dimensions and v is the corresponding velocity vector. As we are focusing on position tracking of the target UAV, we can neglect the tilting-like effects on the body of the UAV. Ignoring the effect of drag and other rotational forces, the velocity of the UAV is influenced by the following factors.

• Gravitational effect along the negative z direction: $-g\widehat{k}$, where g is gravitational constant.
• Thrust vector in inertial frame: ${\displaystyle \frac{1}{m}}\mathbf{R}{T}_b$, where m is the mass of UAV and $\mathbf{R}$ is the rotation matrix
• Lumped term for unmodelled dynamical effects (drag forces) and external disturbances in three dimensions: $\eta$

Considering these factors, we can express the velocity dynamics as:

$$\dot{v}=-g\widehat{z}+{\displaystyle \frac{1}{m}}\mathbf{R}{T}_b+{\displaystyle \frac{1}{m}}\eta$$

(1)

where:

• $\widehat{z}={\left[\begin{array}{ccc}0& 0& 1\end{array}\right]}^{\top }$ is the unit vector in the z-direction.
• $T_b={\left[\begin{array}{ccc}{T}_x& T_y& T_z\end{array}\right]}^{\top }$ represent thrust vector in body frame.
• $\eta ={\left[\begin{array}{ccc}{\eta }_x& {\eta }_y& {\eta }_z\end{array}\right]}^{\top }$ is the lumped uncertainty.

The complete mathematical motion model for UAV considered in this paper can be expressed as:

$$\begin{array}{cc}\hfill & \dot{p}=vs.\hfill \\ \hfill & \dot{v}=\left[\begin{array}{c}0\\ 0\\ -g\end{array}\right]+{\displaystyle \frac{1}{m}}\mathbf{R}\left[\begin{array}{c}{T}_x\\ T_y\\ T_z\end{array}\right]+{\displaystyle \frac{1}{m}}\eta \hfill \end{array}$$

(2)

Problem Definition

We consider a system of two UAVs, one is the target/leader UAV which flies in an unknown trajectory, and the other UAV called the follower has to track and follow the target while maintaining a safe distance from it at all times. Formally we can define this problem through the following safety condition.

$$\parallel {\mathbf{p}}_l-{\mathbf{p}}_f\parallel -d_{safe}\ge 0$$

(3)

Here, ${\mathbf{p}}_l\in {\mathbb{R}}^3$ and ${\mathbf{p}}_f\in {\mathbb{R}}^3$ are the positions of the leader and follower, respectively, and $d_{safe}$ is the minimum distance that needs to be maintained to be considered safe. The ability to maintain this safety constraint at all times under unknown disturbances motivates us to build a safety controller for the follower UAV.

3. Sliding Mode Control Law

In this section, we derive a sliding mode control law, which does not have a barrier certificate. We will present the modifications for a learned CBF in the next section. Given a desired position $p_d$ and desired velocity $v_d$, we consider a PID sliding surface s as:

$$s\left(t\right)=e_v\left(t\right)+{\lambda }_1{e}_p\left(t\right)+{\lambda }_2{\int }_0^t{e}_p\left(\tau \right)d\tau$$

where: $e_p=p-p_d$ is the position tracking error, $e_v=vs.-v_d$ is the velocity tracking error, ${\lambda }_1, {\lambda }_2$ are positive tuning gains, ${\int }_0^t{e}_p\left(\tau \right)d\tau$ is the integral of the position error.

The time derivative of the sliding surface can be derived as:

$$\dot{s}\left(t\right)={\dot{e}}_v\left(t\right)+{\lambda }_1{\dot{e}}_p\left(t\right)+{\lambda }_2{e}_p\left(t\right)$$

Substitute ${\dot{e}}_v=\dot{v}-{\dot{v}}_d$ and ${\dot{e}}_p=vs.-v_d=e_v$:

$$\dot{s}\left(t\right)=(\dot{v}-{\dot{v}}_d)+{\lambda }_1{e}_v+{\lambda }_2{e}_p$$

Using the UAV dynamics, we can obtain (substitute the dynamics of $\dot{v}$ into $\dot{s}$):

$$\dot{s}\left(t\right)=\left(\left[\begin{array}{c}0\\ 0\\ -g\end{array}\right]+{\displaystyle \frac{1}{m}}\mathbf{R}\left[\begin{array}{c}{T}_x\\ T_y\\ T_z\end{array}\right]+{\displaystyle \frac{1}{m}}{d}_{ext}\right)-{\dot{v}}_d+{\lambda }_1{e}_v+{\lambda }_2{e}_p$$

Using a standard Lyapunov function (${\displaystyle \frac{1}{2}}{s}^2$), the control input ${\left[\begin{array}{ccc}{T}_x& T_y& T_z\end{array}\right]}^{\top }$ must be designed to drive $\dot{s}\left(t\right)\le 0$ for the state trajectories to be stable on the sliding surface. It’s not impractical to assume that, the lumped uncertainty is upper bounded by:

$$\parallel \eta \parallel \le {\eta }_{max}.$$

For a positive gain matrix K and a control input $u_{smc}$:

$$\left[\begin{array}{c}{T}_x\\ T_y\\ T_z\end{array}\right]=m{\mathbf{R}}^{-1}(-\left[\begin{array}{c}0\\ 0\\ -g\end{array}\right]+{\dot{v}}_d-{\lambda }_1{e}_v-{\lambda }_2{e}_p-K\mathrm{s}\mathrm{g}\mathrm{n}\left(s\right)),$$

(4)

one can prove that the time derivative of the Lyapunov function $s\dot{s}$ is negative definite, provided $K\ge {\displaystyle \frac{1}{m}}{\eta }_{max}$.

Note that the constants of the sliding surface ${\lambda }_1,{\lambda }_2$ should be positive scalars to make the surface $s=0$ a suitable sliding surface (the state trajectories on the surface remain stable). If these constants are chosen to satisfy ${\lambda }_1^2>4{\lambda }_2$, then the errors $e_p$ and $e_v$ will not show oscillatory behavior on the surface and will converge to zero.

4. Learned CBF for External Wind Disturbances

We aim to design a safe controller for a UAV subjected to disturbances, ensuring the system remains within a predefined safe set.

4.1. Dynamical System Safety

Safety in dynamical systems refers to the system’s ability to remain in a predefined safe set over time. Consider a system $\dot{x}=f\left(x\right)$ where $x\in {\mathbb{R}}^n$, and a safe set $Տ=\{x\in {\mathbb{R}}^n:h\left(x\right)\ge 0\}$, defined by a smooth function $h:{\mathbb{R}}^n\to \mathbb{R}$. Nagumo’s theorem states that the system remains safe if:

$$\dot{h}\left(x\right)\ge 0\forall x\in \partial Տ$$

(5)

4.2. Control Barrier Function

For a nonlinear system:

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

(6)

a Control Barrier Function (CBF) $h\left(x\right)$ ensures that the system remains in the safe set $\mathcal{C}$ defined as,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \mathcal{C}=& \{x\in {\mathbb{R}}^n:h\left(x\right)\ge 0\}\hfill \end{array}\end{array}$$

(7)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \partial \mathcal{C}=& \{x\in {\mathbb{R}}^n:h\left(x\right)=0\}\hfill \end{array}\end{array}$$

(8)

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

(9)

by enforcing the following condition:

$${\displaystyle \frac{\partial h}{\partial x}}(f\left(x\right)+g\left(x\right)u)\ge -\gamma \left(h\left(x\right)\right)$$

(10)

where $\gamma$ is a class $\mathcal{K}$ function.

A Quadratic Program (QP) is used to enforce the CBF condition by filtering a potentially unsafe control input $u_d$ from the feedback control law $k_d\left(x\right)$:

$$\begin{array}{c}\hfill u^{*}=\underset{u}{argmin}{\parallel u-k_d\left(x\right)\parallel }^2\end{array}$$

(11)

$$\begin{array}{c}\hfill \mathrm{s}.\mathrm{t}.{\displaystyle \frac{\partial h}{\partial x}}(f\left(x\right)+g\left(x\right)u)\ge -\gamma \left(h\left(x\right)\right)\end{array}$$

(12)

This forms the basis of the CBF-QP controller, which will be compared with our proposed learned version in later sections.

4.3. Learned CBF

Drones operating under external disturbances such as wind or environmental factors often experience challenges in ensuring safety using traditional Control Barrier Functions (CBFs). To handle these uncertainties, we adapt the CBF framework to account for unknown disturbances using data-driven methods.

For the UAV dynamics, we modify the system to include an external disturbance ${\mathbf{d}}_{ext}\left(t\right)$:

$$\dot{\mathbf{v}}=\left[\begin{array}{c}0\\ 0\\ -g\end{array}\right]+{\displaystyle \frac{1}{m}}\mathbf{R}\left[\begin{array}{c}{T}_x\\ T_y\\ T_z\end{array}\right]+{\displaystyle \frac{1}{m}}{\mathbf{d}}_{ext}\left(t\right)$$

(13)

where ${\mathbf{d}}_{ext}\left(t\right)$ represents external forces like wind disturbances. For our system defined with the safety condition (3) we choose an appropriate CBF candidate as,

$$h(\mathbf{p},\mathbf{v}):=\parallel {\mathbf{p}}_l-{\mathbf{p}}_f\parallel -d_{safe}$$

(14)

Given the CBF $h(\mathbf{p},\mathbf{v})$ defined on the safe set $\mathcal{C}$ (9), the CBF derivative can be written as:

$$\dot{h}(\mathbf{p},\mathbf{v},\mathbf{u})={\dot{h}}_b(\mathbf{p},\mathbf{v},\mathbf{u})+{\displaystyle \frac{\partial h}{\partial \mathbf{v}}}·{\displaystyle \frac{1}{m}}{\mathbf{d}}_{ext}\left(t\right)$$

(15)

Here, ${\dot{h}}_b(\mathbf{p},\mathbf{v},\mathbf{u})$ represents the baseline CBF derivative without disturbances:

$${\dot{h}}_b(\mathbf{p},\mathbf{v},\mathbf{u})={\displaystyle \frac{\partial h}{\partial \mathbf{p}}}·\mathbf{v}+{\displaystyle \frac{\partial h}{\partial \mathbf{v}}}·\left(\left[\begin{array}{c}0\\ 0\\ -g\end{array}\right]+{\displaystyle \frac{1}{m}}\mathbf{R}\left[\begin{array}{c}{T}_x\\ T_y\\ T_z\end{array}\right]\right)$$

(16)

To estimate the unknown disturbance effect, we use a neural network model $\mathcal{N}(\mathbf{p},\mathbf{v},\mathbf{u})$, which approximates:

$$\mathcal{N}(\mathbf{p},\mathbf{v},\mathbf{u})\approx {\displaystyle \frac{\partial h}{\partial \mathbf{v}}}·{\displaystyle \frac{1}{m}}{\mathbf{d}}_{ext}\left(t\right)$$

(17)

Thus, the complete estimated CBF derivative becomes:

$$\widehat{\dot{H}}(\mathbf{p},\mathbf{v},\mathbf{u})={\dot{h}}_b(\mathbf{p},\mathbf{v},\mathbf{u})+\mathcal{N}(\mathbf{p},\mathbf{v},\mathbf{u})$$

(18)

4.4. Training the Neural Network

To train the neural network, we collect data consisting of the state ${\mathbf{p}}_i$, ${\mathbf{v}}_i$, control input ${\mathbf{u}}_i$, and the true CBF derivative ${\dot{h}}_i$. The training target is computed as the difference between the true and baseline CBF derivatives:

$$y_i={\dot{h}}_i-{\dot{h}}_b({\mathbf{p}}_i,{\mathbf{v}}_i,{\mathbf{u}}_i)$$

(19)

The dataset for training the network becomes:

$${\mathcal{D}}_{\mathrm{train}}={\left\{({\mathbf{p}}_i,{\mathbf{v}}_i,{\mathbf{u}}_i,y_i)\right\}}_{i=1}^N$$

(20)

The neural network is trained to minimize the mean squared error (MSE) between the predicted disturbance effect and the true effect:

$$\underset{\mathcal{N}}{min}{\displaystyle \frac{1}{N}}\sum _{i=1}^N{\left(\mathcal{N}({\mathbf{p}}_i,{\mathbf{v}}_i,{\mathbf{u}}_i)-y_i\right)}^2$$

(21)

4.5. Learned CBF-QP with Sliding Mode Control

Once trained, the neural network provides an estimate $\widehat{\dot{H}}(\mathbf{p},\mathbf{v},\mathbf{u})$, which incorporates external disturbances. This estimated CBF derivative can then be used in a CBF-QP combined with the sliding mode control law to ensure safety:

$${\mathbf{u}}^{*}=\underset{\mathbf{u}}{argmin}{\parallel \mathbf{u}-{\mathbf{u}}_{\mathrm{smc}}\parallel }^2$$

(22)

$$\mathrm{s}.\mathrm{t}.\widehat{\dot{H}}(\mathbf{p},\mathbf{v},\mathbf{u})\ge -\gamma \left(h(\mathbf{p},\mathbf{v})\right)$$

(23)

Here, ${\mathbf{u}}_{\mathrm{smc}}$ is the sliding mode control input:

$${\mathbf{u}}_{\mathrm{smc}}=m{\mathbf{R}}^{-1}\left(\left[\begin{array}{c}0\\ 0\\ -g\end{array}\right]+{\dot{\mathbf{v}}}_d-{\lambda }_1{e}_{\mathbf{v}}-{\lambda }_2{e}_{\mathbf{p}}-K\mathrm{sgn}\left(s\right)\right)$$

(24)

5. AirSim Experiments and Results

We opted for Microsoft AirSim [37,38] because AirSim provides an efficient simulation experience with its accurate physics engine and detailed sensor models, allowing for the replication of diverse sensor data under various environmental conditions. Furthermore, its compatibility with widely used robotics frameworks like ROS and PX4 (which are used in real-world UAVs) simplifies the transition from simulation to real-world deployment. AirSim is adaptable to real-world scenarios through its support for custom environments, sensors, and drone models, ensuring it meets the unique requirements of different projects. Moreover, AirSim has been extensively utilized for conducting high-fidelity simulation experiments by many researchers for various robotic platforms that were designed to deploy in the real world as demonstrated in these previous studies [39,40,41,42,43].

We conducted our experiments in the Microsoft AirSim simulator paired with PX4 [44,45] Autopilot in Software In The Loop mode as shown in Figure 1. The simulation setup was installed on a machine with AMD Ryzen 7 4700u APU, and running Windows 11. Microsoft AirSim v1.8.1 was installed with Unreal Engine v4.27 and paired with PX4 Autopilot v1.13.3.

The simulation environment is set with an unknown wind disturbance that is time-varying, which the follower drone does not have prior knowledge of. Another important thing that needs to be kept in mind is the follower drone also does not have prior knowledge of what trajectory the leader drone might take, the follower drone only has access to the translational states of the leader, i.e., the position and velocity which are updated in each control cycle which is roughly 10 Hz.

We constructed the neural network in Python 3.9 using the TensorFlow and Keras libraries, selecting an architecture with two fully connected hidden layers and ReLU activation functions for each hidden layer. ReLU was chosen due to its efficiency and effectiveness in capturing nonlinearity without suffering from vanishing gradients common to other activations such as sigmoid or tanh. The number of neurons per layer was determined empirically through preliminary experiments, balancing model capacity and the risk of overfitting, given the size of our dataset (20). The network was trained offline on an aggregated dataset collected from prior air sim experiments. Training utilized the Adam optimizer, a robust choice for deep networks due to its adaptive learning rates and efficient convergence properties. We applied mean squared error (MSE) as the loss function, reflecting the continuous nature of the CBF regression output. Model selection involved cross-validation on the training data, with a validation split set aside to monitor performance during training and tune hyperparameters. Performance assessment during training was based on validation loss. Further, after training, the model’s generalization ability was evaluated on a separate held out test set.

AirSim has its own low-level PID controller for all the simulated vehicles it offers with the SimpleFlight class, but we decided to go with the open-source PX4 Autopilot flight stack as the low-level flight controller of choice, which takes the desired pitch ${\varphi }_{des}$, roll ${\theta }_{des}$, and thrust $T_{des}$ as the inputs from our proposed controller and sends the motor inputs to AirSim for our experiments. PX4 is a trusted open-source flight controller that is widely used for real-world UAVs and other aerial and ground vehicles. It also has great community support. PX4 can be used with a companion simulator either in Software In The Loop (SITL) or Hardware In The Loop (HITL) mode. For implementation, we need to calculate the desired roll (${\varphi }_{des}$) and pitch (${\theta }_{des}$) using the filtered safe input (Equations (22)–(24)) ${\mathbf{u}}^{*}={\left[\begin{array}{ccc}{T}_x^{safe}& T_y^{safe}& T_z^{safe}\end{array}\right]}^{\top }$, which would be fed to the attitude controller along with the total thrust $T^{safe}$ for generating motor inputs. We use the transformations shown in [46] to calculate the desired roll and pitch.

$${\varphi }_{\mathrm{des}}={\displaystyle \frac{1}{mg}}\left(T_x^{safe}sin\left({\psi }_{\mathrm{des}}\right)-T_y^{safe}cos\left({\psi }_{\mathrm{des}}\right)\right)$$

$${\theta }_{\mathrm{des}}={\displaystyle \frac{1}{mg}}\left(T_x^{safe}cos\left({\psi }_{\mathrm{des}}\right)-T_y^{safe}sin\left({\psi }_{\mathrm{des}}\right)\right)$$

5.1. Results

To showcase and validate the performance of our proposed control strategy, we conducted several simulation experiments and divided them mainly based on the leader drone’s trajectory— Straight Line and Circular Trajectory.

We conduct the experiments and compare the performance of four different types of controllers, which are, (a) Baseline-PDCBF which employs a PD control law for the linearized system along with the baseline CBF-QP (12). (b) Baseline-SMCBF which employs the sliding mode control law (24) along with the baseline CBF-QP. (c) Learned-PDCBF which employs the PD control along with learned CBF-QP (23), and similarly (d) Learned-SMCBF which employs the sliding mode control law along with learned CBF-QP. We chose $d_{safe}=3$ m.

5.2. Straight Line Trajectory

The plots in Figure 2a–d show the trajectory of the leader drone and the follower drone. In this scenario, the leader is flying in a straight line in the +ve x direction. We can see from the plots that all four controllers manage to track the leader drone but to understand if the controllers were able to maintain safety while tracking we need to analyze the CBF $h\left(\mathbf{x}\right)$ vs. Time $\left(t\right)$ plot.

In Figure 3, we can see that all four controllers perform pretty well up until time $t=8$ s, after which the Baseline-PDCBF and Baseline-SMCBF violate the safety condition by going beyond the boundary of the safe set, that is the value of $h\left(\mathbf{x}\right)$ drops below zero. We can also see that Baseline-PDCBF recovers after a while whereas Baseline-SMCBF continues to be in the unsafe region, the reason behind this is the unknown time-varying nature of the wind disturbance. Each simulation has a different wind condition which is completely random and is generated by the AirSim’s wind model. In contrast, our proposed Learned-PDCBF and Learned-SMCBF controllers were able to maintain the safety condition throughout the experiment, with the latter performing slightly better with added robustness than the former.

5.3. Circular Trajectory

Similarly, the plots in Figure 4a–d show the trajectories of the leader and the follower drone. Here the leader flies in a circular trajectory with a radius of $5.5$ m in the clockwise direction starting from the center of the circle and stops after completing a full sweep. We can observe that all four controllers successfully track the leader drone, although from the plots it might look like the tracking performance is not good, we need to keep in mind that the follower drone is not just tracking the leader’s position but also needs to maintain a safe distance while doing so. In this case to maintain the safety condition the controller forces the follower drone to drift inward maintain the safety condition.

However similar to the previous scenario, not all the controllers were able to maintain safety at all times due to the presence of wind disturbance, namely, the Baseline-PDCBF and Baseline-SMCBF. By looking at the $h\left(\mathbf{x}\right)$ vs. Time $\left(t\right)$ plot in Figure 5 we can see that the Baseline-PDCBF and Baseline-SMCBF violate the safety condition approximately at $t=7.5$ s, as the value of $h\left(\mathbf{x}\right)$ goes below 0 for a brief period. Whereas our proposed controllers the Learned-PDCBF and Learned-SMCBF ensure safety all the time and successfully keep the system in the safe region.

For the straight line scenario (Figure 2), the average RMSE between the leader and follower was 0.42 m for Baseline-PDCBF and 0.47 m for Baseline-SMCBF. Our proposed controllers—Learned-PDCBF and Learned-SMCBF—reduced this to 0.28 m and 0.30 m, respectively. Similarly, maximum tracking error decreased from around 1.0 m in the baseline cases to 0.6–0.65 m in the learned cases. For the circular trajectory (Figure 4), the RMSE increased slightly due to curvature, with baselines ranging from 0.55 to 0.60 m and learned controllers improving this to 0.37–0.39 m. Maximum tracking error also dropped from 1.2 m to 0.8 m with the learned approaches. The control framework remain unchanged as the number of UAVs increases. For multi-UAV or swarm scenarios, the method generalizes by applying the control barrier function (CBF) constraints between all relevant pairs or neighbors, supporting higher dimensional and collective behaviors.

6. Conclusions

The paper proposed a novel approach for ensuring safety in UAV pursuit scenarios, where the UAVs are operating under uncertainty like wind. We presented a learning-based Control Barrier Function (CBF) framework that effectively captures the impact of disturbances on the system, enabling accurate estimation of an accurate CBF constraint. Moreover, integrating the learning-based CBF with a sliding mode control law further enhances the system’s robustness while upholding safety guarantees. The experimental results on AirSim demonstrate the effectiveness of our proposed method under varying wind disturbances. In contrast, the baseline controllers experienced safety violations in similar conditions. The combination of learning-based CBF and sliding mode control offers a robust solution for achieving both safety and reliability in UAV pursuit scenarios. Future research will focus on extending the framework to accommodate large-scale UAV swarms and dynamic environments with more complex disturbances, enhancing scalability and adaptability in real-world scenarios.